geometric analysis and applications, proc. nat. res. symposium, canberra

PREFACE

This volume contains the proceeding of the National Research Symposiumon Geometric Analysis and Applications held at the Centre for Mathematicsand its Applications, Australian National University, Canberra, from June26 – 30, 2000. The Symposium celebrated the many significant contributionsof Professor Derek W. Robinson to mathematics, on the occasion of his 65thbirthday. The first day, Monday June 26, in particular was devoted to Derek;speakers with a particularly close connection to Derek, including A. Carey,M. Cowling, D. Evans, P. Jorgensen and T. ter Elst recalled and elaboratedon important aspects of Derek’s work, and the day ended with a banquet inDerek’s honour.

The Symposium brought together researchers working in harmonic anal-ysis, linear and nonlinear partial differential equations, quantum mechanicsand mathematical physics, and included researchers from North America,Europe and Asia as well as Australasia.

We gratefully acknowledge the support of the contributors to this volume.We would like especially thank to Derek W. Robinson for his participation.A short synopsis of Derek’s career, and a full list of his publications to dateare included in this volume.

Contributions for the proceedings were sought from all participants andall papers received were carefully refereed by peer referees.

Alexander Isaev, Andrew Hassell,Alan McIntosh, Adam Sikora

(Editors)

i

At the end of 2000 Derek Robinson retired from his position as Professorof Mathematics in the Institute of Advanced Studies at ANU, a position hehad held for 19 years. Robinson obtained his early training at Oxford witha Bachelor’s degree in mathematics in 1957 and a Doctorate of Philosophyin theoretical nuclear physics in 1960. Subsequently he held various post-doctoral positions in Switzerland, the United States, Germany and Francebefore being appointed Professor of Mathematical Physics at the Universityof Aix–Marseille in 1968. He remained in this position until 1978 at whichpoint he accepted appointment as Professor of Mathematics at the Universityof New South Wales. Soon after he moved to Australia he became a Fellowof the Australian Academy of Science.

In his early years Robinson worked in a wide range of areas of mathemat-ical physics, quantum field theory, statistical mechanics, operator algebras,etc. Operator algebras were introduced into quantum field theory in the1960s as a means to describe the macroscopic observables but their mostfruitful application was to the characterization of the equilibrium states ofstatistical mechanics. The observables of classical statistical mechanics forman abelian C∗-algebra and Robinson realized that the corresponding quantumalgebra was ‘asymptotically abelian’. This observation was of fundamentalimportance since the states of asymptotically abelian algebras could be shownto share many of the good properties of abelian algebras. Therefore extremalstates of the algebra could be identified with pure phases of the system anddecomposition theory of the states could be applied to describe the separa-tion of the phases. Although these results were of a general abstract natureRobinson showed that they could be applied to a broad class of realistic mod-els of quantum spin systems. In particular he demonstrated that a numberof these systems exhibited a phase transition at low temperatures and thepure phases were indeed described by extremal invariant states. Much of thisis described in the two volume monograph Operator Algebras and QuantumStatistical Mechanics which Derek Robinson co-authored with Ola Bratteli.Both volumes of this book appeared in a second edition and continue to beused by research workers in these areas twenty years later.

The majority of applications of operator algebras to quantum statisticalmechanics concerned the equilibrium theory. Robinson and Bratteli realizedthat the description of non-equilibrium phenomena required a dynamicaltheory and this in turn required a theory of unbounded derivations of thealgebras. The foundations of this theory were laid in a series of joint paperswhich are also described in their book. As a direct result of this latter workRobinson’s interests then developed to evolution equations and semigroup

ii

theory in a much broader context. This change of direction coincided withhis move to Australia and most of his subsequent work has been on evolutionequations, in particular equations involving elliptic operators on Lie groups.

One of the issues at that time was the integrability of a Lie algebra rep-resentation. Together with Ola Bratteli, George Elliott and Palle Jørgensen,Robinson gave a sufficient condition in terms of a dissipativity condition andan estimate on the semigroup generated by a Laplacian. Using Lipschitzspaces Robinson could not only weaken the assumptions in the above men-tioned paper, but he could also prove that the analytic vectors associatedwith any representation of a Lie group coincide with the analytic elements ofthe Poisson semigroup. By this stage Robinson became more interested inelliptic operators on Lie groups and decided to write the monograph EllipticOperators and Lie Groups. After completion of the book Robinson begana collaboration with Tom ter Elst and their continuing work can be dividedinto three parts.

First, they developed the theory of complex, weighted, higher order sub-coercive operators on Lie groups, in particular proving Gaussian bounds forthe kernel, and its derivatives, of the semigroup generated by such an op-erator. They established that the Gaussian bounds are equivalent with asubcoercivity condition on the operator, under weak additional assumptions.Secondly, they studied second-order divergence form subelliptic operators onLie groups with complex bounded measurable coefficients and proved, undera variety of conditions, optimal smoothness properties of the kernel. Thirdly,in joint work with Adam Sikora or Nick Dungey, they studied asymptoticproperties of the semigroup and its kernel. They showed that on a Lie groupwith polynomial growth the second-order Riesz transforms associated withthe Laplacian are bounded if, and only if, the group is a direct product ofa compact and a nilpotent Lie group. Their analysis of the asymptotics ofhigher order operators continues.

Robinson has maintained an active involvement in University affairs, hav-ing been Chairman of the Board of the Institute for Advanced Studies atANU (1988-1992), and a member of ANU Council (1997-2000), where henever resiled from robust and provocative debate. He has also been an ac-tive member of the mathematical community, in particular being presidentof the Australian Mathematical Society (1994-1996) and continuing as Vice-president. He also served four years as a member of the Research Trainingand Careers Committee of the ARC (1996-2000) and as Chair of the NationalCommittee for Mathematics (1997-2001).

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LIST OF PUBLICATIONS

Books

[1] The Thermodynamic Pressure in Quantum Statistical Mechanics - Springer Verlag Lecture

Notes in Physics 9, (1971) 115 pages.[2] Operator Algebras and Quantum Statistical Mechanics Vol I (with O Bratteli) 500 pages,

Springer-Verlag (1979), 2nd edition 505 pages, Springer-Verlag (1979). Russian translationof first edition (1981).

[3] Operator Algebras and Quantum Statistical Mechanics Vol II (with O Bratteli) 501 pages,

Springer-Verlag (1981), 2nd edition 517 pages, Springer-Verlag (1996).[4] Basic Semigroup Theory Proceedings of the Centre for Mathematical Analysis, ANU, Can-

berra Volume 2, (1983) 138 pages.[5] Elliptic Operators and Lie Groups 556 pages, Oxford University Press (1991).

Research papers

[1] Multiple Coulomb Excitations of Vibrational Nuclei Nuclear Physics 45 (1961) 459–470.

[2] Zero-Mass Representations of the Inhomogeneous Lorentz Group Helv Phys Acta 35 (1962)98–112.

[3] Support of a Field in Momentum Space Helv Phys Acta 35 (1962) 403–413.[4] Multiple Coulomb Excitations of Deformed Nuclei Helv Phys Acta 36 (1963) 140–154.

[5] On a Soluble Model of Relativistic Field Theory Physics Letters 9 (1964) 189–191.

[6] A Theorem Concerning the Positive Metric Commun Math Phys 1 (1965) 89–95.[7] The Ground State of the Bose Gas Commun Math Phys 1 (1965) 159–174.

[8] Conserved Currents and Associated Symmetries (with D Kastler and A Swieca) CommunMath Phys 2 (1966) 108–120.

[9] Covariance Algebras in Field Theory (with S Doplicher and D Kastler) Commun Math

Phys 3 (1966) 1–28.[10] Invariant States in Statistical Mechanics (with D Kastler) Commun Math Phys 3 (1966)

151–180.[11] Extremal Invariant States (with D Ruelle) Ann Inst Henri Poincare 6 (1967) 299-310.

[12] Mean Entropy of States of Classical Statistical Mechanics (with D Ruelle) Commun Math

Phys 5 (1967) 288–300.[13] Mean Entropy of States of Quantum Statistical Mechanics (with O E Lanford) J Math

Phys 9 (1968) 1120–1125.[14] Statistical Mechanics of Quantum Spin Systems Commun Math Phys 6 (1967) 151–160.

[15] Asymptotic Abelian Systems (with S Doplicher, R Kadison and D Kastler) Commun MathPhys 6 (1967) 101–120.

[16] Analyticity Properties of a Lattice Gas (with G Gallavotti and S Miracle-Sole) Physics

Letter 25A (1967) 443–444.[17] Statistical Mechanics of Quantum Spin Systems II Commun Math Phys 7 (1968) 337–346.

[18] Statistical Mechanics of Quantum Spin Systems III (with O E Lanford) Commun Math

Phys 9 (1968) 327–338.[19] Analyticity Properties of the Anisotropic Heisenberg Model (with G Gallavotti and S

Miracle-Sole) Commun Math Phys 10 (1968) 311–324.[20] Proof of the Existence of Phase Transitions in the Anisotropic Heisenberg Model Commun

Math Phys 14 (1969) 196–204.

[21] Physical States of Fermi Systems (with S Miracle-Sole) Commun Math Phys 14 (1969)235–270.

[22] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores I (with S Miracle-Sole) Commun Math Phys 16 (1970) 290–309.

[23] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores II (with S Miracle-

Sole) Commun Math Phys 19 (1970) 204–218.[24] Normal and Locally Normal States Commun Math Phys 19 (1970) 219–234.[25] Normal States and Representations of the Canonical Commutation Relations (with M

Courbage and S Miracle-Sole) Ann Inst Henri Poincare XIV (1971) 171–178.

iv

[26] Approach to Equilibrium of Free Quantum Systems (with O E Lanford) Commun Math

Phys 24 (1972) 193–210.[27] The Finite Group Velocity of Quantum Spin Systems (with E H Lieb) Commun Math Phys

28 (1972) 193–210.

[28] Return to Equilibrium Commun Math Phys 31 (1973) 171-189.[29] Scattering Theory with Singular Potentials I Ann Inst Henri Poincare XXI (1974) 185–215.

[30] Scattering Theory with Singular Potentials II (with P Ferrero and O Depazzis) Ann InstHenri Poincare XXI (1974) 216–231.

[31] A Characterisation of Clustering States Commun Math Phys 41 (1975) 79–88.

[32] Dynamical Stability and Pure Thermodynamic Phases (with H Narnhofer) Commun MathPhys 41 (1975) 89–97.

[33] Unbounded Derivations of C∗-Algebras I (with O Bratteli) Commun Math Phys 42 (1975)253–268.

[34] Unbounded Derivations of C∗-Algebras II (with O Bratteli) Commun Math Phys 46 (1976)

11–30.[35] Unbounded Derivations of Von Neumann Algebras (with O Bratteli) Ann Inst Henri

Poincare XXV (1976) 139–164.[36] Unbounded Derivations and Invariant Trace States (with O Bratteli) Commun Math Phys

46 (1976) 31–35.

[37] The Approximation of Flows J Funct Anal 24 (1977) 280–290.[38] Green’s Functions, Hamiltonians and Modular Automorphisms (with O Bratteli) Commun

Math Phys 50 (1976) 53–59.[39] Bose-Einstein Condensation with Attractive Boundary Conditions Commun Math Phys 50

(1976) 53–59.

[40] Properties of Propagation of Quantum Spin Systems J Aust Math Soc Vol XIX (Series B)19 (1976) 387–399.

[41] Quasi Analytic Vectors and Derivations of Operator Algebras (with O Bratteli and R Her-man) Math Scan 39 (1976) 371–38.

[42] Perturbations of Flows on Banach Spaces and Operator Algebras (with O Bratteli and R

Herman) Commun Math Phys 59 (1978) 167–196.[43] Stability Properties and the KMS Condition (with A Kishimoto and O Bratteli) Commun

Math Phys 61 (1978) 209–238.[44] Ground States of Quantum Spin Systems (with A Kishimoto and O Bratteli) Commun

Math Phys 64 (1978) 41–48.

[45] Propagation Properties in Scattering Theory J Aus Math Soc (Series B) 21 (1979) 474–485.[46] Lie and Jordan Structure in Operator Algebras (with E Stormer) J Aus Math Soc (Series

A) 29 (1980) 129–142.[47] Equilibrium States of a Bose Gas with repulsive interactions (with O Bratteli) J Aus Math

Soc (Series B) 22 (1980) 129–147.

[48] Positivity and Monotonicity Properties of C0-Semigroups I (with O Bratteli and A Kishi-moto) Commun Math Phys 75 (1980) 67–84.

[49] Positivity and Monotonicity Properties of C0-Semigroups II (with A Kishimoto) CommunMath Phys 75 (1980) 85–101.

[50] Subordinate Semigroups and Order Properties (with A Kishimoto) J Aus Math Soc (Series

A) 31 (1981) 69–76.[51] Positive C0-Semigroups on C∗-Algebras (with O Bratteli) Math Scand 49 (1981) 259–274.

[52] Strictly Positive and Strongly Positive Semigroups (with A Majewski) J Aus Math Soc(Series A) 34 (1983) 36–48.

[53] Strongly Positive Semigroups and Faithful Invariant States Commun Math Phys 85 (1982)

129–142.[54] On Unbounded Derivations Commuting with a Compact Group of ∗-Automorphisms (with

A Kishimoto) Publ RIMS Kyoto Univ 18 (1982) 1121–1136.[55] Positive Semigroups on Ordered Banach Spaces (with O Bratteli and T Digernes) J Op

Theor 9 (1983) 371–400.[56] Addition of an Identity to an Ordered Banach Space (with S Yamamuro) J Aus Math Soc

(Series A) 35 (1983) 200–210.[57] Hereditary Cones, Order Ideals, and Half-norms (with S Yamamuro) Pac J Math 110 (1984)

335–343.

v

[58] The Jordan Decomposition and Half-norms (with S Yamamuro) Pac J Math 110 (1984)

345–353.[59] The Canonical Half-norm, Dual Half-norms, and Monotonic Norms (with S Yamamuro)

Tohoku J Math 35 (1983) 375–386.

[60] Continuous Semigroups on Ordered Banach Spaces J Funct Anal 5 (1983) 268–284.[61] On Positive Semigroups Publ RIMS Kyoto Univ 20 (1984) 213–224.

[62] Extending Derivations (with C J K Batty, A Carey and D E Evans) Publ RIMS KyotoUniv 20 (1984) 119–139.

[63] Positive One-parameter Semigroups on Ordered Banach Spaces (with C J K Batty) Acta

Appl Math 2 (1984) 221–296.[64] A C∗-algebraic Schoenberg Theorem (with O Bratteli, P Jørgensen and A Kishimoto) Ann

Inst Fourier (Grenoble) XXXIV (1984) 155–187.[65] Relative Locality of Derivations (with O Bratteli and T Digernes) J Funct Anal 59 (1984)

12–40.

[66] Derivations, Dynamical Systems, and Spectral Restrictions (with A Kishimoto) Math Scand56 (1985) 83–95.

[67] Dissipations, Derivations, Dynamical Systems and Asymptotic Abelianness (with A Kishi-moto) J Op Theor 13 (1985) 237–253.

[68] Derivations of Simple C∗-algebras Tangential to Compact Automorphism Groups (with E

Stormer and M Takesaki) J Op Theor 13 (1985) 189–200.[69] Strong Topological Transitivity and C∗-dynamical Systems (with O Bratteli and G A El-

liott) J Math Soc Japan 37 (1985) 115–133.[70] The Characterisation of Differential Operators by Locality; Classical Flows (with O Bratteli

and G A Elliott) Comp Math 58 (1985) 279–319.

[71] The Characterisation of Differential Operators by Locality; Abstract Derivations (with CJ K Batty) Erg Theor and Dyn Syst 5 (1985) 171–183.

[72] The Characterisation of Differential Operators by Locality; Dissipations and Ellipticity(with O Bratteli and G A Elliott) Publ RIMS Kyoto Univ 21 (1985) 1031–1049.

[73] The Characterization of Differential Operators by Locality: C∗-algebras of Type I (with O

Bratteli and G A Elliott) J Op Theor 16 (1986) 213–233.[74] Integration in Abelian C∗-dynamical Systems (with O Bratteli, T Digernes and F M Good-

man) Publ RIMS Kyoto Univ, 21 (1985) 1001–1030.[75] Smooth Cores of Lipschitz Flows Publ RIMS Kyoto Univ 22 (1986) 659–669.

[76] Smooth Derivations on Abelian C∗-dynamical Systems J Aus Math Soc (Series A) 42 (1987)

247–264.[77] Commutators and Generators (with C J K Batty) Math Scand 62 (1988) 303–326.

[78] Commutators and Generators II Math Scand 64 (1989) 87–108.[79] Return to Equilibrium in the X−Y Model (with L R Hume) J Stat Phys 44 (1986) 829–848.

[80] Embedding Product Type Actions into C∗-dynamical Systems (with O Bratteli and A

Kishimoto) J Funct Anal 75 (1987) 188–210.[81] Fractional Powers of Generators of Equicontinuous Semigroups and Fractional Derivatives

(with O E Lanford) J Aus Math Soc (Series A) 46 (1989) 1–32.[82] Commutator Theory on Hilbert Space Can J Math 5 (1987) 1235–1280.

[83] The Differential and Integral Structure of Continuous Representations of Lie Groups J Op

Theor 19 (1988) 95–128.[84] The Heat Semigroup and Integrability of Lie Algebras (with O Bratteli, F M Goodman and

P E T Jørgensen) J Funct Anal 79 (1988) 351–397.[85] An Index for Continuous Semigroups of ∗-endomorphisms of B(H) (with R T Powers) J

Funct Anal 83 (1989) 1–12.

[86] Lie Groups and Lipschitz Spaces Duke Math Jour 57 (1988) 357–395.[87] The Heat Semigroup, Derivations, and Reynold’s Identity (with O Bratteli and C J K

Batty) Operator algebras and applications. London Math Soc Lecture Notes Series 136(1988) 22–47.

[88] The Heat Semigroup and Integrability of Lie Algebras: Lipschitz Spaces and SmoothnessProperties Commun Math Phys 132 (1990) 217–243.

[89] Lipschitz Operators J Funct Anal 85 (1989) 179–211.[90] Comparison of Commuting One-parameter Groups of Isometries (with O Bratteli and H

Kurose) Trans Amer Math Soc 320 (1990) 677–694.

vi

[91] Unitary Representations of Lie Groups and Garding’s Inequality (with O Bratteli, F M

Goodman and P E T Jørgensen) Proc Amer Math Soc 107 (1989) 627–632.[92] Positive Semigroups Generated by Elliptic Operators on Lie Groups (with W Arendt and

C J K Batty). J Op Theor 23 (1990) 369–407.

[93] Elliptic Differential Operators on Lie Groups J Funct Anal 97 (1991) 373–402.[94] 2nd order Elliptic Operators and Heat Kernels on Lie Groups (with O Bratteli) Trans Amer

Math Soc 325 (1991) 683–713.[95] Subelliptic Operators on Lie Groups: Regularity (with A F M ter Elst) J Aus Math Soc 57

(1994) 179–229.

[96] Subelliptic Operators on Lie Groups: Variable Coefficients (with O Bratteli) Acta ApplMath 42 (1996) 1–104.

[97] Subcoercivity and Subelliptic Operators on Lie Groups I: Free Nilpotent Groups (with AF M ter Elst) Pot An 3 (1994) 283–387.

[98] Subcoercivity and Subelliptic Operators on Lie Groups II:The General Case (with A F M

ter Elst) Pot An 4 (1995) 205–243.[99] Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients (with A F M

ter Elst) Publ RIMS Kyoto Univ 5 (1993) 745–801.[100] Lp-regularity of Subelliptic Operators on Lie Groups (with R J Burns and A F M ter Elst)

J Op Theor 31 (1994) 165–187.

[101] Functional Analysis of Subelliptic Operators on Lie Groups (with A F M ter Elst) J OpTheor 31 (1994) 277–301.

[102] Asymptotics of Periodic Subelliptic Operators (with C J K Batty, O Bratteli and P E TJørgensen) J Geom Anal 5 (1995) 427–443.

[103] Weighted Strongly Elliptic Operators on Lie Groups (with A F M ter Elst) J Funct Anal

125 (1994) 548–603.[104] On Positive Rockland Operators (with P Auscher and A F M ter Elst) Colloq Math LXVII

(1994) 197–216.[105] Spectral Estimates for Positive Rockland Operators (with A F M ter Elst) Algebraic groups

and Lie groups (Ed G I Lehrer), Aus Math Soc Lecture Series 9. Cambridge Univ Press

(1997).[106] Reduced Heat Kernels on Nilpotent Lie Groups (with A F M ter Elst) Commun Math Phys

173 (1995) 475–511.[107] Semigroup Kernels, Poisson bounds and Holomorphic Functional Calculus (with X T

Duong) J Funct An 142 (1996) 89–129.

[108] Weighted Subcoercive Operators on Lie Groups (with A F M ter Elst) J Funct Anal 156(1998) 88–163.

[109] On Kato’s Square Root Problem (with A F M ter Elst) Hokkaido Math J 26 (1997) 1–12.[110] Analytic Elements of Lie Groups (with A F M ter Elst) Helv Phys Acta 69 (1996) 655–678.

[111] Abundance of Invariant and Almost Invariant Pure States of C∗-dynamical Systems (with

O Bratteli and A Kishimoto) Commun Math Phys 187 (1997) 491–507.[112] Second-order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Prin-

cipal Coefficients (with A F M ter Elst) Acta Appl Math 59 (1999) 299–331.[113] High Order Divergence-form Elliptic Operators on Lie Groups (with A F M ter Elst) Bull

Aus Math Soc 55 (1997) 335-348.

[114] Second-order Subelliptic Operators on Lie Groups II: Real Measurable Coefficients (withA F M ter Elst) Progess in Nonlinear Differential Equations 42 Birkhauser Verlag (2000)

103–124.[115] Second-order Strongly Elliptic Operators on Lie Groups with Holder Continuous Coeffi-

cients. (with A F M ter Elst) J Aus Math Soc (Series A) 63 (1997) 297–363.

[116] Heat Kernels and Riesz Transforms on Nilpotent Lie Groups. (with A F M ter Elst and ASikora) Colloq Math LXXIV (1997) 191–218.

[117] Second-order Subelliptic Operators on Lie Groups III: Holder Continuous Coefficients (withA F M ter Elst) Calc Var P D E 8 (1999), 327–363.

[118] Spectral Asymptotics of Periodic Elliptic Operators (with O Bratteli and P E T Jørgensen)Math Z 232 (1999), 621–650.

[119] Local Lower Bounds on Heat Kernels (with A F M ter Elst) Positivity 2 (1998) 123–151.[120] Riesz Transforms and Lie Groups of Polynomial Growth. (with A F M ter Elst and A

Sikora) J Funct Anal 162 (1999), 14–51.

vii

[121] Asymptotics of Semigroups on Nilpotent Lie Groups (with N Dungey, A F M ter Elst and

A Sikora) J Op Theor (to appear)[122] Asymptotics of Sums of Subcoercive Operators (with N Dungey and A F M ter Elst) Colloq

Math 82 (1999) 231–260.

[123] On second-order Periodic Elliptic Operators in Divergence Form (with A F M ter Elst andAdam Sikora) Math Z (to appear)

[124] On Anomalous Asymptotics of Heat Kernels (with A F M ter Elst) Evolution equationsand their applications to Physical and Life Sciences, Lecture Notes in Pure and Applied

Mathematics, Vol. 215, (2001) 89–103, Eds. G Lumer, L Weis, Marcel Dekker, New York

[125] On Second-order Almost-periodic Elliptic Operators (with N Dungey and A F M ter Elst)J London Math Soc (submitted)

[126] On Anomalous Asymptotics of Heat Kernels on Groups of Polynomial Growth (with NDungey and A F M ter Elst) Comp Math (submitted)

Lecture notes, reviews and conference papers

[1] Algebraic Aspects of Relativistic Field Theory - Brandeis Lectures, Gordon and Breach,

New York, Vol 1 (1966).[2] Symmetries, Broken Symmetries, Charges and Currents - Istanbul Lecture, Freeman and

Co, San Francisco, (1967) 463–513.

[3] Conserved Currents and Broken Symmetries - Colloque du CNRS (1966), Eds du CNRS 15(1968) 131–135.

[4] Physical States and Finiteness Restrictions - Proceeding of the CNRS Conference at Gif-sur-Yvette, CNRS (1969).

[5] Existence Theorems in Quantum Statistical Mechanics - Proceedings of the Cargese Summer

School (notes by R Lima and S Miracle) Gordon and Breach (1970).[6] Approach and Return to Equilibrium of Free Fermi States - Colloquium of the American

Math Soc 1971 - Siam AMS Proceeding 5 55 (1972).[7] C∗-Algebras and Quantum Statistical Mechanics - Int School of Physics Enrico fermi

Varenna-Editrice Comp Bologna (1975) 235–252.

[8] Time Dependent Scattering Theory III - Int Colloq on Group Theoretical Methods, Mar-seille (1974).

[9] Unbounded Derivations of C∗-Algebras - Int Symp on Math Problems in Theor Phys,Springer-Verlag (1975).

[10] Dynamics in Quantum Statistical Mechanics - Int Summer School Bielefeld 1976, Plenum

Press (1978).[11] Review of Derivations - Int Conference on Mathematical Physics, Lausanne 1979, Springer-

Verlag (1980).[12] Commutator Theory and Partial Differential Operators on Hilbert Space - Proceedings of

the Centre of Mathematical Analysis, ANU, Canberra 14 (1986) 295–302.

[13] Differential Operators on C∗-algebras - Contemporary Mathematics, Amer Math Soc 62(1987) 367–384.

[14] C∗-algebras and a Single Operator - Surveys of some recent results in operator theory IIPitman Research 171 (1988) 235–266.

[15] Integration of Lie Algebras - Proceedings of the Centre of Mathematical Analysis, ANU,

Canberra 15 (1987) 255–278.[16] Schauder Estimates on Lie Groups- Proceedings of the Centre of Mathematical Analysis,

ANU, Canberra 24 (1989) 1–9.[17] Subelliptic Operators on Lie Groups (with A F M ter Elst) - Proceedings of the Centre of

Mathematical Analysis, ANU, Canberra 29 (1992) 63–72.[18] Strongly Elliptic and Subelliptic Operators on Lie Groups - Quantum and Non-

Commutative Analysis, Eds H Araki et al., Kluwer Academic Publishers (1993) 435–453[19] Basic Semigroup Theory - Proceedings of the Centre of Mathematical Analysis, ANU,

Canberra 34–Part III (1996) 1–33.

[20] Elliptic Operators on Lie Groups (with A F M ter Elst) Acta Appl. Math. 44 (1996) 133–

150.[21] C∗-Algebras and Mathematical Physics - Fields Institute Monograph volume, Eds R Bhat,

G Elliott and P Fillmore, American Mathematical Society, Providence, Rhode Island.

viii

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO

NONLINEAR PDE

HILARY BOOTH

Abstract. The Maxwell-Dirac equations give a model of an elec-tron in an electromagnetic (e-m) eld, in which neither the Diracor the e-m elds are quantized. The two equations are coupledvia the Dirac current which acts as a source in the Maxwell equa-tion, resulting in a nonlinear system of partial dierential equations(PDE's). In this way the self-eld of the electron is included.

We review our results to date and give the four real consistencyconditions (one of which is conservation of charge) which applyto the components of the wavefunction and its rst derivatives.These must be met by any solutions to the Dirac equation. Theseconditions prove to be invaluable in the analysis of the nonlinearsystem, and generalizable to higher dimensional supersymmetricmatter.

In earlier papers, we have shown analytically that in an isolatedstationary system, the surrounding electon eld must be equal andopposite to the central (external) eld. The nonlinearity forceselectric neutrality, at least in the static case. We illustrate theseproperties with a numerical family of orbits which occur in the(static) spherical and cylindrical ODE cases. These solutions arehighly localized and die o exponentially with increasing distancefrom the central charge.

1. The Maxwell-Dirac Equations and QED

The coupled Maxwell-Dirac equations can be written as follows:

(@ ieA) + im = 0 where = 0; : : : ; 3

F = A; A;

@F = 4ej

where j = :(1)

Note that 2 C4 is the Dirac wave-function or 4-spinor and theDirac conjugate. These are acted upon by which are the usualgamma matrices (representations of a Cliord algebra), and A is the4-potential.These equations model an electron in an electromagnetic eld. The

two equations are coupled via the Dirac current j i.e. we include1

2 HILARY BOOTH

the nonlinearity of the self-eld, see Equation (1). These equationsform the foundation of quantum electrodynamics (QED), the theoryof electrons interacting with elds. QED is one of the most successfulphysical theories, explaining the Lamb shift and the anomalous mag-netic moment of the electron. The calculations of QED are acheived byquantizing the elds and using perturbation theory. In so doing, well-known mathematical problems occur, which have yet to be resolved atthe fundamental level.It is possible that the full nonlinearized equations must be analysed

rigorously before we can hope to resolve these deep problems. Forexample, in [23], Lieb and Loss observe that the stability of matter re-quires that the electron be dened with a Dirac operator with the mag-netic vector potential A(x), instead of the free Dirac operator (withoutA(x)). That perturbation theory must start from the dressed electrons(including their own self-eld) might be \fundamentally important ina non-perturbative QED".This view is shared by many analysts working on this problem in-

cluding Flato, Simon and Ta in who established global existence forthe M-D equations as recently as 1997 [18], following many years ofsustained interest [21] [9] [20] [19] in the problem. In [18], Flato et alshowed that the nonlinear representation is integrable to a global non-linear representation of the Poincare group on a dierential manifold,U1 of small initial conditions. This established the existence of globalsolutions for initial data in U1 at t = 0. They go on to show that theasymptotic representations are also nonlinear and draw conclusions forthe infrared tail of the electron. Their results show that \in the clas-sical case (also), one obtains infrared divergencies, if one requires freeasymptotic elds as it is needed in QED". In other words, the eldsmust remain coupled via the self-eld if we are to resolve the infraredproblem.In the case where we assume that the system is static and/or station-

ary (see Section 2), we can make some simplications. The system iselliptic in the stationary case (no time dependence). Esteban, Georgievand Sere showed existence of soliton-like solutions (that is, solutionswhich are spatially localized) in this case [16]. Furthermore, the wave-function, together with all its derivatives decreases exponentially atinnity.

2. Some simplified versions of the problem

The full 4-dimensional nonlinear problem is somewhat intractable as stated in Section 1, global existence has only been establishedrecently [18] (and references therein). If we want to get some idea of

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 3

the types of behaviour we might expect in the 4-dimensional problem,we might begin by looking at various subcases.Some simplied versions of the problem are as follows:

1. The static case in which we assume that there exists a Lorentzframe in which there is no current \ ow" i.e. j = 0j0 [29] [4],

2. The stationary case in which we assume (x0;x) = ei!x0 (x)[16],

3. The static spherically symmetric case [29],4. The static cylindrically symmetric case [4] [8],5. The static case with z dependence only [6],6. The 1 + 1 case which was solved exactly for a massless electron

by Schwinger in [31],7. The static axi-symmetric case,8. The circular current case in which we assume (in spherical coor-

dinates) that j = (j0; 0; j; 0):

It appears that the static and stationary assumptions are rather strong,since the resulting system becomes elliptic rather than hyperbolic. In[5] and [30] we proved electric neutrality in the static case (for an iso-lated system). While this is interesting in that it implies that solutionsof this type must consist of an inner charge, say, surrounded by anequal and oppositely charged electron eld i.e. the solutions must beatom-like, it raises the problem of nding a solution that representsa single charged particle. We need to nd a weaker ansatz, perhapsthe circular current assumption, with which we have sucient simpli-cation without losing the important properties of the 4-dimensionalsystem. It is possible of course, that electric neutrality could be shownto be fully general, which would supporting the conjecture that thetotal charge of the universe is zero. The larger (quantum cosmologi-cal) problem here is the Einstein-Maxwell-Dirac problem, and relatedproblems such as Einstein-Dirac [17] and Einstein-Yang-Mills [3].Following the methods of the analysts working in relativity theory,

our aim in [29] [4] etc was to enumerate the subcases, starting with allpossible ODE cases (see Section 4), and then progressing to (static)axi-symmetric and other two-dimensional cases. Meanwhile, the threedimensional static case proved to be somewhat tractable.

3. Using the Clifford Algebra

to derive some constraints

In this section, we make use of the properties of the Cliord algebra(which the represent as 4 4 matrices), to solve the Dirac equationfor the potential and to show that there are some useful consisitency

4 HILARY BOOTH

conditions upon the wavefunction and its rst derivatives. We wantto write the potential in terms of the wavefunction, so that we cansubstitute it into the Maxwell equation.A complex Cliord algebra A(n) consists of all possible products of

the n basis vectors (of an n-dimensional vector space), ei, which obeythe following:

(ei)2= 1; if i 2 f0; : : : ; n 1g;

eiej + ejei = 0; if i 6= j:(2)

Using the isomorphism [11]

A(n+ 2) = A(n)A(2)(3)

we can construct the representation of A(4), the gamma matrices, fromthe Pauli matrices which represent A(2). For the construction of higherdimensional Cliord algebras from lower dimensions see for example[14]. One possible representation of A(4) is:

0 =

2664

0 0 0 10 0 1 00 1 0 0

1 0 0 0

3775 1 = i

2664

0 0 0 10 0 1 00 1 0 0

1 0 0 0

3775

2 = i

26641 0 0 00 1 0 00 0 1 00 0 0 1

3775 3 = i

2664

0 0 1 00 0 0 1

1 0 0 00 1 0 0

3775(4)

We note that in this representation, 0 is anti-symmetric (a-s), and i; i = 1; 2; 3 are symmetric.Whatever representation we use, we can always invert the Dirac

equation to express the spatial components of the potential, Ai; i =1; 2; 3 in terms of A0, the wavefunction, and its rst derivatives.Multiplying (1) on the left t i (where t is the transpose of ) wehave

t i A =1

ie t i ( @ + im) :(5)

But the product i j; i 6= j is (a-s), i i = 1, and i 0 is symmetric,so that, using the argument following (7) we can write:

t Ai = t 0 iA0 +1

ie t i ( @ + im) :(6)

In a similar way (by multiplying the Dirac equation by 0 1 2 3

on the left, and subtracting the conjugate equation premultiplied by


0 1 2 3) we can solve for A0. There is a condition to be met here see[29] i.e. that j is not a null vector which has the physical interpretationthat the electron is not travelling faster than the speed of light.In 1957, one of Dirac's students, Eliezer solved for the potential in

this way [15]. He went on to show that when we solve for the potential,we must also adopt a consistency condition which applies to . Inthat paper there was a contribution by Dirac who streamlined some ofEliezer's calculations. Although they used a dierent representation ofthe algebra, the argument was essentially the following.If is an antisymmetric (a-s) (4 4) matrix then

t = 0; since this quantity is an a-s scalar.(7)

If we can nd a such that are all a-s, then

t ( A) = 0;

since A is a scalar, so that if satises (1) then

t ( @ + im) = 0;(8)

which gives us a consistency condition on and its rst derivatives.The same is true of the complex Dirac equation and we have

t ( @ im) = 0:(9)

We can extend Dirac's argument one step further by premultiplyingthe Dirac equation by and the complex equation by and notingthat since is a-s,

t A

t= t A :

This gives us another condition on namely t ( @ + im)

t+ t ( @ im) = 0:(10)

There is only one possible element of the Cliord algebra which whenpremultiplying all of the yields an a-s matrix. In the representa-tionused here, this is the product

= 1 2 3:

Using this in (8) (9) and (10), we have two complex conditions (or fourreal conditions) on the components of .As is well-known (see [11] for example), even dimensional complex

Cliord algebras are simple, that is, they can not be decomposed intothe direct sum of two nontrivial subspaces which obey closure underalgebraic multiplication. If we want to decompose the 4-dimensionalinto a 2-dimensional algebra and use 2-spinors, then when must accept

6 HILARY BOOTH

the additional structure in which the two 2-spinor spaces are conju-gate dual spaces. See for example [28] in which the 2-spinor formalismis given in terms of Ineld-van der Waerden symbols. An argument,equivalent to Dirac/Eliezer's but giving all of the consistency condi-tions, was given by Radford using the 2-spinor formalism in 1996 [29],and subsequently in [4], although Radford referred to them as \realityconditions", in keeping with the conventions in [28].We can think of the Dirac equation and its conjugate equations as

eight equations in four unknowns (the four real scalars, A). If we solvefor these A then we must have four additional (real) constraints, whichcorrespond to the two complex consistency conditions, conrming thatthere are no further constraints upon the system. In [7] we outlinethese conditions and then go on to generalize to higher dimensionalcases. Allowing these higher Cliord algebras enables us to pursue thesame arguments when applied to supersymmetric matter [7]. See [22]and references therein.

4. ODE solutions

Within the static system there are three interesting ODE cases,spherically symmetric, cylindrically symmetric and dependence on z

only. The spherical and cylindrical cases were examined extensively in[29] [4]. The case where dependence is on z only, is similar in somerespects [5]. We rst apply our consistency conditions to the elec-tromagnetic potential A which has been expressed in terms of Diracspinors and their rst derivatives (by solving the Dirac equations forthe potential as outlined in Section 3). These reality conditions allowus some simpler expressions which are then inserted into the Maxwellequation, resulting in fourth order ODE's. We will also note here thatin the 1 + 1 case [12] [13] [6] the system was also reduced to fourthorder ODE's, which in some cases are solved explicitly [12] [13], whilstin others we are currently developing more numerical results [6].When we assume that the Dirac current is static, we lose three (real)

degrees of freedom in , since three components of j are set to zero.See [29] and [4] in which we also x the gauge by choosing

0 = Y ei2() 2 = Y e

i2(+)

1 = Xei2(+) 3 = Xe

i2() ;(11)

where , , X and Y are real functions. These expressions were substi-tuted into the potential (which was solved for in terms of Dirac spinors


and their rst derivatives), yielding:

A0 = cos +(X2 Y 2)

(X2 + Y 2)

@

@t+

(r):l

(X2 + Y 2)(12)

A =1

(X2 + Y 2)

@

@tl + (X2 Y 2)r r l

:(13)

We now use the consistency conditions, one of which is conservationof charge, which is obeyed automatically as stated in [29]. The otherthree conditions, in the variables required for the static case, are givenbelow.

@

@t(X2 + Y 2) = 0(14)

r:l = (X2 + Y 2) sin(15)

@l

@t+ (r) l = 0:(16)

where l = (2XY cos ; 2XY sin ;X2 Y 2).The Maxwell equations act uponA, as dened above, and the current

vector becomes j = (2(X2 + Y 2); 0; 0; 0) :We showed in [5] that the static equations, in the gauge given by

Eq. 11, are stationary if and only if @

@t= 0 and @X

@t= 0 (or @Y

@t= 0). In

the stationary case, @

@t= 0 and @l

@t= 0. Now in the stationary case, the

third reality condition Eq. 16 tells us that r is proportional to l andwe choose the function such that l =

r sin r: Substituting this into

the expression for the potential Eq. 12 and noting that X2 + Y 2 = jlj

then

A0 = cosr:l

(X2 + Y 2)= cos

s2r +

2

r2+

2

r2 sin2 :(17)

From here, it is a straightforward calculation to apply symmetry argu-ments and calculate the resulting ODE's [29] [4] [5].Then, in dimensionless variables [29] [4], the equations reduce to:

d

dx= A cos

dF

dx= Z

dA

dx=

F

f(x)

dZ

dx= Z sin;(18)

8 HILARY BOOTH

where

f(x) =

8<:

x2 in the spherical casex in the cylindrical case1 in the z dependent case

and

x =

8<:

r in the spherical case in the cylindrical casez in the z dependent case,

r being the distance from the origin and the distance from the centralaxis.It is easily shown that these four rst order equations can be writtenas the fourth order,

d2

dx2

f(x)

d2

dx2 sin

d

dx

+d

dx

f(x)

d2

dx2 sin

d

dx

sin = 0:(19)

In the cylindrical case, Z() is the charge per unit ring radius , F () isthe charge within a radius , and A is the scalar potential, A0. See [4].Similar physical quantities are represented in the spherical and z cases.Most importantly, we are looking for solutions whose charge densityZ() decreases rapidly towards innity, so that we can nd solutionswhich are localized or particle-like. Our zero total charge result [5] tellsus to expect that F ! 0 as !1 and likewise, A! const as !1.As pointed out by Chris Cosgrove (private communication), equa-

tion (19) has non-integer resonance numbers [1] [2] and we do notexpect to nd an integrable system (in the soliton sense) here. In-stead, we show that there are a family of orbits (in the sense of [10])all of which approach the trivial (constant) solution at innity. As anexample, we look at the cylindrical case, noting that similar resultshold in the spherical and z case. The numerical orbits in the Figure 1complete the results in [4] in which a single member (analytic in 1

) of

these families was shown to exist and calculated numerically.

5. Vacuum Maxwell singularities near the origin

In [4] it was shown that in the cylindrical case

Lemma 1. Suppose (; F; A; Z) is a solution to Equation(18) on I =(0; 1), for some 1; 0 < 1 < 1. Suppose also that Z 0 is continuous

and bounded on I. Then,


(i) F is C1 on I and has a well dened, nite limit as ! 0. Z hasa well-dened limit as ! 0.

(ii) if F (0) 6= 0 then A is unbounded as ! 0. In particular, A =() ln(), where is C2 and bounded on I, ! F (0) as ! 0.Also, is bounded as ! 0.

Lemma 2. Suppose (; F; A; Z) is a solution to Equation(18) on 2

(0;1). Suppose also that Z 0 with F continous and bounded on theinterval. Then

(i) If F (1) 0 for some 1 2 [0;1), then ! 1, A ! 1, andF !1 as !1.

(ii) If F < 0 on (0;1) then F ! 0 as !1. In addition, if A and

Z have well-dened limits as ! 1 then Z ! 0 and A ! A1as !1, with 1 A1 1.

Similar results were established for the spherical case in [29]. In[29][4], Radford and the current author have shown that the behaviourof solutions near the origin resembles the vacuum solutions of theMaxwell equations. Given that the Maxwell equation can be writtenas the square of a Dirac operator (see for example [26]), it should bepossible to formulate these results, together with exponential decreasein terms of the properties of k-monogenic functions | those functionswhich are solutions to

Dk = D + k0 = 0;

where 0 is a basis vector (corresponding to the time coordinate) ofa complex Cliord algebra. The fundamental solution of the k-Diracoperator has the same singularity at the origin and decreases exponen-tially as x ! 1. In this case, the operator has coecients in 1, thesubspace of vectors . We are currently working on a clarication ofthis point.

6. Numerical Solutions

In [29][4] [8] we showed examples of numerical solutions exhibitingthe characteristics referred to in Section 5. Earlier attempts at ndingnumerical solutions[32] [25] were marred by a \simplifying assumption"which proved to be invalid solutions did not exist in that case. See[29].In [8] we noted that there are families of orbits parameterized by

the constant cjxj where we assume that solutions are of the form

ecjxjg(jxj), with g(jxj) an analytic function. By varying the valueof c, the boundary conditions are perturbed to neighboring solutions

10 HILARY BOOTH

(orbits), all decreasing exponentially at innity. See Figure 1. All val-ues of c yield solutions which satisfy the two Lemmas, and the zerototal charge result. That is, all solutions surround a central wire alongthe axis of symmetry.Similar numerical solutions can be found in the spherical case [6].

The spherical solutions surround a central Coulomb eld, but the staticcondition forces a monopole at the origin [29] [5] [30], that is, an un-bounded A component. Further ODE solutions occur in the z case[5] and 1 + 1 case, neither of which have been fully examined in theprevious literature [12] [13]. These results will be forthcoming also in[6].Similar spherically-symmetric solutions were found in [27]. This time

the Schrodinger-Newton equations provided an identical coupling asin the static Maxwell-Dirac case, which is essentially an elliptic sys-tem. More work must be done to investigate the stability of both ofthese systems. In [27] implications have been developed in the contextof quantum gravity, but work in this area is far from complete. Weare currently considering the problem in this context. The solutionsfound in [27] blow-up at larger distances from the origin. However, inthe Maxwell-Dirac case, this behaviour appeared only as a numericalanomaly. When the total charge became slightly positive (due to thestep-size of the numerical solution), we entered a regime described inLemma 2 of [4] in which all solutions become unbounded as ! 1.These solutions were illustrated numerically in [4], but discarded asbeing of less interest than the bounded solutions.The solution shown in Figure 1 was calculated using the MATLAB

ODE solver ODE113. The relative error tolerance was set at 1e 4and the absolute error tolerance at 1e 8. The same behaviour wasobserved when the tolerances were decreased to 1e 6 and 1e 12 orincreased to 1e 3 and 1e 6. All solutions remained stable whethercalculated as a function of increasing or decreasing radius (i.e. shootingaway from or towards the central axis).The two dimensional cases (static axi-symmetric, circular current

axi-symmetric, the massive 1 + 1 case) still require reliable numericalresults. Those available to date [32] [25] have been awed by the im-position of \approximations" which were shown in [29] to be possibleonly in trivial cases (in which the equations are no longer coupled).

7. Vanishing Total Charge

In the solutions in Figure 1 the variable F () representing the totalcharge within a ring, radius , tends towards zero at 1. Lemma 1also states that the potential must tend towards a solution which is


0 5 10 150

2

4

6

8

10

12

14

16

18

Orbits of charge on ring radius r

r

Z

0 5 10 15

−300

−250

−200

−150

−100

−50

0

chi vs r

r

chi

10−1

100

101

0

100

200

300

400

500

600

700

Potential vs log r

A

0 5 10 15

−90

−80

−70

−60

−50

−40

−30

−20

−10

0Total charge within radius r

r

F

Figure 1. A family of orbits exhibiting the propertiesof Lemma 1 and Lemma2.

logarithmic in , near the central axis. This corresponds to a centralcharged wire (along the z axis) which is a solution to the vacuum

Maxwell equations. As such, we can think of this part of the solution asrepresenting an external eld. (The scalar potential could be separatedat this point into Aexternal + Afermion+interaction, since the Aexternal, asthe homogenous solution, contibutes nothing to the coupling betweenthe Maxwell and Dirac equations.) Physically, this means that the(inner) external eld must be surrounded by an equal and oppositelycharged eld. The total electric charge of the system

lim!1

1

4

ZS

rA0

:dS; with S the unit ring of radius ;

must vanish.

12 HILARY BOOTH

In [5], we were able to show that this is the case for all station-ary solutions, given that they are in a \reasonable" function space. Ifwe dene an isolated system as one for which all sources are con-tained in some ball Bk (k < 1) and for which the elds die o asjxj = r ! 1, then we showed in [5] and [30] that an isolated, sta-tionary, static Maxwell-Dirac system is electrically neutral. A similarresult, for the total electric charge per unit length in the z direction (orunit ring), holds in the cylindrical case. This shows mathematically,that a (stationary) solution must be atom-like (in the sense that anycentral charge must be surrounded by an equal and opposite charge.In addition it shows that there cannot be a stationary solution repre-senting an isolated electron (or even one that rotates around the axiswith constant velocity).Note also, that the signs of the charges can be reversed, giving us

negatively charged singularites and positively charged \electron" elds.All results remain unchanged under such a reversal of sign. The totalcharge must vanish overall. A more dicult problem presents itself if weare to allow for two fermion elds of opposite charge, for example, thesolution corresponding to positronium. We are currently consideringways in which we can have have solutions of opposite charge interactingtogether.

8. Discussion and Conclusions

Until the mathematical problems of QED have been resolved, allmethods of addressing the equations governing the interaction betweenelectrons and elds are signicant. A non-perturbative QED must con-sist of classical and/or semi-classical arguments which aim to justify theFeynman integral method, and the quantized eld approach.For the coupled M-D system, we have existence (for small initial

data), existence for the stationary case, and descriptions of varioussubcases. But analysis of the system is far from complete. We haveindicated, within this review paper, the directions we are currentlypursuing.

References

[1] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution

equations and ordinary dierential equations of P-type.I J.Math.Phys. 21 (4),715-721 (1980)

[2] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution

equations and ordinary dierential equations of P-type.II J.Math.Phys. 21 (5),1006-1018 (1980)


[3] R.Bartnik, J McKinnon, Particlelike solutions of the Einstein-Yang-Mills equa-

tions Phys. Rev. Lett., 61 141-144, 1988.[4] H.S.Booth, C.J.Radford, The Dirac-Maxwell equations with cylindrical sym-

metry J. Math. Phys. 38 (3), 1257-1268 (1997).[5] H.S.Booth, The Static Maxwell-Dirac Equations PhD Thesis, University of

New England (1998).[6] H.S.Booth Various ODE Solutions to the static and 1+1 Maxwell-Dirac Equa-

tions (in preparation).[7] H.S.Booth,P.Jarvis,G.Legg, Algebraic solution for the vector potential in the

Dirac equation (in preparation).[8] H.S.Booth, Nonlinear electron solutions and their characteristics at innity

The ANZIAM Journal (formerly J.Aust.M.S (B)) (to appear).[9] J.Chadam Global solutions of the Cauchy problem for the (classical) cou-

pled Maxwell-Dirac system in one space dimension J.Funct.Anal. 13, 173-184(1973).

[10] K.Cieliebak and E.Sere, Pseudoholomorphic curves and multiplicity of homo-

clinic orbits, Duke Mathematical Journal 77, (2) 483-518 (1995).[11] R.Coquereaux, Modulo 8 periodicity of real Cliord algebras and particle

physics, Phys Lett 115B (5), 389-395 (1982).[12] A.Das, D.Kay, A class of exact plane wave solutions of the Maxwell-Dirac

equations, J.Maths.Phys. 30 (10) (1989) 2280-2284.[13] A.Das, General Solutions of Maxwell-Dirac equations in 1+1 dimensional

space-time and a spatially conned solution, J.Maths.Phys. 34 (9) (1993) 3986-3999.

[14] R.Delanghe,F.Sommen,V.Soucek, Cliord Algebra and Spinor-Valued Func-

tions, KLUWER (1992).[15] C.J. Eliezer, A Consistency Condition for Electron Wave Functions,

Camb.Philos. 54, 2 (1957).[16] M.Esteban, V.Georgiev, E. Sere, Stationary solutions of the Maxwell-Dirac

and the Klein-Gordon-Dirac equations, Calc. Var. 4, 265-281, (1996).[17] Felix Finster, Niky Kamran, Joel Smoller, Shing-Tung Yau, Nonexistence of

time-periodic solutions of the Dirac equation in an axisymmetric black hole

geometry CPAM 53,7, 902-929 (2000).[18] M. Flato, J.C.H. Simon, E Ta in, Asymptotic Completeness, Global Existence

and the Infrared Problem for the Maxwell-Dirac Equations Memoirs of theAmerican Mathematical Society, (May 1997).

[19] V.Georgiev, Small amplitude solutions of the Maxwell-Dirac equations IndianaUniv. Math. J. 40 (3), 845-883 (1991).

[20] R.T.Glassey, W.A.Strauss Conservation Laws for the Maxwell-Dirac and

Klein-Gordon-Dirac equations, J.Math.Phys., 20, (1979).[21] L.Gross The Cauchy problem for the coupled Maxwell and Dirac equations

Comm. Pure Appl. Math. 19 1-5 (1966).[22] P.S.Howe, G.Sierra, P.K.Townsend Supersymmetry in six dimensions Nuclear

Phys B221 331-348 (1983).[23] E.H. Lieb and M. Loss, Self-Energy of Electrons in Non-perturbative QED ,

in Dierential Equations and Mathematical Physics, University of Alabama,

Birmingham, 1999, R. Weikard and G. Weinstein, eds. 255-269 Internat. Press(1999). arXiv math-ph/9908020, mp arc 99-305.

14 HILARY BOOTH

[24] E.H. Lieb and M. Loss, Remarks about the ultraviolet problem in QED, (inpreparation).

[25] A.G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J.Phys.A28 No.18, 5385-5392 (1995).

[26] A.McIntosh, Cliord algebras, Fourier theory, singular integrals, and harmonicfunctions on Lipschitz domains in: Cliord Algebras in Analysis and Related

Topics (John Ryan ed), 33-88 CRC Press Boca Raton, FL, 1996.[27] I.M.Moroz,R.Penrose,P.Tod, Spherically-symmetric solutions of the

Schrodinger-Newton equations, Class. Quantum Grav. 15 2733-2742 (1998).[28] R.Penrose,W.Rindler, Spinors and Space-Time I (Two Spinor Calculus and

Relativistic Fields), CAMBRIDGE UNIVERSITY PRESS, (1984).[29] C.J. Radford, Localised Solutions of the Dirac-Maxwell Equations, J. Math.

Phys. 37 (9), 4418-4433 (1996).[30] C.J.Radford, H.S.Booth Magnetic Monopoles, electric neutrality and the static

Maxwell-Dirac equations J.Phys.A:Math.Gen. 32, 5807-5822 (1999).[31] J.Schwinger Gauge Invariance and Mass. II Phys. Rev. 128 (5), 2425-2429

(1962).[32] M. Wakano, Intensely Localized Solutions of the Dirac-Maxwell Field Equa-

tions, Prog.T.Phys. 35, 1117-1141, (1966).

Centre for Mathematics and it Applications, Australian National

University, Canberra ACT 0200 Australia

E-mail address : [email protected]

QUANTUM MECHANICS AS AN INTUITIONISTIC

FORM OF CLASSICAL MECHANICS

JOHN V. CORBETT AND MURRAY ADELMAN

Abstract. Intuitionistic real numbers are constructed as sheaves

on the state space of the Schrodinger representation of a CCR-

algebra with a nite number of degrees of freedom. These num-

bers are used as the values of position and momentum variables

that obey Newton's equations of motion. Heisenberg's operator

equations of motion are shown to give rise to numerical equations

that, on a family of open subsets of state space, are local approxi-

mations to Newton's equations of motion for the intuitionistically

valued variables.

Introduction

Do sub-atomic particles have positions and momenta at all times?Can the numerical value of the position of such a particle always begiven a triplet of real numbers? What real numbers should be used?

In this paper we argue that there is a class of real numbers that canbe used to label the positions and momenta of sub-atomic particles atall instances of time. Furthermore if these numbers, as values of po-sitions and momenta, are assumed to satisfy the equations of motionof classical mechanics, then these classical equations are approximated

locally by the quantum mechanical operator equations of motion, re-stricted to act on certain subsets of state space.The real numbers that we use are real numbers in the topos of sheaves

on the state space of the Schrodinger representation of a CCR-algebra

with a nite number of degrees of freedom. In the standard Hilbertspace framework for quantum mechanics[1] the algebra of the canonicalcommutation relations (CCR) is generated by the operators Qj = Q

j ,Pj = P

j , j = 1; : : : ; n, and the identity operator I which satisfy the

relations [Pj; Qj] = iI, [Pj; Qk] = 0 if k is dierent from j, [Qj; Qk] =0, [Pj; Pk] = 0 for all j; k. We have put Planck's constant divided by2 equal to one.

We will further simplify the notation by taking n = 1.15

16 JOHN V. CORBETT AND MURRAY ADELMAN

1. The Schrodinger representation

The Schrodinger representation of the CCR-algebra M is the rep-resentation in which the Hilbert space is L2(R). Q is representedby multiplication by the real variable x and P by (1=i) times theoperator of dierentiation with respect to x. Let S(R) denote theSchwartz space of innitely dierentiable functions of rapid decrease

on R. Then the physical quantities are represented by self-adjointelements in the closure M of the CCR-algebra M , where M is thesmallest closed extension of M . M =

X

X 2 M and X

is the restriction to S(R) of the Hilbert space closure of X. Follow-

ing the denitions of Powers[7], M is essentially self-adjoint becausethe adjoint M of M equals the closure M of M .

2. The State Space ES

We give only a resume of the results, for more details see Inoue[8]. Alinear functional f on M is strongly positive i f(X) 0 for all X 0

in M .

Denition 1. The states on M are the strongly positive linear func-

tionals on M that are normalised to take the value 1 on the element Iof M .

Theorem 1 (Inoue[8]). Every strongly positive linear functional onMis given by a trace functional.

Denition 2. The state space ES of the Schrodinger representation of

M is the set of all strongly positive linear functionals on M that are

normalised.

The state space ES is contained in the convex hull of projections Ponto one-dimensional subspaces spanned by unit vectors u 2 S(R3).Recall that each state 2 ES is a trace class positive bounded operatorwith trace 1, which can be written as =

PnPn, where the sum

over n may go from 1 to 1. For all n, n 0,P

n = 1 and the Pnare orthogonal projections. The following result demonstrates that allstates in ES must satisfy the further condition that as n approachesinnity the sequence fng converges to zero faster than any power of

1=n.

Theorem 2. Given =P

nPn 2 ES, where the projections Pnproject onto one-dimensional subspaces spanned by unit vectors un 2

S(R3). tr A is nite for any self-adjoint operator A in the Schrodinger

representation of the CCR-algebra M if and only if limnkn = 0, for

all integers k > 0.

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS 17

Proof. Let A be a self-adjoint operator inM and is a state onM whichbelongs to ES . Then for any orthonormal basis ofH, fumg,m = 1 to1,

which is composed from elements in S, tr A =P(um ; Aum) .

Now using =P

nPn the trace becomes a double sum

tr A =X

um ;X

nPnAum:

Because the trace is independent of the basis used to calculate it, wecan choose the ortho-normal basis fumg to be in the ranges of the Pnso that the double sum reduces to

tr A =X

mum ; Aum

:

Since A is in the algebra M , it is a polynomial of some nite degreek in the self-adjoint operators P , Q. By Lemma 18 in Jae[10], Ais majorized by a polynomial of degree k in the self-adjoint operator

H = 12

P(P 2 +Q2),um; Aum2

um; pk(H)um:

When the fumg are the eigenfunctions of H, pk(H)um = pk(m)um,

where m = 12(2m + 1) are the eigenvalues of H, so that the absolute

value of the mth term of the seriesP

m(um; Aum) satises

0 m(um; Aum)

mum; H(m)um

12 = mpk(m)

1

2 :

Therefore if limnkn = 0, for all k > 0, then 9 a positive integer Nand a positive constant C such that, for all m > N ,

mpk(m)1

2 <C

m2

and so the seriesP

m(um; Aum) converges absolutely by the compar-

ison test. On the other hand, if tr A is nite for all self-adjoint opera-tors A then the series

Pm(um; Aum) must converge. If A is positive

then the convergence is absolute so that when A is a monomial of de-

gree k in the Hamiltonian operator H,then limm(m)1

2(k) = 0 for all

positive k which shows the converse is true.The same results hold even when the fumg are not the eigenfunctions

of H. The minimax principle for eigenvalues of symmetric operators,

see for example, Kato[6] section I.6.10, implies thatum; pk(H)um

pk(m)

because it states that the maximum ofv; pk(H)v

, when v is a unit

vector and (v; ei) = 0, i = 1; ; n 1, for any orthonormal set of

vectors feig, is the eigenvalue pk(m) obtained when the feig are the


rst n 1 eigenvalues of H. This result is applicable here to the set ofvectors fuig, i = 1; ; m, with v = um.

We have shown that tr A is nite for any self-adjoint operator A inthis representation of the CCR-algebra M if and only if limnkn = 0,for all integers k > 0.

The topology on ES is chosen to make the functions A, where A() =

tr A, continuous when A is an essentially self-adjoint continuous op-erator on S(R).

3. The topos Shv(ES)

A sheaf Y on a topological space X can be described[3] by a rule

which assigns to each point x of X a set Y (x) consisting of the germsof a prescribed class of functions, where the germs of the functionsare dened in neighborhoods of the point x. The collection of setsY (x) which are labelled by points x in X can be glued together toform a space Y in such a way that the projection from Y onto X is a

local homeomorphism; that is, for each x in X and each y on the berabove x (i.e. for each such y the projection of y onto X is x) thereis a neighborhood N of y such that the projection of N onto X is aneighborhood of x.

A section of the sheaf Y over the open subset U of X is a functions from U to Y that belongs to the prescribed class of functions andsatises the condition that, for all x in U , the projection of s(x) ontoX is x. The sheaf construction allows a section f dened on the open

set U to be restricted to sections fVon open subsets V contained in

the open set U and, conversely, the section f on U can be recovered bypatching together the sections f

V 0

where V 0 belongs to an open cover

of U .A spatial topos is a category of sheaves on a topological space. The

objects of this category are sheaves over the topological space and thearrows are sheaf morphisms, that is, an arrow is a continuous function

that maps a sheaf Y to a sheaf Y 0 in such a way that it sends bers inY to bers in Y 0, equivalently, sections of Y over U to sections of Y 0

over U , where U is an open subset of X.The topos Shv(ES) of sheaves on the topological space ES is con-

structed in this way.In 1970, Lawvere[2] showed that toposes can be viewed as a \vari-

able" set theory whose internal logic is intuitionistic. The propositionalcalculus of the logic of the spatial topos of sheaves over X is the Heyt-

ing Algebra[3] of the open subsets of X. This means that as well as


being true or false, propositions in this logic can be true to intermedi-ate extents which are given by open subsets of X. True corresponds

to the whole set X. False corresponds to the empty set. There existBoolean algebras in which propositions can be true to varying extentsbut, in addition, the Heyting algebra of open sets does not satisfy allthe laws of classical logic. Two of the most striking dierences between

classical and intuitionistic logics are that the law of the excluded mid-dle and the Axiom of Choice do not hold for intuitionistic logic. Wehave argued elsewhere[5] that aspects of quantum mechanics, such asthe two slit experiment,that are dicult to understand with Boolean

logic are better described using intuitionistic logic.There is an analogy between the language of toposes and that of sets

which makes it easier to work in toposes. Sheaves in a topos correspondto sets, subsheaves of a sheaf to subsets and local sections to elementsof a set. Then as long as a proof in set theory does not use the law of

excluded middle or the Axiom of Choice then it can be translated intoa proof in topos theory.

4. Real Numbers in Spatial Toposes

Dedekind numbers are dened to be the completion of the rational

numbers obtained by using cuts, and Cauchy numbers are dened asthe completion of the rationals obtained by using Cauchy sequences.These dierent constructions can only be shown to be equivalent byusing either the Axiom of Choice or the law of the excluded middle.[3]

Therefore, when intuitionistic logic is assumed, we expect that thesetwo types of real numbers are not equivalent.It has been shown[4] that in a spatial topos the sheaf of rational

numbers Q is the sheaf whose sections over an open set U are given bylocally constant functions from U with values in the standard rationals

while the sheaf of Cauchy reals RC is the sheaf whose sections over anopen set U are given by locally constant functions from U with valuesin the standard reals. A function is locally constant if it is constant oneach connected open subset of its domain.

On the other hand the sheaf of Dedekind reals RD is the sheaf whosesections over U are given by continuous functions from U to the stan-dard reals.The Cauchy reals form a proper sub-sheaf of the Dedekind reals

unless the underlying topological space X is the one point space.


5. The Quantum Reals

By the construction of the topology on ES for any self-adjoint opera-tor A in the Schrodinger representation of the CCR, the function tr Ais a globally dened continuous function and therefore in RD . We inter-pret the functions A() = tr A, with domains given by open subsetsof state space, to be the numerical values of the physical quantity that

is represented by the self-adjoint operator A. Not every Dedekind realnumber is of this form. Real numbers of this form are a proper sub-sheaf of the Dedekind real numbers in the spatial topos of sheaves onthe state space ES. We will call them the quantum reals, they belong

to the sheaf of locally linear functions.[5]

Denition 3. If U is an open subset of ES then the function f from

U to the standard reals is locally linear at in U and there is an open

neighborhood U 0 of , with U 0 inside U , and a bounded self-adjoint

operator A such that fU 0

= AU 0.

Denition 4. The sheaf of locally linear functions, A , is dened by

its sections over any open subset U of ES as the set of all locally linear

functions on U with the requirement that if the open set V is contained

in U then the sheaf of locally linear functions over V is obtained by

restricting the locally linear functions over U .

The global elements of the locally linear functions are given by the

functions A, where A is a self-adjoint operator, that are continuouson S. It suces to dene algebraic relations between elements of Aglobally, because ES is locally connected and so we can treat functions

which are dened on disjoint connected components as if they wereglobally dened.When U is an open neighbourhood of the state then the quantum

real numbers belonging to the sections of A over U can be thought ofas real numbers tangent to those Dedekind reals that have a tangent

space at .

6. The Dedekind Reals RD

Stout[4] has shown that the usual order on the rational numbers Qcan be extended to the following order on RD .

Denition 5. The order relation < on the Dedekind reals, RD , is givenby the denition:

x < y if and only if 9 q 2 Q(q 2 x+) ^ (q 2 y)

where x+ is the upper cut of x and y is the lower cut of y. The relation

< is the subobject of RD RD consisting of such pairs (x; y).


Trichotomy does not hold universally for the order < on RD .The order on RD has the property that x y is not the same as

(x < y) _ (x = y).

Denition 6. The order relation x y is the subobject of RD RDconsisting of the pair (x; y) with x+ y+ and y x, where x

+ is

the upper cut of x and x is the lower cut of x, and similarly for y.

Stout[4] also showed that the statement (x y) is equivalent to thestatement :(y < x).

Denition 7. The open interval (x; y) for x < y is the subobject of

RD consisting of those z in RD that satisfy x < z < y.

The closed interval [x; y] for x < y is the subobject of RD consisting

of those z in RD that satisfy x z y.

The open intervals can be used to construct an interval topology T onRD analogously to the interval topology on the standard real numbersthat is generated from the open intervals by nite intersection and

arbitrary union.The topology T on RD is such that Q is dense in RD with respect to

T .If the max function is dened using the order by the conditions:

(i) x max(x; y) and y max(x; y), and (ii) if z x and z y

then z max(x; y) and the norm function: j j : RD ! RD is denedby jxj = max(x;x), then the norm j j satises the usual conditionsof non-negativity, that only 0 has norm zero and that the triangleinequality holds.

(RD ; T ) is a metric space with the metric d(x; y) = x y

. It is

both complete and separable[4].RD is a eld in the sense that for all a in RD ,if a does not belong to

the sheaf of germs of invertible functions, Unit(RD),then a = 0.

7. Properties of the Quantum Reals

Theorem 3. A is a proper sub-sheaf of the sheaf of Dedekind numbers

RD and is dense in RD in the metric topology T .

The sheaf A inherits the orders and < from RD . On the otherhand A can be ordered as a consequence of the orders on the self-

adjoint operators:

1. A is strictly positive, A > 0, if (Au; u) > 0 for u 6= 0, u 2 D(A).2. A is non-negative, A 0, if (Au; u) 0 for all u 2 D(A).

Lemma 8. The orders and < on A inherited from RD are equivalent

to those obtained from those on continuous self-adjoint operators.


Proof. A is a non-negative self-adjoint operator i tr A 0 for all in

ES, i.e. A 0 globally. When, as a Dedekind real number, a = A 0then 0+ a+ and a 0. Globally, 0+ = fq 2 Q j q > 0g and

0 = fq 2 Q j q > 0g so that, if a = A 0 then A is a non-negative

operator and if A is a non-negative operator then a = A 0.The positivity order for a continuous self-adjoint operator A is equiv-

alent to A being bounded away from zero, i.e. there exists a rationalnumber q > 0 such that (u;Au) > q for all u 6= 0. This gives the

equivalence of the operator > with the > for Dedekind reals restrictedto A , because for the latter a > 0 means globally that 9q 2 Q withq 2 0+ and q 2 a.

If a = A and b = B then the RD distance between a and b is given

by the metric ja bj on RD . There is another metric on the quantum

numbers A , it is given by the number \jA Bj, where jCj is the operator

jCj = (C2)1

2 , when C is self-adjoint.

Proposition 9. The two metrics coincide on A , that is, \jA Bj =

ja bj, for all pairs of quantum numbers a = A and b = B.

Proof. It is sucient to let B = 0. We will only consider the globalsections.It is well-known, see for example section VI.2.7 of Kato[6], that(u;Au) (u; jAju) for all u in the domain D(A) = D(jAj), whencetr A tr jAj for all in ES, i.e. jAj

cjAj.As on elements of RD , jAj = max(A;A) and

cjAj = cjAj. The

lower cut of jAj is the union of the lower cuts of A and of A, that is

jAj = (A) [ (A), which means that jAj cjAj

. The upper

cut of jAj is the intersection of the upper cuts of A and of A, that is

q 2 Q belongs to jAj+ if q is greater than or equal to both A and of

A. Thus if q 2cjAj+ then q 2 jAj+, therefore jAj+

cjAj+.This shows that cjAj A and therefore cjAj = A.The metric is used to dene Cauchy sequences in RD , this result

means that we can dene Cauchy Sequences in A in the same way. Inorder to ensure uniqueness of the limits of a Cauchy sequence we needa concept of apartness, ><, that is stronger than that of not equal to,

6=.

Denition 10. The pair (a; b) of Dedekind reals are apart, a >< b, i(a > b) _ (a < b).

Proposition 11. a >< b i ja bj > 0.


8. Calculus of Functions

The concept of limit is available for functions: RD ! RD and dier-ential calculus for such functions can be developed. The denition ofthe limit of a function G 2 RD is modeled on the standard denition;the limit of G as x tends to b is L i 8n > 0; 9 m > 0; 8x 2 RD ;0 < jx bj < 1=m =)

G(x) L < 1=n .

In this denition both m and n are in Q . The uniqueness of the limitL holds in the sense that if L and M are two limits of G as x tends tob then L and M are apart,

LM = 0.

For the requirements of this paper it is enough to consider only

polynomial functions. For any b 2 RD all powers bn of b are in RDfor n a natural number because products of continuous functions arecontinuous. Let both b and c 2 RD , then bc = cb 2 RD for the samereason. The product, however, is dened only on the intersection of

the domains of the continuous functions so that care has to be takenwith the extent to which the product exists. Sums of numbers b + cexist to the extents given by the intersection of the extents of b and c.Hence we can construct polynomials

F (b) =X

ambm

where the sum is over nitely many terms and the coecients am 2

RD . We can construct power series by dening the convergence of thesequence of partial sums in the metric on RD .The derivative of bm = mbm1 which implies that the derivative of a

monomial is dened to the same extent as the monomial. This allowsus to obtain the derivatives of polynomials and power series.We dene continuity of a function F : RD ! RD at b in its domain

by the requirement that

(1) 8n > 0; 9m > 0; 8x 2 RD ;

0 < jx bj <1

m=)

F (x) F (b) < 1

n:

In this denition we have taken m and n to be in Q .A function is continuous on an interval I in RD i it is continuous

at each number in I.Interesting phenomena occur in this calculus and they deserve more

study. However we have enough structure now to return to the dynam-

ics of quantum mechanics.


9. Comparing the Dynamics

Consider the example of a non-relativistic quantum particle of pos-itive mass that moves in a central force eld which is derived from

a potential function V . We assume that the quantum values X of theposition of the particle satisfy Newton's equations of motion globally,that is,

DQ =1

P(2)

and

DP = F (Q)(3)

or

D2(Q) = F (Q)(4)

where P represents the momentum of the particle and D denotes dier-entiation with respect to time. F represents the force, it is the negativegradient of the potential function V as a function: RD ! RD .We will now prove that standard quantum mechanics is a local ap-

proximation to a global classical mechanics. More precisely, the the-orem states that if the quantum values of the components of positionand momentum of a particle are assumed to satisfy Newton's equa-tions of motion globally then the self-adjoint operators corresponding

to these values locally satisfy equations that well approximate Heisen-berg's equations of motion.The theorem relates a set of operator equations (Heisenberg's equa-

tions) to a set of numerical equations (Newton's equations). Thestraightforward way to get a numerical equation from an operator equa-

tion is to multiply each side of the operator equation by a suitabletrace class operator (a state) and then take the trace of each side. Theoriginal operator equation has become a family of numerical equationswhich is labelled by the states.

Recall that Heisenberg's equations for an operator A are

DA = i[A;H](5)

where H is the Hamiltonian operator of the system and the squarebracket denotes the operator commutator.

Therefore, we get Heisenberg's equations of motion, when Q and Prepresent operators and the Hamiltonian operator is H = (1=2)P 2 +V (Q), to be

DQ =1

P(6)


and

DP = i[P;H] = F (Q) :(7)

If we multiply each side of equations (6) and (7) by the operator and then take the trace of each side we get Heisenberg's numericalequations:

D(tr Q) =1

tr P(8)

and

D(trP ) = tr (DP ) = tr i[P;H]

= tr

F (Q)

:(9)

Equation (8) is just Newton's equation (2) at the state . Equation(9) is similar to Newton's equation (3) at the state . The dierencebetween equations (3) and (9) is the same as that which restricts thevalidity of the result known as Ehrenfest's Theorem, namely, that, in

general,

tr F (Q) 6= F (tr Q):(10)

To simplify the discussion we remove the explicit dependence of theequations (2), (3), (8), and (9) on the operators P and use equationsof motion in the form of the second order dierential equations:Newton's equations to the extent W are

D2QW

= F

QW

:(11)

Heisenberg's numerical equations to the extent W are

D2QW

= [F (Q)

W:(12)

It is possible, however, that the dierence between the two sides of

(10) is small at some state a and remains small for all states in anopen neighborhood of a . In this case the equations (11) and (12) areapproximately the same in that open set. We deduce that Heisenberg'sequations approximate Newton's equation in that open set because if

(11) holds in an open set W then it holds at each in W .We claim that for a suitable class of functions F , Heisenberg's equa-

tions approximate Newton's equations locally. The localness of theassertion is twofold, we mean that for every standard real number r we

can nd an open set,W , on which both the number QW

is arbitrarily

close to r, and [F (Q)W

is arbitrarily close to FQW

. The physical

interpretation is that if an observer's measurement apparatus is locatedin a neighbourhood of the position r (in this world with one spatial di-mension) then the dierence between the accelerations of the particle

due to the two forces cannot be distinguished with this apparatus.


The class of suitable functions is dened through the concept of S-continuity.

Denition 12. We call functions G, S-continuous, if they are real-

valued continuous functions on R such that,for the position operator

Q, G(Q) : S ! S, is continuous in the standard countably normed

topology on the Schwartz space S.

Theorem 4. If the force F is S-continuous, then given " > 0, Heisen-berg's equations of motion approximate Newton's equations of motion

to within " on each member of a collection of open sets fW (r; ")g of

state space ES, indexed by the standard real numbers r and " > 0. Thatis, on each fW (r; ")g,[F (Q)

W F

QW

< " :(13)

Proof. The idea behind the proof is to nd states r on which F (tr rQ)

closely approximates tr rF (Q), then F (Q) will be close to [F (Q) when

F (Q) is close to FQ(r)

11 and[F (Q) is close to tr rF (Q)11 as Dedekind

numbers.The following pair of lemmas complete the proof.

Lemma 13. If Q is a self-adjoint operator which has only absolutely

continuous spectrum, then for any real number r in its spectrum we

can construct a sequence of pure states fng such that for the given

S-continuous function F ,

lim tr nF (Q) = F (r) :

Proof. Weyl's criterion for the spectrum of self-adjoint operators[9] im-plies that, for any number r in the spectrum of Q, there exists a se-quence of unit vectors fung, in the domain of Q, such that

lim (Q r)un

= 0 :

Take n to be the projection onto the one dimensional subspace

spanned by un. It is easy to check that

limtr nQ = r :

The vectors fung can be chosen to be in S. For example, for anypositive integer n, let

un(x) = n12

14 exp

1

2n2(x r)2

The sequence fung satises the requirements of Weyl's lemma for the

operator Q and the number r.


Furthermore we can nd a sequence of vectors fung in S so that forn large enough the support of fung lies in a narrow interval centred on

r. Then, by the spectral theorem for Q, the corresponding pure statesn form a sequence so that both

limtr nF (Q) = F (r)

and

limtr nQ = r :

From Lemma 13, given a real number r in the spectrum of Q, the S-continuous function F and a real number " > 0, there exists an integerN such that, for all j > N , bothtr jF (Q) F (r)

< 16"

and (tr jQ) r <

where is such thatF (r) F (x) < 1

6" when jr xj < :(14)

We choose r = j, for some j > N , and deduce thattr rF (Q) F (tr rQ) < 1

3"(15)

because tr rF (Q) F (tr rQ)

=tr rF (Q) F (r) + F (r) F (tr rQ)

tr rF (Q) F (r)+ F (r) F (tr rQ)

:With this choice of r,the open set W (r; ") can be dened as

W (r; ") = N(r; Q; ) \N(r; F (Q);13")

where satises the requirements (14), and for any S-continuous func-tion F and " > 0 we have that N(r; F (Q); ") is given by

Nr; F (Q); "

=n ;tr F (Q) tr rF (Q)

o < " :

Lemma 14. When r is chosen so that equation (15) holds, then for

all in W (r; "), tr F (Q) F (tr Q) < " :(16)


That is, for W =W (r; "),[F (Q)W F (Q

W) < " :

In the denition of the open neighborhood W, may depend upon r,as well as on F and ".

Proof. For any pair of states, and r, we havetr F (Q) F (tr Q) tr F (Q) tr rF (Q)

+tr rF (Q) F (tr rQ)

+ F (tr rQ) F (tr Q) :

If is in N(r; F (Q);13") the rst summand is < 1

3", as is the second

by choice of r. The nal summand is also because by assumption thefunction F : R ! R is continuous everywhere in the usual topology onR. Because given " > 0, there exists a (xa) > 0, such that

F (x) F (xa)

< 13", whenever jx xaj < . Apply this to x = (tr Q) and

xa = (tr rQ).Therefore, given r, for any in N(r; Q; ) \ N(r; F (Q);

13"), the

inequality (16) holds.

The question remains whether we can construct suciently many ofthese open sets. In general, for a given smooth function F , the familyof open sets

W (r; ")

, does not form an open cover of state space ES.

However for every standard real number r, and hence for every point

in classical coordinate space, all the standard real numbers that liewithin of r in the standard norm topology on R also lie in

W (r; ")

as Cauchy real numbers RC . In this sense, the family of open setsW (r; ")

covers the classical coordinate space of the physical system.

10. Conclusion

It is important to be clear about the meaning of this result. It in-volves the comparison of two dierent theories. Assume that Newton's

equations of motion hold for the position and momentum variables ofa non-relativistic massive quantum particle when they are expressed interms of the real numbers RD . Assume also that these real numbersRD are given by sheaves of continuous functions over the state space

ES and include quantum numbers like Q = tr Q and P = tr P for

2 fopen subsets of ESg, where P and Q are self-adjoint operators onan underlying Hilbert space. Then we can nd open subsets of ES such

that on each open subset, the restrictions of the functions P and Q canbe reinterpreted as the average values of the operators P and Q whichalmost satisfy the Heisenberg's equations of motion with the analogous

Hamiltonian operator. This result only involves a comparison of the


dynamical equations of motion. Comparison of the trajectories requiresthat the initial data be compatible which leads to further constraints

on the allowable trajectories. Nevertheless, there are trajectories thatcan be described with these real numbers.

References

[1] J. von Neumann, Mathematical Foundations of Quantum Mechan-

ics (Princeton University Press, 1955), p. 212.[2] W. Lawvere, \Quantiers as Sheaves", Actes Congres Intern.

Math. 1, (1970).[3] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic

(SpringerVerlag, New York, 1994).[4] L. N. Stout, Cahiers Top. et Geom. Di. XVII, 295 (1976); C.

Mulvey, \Intuitionistic Algebra and Representation of Rings" in

Memoirs of the AMS 148 (1974).[5] M. Adelman and J. V. Corbett, Applied Categorical Structures 3,

79 (1995).[6] T. Kato, Perturbation Theory for Linear Operators (Springer-

Verlag, New York, 1966).[7] R.T. Powers, Commun. Math. Phys. 21, 85124 (1971).[8] A. Inoue, TomitaTakesaki Theory in Algebras of Unbounded Op-

erators (Lecture Notes in Mathematics,1699, Springer, 1998)

[9] W. Reed and B. Simon, Methods of Mathematical Physics I: Func-

tional Analysis (Academic Press, New York, 1972).[10] A. Jae, \Dynamics of a cut-o 4-eld theory", Ph.D. Thesis,

Princeton University, 1965; see also B. Simon, J. Math. Phys. 12,140 (1971).



Department of Mathematics,, Macquarie University, Sydney, N.S.W.

2109, Australia

VILENKIN BASES IN NON-COMMUTATIVELp-SPACES

P. G. DODDS AND F. A. SUKOCHEV

Abstract. We study systems of eigenspaces arising from the rep-

resentation of a Vilenkin group on a seminite von Neumann alge-

bra. In particular, such systems form a Schauder decomposition in

the re exive non-commutative Lpspaces of measurable operators

aliated with the underlying von Neumann algebra. Our results

extend classical results of Paley concerning the familiar Walsh-

Paley system to the non-commutative setting.

1. Introduction

It is a classical theorem of Paley [Pa] that the Walsh system, taken inthe Walsh-Paley ordering, is a Schauder basis in each of the re exiveLp-spaces on the unit interval. Although the Walsh basis in not un-conditional, except in the case p = 2, it was further proved by Paleythat partitioning the Walsh system into dyadic blocks yields an un-conditional Schauder decomposition. This paper provides an overviewof recent work by the present authors (and collaborators) which de-velops the theory of orthogonal systems in the setting of re exive non-commutative Lp-spaces of measurable operators aliated with a semi-nite von Neumann algebra. The work nds its roots in that of Paleyand in the subsequent development of Paley's ideas in the more generalsetting of Vilenkin systems by Watari [Wa], Schipp [Sc], Simon [Si1,2]and Young [Y].

In the classical setting, the Walsh and Vilenkin systems arise as thesystem of characters of the familiar dyadic and Vilenkin groups. Thepresent approach exhibits non-commutative orthogonal systems as eigen-vectors corresponding to the action of an ergodic ow on the underlyingnon-commutative Lp-space. The classical results are recovered by spe-cialisation to the case where the ow is given by the action of righttranslation on the space Lp(G), with G an arbitrary Vilenkin group.In this case, the eigenspaces are one-dimensional and are spanned bythe characters of G.

Research supported by the Australian Research Council (ARC).

30

VILENKIN BASES IN NON-COMMUTATIVE Lp-SPACES 31

2. Preliminaries

There is an extensive literature concerning harmonic analysis on com-pact Vilenkin groups that is related to the theme of present article.In particular we refer the reader to [AVDR], [BaR], [Gos], [SWS], [Sc],[Si1,2], [Vi], [W], [Wa], [Y] and the references contained therein.

We shall be concerned here with Vilenkin groups of the form Gm =Q1

k=0Zm(k)

, equipped with the product topology and normalized Haar

measure. Here m = fm(k) j k 2 N [ f0gg is a sequence of naturalnumbers greater than one and

Zm(k)

= f0; 1; 2 : : : ; m(k) 1gis the discrete cyclic group of order m(k). The dual group dGm of Gm

can be identied with the sum1ak=0

[Zm(k)

=1ak=0

Zm(k)

consisting of all sequences n = (n0; n1; : : : ) with nk 2 Zm(k)for all k

and nk 6= 0 for at most nitely many k. See for example [SWS]. The

pairing between Gm and dGm is given by

ht;ni = n(t);

where

n(t) =1Yk=0

nkk (t); 8n = (n0; n1; : : : ) 2 dGm

and

k(t) := exp(2i

tk

m(k)); 8t = (t0; t1; : : : ) 2 Gm:

The dual group dGm is linearly ordered by the (reverse) lexicographical

ordering: for n;p 2 dGm we dene n < p if and only if there existsk 2 N [ f0g such that nj = pj for all j > k and nk < pk. We shall

always consider the system of characters f n : n 2 dGmg with the

enumeration induced by the reverse lexicographical ordering of dGm.

For k = 1; 2; : : : and 1 j m(k) 1, dene d0; dk; d(k;j) 2 dGmvia

d0 := 0; dk := fjkg1j=1; d(k;j) := fj

ikg1i=1:

dk; d(k;j) correspond to k; j

krespectively.

The system of characters f n : n 2 dGmg forms a complete orthonormalsystem in L2(Gm; dtm), called the Vilenkin system corresponding tom. Here dtm denotes normalised Haar measure on Gm.

32 P. G. DODDS AND F. A. SUKOCHEV

We set M0:= 1 and M

k:= m(k 1)M

k1and to each n 2 dGm we

assign the natural number

n =1Xk=0

nkM

k; 0 n

k< m(k):

This denes an order preserving bijection between dGm and N [ f0g. Ifwe denote the character n by n, with n corresponding to n underthis bijection, then we may write the Vilenkin system as f ng1n=0

.

There is a well-known and natural measure preserving identicationbetween the Vilenkin group Gm and the closed interval [0; 1] given bythe map t! t(t) 2 [0; 1] where

t(t) :=1Xk=0

tk

Mk+1

:

In the special case that m(k) = 2 for all k 2 N [ f0g, the Vilenkingroup Gm is the dyadic group D , and the characters fkg1k=0

may be

identied with the usual Rademacher system frkg1k=0on [0; 1]. In this

case, the Vilenkin system f ng1n=0coincides with the familiar Walsh

system fwkg1k=0, taken in the Walsh-Paley ordering.

We begin with the following classical results concerning Vilenkin sys-tems. If f 2 Lp (Gm; dtm), we set

ck(f) :=ZGm

f kdt

m; k 2 dGm:

Theorem 1 Suppose 1 < p <1.

(i) 9Kp; 8f 2 Lp (Gm; dtm) ; 8n 2 dGm Xk<n

ck(f) k

p

Kpkfkp: (1:1)

(ii) 9Kp; 8k = 1; 8f 2 Lp (Gm; dtm) 1Xk=0

k

0@ Xdkn<dk+1

cn n

1A p

Kpkfkp: (1:2)


(iii) If supkm(k) <1, 9Kp; 8k;j = 1; 8f 2 Lp (Gm; dtm)

1Xk=0

m(k)1Xj=1

k;j

0@ Xdk;jn<dk;j+1

cn n

1A p

Kpkfkp: (1:3)

Remarks In the terminology of Banach spaces (see, for example[LT1,2]), the assertion (1.1) is equivalent to the statement that the

Vilenkin system f n;n 2 dGmg is a Schauder basis in Lp (Gm; dtm).

Similarly, the estimate (1.2) asserts that the system

Xk = clmf n : dk n < dk+1g; k = 0; 1; 2; : : :

is an unconditional Schauder decomposition of Lp (Gm; dtm). Thestatement of (1.3) is that the system

Xk;j = clmf n : dk;j n < dk;j+1g; 1 j m(k1); k = 0; 1; 2; : : :

is a (ner) unconditional Schauder decomposition of Lp (Gm; dtm),provided that the Vilenkin group Gm is bounded in the sense thatsupkm(k) <1. Here, clm(A) denotes the closed linear manifold gen-erated by A.

We remark further that if we set

k :=X

dkn<dk+1

cn n; k = 0; 1; 2; : : : ;

then (1.2) equally asserts that the sequence fk; k = 0; 1; 2; : : :g isan unconditional martingale dierence sequence in Lp (Gm; dtm).

In the case of the classical Walsh-Paley system, that is, for m(k) =2; k = 0; 1; 2; : : : , Theorem 1 is due to Paley [Pa, 1931], who showedfurther that statements (i), (ii) are equivalent. In the case of moregeneral but bounded Vilenkin systems, Theorem 1 was established byWatari [Wa,1958]. In this case, (ii) is valid, but is not equivalent to(i). Watari showed that (i) is equivalent to (iii), but that (iii) is notequivalent to (i) if the Vilenkin group is not bounded. For generalVilenkin systems that are not necessarily bounded, Theorem 1(i) isdue independently to Young, Schipp and Simon [Y,Sc,Si1, 1976].


3. Non-commutative Vilenkin systems

In order to establish a general framework, it will be convenient to recallsome basic facts concerning representations of compact Abelian groupson Banach spaces.

Let X be a Banach space, let G be a compact Abelian group with dual

group G and normalized Haar measure dm. Let = ftgt2G be anaction (strongly continuous representation) of G by automorphisms ofX. We set

c := supt2G

ktkX!X<1: (1:1)

For each 2 G, we dene the eigenspace X corresponding to andthe representation by setting

X := fx 2 X : t(x) = ht; ix; 8t 2 Gg:It should be noted that the eigenspace X may be f0g, that X \X 0 =

f0g if 6= 0, and that clmfX : 2 Gg = X:

It is important to note that if X = Lp(G; dm) and if is dened bysetting

t(x)(s) := x(s t); s; t 2 Gthen

X = fc : c 2 C g:In other words, the eigenspaces corresponding to the action of G onLp(G; dm) given by forward translation are precisely the one-dimensionaleigenspaces spanned by the characters of G.

The non-commutative framework that we require is provided by thewell-known theory of non-commutative integration with respect to asemi-nite trace, introduced by Segal [Se] and Dixmier[Di]. We sup-pose that (M; ) is a semi-nite von Neumann algebra equipped with afaithful, normal semi-nite trace , and unit 1. For relevant denitionssee [Ta]. A closed densely dened operator x is said to be aliated withM if ux = xu for any unitary u in the commutantM0 ofM. An opera-tor x aliated withM is said to be -measurable if (e(jxj)) <1 forsome > 0, where e(jxj) denotes the spectral projection (1;]

(jxj).If 1 p < 1, we denote by Lp(M; ) the space of all -measurableoperators x aliated with M for which

kxkp := (jxjp) 1p <1


where jxj = pxx. The precise denitions and relevant properties may

be found in [FK],[GK1]. Let be an action of the Vilenkin group Gm

on Lp(M; ). The system of eigenspacesnLp(M; )n : n 2 dGm

o

taken in the (reverse) lexicographical ordering will be called a Vilenkin

decomposition of Lp(M; ). An important special case is given when(1) < 1 and the representation is given by an ergodic Gm- ow,that is is an ultraweakly continuous group of trace preserving -automorphisms ofM, and which is ergodic in the sense that the xed-point algebra consists of scalar multiples of the identity (see [OPT]).In this case, extends to an isometric action on Lp(M; ) and eacheigenspace Lp(M; )n for the exended action coincides the correspond-ing eigenspace Mn for the original action of on M. Further, eacheigenspace Mn is the one-dimensional span of some unitary operatorWn 2 M and the sequence fWng is an orthonormal basis (in the nat-ural sense) in the space L2(M; ).

We consider the special case obtained by taking M to be L1(Gm; dt)acting by multiplication on L2(Gm; dt), with trace given by integrationwith respect to Lebesgue measure. If is given by right translation,

then the Vilenkin decomposition fMng = f n : n 2 dGmg is simplythat given by the classical Vilenkin basis. This observation shows thatTheorem 1 is actually a special case of the following Theorem.

Theorem 2 [DFPS] (i) If 1 < p <1, then each Vilenkin decom-

position fLp(M; )n;n 2 dGmg is a Schauder decomposition that is, for

all x 2 Lp(M; ) there exists a unique sequence xn 2 Lp(M; )n such

that x =Pn2cGm xn and there exists a constant Kp, depending only on

p and the bound c of the representation such that Xk<n

xk

p

Kpkxkp (2:1)

(ii) [SF1,2, FS] There exists a constant Kp depending only on p

such that for all k = 1, 1Xk=0

k

0@ Xdkn<dk+1

xn

1A p

Kpkxkp: (2:2)


(iii)[DS] If supm(k) <1, there exists a constant Kp depending onlyon p such that for all k;j = 1,

1Xk=0

m(k)1Xj=1

k;j

0@ Xdk;jn<dk;j+1

xn

1A p

Kpkxkp: (2:3)

We remark rst that the starting point for the proof of Theorem 2 maybe taken to be the assertion of (ii). This is a consequence of the factthat re exive non-commutative Lp-spaces have the (so-called) UMD-property combined with an application of the well- known transferenceprinciple. Details may be found in [SF1,2]. In the case of boundedVilenkin groups and nite von Neumann algebras, Theorem 2 was es-tablished on [DS]. The case that the von Neumann algebra has innitetrace follows from [DS] and [CPSW]. The general theorem is proved in[DFPS].

As observed above, Theorem 2 contains Theorem 1 as a special case.However, the proof of Theorem 2 is new even in its commutative spe-cialisation and is based on essentially non-commutative techniques. Toillustrate some of the ideas on which the proof of Theorem 2 is based,we will consider some very special cases.

We begin with the commutative example given by takingm=(n; 0; 0; : : :)for some natural number n 2 so that Gm may be identied with Zn.We take M to be L1(Zn) = l1n , where Zn is equipped with countingmeasure and acting on L2(Zn) = l2n by pointwise multiplication. We

identify k 2 cZn with the character k 2 L1(Zn) given by

k(j) = ("k)j; " = e2in ; j = 0; 1; : : : ; n 1

with " = e2in . The system fkgk2cZn is a vector basis of lpn. The

assertion that the system fkgk2cZn is a Schauder basis of lpn with basis

constant independent of n is the assertion that there exists a constantKp such that

klX

k=0

ckkkp Kpkn1Xk=0

ckkkp; l = 0; 1; : : : ; n 1; (2:4)

for all scalars c0; c1; : : : cn1. To see what is involved in the proof of

(2.4), we consider the representations ; of cZn;Zn respectively on l2ngiven by

(k) =n1Xi=0

("k)iei;i; (j) =n1Xi=0

ei;(i+j)mod n; k 2 cZn; j 2 Zn


Here fei;lg denotes the usual system of matrix units in Mn(C ). Note

that the linear span of the system (k); k 2 cZn is precisely the imageof L1(Zn), identied as acting by multiplication on L2(Zn). At thesame time, the mapping

Xk2Zn

ck(k)!Xk2cZn

ck(k)

is an algebraic, trace-preserving -isomorphism from L1(Zn) toMn(C ).This implies that

lXk=0

ckk

Lp(Zn)

=

lX

k=0

ck(k)

Cp

=

lX

k=0

ck(k)

Cp

for all 0 l n 1. Here k kCp

denotes the usual Schatten p-norm

on Mn(C ) given by

kxkpCp

= Tr(jxjp); x 2Mn(C ):

Accordingly to establish (2.4), it suces to show the existence of aconstant Kp, independent of n such that

lX

k=0

ck(k)

Cp

Kp

n1Xk=0

ck(k)

Cp

; 0 l n 1; (2:5)

A simple proof of the estimate (2.5) may be based on the well-knowntheorem of Macaev: if 1 < p < 1, then there exists a constant Kp,independent of n such that

kT (x)kCp CpkxkCp

where T is the operator of triangular truncation given by

T (n1Xi;j=0

xijei;j) =Xi<j

xijei;j;

for all x =Pn1

i;j=0 xijei;j 2Mn(C ) (see [GK1,2]). In the above argument,the use of the operator of triangular truncation introduces ideas thatare essentially non-commutative in order to prove an estimate in asetting that is commutative.

The general counterpart to the estimates (2.4), (2.5) given in the proofof Theorem 2(i) [DFPS] is based on the preceding ideas but imple-mented via the introduction of a generalised operator of triangulartruncation. Boundedness of this generalised truncation operator isproved by applying a very general version of the Macaev theorem dueto Zsido [Zs 80] (see also [DDPS]), which contains as a special case the


classical Riesz projection theorem, and which further implies the factthat replexive non-commutative Lp-spaces have the UMD -property.A further essential ingredient in the proof of Theorem 2 is the applica-tion of general non-commutative Khintchine inequalities [LP 86], [LPP91].

4. Non-commutative Examples

We will indicate here how the ideas of the preceding section yield anexplicit Schauder basis in the non-commutative Lp-spaces associatedwith the hypernite von Neumann II1 factor R. In a certain sense,this factor may be regarded in a natural manner as a non-commutativeextension of the commutative von Neumann algebra L1([0; 1]).

To set ideas, we consider a simple non-commutative example. We takem = (n; n; 0; 0; : : : ) so that Gm may be identied with ZnZn. We let

(M; ) be Mn(C ), equipped with normalised trace tr. If k 2 cZn, thenthe equality

k(s j) = hj; kik(s); s 2 Zn (3:1)

identies fkgk2cZn as a Vilenkin system for the action of Zn on L1(Zn)

by right translation. If we now identify L1(Zn) as the commutativediagonal subalgebra in Mn(C ), then the relation (3.1) in matrix formbecomes

(j)(k)(j) = hj; ki(k) (3:2)

for all j 2 Zn; k 2 cZn. These identities are of course easily checkedby direct calculation and are a special case of the (so-called) Weyl-Heisenberg relations. The equalities (3.2) identify the matrices f(k) :k 2 cZng as a system of eigenvectors for the action

x! (j)x(j); x 2Mn(C ):

A corresponding Vilenkin decomposition for this action is easily seento be

(Mn(C ))k = span f(k)(m) : 0 m n 1g :The equalities given in (3.2) may be written equivalently in the form

(k)(j)(k) = hj; ki(j) (3:3)

The equalities (3.3) identify the matrices f(j) : j 2 Zng as a systemof eigenvectors for the action

x! (k)x(k); x 2Mn(C ):

This yields a corresponding Vilenkin decomposition

(Mn(C ))j = span f(l)(j) : 0 l n 1g ; j 2 cZn


If we now successively apply (3.2),(3.3), we obtain that

(m)(l) ((k)(j))(l)(m) = hl; kihm; ji(k)(j) (3:4)

for all (l; m) 2 ZnZn and (k; j) 2 cZn cZn = \Zn Zn The equalities(3.4) identify the matrices f(k)(j) : (k; j) 2 \Zn Zng as a Vilenkinbasis in Lp(Mn(C ); tr) for the action

x! (m)(l)x(l)(m); x 2 Mn(C ):

We remark that span f(k)(0) : 0 k n 1g may be identied inthe obvious way with Lp(Zn). We summarise as follows.

Corollary 1 For every n 2 N , the system f(k)(j)g(k;j)2\ZnZn

,

taken in the reverse lexicographical order, forms a Schauder basis of

Lp(Mn(C ); tr) with basis constants independent of n.

To proceed further, identify L1[0; 1] with the space L1(Gm; dtm) inthe sense of measure spaces. We observe that the equality

L1(Gm; dtm) = L1(Zm(0)) L1(Zm(1)) holds in the sense of innite product of measure spaces. The hypernitevon Neumann II1 factor (R; ) is given by the tensor product

(R; ) = (Mm(0)(C ); tr) (Mm(1)(C ); tr) with product trace given by

(x0 x1 ) = tr(x0)tr(x1) Details of this construction may be found in [Sa]. Observe thatL1(Gm; dtm)may be identied with the commutative \diagonal "subalgebra

(Mdm(0)(C ); tr) (Md

m(1)(C ); tr) where Md

n(C ) denotes the subalgebra of Mn(C ) consisting of all diago-nal n n matrices.

If l = (l0; l1; : : : ); n = (n0; n1; : : : ) 2 dGm, we set

(l) = (l0) (l1) : : : ; (n) = (n0) (n1) : : :

It may be shown that the mapping

x! (l)(n)x(n)(l)

extends to an ergodicGmGm- ow onR, with eigenspace (Lp(R; ))(k;j)given by the one-dimensional span of the unitary operator (k)(j) for

each (k; j) 2 \Gm Gm = dGm dGm. We obtain the following


Corollary 2 The system of unitary operators f(k)(j)g(k;j)2 \GmGm

taken in the reverse lexicographic ordering, is a Schauder basis in

Lp(R; ); 1 < p <1:

We note nally that the basis exhibited in the preceding Corollarycontains the classical Vilenkin systems via the \diagonal"subalgebra

(k)(0);k 2 dGm:

References

[AVDR] G.N. Agaev, N.Ya. Vilenkin, G.M. Dzafarli and A.I. Rubinstein, Mul-

tiplicative systems and harmonic analysis on zero-dimensional groups,

Elm, (1981) (Russian).

[CPSW] Ph. Clement, B. de Pagter, F.A. Sukochev and H. Witvliet, Schauder

decompositions and multiplier theorems, Studia Math. 138 (2000), 135-

163.

[Di] J. Dixmier, Von Neumann algebras, North-Holland mathematical library

27, North-Holland publishing company, (1981).

[DDPS] P.G. Dodds, T.K. Dodds, B. de Pagter and F.A. Sukochev, Lipschitz

continuity of the absolute value and Riesz projections in symmetric op-

erator spaces, J. Functional Analysis 148 (1997), 28-69.

[DFPS] P.G. Dodds, S.V. Ferleger, B. de Pagter and F.A. Sukochev, Vilenkin

systems and a generalised triangular truncation operator, Integral Equa-

tions Operator Theory (to appear).

[DS] P.G. Dodds and F.A.Sukochev, Non-commutative bounded Vilenkin sys-

tems, Math. Scandinavica 87 (2000), 73-92.

[FK] T. Fack and H. Kosaki,Generalized s-numbers of -measurable operators,

Pacic J. Math. 123 (1986), 269-300.

[FS] S.V. Ferleger and F.A. Sukochev,On the contractibility to a point of

the linear groups of re exive non-commutative Lp-spaces, Math. Proc.

Camb. Phil. Soc. 119 (1996), 545-560.

[GK1] I.C. Gohberg and M.G. Krein, Introduction to the theory of linear non-

selfadjoint operators, American Mathematical Society (1969).

[GK2] I.C. Gohberg and M.G. Krein, Theory of Volterra operators in Hilbert

spaces, American Mathematical Society (1970).

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Trans. Amer. Math. Soc. 185 (1973), 345-370.

[LP] F. Lust-Piquard, Khintchine inequalities in Cp (1 < p < 1),

C.R.Acad.Sc.Paris 303 (1986), 289-292.

[LPP] F. Lust-Piquard and G. Pisier, Non-commutative Khintchine and Paley

inequalities, Ark. Mat. 29 (1991), 241-260.

[LT1] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence

spaces, Springer-Verlag, (1977).

[LT 2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function

spaces, Springer-Verlag, (1979).

[OPT] D. Olesen, G.K. Pedersen and M. Takesaki, Ergodic actions of compact

abelian groups, J. Operator Theory 3 (2) (1980), 237-271.


[Pa] R.E.A.C. Paley, A remarkable system of orthogonal functions, Proc.

Lond. Math. Soc. 34 (1932), 241-279.

[Sa] S.Sakai, C-algebras and W -algebras, Springer-Verlag, (1971).

[Se] I.E. Segal, A non-commutative extension of abstract integration,

Ann.Math. 57 (1953), 401-457.

[Sc] F. Schipp, On Lp-convergence of series with respect to product systems,

Analysis Mathematica, 2 (1976), 49-64.

[Si1] P. Simon, Verallgemeinerte Walsh-Fourierreihen. II, Acta Math. Acad.

Sci. Hungar. 27 (1976), 329-341.

[Si2] P. Simon, Remarks on Vilenkin bases, Annales Univ. Sci. Budapest.,

Sect. Comp. 16 (1996), 343-357.

[SF1] F.A. Sukochev and S.V. Ferleger, Harmonic analysis in symmetric spaces

of measurable operators, Russian Acad. Sci. Dokl. Math 50 (1995), no.

3, 432-437.

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application to the theory of bases, Mathematical Notes, 58 (1995), 1315-

1326.

[SWS] F. Schipp, W.R. Wade and P. Simon, Walsh Series: an introduction to

dyadic harmonic analysis, Adam Hilger, (1990).

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Transl. (2) 28, 1-35, (1963).

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approximation theory and applications (Aligarh, 1993), New Age, New

Delhi (1997), 339-369.

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(1958), 211-241.

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P. G. Dodds and F. A. Sukochev, Department of Mathematics and

Statistics, School of Informatics and Engineering, The Flinders Uni-

versity of South Australia, GPO Box 2100, Adelaide, SA 5001, Aus-

tralia

ORBITAL CONVOLUTIONS, WRAPPING MAPS AND

e-FUNCTIONS

A.H. DOOLEY

Abstract. We survey the theory of wrapping maps as applied

to compact groups and vector times compact semidirect products,

and give an explicit description of the e-function for compact sym-

metric spaces. The latter is globally dened.

1. Introduction

Let G be a Lie group. The Kirillov orbit method gives a heuristicmethod which relates the Euclidean Fourier transforms of coadjointorbits in g

to the innitesimal characters of the irreducible represen-tations. At its simplest, it has the following form

j(X)tr (exp X) =

ZO

ei(X)dO();(1.1)

where is an irreducible representation of G related to the coadjointorbit O, O is G-invariant Liouville measure on O and j is the squareroot of the Jacobian of the exponential map.In the case of a compact Lie group, this formula is exact | for other

groups, where is innite dimensional and O need not have compactsupport, the formula needs more careful interpretation | in general,it should be seen as an equality of distributions. The reader shouldconsult Kirillov's recent survey article [11] for a detailed discussion ofthe orbit method.In [4], the author and Norman Wildberger remarked that, for com-

pact groups, the Kirillov formula follows in a simple way from the factthat the wrapping map is a homomorphism of Banach algebras between(say)MG(g) andMG(G). HenceMG is the set of G-invariant measures(Ad-invariant on g and central on G). For an Ad-invariant distribution of compact support on g, let be a distribution on G dened forf 2 C1(G), by

h; fiG = h; j:f expig:

The wrapping formula then states

G = ( g ):(1.2)

42

ORBITAL CONVOLUTIONS, WRAPPING MAPS AND e-FUNCTIONS 43

From this formula, it follows that the adjoint mapping 0 :MG(G)0 !MG(g)0 is an injection of Gelfand spaces; thus, to each irreducible char-acter of G, one obtains a character of g averaged over adjoint orbits,that is, a mapping of the form 7!

R ROei(x)dO()d(x). From this

the Kirillov character (1.1) follows easily.The above theorem allows us to relate the central harmonic analysis

of G to Euclidean harmonic analysis of g. This becomes particularlyinteresting when one realizes that the latter can be described explicitly.In fact, in [3], we gave the following formula for Ad-invariant convo-lution on g. Recall that each adjoint orbit O intersects the positiveWeyl chamber t

+ of the Cartan subalgebra t in a unique point | say. Then we have

=

Zt+

N(; ; ) d;

where

N(; ; ) =Xw2W

sgnw ew T ();

T() being the projection on t of O, given by

T =1

jW j

Xw;w02W

sgnww0ewY2+

Fw0;

where F is the distribution on t given by Lebesgue measure on the raythrough .In work currently in progress, these results are being extended in two

directions | to some non-compact groups, and to compact symmetricspaces. I will describe these results in the next section and in section4.

2. Semi-direct product groups

In [6], we extend the wrapping map formula to G = V o K, V avector group, K a compact group. Here already, there is a substan-tial technical hurdle to be overcome, in the sense that formula (1.2)requires the convolution of Ad-invariant distributions (or measures) tobe dened, and there are no non-trivial G-invariant distributions ofcompact support, as the G-orbits in g (and in g

) are not compact.This problem can be overcome by the following device. Notice that

conjugacy classes, adjoint orbits and coadjoint orbits are all bredspaces over K-orbits.I will illustrate this for adjoint orbits only. For each A 2 k, split

V into two subspaces, VA = fv 2 V : A v = vg and V A = V ?A (the

44 A.H. DOOLEY

orthogonal complement with respect to a K-invariant inner product)| where A v is the derivative of the K-action. Then for v 2 V theorbit G A is bred over the compact orbit K v, the bre at k A beingV kA.Now we replace the G-invariant distributions of compact support on

G with a family of distributions on g = k + V which are K-invariant,compactly supported in the k-direction and for each A in this support,are given by an invariant mean (suitably normalized) on V A. It turnsout that such distributions:

(i) wrap to similarly dened distributions on conjugacy classes,(ii) belong to the dual of a suitable G-stable family of functions on g

| they are C1 in the k-direction and for each A 2 k, are almostperiodic in the V A direction,

(iii) have a natural notion of convolution (using the above duality ina standard way),

(iv) have as Fourier transforms, similar distributions on g.

The formula (1.2) continues to hold for V o K with the above de-nitions. This leads to a new proof of the Lipsman character theorem[13]. The full details are somewhat technical. We may interpret theabove results as follows. Each of the conjugacy classes, adjoint orbitsand coadjoint orbits possesses a natural convolution on hypergroupstructures. Denote the associated hypergroups as Conj, Adj and Coadjrespectively. Now provides a homomorphism : Adj ! Conj andwe may identify Coadj as the dual hypergroup of Adj. The Lipsmancharacter may thus be interpreted as the mapping 0 from

Conj ! Coadj = (Adj):

It is possible also to give very explicit formulae for the hypergroupstructures of Adj and Coadj: they are no longer identical, in contrastto the compact case. The gist of this structure is that the compactorbits in V (or V ) are convolved as in section 1, and one forms thesums of the bres. Full details are in [6].

3. Generalizations of the Duflo Isomorphism

The wrapping map formula (1.2) can be considered as a global ver-sion of the Du o isomorphism. To see this, notice that the Ad-invariantdistributions of support f0g in g wrap to central distributions of sup-port feg in G. The latter may be identied with zu(g), the centre ofthe universal enveloping algebra; the former with SG(g), the centre ofthe symmetric algebra of g.


Recently, far reaching generalizations of the Du o isomorphism havebeen proved by Maxim Kontseivich [12]. He proves that for a Poissonmanifold (X; ), there is a family r of star products which deformproducts (at r = 0) to convolutions (at r = 1).However, this series can only be shown to converge near f0g, and

it is of considerable interest to ask how far they can be extended, andif global versions such as (1.2) are available in special cases. Andler,Dvorsky and Sahi [1] recently showed, using [12], that every bi-invariantdierential operator on a Lie group is locally solvable.Actually, in the case where X = G=H, G a Lie group and H the set

of xed points of an involution , Kontseivich's construction coincideswith that of Rouviere [14].Rouviere's theory is as follows. We may write g = h+s, where h and

s are the eigenspaces of (by which I denote also the dierential of theinvolution, by abuse of notation) of eigenvalues +1 and 1 respectively.Then h is the Lie algebra ofH, and pmay be identied with the tangentspace of G=H. There is, furthermore, an exponential map Exp : p !

G=H. We take an H-invariant neighbourhood s of o in p on whichExp is a dieomorphism. For X 2 s, let j(X) be a suitable squareroot of J;0(Exp)(X). (We leave aside temporarily the existence of asmooth real-valued square root | in the case of interest below, it canbe explicitly calculated.) Then AdjH : p ! p and j(Ad(h)X) = j(X)for all x 2 p, h 2 H. We may thus dene a version of wrapping usingthe same ideas as above: for an H-invariant distribution (of compactsupport) on p, and for f 2 C1

c (G=H), let

h; fiG=H = h; j:f Expip:

If now ; are supported in s, and are such that p is also sup-ported in s, we can ask whether we have a formula such as \( p) =() G=H ()". It turns out that the formula requires some modi-cation and that it should read

( p;e ) = () G=H ():(3.1)

In this formula, e(X; Y ) is a certain function of two variables on s s,and p;e is \twisted" convolution given, for , H-invariant andlocally integrable, by

( p;e )(X) =

Zp

(Y )(X Y )e(X; Y )dY:(3.2)

(This formula needs an obvious adaptation in order for it to work fordistributions | see [14].)It is instructive to understand where the e-function comes from, as

we will be calculating it in some special cases in the next section. We

46 A.H. DOOLEY

write the right-hand side of (3.1), for u 2 C1(G=H), as

h() G=H (); ui =

ZG=H

ZG=H

()(x)()(y)u(xy)dxdy

=

Zs

Zs

(X)(Y )j(X)j(Y )u(ExpX ExpY )dXdY:

We now claim that there exist h; k 2 H so that ExpX Exp Y =Exp(h:X + k:Y ). Accepting this for the moment, consider the changeof variables (h:X; k:Y ) 7! (X; Y ). Denote the Jacobian of this changeof variables by (X; Y ). Then the integral transforms toZ

s

Zs

(h1X)(k1Y )j(h1X)j(k1Y ) (X; Y )u(Exp(X + Y ))dXdY:

Now j, and are all H-invariant, so we obtainZs

Zs

(X)(Y )j(X)j(Y )

j(X + Y ) (X; Y )(j:u Exp)(X + Y )dXdY:

Letting

e(X; Y ) =j(X)j(Y )

j(X + Y ) (X; Y );

we obtain the right-hand side.Rouviere is able to calculate e(X; Y ) as an innite power series, which

converges in a neighbourhood of o in s.To see why the elements h and k above exist, consider the Campbell-

Baker-Hausdor series for Exp.

ExpX Exp Y =Exp(X+Y +1

2[X; Y ] +

1

6([X[X; Y ]]+[Y [Y;X]])+ : : : ):

Now the hypothesis that is an involutive automorphism of g impliesthat [h; p] p, and [p; p] h. Thus we may rearrange the above seriesto get

X +1

6[[X; Y ]; X] + + Y

1

6[[X; Y ]; Y ] + + +H(X; Y )

= (I +1

6[X; Y ] + +)X + (I

1

6[X; Y ] + +)Y +H(X; Y );

= h:X + k:Y +H;

where H 2 h.The result now follows. (For a fuller proof, and for computations of

the series, see [14].)


4. Symmetric spaces of the compact type

The question I would like to address in this section is: can we nd aglobal version of Rouviere's formula in the case of a symmetric spaceof the compact type?Let (G;K) be a Riemannian symmetric pair of the compact type.

Then K is the set of xed points of the Cartan involution , andg = k + p is the Cartan decomposition. Choose a maximal abeliansubalgebra q of p and let A be the corresponding subgroup of G. Wethen have the Cartan decomposition of G, G = KA+K, where A+ =exp(a+). Here a+ is the positive Weyl chamber for a set of positiverestricted roots +

r . Let m denote the multiplicity of 2 +r .

We may identify p as the tangent space at eK of X = G=K, andhave the standard exponential map Exp : p ! X. The square root ofthe Jacobian of Exp is AdK invariant of p, and is given by

j(H) =Y2+

r

sin(H)

(H)

m=2

; (H) 2 a+:

For X; Y 2 p, let X = Ad (k1)H1, Y = Ad(k2)H2 and X + Y =Ad(k3)H3. We dene

e(X; Y )

= j(H1)j(H2)j(H3)

Q2+

r

Qw;w02Wr

cos 1

2((H1)+w(H2)+w

0

(H3))

(H1)+w(H2)+w0 (H3)

m=2

;

where Wr denotes the restricted Weyl group and w(H) is the imageof the root by the Wr-action. The following theorem then holds.Theorem (i) e is dened on all of p.(ii) Let and be K-invariant distributions of compact support on p.Then

X = ( p;e ):

Details of the proof of this theorem will appear in [7]. In order toprove it, we need to nd explicitly the hypergroup convolution of K-orbits in p and of K-orbits in X. The e-function is then found bycomparing the two structures.The gist of the calculation is already present in the case of the sphere

X = SO(3)=SO(2), which is discussed in [15]. We brie y describe thiscalculation here.Let us discuss the convolution of two circles (radii r1 and r2) in the

plane. (This corresponds to K-orbits in p.) One needs to write

r1 + r2ei = rei

48 A.H. DOOLEY

and then compute \dr"in terms of \d". (It is obvious that the resultingmeasure is rotationally invariant.) A little rst year calculus yields

2r

2r1r2 sin

dr

=d

and one identies the denominator on the left-hand side as the areaof a triangle in the complex plane with vertices at 0, r1 and r2e

i. By

Heron's formula, this is given also by [(r2(r1r2)2)((r1+r2)

2r2)]1

2 ,which we may write as Y

(r r1 r2)

1

2

;

where the product is over all choices of signs.Thus, the convolution of two circles of radius r1, r2 is a rotationally

invariant density given by

fr1;r2(r) =2rY

(r r1 r2)1

2

[jr1r2j;r1+r2]

(r):

If one now carries out the same calculation on the surface of the sphereS2 | there is a convenient version of Heron's formula for sphericalgeodesic triangles | one obtains the density function:

gr1;r2(r) =sin r

[(cos(r1 r2) cos r)(cos r cos(r1 + r2))]1

2

sin r1 sin r2

=1

sin r

sin r1 sin r2

Y

cos1

2(r r1 r2)

1

2

using the half-angle formula for cosine.The e-function is now given by the ratio g=f

e =sin r

sin r1 sin r2

Y

12cos 1

2(r r1 r2)

12(r r1 r2)

1

2

:

In essence, the proof for the symmetric space case is to reduce ev-erything to two dimensions and to use these elementary ideas.For the n-dimensional sphere, one obtains

e(X; Y ) =

"Y

2 cos(r r1 r2)=2

(r r1 r2)

#n3

2

(jr1r2j;r1+r2)

(r):

These formulae will have interesting consequences for harmonic anal-ysis | for example in nding fundamental solutions of K-invariantdierential operators on X.


One can also prove a compact version of the Gindikin-Karpilevicformula for the Harish-Chandra c-function.

c() =Y2+

r

c();

where the right-hand side is the c-function for each of the symmetricspaces G=K, 2 +

r . It seems most likely that the proof will alsogo through in the non-compact case and yield an elementary proof ofthis formula.

References

[1] M. Andler, A. Dvorsky, S. Sahi, Deformation quantization and invariant dis-

tributions, E-print Math 9.9./9905065, 12 May 1999

[2] K.H. Chung, Compact group actions and harmonic analysis, thesis, UNSW

1999

[3] A.H. Dooley, N.J. Wildberger and J. Repka, Sums of adjoint orbits, Lin.

Multilin. Alg. 36 (1993), 79-101

[4] A.H. Dooley and N.J. Wildberger, Harmonic analysis and the global expo-

nential map for compact Lie groups, Funct. Anal. Appl. 27 (1993), 25-32

[5] A.H. Dooley and N.J. Wildberger, Global character formulae for compact Lie

groups, Trans. Amer. Math. Soc. 351 (1999), 477-495

[6] A.H. Dooley and N.J. Wildberger, Orbital convolutions for semi-direct prod-

ucts, in preparation

[7] A.H. Dooley, Global versions of the e-function for compact symmetric spaces,

in preparation.

[8] M. Du o, Caracteres des algebres de Lie resolubles, C.R. Acad. Sci. (Paris)

269 (1969) serie a, 437-438

[9] S. Helgason, Groups and Geometric Analysis, Academic Press (1984)

[10] M. Kashiwara, M. Vergne, The Campbell-Hausdor formula and invariant

hyperfunctions, Invent. Math. 47 (1978), 249-272

[11] A. Kirillov, Merits and Demerits of the Orbit method, Bull. Amer. Math.

Soc. 36 (1999), 433-488

[12] M. Kontsevich, Deformation Quantization of Poisson manifolds I, EIMS q-

alg/9709040 (1997)

[13] R. Lipsman, Orbit theory and harmonic analysis on Lie groups with cocom-

pact nilradical, J. Math. Pures et Appl. 59 (1980), 337-374.

[14] F. Rouviere, Invariant analysis and contractions of symmetric spaces, Com-

positio Math. 80 (1991), 11-136

[15] N.J. Wildberger, Hypergroups, symmetric spaces and wrapping maps, in

Probability Measures on Groups and related Structures, XI, World Scientic

Singapore, 1995, 406-425

School of Mathematics, UNSW Sydney NSW 2052, Australia

NORMS OF 01 MATRICES IN CP

IAN DOUST

To Derek Robinson on his 65th birthday

Abstract. We announce a new result (proved in collaboration

with T.A. Gillespie) on the boundedness of a class of Schur mul-

tiplier projections on the von Neumann-Schatten ideals Cp. We

also show that for 1 p 2 the average Cp norm of a 01 matrix

grows just as quickly as the largest norm of such a matrix.

1. Introduction

Calculating the norm of an operator which acts on one of the von Neu-mann Schatten ideals Cp is often rather dicult | even for rather sim-ple operators | because of the scarcity of elements T 2 Cp for whichone can easily calculate (or even approximate) kTk

p= kTk

Cp. Even

for the algebraically simple Schur projections which act by replacingcertain xed entries of the matrices of elements of Cp by 0, provingboundedness results is typically quite hard.To x some notation, for 1 p < 1, let Cp denote the von Neu-

mann Schatten ideal of compact operators on `2, with norm kTkp=

trace((T T )p=2)1=p. We take C1

to be the set of all compact operatorson `2 with the usual operator norm. We will let Cn

pdenote the n n

matrices equipped with the corresponding norm. We will, as usual,

think of elements of Cp as being innite matrices.Let Z denote the set of all zero-one arrays [ai;j]

1

i;j=1. We are inter-

ested in the norms of projections dened by Schur multiplication ofsuch arrays. If A = [ai;j] 2 Z, dene the Schur projection corre-

sponding to A to be the map PA : T 7! A T , where denotes Schuror elementwise multiplication of matrices. Let

Bp = fA 2 Z : PA 2 B(Cp)g :

The set of n n matrices with zero-one entries will be denoted by Zn.By requiring that each entry in an array is either zero or one with equal

2000 Mathematics Subject Classication. Primary 47B10. Secondary 43A22,

46B20, 47B49.

Key words and phrases. Von Neumann-Schatten ideals of compact operators,

Schur multiplier projections.50

NORMS OF 01 MATRICES IN CP 51

probability we may regard Z and Zn as being probability spaces withthe appropriate product measures.

If 1 p <1 and 1

p+ 1

q= 1, then C

p= Cq, under the natural pairing

hS; T i = trace(ST ). If A 2 Bp then trace(PA(S)T ) = trace(SPA(T ))

for all S 2 Cp, T 2 Cq and so it follows that kPAkp = kPAkq. In

particular note that B2 = Z with kPAkp = 1 for all A 2 Z and that

Bp = Bq.It is a trivial consequence of the ideal inequalities for Cp that ifA 2 Z

is a nonzero array which is constant on each row (or on each column)then kPAkp = 1 for all p. Proving boundedness for other types of arrays

has been signicantly more dicult. The rst major result was due toMacaev (see [7]) who showed that the upper triangular truncation mapis bounded on Cp for 1 < p <1 (but not for p = 1 or p =1). In the1980s Bourgain [3] showed boundedness results for a class of `Toeplitz'

arrays which are analogous to multiplier results from Fourier analysis.In particular, for 1 < p <1 there is a constant Kp such that if A 2 Zis any array which is constant on diadic blocks of (long) diagonals, thenkPAkp Kp.

It is still an open question as to whether if 2 < r < p < 1 there isalways a Schur multiplier projection which is bounded on Cr, but noton Cp. An important special case when r and p are even integers hasrecently been solved (in the armative) by A. Harcharras [8].

Recently, Alastair Gillespie and I have proved boundedness for a newclass of such projections.

Denition 1.1. A zero-one array [aij] will be said to be obtainableif for all indices i1; i2; j1; j2,

ai1j1 ai1j2ai2j1 ai2j2

6=

1 0

0 1

:

Theorem 1.2. For all p 2 (1;1) there exists a constant Kp such that

for every obtainable array A, kPAkp Kp.

The proof, which depends on the fact that for 1 < p < 1, Cp is a

UMD space, uses a new result from spectral theory proved in [5]; thesum of two commuting real scalar-type spectral operators on a UMDspace is a well-bounded operator. (Since completing this work we havebeen made aware of some closely related work by Clement, de Pagter,

Sukochev and Witvliet [4] [11] [12].)As we show in [6], it is relatively easy to use this result to recover

Bourgain's theorem (at least as it applies to 01 arrays). One can alsoprove Littlewood-Paley type decomposition results in Cp, of which we

shall just give two simple examples here.

52 IAN DOUST

Split Z+Z+ into rectangular subarrays as in either of the diagramsbelow.

(i)

0BBBBBBBB@

B0 B1

B4 B7B2 B3

B5 B6

B8 B9

: : :

.... . .

1CCCCCCCCA

(ii)

0BBBBBBBB@

B0 B1

B2

B3 B4

B5

B6 B7

B8B9

: : :

.... . .

1CCCCCCCCA

The actual sizes of the subarrays is not important.

Let Pk denote the projection given by Schur multiplication by thecharacteristic function of Bk.

Theorem 1.3. Suppose that 1 < p <1 and that ; 6= J N . ThenPk2J

Pk converges in the strong operator topology and Xk2J

Pk

p

2Kp + 1

where Kp is the constant from Theorem 1.2.

Actually, if all the `diagonal' subarrays B3k are just 1 1 subarrays,then this result is an easy consequence of Macaev's result. Even for themore general splittings of this form described above, it is probably nottoo hard to deduce this bound from known estimates (see, for example,

Section 4 of [1]). Our techniques however cover a rather wider range ofdecompositions including some which are not at all related to triangulartrunctions. Details will appear in [6].

2. Norms of 01 matrices

In proving the above results we were lead to considering what one

can say about the bounds of Schur multiplier projections on Cn

p. The

following example is the standard way of showing that these projectionscan have bad norms.

Example 2.1. Let B1 =

1 11 1

. For m 2, dene Bm = Bm1

B1. For example

B2 =

0BB@

1 1 1 11 1 1 11 1 1 11 1 1 1

1CCA :


Thus Bm is a 2m 2m orthogonal matrix. Let n = 2m. Then BmB

is just nI. Clearly then kBmkp = (nnp=2)1=p = n1

2+

1

p . Let Om be the

nn matrix all of whose entries are 1. This is just n times a projection,

so kOmkp = n for all p. Let Am be the nn array formed by replacing

the 1's in the matrix Bm by zeros. Thus Am = 1

2(Bm + Om). Note

that Am = PAmOm. If 1 p < 2 then

kPAmkp kAmkpkOmkp

1

2

n(

1

2+

1

p) n

n

=1

2n(

1

p

1

2)

1

2:

Using the earlier remarks about duality we see that there exists c > 0

such that for 1 p 1 and large n,

kPAmkp cnj1

2

1

pj:

2

This order of growth is the worst that one can get from nnmatrices.

Theorem 6.2 of [2] shows that for any A 2 Zn, kPAk1

n1=2. Using

interpolation and duality gives that kPAkp nj1

2

1

pj.This upper bound

also follows from the following upper bound due to Ong [10]:

kPAk1

minmax `2 norm of a column; max `2 norm of a row

:

We shall say that A1 2 Zn is a subarray of A 2 Z if A1 is formedby deleting all but n of the rows and all but n of the columns of A.If A1 is a subarray of A then kPA1

kp kPAkp. With probability one,

a randomly chosen innite string of binary digits contains all nitestrings as substrings. In other words, with probability one, an elementof Z contains every array Am (from Example 2.1 above) as a subarray.It follows immediately that if p 6= 2 then Prob(kPAk <1) = 0.

A more delicate question concerns whether this `bad' array is typicalof n n arrays. The answer is Yes. This is undoubtedly known tothe experts, although as far as we are aware, this does not appear inthe literature. (Indeed, throughout this area, much more seems to be

known than is written down!)To simplify notation we shall write g(n) f(n) if there exist con-

stants 0 < c1 < c2 < 1 such that c1f(n) < g(n) < c2f(n) for all(suciently large) n. Throughout we shall use c as a generic absoluteconstant whose value may change from one line to the next.

Theorem 2.2. E (kPAkp : A 2 Zn) nj1

2

1

pj.

Proof. For each n let Xn denote the matrix-valued random vari-

able where each entry is an independent Gaussian variable with mean0 and variance 1. That is, Xn =

Pn

i;j=1gijEij where each gij is an

54 IAN DOUST

independent N(0; 1) random variable and fEijgn

i;j=1are the standard

matrix units. Let frijgn

i;j=1be a family of independent identically dis-

tributed random variables which take the values 1 and 1 with equalprobability.Suppose rst that 1 p 2. Some rather deep estimates of Szarek

[14] give that

E (kXnkp) n1

2+

1

p :

Then, since Cp has cotype 2 for p in this range, [13, Theorem 3.9]implies that

cn1

2+

1

p E

nX

i;j=1

gijEij

E

nX

i;j=1

gijEij

21=2

c

E

nX

i;j=1

rijEij

21=2

But by Kahane's inequality [9, Theorem 1.e.13]

E

nX

i;j=1

rijEij

21=2 c E

nX

i;j=1

rijEij

:Following the same argument as in Example 2.1,

E (kAkp: A 2 Zn)

1

2

E

nX

i;j=1

rijEij

n

c n

1

2+

1

p ;

(for large n) and so

E (kPAkp : A 2 Zn) cn1

p

1

2 :

Since for any A 2 Zn, kPAkp n1

p

1

2 the result is proved. The case

when p > 2 follows from duality. 2

I would like to thank Alastair Gillespie and Quanhua Xu for several

interesting discussions on these matters, and the referee for bringing

my attention to some additional references.

References

[1] Arazy, J., Some remarks on interpolation theorems and the boundedness of

the triangular projection in unitary matrix spaces, Integral Equations Oper-

ator Theory 1 (1978) 453495.

[2] Bennett, G., Schur multipliers, Duke Math. J. 44 (1977), 603639.


[3] Bourgain, J., Vector-valued singular integrals and the H1-BMO duality, in:

Probability Theory and Harmonic Analysis (Mini-conference on Probability

Theory and Harmonic Analysis, Cleveland, 1983), J.-A. Chao and W.A. Woy-

czynski (eds), Monographs and Textbooks in Pure and Appl. Math. 98, Mar-

cel Dekker, New York, 1986, 1-19.

[4] Clement, P., de Pagter, B., Sukochev, F.A. and Witvliet, H., Schauder de-

composition and multiplier theorems, Studia Math. 138 (2000), 135163.

[5] Doust, I. and Gillespie, T. A., Well-boundedness of sums and products of

operators, in preparation.

[6] Doust, I. and Gillespie, T. A., Schur multipliers on Cp spaces, in preparation.

[7] Gohberg, I. C. and Kren, M. G., Introduction to the theory of linear non-

selfadjoint operators, Translations of Mathematical Monographs, Vol. 18,

American Mathematical Society, Providence, R.I. 1969.

[8] Harcharras, A., Fourier analysis, Schur multipliers on Sp and non-commut-

ative (p)-sets, Studia Math. 137 (1999), 203260.

[9] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II, Function spaces,

Springer Verlag, BerlinNew York, 1979.

[10] Ong, S.C., On the Schur multiplier norm of matrices, Lin. Algebra Appl. 56

(1984), 4555.

[11] de Pagter, B., Sukochev, F.A. and Witvliet, H., Unconditional decompo-

sitions and Schur-type multipliers, to appear in Proceedings of the 1998

IWOTA conference in Groningen, Birkhauser.

[12] de Pagter, B. and Witvliet, H., Unconditional decompositions and UMD-

spaces, in: Semi-groupes d'operateurs et calcul fonctionnel, Ecole d'Ete (June

1998), Publications Mathematiques de l'UFR Sciences et Techniques de Be-

sancon, Vol. 16, Besancon, 1998, 79111.

[13] Pisier, G., Factorization of linear operators and geometry of Banach spaces,

CBMS Regional Conference Series in Mathematics, 60, American Mathemat-

ical Society, Providence, R.I., 1986

[14] Szarek, S.J., Condition numbers of random matrices. J. Complexity 7 (1991),

131149.

School of Mathematics, University of New South Wales,

Sydney, NSW 2052, Australia

E-mail address: [email protected]

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT

OPERATORS

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

Abstract. In this article, we give a survey of spectral multipliers

and present (without proof) sharp Hormander-type multiplier the-

orems for a self adjoint operatorA under the assumption that A has

Gaussian heat kernel bounds and satises appropriate estimates of

the L2 norm of the kernels of spectral multipliers. Our theorems

imply several important, previously known results on spectral mul-

tipliers and give new results for sharp estimates for the critical

exponent for the Riesz means summability.

1. Introduction

This paper contains discussion and survey of the topic of spectralmultipliers and main results of [DOS] without giving proofs. Readersare referred to [DOS] for their proofs and more applications.Suppose that A is a positive denite self-adjoint operator acting on

L2(X), where X is a measure space. Such an operator admits a spectraldecomposition EA() and for any bounded Borel function F : [0;1)!C, we dene the operator F (A) by the formula

F (A) =

Z 1

0

F () dEA():(1)

By the spectral theorem the operator F (A) is continuous on L2(X).Spectral multiplier theorems investigate sucient conditions on func-tion F which ensure that the operator F (A) extends to a boundedoperator on Lq for some q, 1 q 1.Spectral multiplier has been a very active topic of Harmonic anal-

ysis. Roughly, it is sucient that F is dierentiable to some orderwith appropriate bounds on its derivatives. The more information weknow about the underlying space X and the operator A, the sharpermultiplier results can be obtained, i.e. less derivatives on F are neededfor F (A) to be bounded. For example, when X is the Euclidean space

Rd and A is the Laplacian d =Pd

k=1@2k a sucient condition is that

F possesses [d=2] + 1 derivatives which satisfy certain size estimateswhere [d=2] denotes the integral part of d=2. Recent results extend

56

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS 57

this to more general underlying spaces and more general self adjointoperators. For example, see, [He3, He2, DO, CS, Ale2] when A is anabstract positive self-adjoint operator which has heat kernel bounds (ornite propagation speed) and the underlying space X satises doublingvolume property. (See Assumption 2.1).Let us discuss two important examples of spectral multiplier theo-

rems concerning group invariant Laplace operators acting on Lie groupsof polynomial growth. Let G be a Lie group of polynomial growth andlet X1; : : : ; Xk be a system of left-invariant vector elds on a G satis-fying the Hormander condition. We dene Laplace operator L actingon L2(G) by the formula

L =

kXi=1

X2i :(2)

If B(x; r) is a ball dened by the distance associated with systemX1; : : : ; Xk (see e.g. [VSC, xIII.4]), then there exist natural numbersd0; d1 0 such that (B(x; r)) rd0 for r 1 and (B(x; r)) rd1

for r > 1 (see e.g. [VSC, xVIII.2]). We call G a homogeneous group ifthere exists a family of dilations on G. A family of dilations on a Lie

group G is a one-parameter group (~t)t>0 (~t ~s = ~ts) of automor-phisms of G determined by

~tYj = tdjYj;(3)

where Y1; : : : ; Yl is a linear basis of Lie algebra of G and dj 1 for1 j l (see [FS]). We say that an operator L dened by (2) is

homogeneous if ~tXi = tXi for 1 i k. For a homogeneous Laplace

operator d0 = d1 =Pl

j=1dj (see [FS]).

Spectral multiplier theorems for homogeneous Laplace operators act-ing on homogeneous groups were investigated by Hulanicki and Stein[HS] (see also [FS, Theorem 6.25]), De Michele and Mauceri [dMM].The following theorem was obtained independently by Christ [Ch2] andMauceri and Meda [MM]. See also [Si3]. Its proof relies on heat kernelbounds, L2 estimates from Plancherel theorems, translation and dila-tion invariant structures of homogeneous groups, Calderon Zygmundoperator theory and interpolation theory.

Theorem 1. Let L be a homogeneous operator dened by the formula

(2) acting on a homogeneous group G. Denote by d = d0 = d1 ho-mogeneous dimension of the underlying group G. Next suppose thats > d=2 and that F : [0;1)! C is a bounded Borel function such that

supt>0

k tFkW 2s<1;(4)

58 XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

where tF () = F (t), kFkW ps= k(I d2= dx2)s=2FkLp and 2

C1c (R+) is a xed function, not identically zero. Then F (L) is of

weak type (1; 1) and bounded on Lq when 1 < q <1.

Note that condition (4) is independent of the choice of 2 C1c (R+).

The Hormander multiplier theorem describes the Fourier multiplieron Rd (see [Ho1]). If we apply Theorem 1 to Rd we obtain a resultequivalent to the Hormander multiplier theorem restricted to radialFourier multipliers. Therefore we call Theorem 1 the Hormander-typemultiplier theorem and condition (4) the Hormander-type condition.Theorem 1.1 is optimal for a general homogeneous group, see esti-

mate 1.6. However, for specic groups such as Heisenberg and relatedgroups, it is possible to obtain multiplier results where the number ofderivatives needed is roughly half of the topological dimension n. Oftenthe homogeneous dimension d is strictly greater than the topologicaldimension n, hence the number of derivatives needed could be less thand=2. See for example, [MS, He4, Du, CS].In the setting of general Lie groups of polynomial growth spectral

multipliers were investigated by Alexopoulos. Note that in this setting,the local dimension d0 and dimension at innity d1 are dierent ingeneral and the group G does not have dilation invariants as in thecase of a homogeneous group. Using nite propagation speed property,estimates on upper bounds of heat kernels and their space gradients,Alexopoulos proved the following (see [Ale1]).

Theorem 2. Let L be a group invariant operator acting on a Liegroup of polynomial growth dened by (2). Suppose that s > d=2 =max(d0; d1)=2 and that F : [0;1) ! C is a bounded Borel functionsuch that

supt>0

k tFkW1

s<1;(5)

where tF () = F (t) and kFkW ps= k(I d2= dx2)s=2FkLp. Then

F (L) is of weak type (1; 1) and bounded on Lq when 1 < q <1.

Condition (5) is also independent of the choice of . We note thatTheorem 1.2 does not appear exactly the same but is essentially equiv-alent to the results of [Ale1].In [He3] Hebisch extended Theorem 2 to a class of abstract operators

acting on spaces satisfying \doubling condition" (see also [Ale2]). Theorder of dierentiability in the Alexopoulos-Hebisch multiplier theoremis optimal. This means that for any s < d=2 we can nd a functionF such that F satises condition (5) but F (A) is not of weak type(1; 1). Indeed, let A be a uniformly elliptic, self-adjoint second-order


dierential operator on Rd, e.g. A = d, where d is the standardLaplace operator. One can prove that

C1(1 + jj)d=2 kAikL1!L1;1 C2(1 + jj)d=2(6)

(see [SW]). (See also [St1, pp. 52] and Christ [Ch2]). However, if weput F() = jji, then

C 01(1 + jj)s=2 sup

t>0

ktFkW1

s C 0

2(1 + jj)s=2:

Therefore for any s < d=2 Theorem 2 does not hold.If X is a manifold with exponential volume growth, i.e. V (x; r)

cekr and L is the Laplace-Beltrami operator, spectral multipliers wasinvestigate by M. Taylor in [Tay] where it was shown that a sucient

condition is that F is holomorphic on a strip of width k for F (pL) to

be bounded on Lp for 1 < p < 1. For more specic spaces such ascertain Iwasawa AN groups, see [He5, CGHM] where it was shown thatonly a nite number of derivatives are required for F (L) to be boundedon L1 or to be of weak type (1,1). Note that the exponential volumegrowth is like the dimension d =1.The theory of spectral multipliers is related to and motivated by

the study of convergence of the Riesz means or convergence of othereigenfunction expansions of self-adjoint operators. To dene the Rieszmeans of the operator A we put

R() =

(1 =R) for R

0 for > R:(7)

We then dene the operator R(A) using (1). We call R(A) the Rieszor the Bochner-Riesz means of order . The basic question in thetheory of Riesz means is to establish the critical exponent for the con-tinuity and convergence of the Riesz means. More precisely we wantto study the optimal range of for which the Riesz means R(A) areuniformly bounded on L1(X) (or other Lq(X) spaces).Since the publication of Riesz paper [Ri] the summability of the Riesz

means has been one of the most fundamental problems in HarmonicAnalysis (see e.g. [St2, IX.2 and xIX.6B]). Despite the fact that theRiesz means have been extensively studied we do not have the fulldescription of the optimal range of even if we study only the spaceL1(X). On one hand we know that for the Laplace operator d =Pd

k=1@2k acting on Rd and the Laplace-Beltrami operator acting on

compact d-dimensional Riemannian manifolds the critical exponent isequal (d1)=2 (see [So1]). This means that Riesz means are uniformlycontinuous on L1(X) if and only if > (d 1)=2 (see also [ChS, Ta]).On the other hand, if we consider more general operators like e.g.


uniformly elliptic operators on Rd it is only known that Riesz meansare uniformly continuous on L1(X) if > d=2 (see [He1]). One ofthe main points of our work is to investigate the summability of Rieszmeans for d=2 > (d 1)=2.The Alexopoulos-Hebisch multiplier theorem discussed above gives

optimal value for the exponent d=2 of the number of derivatives neededin spectral multipliers, but it does not give the optimal range of theexponent for the Riesz summability. Indeed, if k1 kW1

s< 1, then

s. However, kR1kW 2

s<1 if and only if > s 1=2. This means

that in virtue of Theorem 2 one obtains uniform continuity of Rieszmeans on Lq for any > d=2 and for all q 2 (1;1), whereas Theorem 1shows Riesz summability for > (d1)=2 (see also [Ch2, pp. 74]). Aswe mentioned earlier (d1)=2 is a critical index for Riesz summabilityfor standard Laplace operator on Rd and Laplace-Beltrami operatoron compact manifolds. To conclude we see that the optimal numberof derivatives in multiplier theorems is d=2. However, in condition (5)we required d=2 derivatives in L1. In the Hormander-type condition(4) we required d=2 derivatives in L2. Note that functions tF arecompactly supported so condition (5) is strictly stronger than (4).We would like to investigate when it is possible to replace condi-

tion (5) in the Alexopoulos-Hebisch multiplier theorem by condition(4) from Theorem 1. As we investigate spectral multipliers in a generalsetting of abstract operators rather than in a specic setting of groupinvariant operators acting on Lie groups, we do not have certain es-timates which are consequences of invariant structures of Lie groups.Also we only assume suitable bounds on heat kernels but not pointwisebounds on their space derivatives.The subject of Bochner-Riesz means and spectral multipliers is so

broad that it is impossible to provide comprehensive bibliography of ithere. Hence we quote only papers directly related to our investigationand refer reader to [Ale1, Ch2, Ch1, ChS, CS, dMM, Du, He1, He3, Ho1,Ho3, HS, MM, SeSo, So1, So2, Si2, St1, St2, Ta] and their references.

2. Main results

In this section we rst introduce some notation and describe thehypotheses of our operators and underlying spaces. We then state ourmain results.

Assumption 1. Let X be an open subset of eX, where eX is a topo-logical space equipped with a Borel measure and a distance : Let

B(x; r) = fy 2 eX; (x; y) < rg be the open ball (of eX) with centre at

x and radius r. We suppose throughout that eX satises the doubling


property, i.e., there exists a constant C such that

(B(x; 2r)) C(B(x; r)) 8x 2 eX; 8r > 0:(8)

Note that (8) implies that there exist positive constants C and d such

that

(B(x; r)) C(1 + )d(B(x; r)) 8 > 0; x 2 eX; r > 0:(9)

In a sequel we always assume that (9) holds.

We state our results in terms of the value d in (9). Of course for anyd0 d (9) also holds. However, the smaller d the stronger multipliertheorem we will be able to obtain. Therefore we want to take d assmall as possible. Note that in the case of the group of polynomialgrowth the smallest possible d in (9) is equal to max(d0; d1). Henceour notation is consistent with statements of Theorems 1 and 2.Note that we do not assume that X satises doubling property. This

poses certain diculties which we overcome by using results of singularintegral operators of [DM]. An example of such a space X is a domainof Euclidean space Rd. If we do not assume any smoothness on itsboundary, then doubling property fails in general.Now we describe the notion of the kernel of the operator. Suppose

thatT : L1(X; ) ! Lq(X; ) for q > 1. Then by KT (x; y) we denote thekernel of the operator T dened by the formula

hTf1; f2i =ZX

Tf1f2 d =

ZX

KT (x; y)f1(y)f2(x) d(x) d(y):(10)

for all f1; f2 2 Cc(X). Note that

kTkL1(X;)!Lq(X;) = supy2X

kKT ( ; y)kLq(X;):

Hence if kTkL1(X;)!Lq(X;) < 1, then its kernel KT is a well denedmeasurable function. Vice versa, if supy2X kKT ( ; y)kLq(X;) < 1,

then KT is a kernel of the bounded operator T : L1(X; )! Lq(X; ),even if q = 1.Next we denote the weak type (1; 1) norm of an operator T on a

measure space (X; ) by kTkL1(X;)!L1;1(X;) = sup (fx 2 X :jTf(x)j > g), where the supremum is taken over > 0 and func-tions f with L1(X; ) norm less than one.

Assumption 2. Let A be a self-adjoint positive denite operator. We

suppose that the semigroup generated by A on L2 has kernel


pt(x; y) = Kexp(tA)(x; y) which for all t > 0 satises the followingGaussian upper bound

jpt(x; y)j C(B(y; t))1=m exp b

(x; y)m=(m1)

t1=(m1)

(11)

where C; b and m are positive constants and m 2.

Such estimates are typical Gaussian estimates for elliptic or sub-elliptic dierential operators of order m (see e.g. [Da1, Ro, VSC]). Wewill call pt(x; y) the heat kernel associated with A. When order m = 2,Gaussian estimates (2.4) is equivalent to nite propagation speed, see[Si1]. When m 6= 2, we can have (2.4) but nite propagation speedproperty does not hold.In our following main results, we suppose that Assumptions 1 and 2

hold. The values d and m always refer to (9) and (11).

Theorem 3. Suppose that s > d=2 and assume that for any R > 0and all Borel functions F such that suppF [0; R]Z

X

jKF (mpA)(x; y)j2 d(x) C(B(y; R1))1kRFk2Lp(12)

for some p 2 [2;1]. Then for any Borel bounded function F such thatsupt>0

ktFkW ps< 1 the operator F (A) is of weak type (1; 1) and is

bounded on Lq(X) for all 1 < q <1. In addition

kF (A)kL1(X;)!L1;1(X;) Cs

supt>0

ktFkW ps+ jF (0)j

:(13)

Note that if (12) holds for p <1, then the pointwise spectrum of Ais empty. Indeed, for all p <1 and all y 2 X

0 = Ckf1=2gkLp = Ck2afagkLp (B(y;1

2a)kK

fag(mpA)( ; y)kL2(X;)

(14)

so fag(mpA) = 0. Hence for elliptic operators on compact manifolds,

(12) cannot be true for any p <1. To be able to study these operatorsas well we introduce some variation of assumption (12). Following [CS]for a Borel function F such that suppF [1; 2] we dene the normkFkN;p by the formula

kFkN;p =1

N

2NXl=1N

sup2[ l1

N; lN)

jF ()jp1=p

;

where p 2 [1;1) and N 2 Z+. For p = 1 we put kFkN;1 = kFkL1.It is obvious that kFkN;p increases monotonically in p. The next the-orem is a variation of Theorem 3. This variation can be used in the


case of operators with nonempty pointwise spectrum (compare [CS,Theorem 3.6]).

Theorem 4. Suppose that is a xed natural number, s > d=2 and

that for any N 2 Z+ and for all Borel functions F such that suppF [1; N + 1]Z

X

jKF (mpA)( ; y)j2 d(x) C(B(y; 1=N))1kNFk2N;p(15)

for some p 2. In addition we assume that for any " > 0 there existsa constant C" such that for all N 2 Z+ and all Borel functions F suchthat suppF [1; N + 1]

kF ( mpA)k2L1(X;)!L1(X;) C"N

d+"kNFk2N;p:(16)

Then for any Borel bounded function F such that supt>1 ktFkW ps<1

the operator F (A) is of weak type (1; 1) and is bounded on Lq(X) forall q 2 (1;1). In addition

kF (A)kL1(X;)!L1;1(X;) Cs

supt>1

ktFkW ps+ kFkL1

:(17)

Remarks 1. It is straightforward that (12) always holds with p =1as a consequence of spectral theory. This means that Alexopoulos' mul-tiplier theorem i.e. Theorem 2 follows from Theorem 3. Theorem 1 alsofollows from Theorem 3. Indeed, it is easy to check that for homoge-neous operators (12) holds for p = 2 (see Section 6 [DOS] or [Ch2,Proposition 3]).2. The main point of our theorems is that if one can obtain (12)

or (15) then one can prove stronger multiplier results. If one shows(12) or (15) for p = 2, then this implies the sharp Hormander-typemultiplier result. Actually we believe that to obtain any sharp spectralmultiplier theorem one has to investigate conditions of the same typeas (12) or (15), i.e. conditions which allow us to estimate the normkKF (

mpA)( ; y)kL2(X;) in terms of some kind of Lp norm of the function

F .3. We call hypotheses (12) or (15) the Plancherel estimates or the

Plancherel conditions. In the proof of Theorems 3 and 4 one does nothave to assume that p 2 in estimates (12) or (15). However (12) or(15) for p < 2 would imply Riesz summability for < (d 1)=2 andwe do not expect such a situation.Note that (12) is weaker than (15) and we need additional hypothesis

(16) in this case. However, in practice once (15) is proved, (16) isusually easy to check and we can often put " = 0.


4. We conclude this paper with a theorem on Riesz summability ford=2 > (d1)=2. Theorem 3 with p = 2 implies Riesz summabilityfor all > (d1)=2 and that in addition it seems that Theorem 3 withp = 2 is essentially stronger than sharp Riesz summability. However,one can obtain only weak type (1; 1) estimates in virtue of Theorem 3and formally Theorem 3 does not imply continuity and convergence ofRiesz means on L1(X; ). However, Theorem 3 and 4 can be modiedto prove that uniform continuity of Riesz means of order greater than(d=21=p) on all spaces Lq(X; ) for q 2 [1;1]. We claim the followingTheorem.

Theorem 5. Suppose that operator A satises condition (12), or (15)and (16) for some p 2 [2;1]. Then for any > d=2 1=p and

q 2 [1;1]

supR>0

kR(A)kLq(X;)!Lq(X;) C <1:

Hence for any q 2 [1;1) and f 2 Lq(X; )

limR!1

kR(A)f fkLq(X;)!Lq(X;) = 0;

where R is dened by (7).

For the proofs of our Theorems, we refer reader to [DOS]. Here let usonly mention that the proofs of Theorems 5 and 3 are less complicatedthan most of earlier spectral multiplier results. Our strategy is touse the complex time heat kernel bounds (see [Da1, DO]) to showW 2

(d+1)=2 functional calculus for the considered operator A. Then we use

Mauceri-Meda interpolation trick (see [MM]) and our Plancherel typeassumption (12) to obtainW p

d=2+"functional calculus. This is enough to

show Riesz summability (i.e. Theorem 5. To prove Theorem 3 we needalso some Calderon-Zygmund singular integral techniques. However incontrast to the standard Calderon-Zygmund singular integral estimateswe do not use estimates for the gradient of the kernel of singular integraloperators. Instead of that we follow the ideas of [DM, He3, He2].

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Proc. Amer. Math. Soc., 120(3):973979, 1994.

[Ale2] G. Alexopoulos. Lp bounds for spectral multipliers from Gaussian estimates

of the heat kernel. preprint, 2000.

[ChS] F. M. Christ and C. D. Sogge. The weak type L1 convergence of eigenfunc-

tion expansions for pseudodierential operators. Invent. Math., 94(2):421453,

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[Ch1] Michael Christ. Weak type (1; 1) bounds for rough operators. Ann. of Math.

(2), 128(1):1942, 1988.


[Ch2] Michael Christ. Lp bounds for spectral multipliers on nilpotent groups.Trans.

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[CS] Michael Cowling and Adam Sikora. A spectral multiplier theorem on SU(2).

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[CGHM] Michael Cowling, Saverio Giulini, Andrzej Hulanicki, and Giancarlo

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[Da1] E. B. Davies. Heat kernels and spectral theory. Cambridge University Press,

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[Da2] E. B. Davies. Uniformly elliptic operators with measurable coecients. J.

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[dMM] Leonede De-Michele and Giancarlo Mauceri. Hp multipliers on stratied

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[DO] Xuan Thinh Duong and El-Maati Ouhabaz. Heat kernel bounds and spectral

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[Du] Xuan Thinh Duong. From the L1 norms of the complex heat kernels to a

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[DM] Xuan Thinh Duong and Alan McIntosh. Singular integral operators with non-

smooth kernels on irregular domains.Rev. Mat. Iberoamericana, 15(2):233265,

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[DOS] Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora. Plancherel type

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) dierential operators on compact manifolds. Duke Math. J., 59(3):709736,

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ACT 0200, Australia


FROM XY TO ADE

DAVID E. EVANS

Abstract. We survey the role of non-commutative operator al-

gebras in statistical mechanics and the relation between the classi-

cation of modular invariant partition functions in conformal eld

theories and braided subfactors.

As exposed in the treatises of Bratteli and Robinson [21] non-commu-

tative operator algebras have a long tradition of providing a framework

for understanding quantum statistical mechanics. For example, the

one-dimensional XY -model is studied in the Pauli or Fermion algebraNZM2 with the Hamiltonian

H = Xj2Z

f(1 + )jxj+1x + (1 )jy

j+1y + 2jzg:

Here j; = x; y; z are the usual Pauli matrices placed at the jth

site of the tensor product. Typically one studies time evolution on

the Pauli algebra via the one-parameter group of *-automorphisms

t = eiHt()eiHt suitably dened. In such lattice models one is in-

terested in determining the set of equilibrium states, using the Gibbs

conditions, KMS condition or a variational principle, minimizing the

thermodynamic quantity (energy - temperature.entropy), as well as the

return to equilibrium of locally perturbed models. Robinson played a

seminal role in this theory, which is described in detail in [21]. Amongst

other things, this led to the development of the theory of derivations on

operator algebras, the innitesimal generators of time evolution, which

is still relevant today with the Powers-Sakai conjecture [58] a particu-

larly challenging open problem. This led Robinson to working on the

innitesimal generators of (completely) positive semigroups on operator

algebras and subsequently his most recent work on heat kernel methods.

The Powers-Sakai conjecture asks whether every one-parameter dynam-

ics on a UHF algebra (an innite tensor product of matrix algebras) or

more generally on a simple AF algebra (an inductive limit of nite di-

mensional algebras) is approximately inner, obtained as a limit of inner85

86 DAVID E. EVANS

one-parameter groups as in the above XY -example. Kishimoto [49] has

recently shown that a stronger form of the conjecture is false, namely

the core problem of whether the innitesimal generator has an AF-core

in a suitable sense. If is a strongly continuous one-parameter group of

*-automorphisms of a simple AF algebra, where the domain of the gen-

erator is AF, then is approximately inner. However, [49] constructs

on any non type I simple AF C-algebra examples of approximately

inner one-parameter groups of *-automorphisms where the domain of

the generator is not AF. These can be regarded as one-parameter con-

tinuous analogues of the exotic examples of compact group actions on

AF algebras whose xed points are not AF (rst shown by Blackadar

[5] for Z=2 on the Pauli algebra, and latter by Bratteli et al [18] for

nite groups and Evans and Kishimoto [33] for compact groups).

Returning to our original starting point of this paper, the XY -model,

notice that it degenerates at certain values of (; ), namely at (0;1),to the Ising nearest neighbour model. This is a classical Hamiltonian,

and it would therefore appear to be articial to study it via a non-

commutative framework, the Pauli algebra. Nevertheless, there is a

natural role for non-commutative operator algebras in the study of

such classical statistical mechanical models which is the point of this

present survey.

This connection begins with the transfer matrix method. Let us take

a two dimensional nearest neighbour Ising model on a square lattice Z2

with Hamiltonian

H = X

; nn

J

with the summation over the vertices or sites ; on Z2 which are near-

est neighbours (nn). We switch from one to two dimensions because

the one dimensional version does not have a phase transition at a non

zero temperature. At each site or vertex point of the lattice we have

a variable, a spin or magnetization with either a positive or negative

orientation or value represented by +1 or 1. Then a state = () of

the Ising model is a distribution of pluses and minuses over the square

lattice L = Z2, so any conguration is represented by a point in con-

guration space the compact Hausdor space P = f1gL. Thus the

natural home to study this Ising model is the space C(P ) =NZ2C2

FROM XY TO ADE 87

the commutative C-algebra of all continuous functions on the com-

pact conguration space. At each inverse temperature we may be

interested in the simplex K of equilibrium states, given say by solu-

tions to the equations of Dobrushin, Lanford and Ruelle [30][50] or the

variational principle: minimize(energy - temperature.entropy). In the

algebraic approach, one uses the transfer matrix formalism to transform

the setting to that of a one dimensional quantum model, represented

by a non commutative "one dimensional" C-algebra and time evolu-

tion t. The transfer matrix T is obtained for the partition function of

a strip of nite length M and width length one. With boundary con-

ditions ; along the two lengths the corresponding partition function

T denes us the transfer matrix T . The partition function Z of a

nite rectangular lattice of length M and width N is then obtained by

multiplying the strip partition functions, namely transfer matrix entries

and summing over internal edges. For periodic boundary conditions we

obtain

Z =X

exp(H()) =X

T 12T 23 : : : TN 1 = trace TN :(1)

In this way we move from the commutative algebra C(P ) =NZ2C2

to the non-commutative Pauli algebra A =NZM2 where the local

transfer matrices T generate the even part A+. Time evolution can be

formally written as t = T it()Tit, i.e. we consider T = eH where

H is now a quantum Hamiltonian which is no longer a (one dimen-

sional) Ising Hamiltonian. Spatial translation by Z2 in the classical

modelNZ2C2 corresponds to spatial translation in AP =

NZM2 to-

gether with an evolution fT n()Tn : n 2 Zg in the orthogonal transferdirection.

For each inverse temperature one looks for a map F ! F from

(local) classical observables in C(P ) to the quantum algebra A, and a

map ! ' from states on C(P ) (or measures on P ) to linear func-

tionals on the local observables in AP such that one can recover the

classical expectation values or correlation functions from a knowledge

of the quantum ones alone: (F ) = '(F). Fixing some bound-

ary conditions, then for each inverse temperature , let ' denote the

corresponding state on A. [In general positivity of ' is not auto-

matic but follows from re ection positivity of ]. Then if c denotes

the inverse critical temperature of Onsager, there exist automorphisms

88 DAVID E. EVANS

f : 6= cg of A [34] which do not depend on boundary conditions,

and real analytic in 6= c such that

' =

('1 > c

'0 < c:

Here '0 =NZ!, where =

1

1

p2 is the disordered state, and for

+ or - boundary conditions, '1 =NZ! where + =

1

0

; =

0

1

respectively and '1 = ('+

1 + '1) =2 for free or periodic boundary con-

ditions. Thus with free or periodic boundary conditions, we conclude

that ' is pure for 0 < c (also for = c by dierent methods

[2]) and is a mixture of two inequivalent pure states for ' for > c.

If h i denote the classical states corresponding to + and - boundary

conditions respectively, then we can deduce that hF i is real analytic

in > c when F is a local classical observable, as hF i = '1(F)

using analyticity of . A dynamical system t on AP is formally given

as t = T it()Tit which has a unique ground state for < c and two

extremal ground states ' for > c.

The Ising model can be generalized to the possibility of having more

than two values or spins possible at any lattice site, and moreover

some constraints or rules to determine allowable congurations. A

particular value at one site may force only restricted choices at nearest

neighbours. This would be achieved by distributing values of a xed

graph G at sites of the lattice L in such a way that if and are

nearest neighbours in L, then the corresponding values and are

joined in the graph. The state space P can be dened for any graph,

but if G contains some multiple edges, we consider distributions of

edges of G on edges of L. For the Dynkin diagram A3 with vertices

labelled by f;g and square lattice L one obtains two copies of the

Ising model as in Fig. 1 by placing the frozen spin on the even or odd

sublattices of L. This graph may be generalised to the Dynkin diagrams

of Fig. 1 for the models of Andrews, Baxter and Forrester [1]. These

in turn can be generalised by considering the Weyl alcove A(n;k) of the

level k integrable representations of the Kac-Moody algebra SU(n)^:

Boltzmann weights associated to a local conguration around minimal

squares of the lattice can be chosen to satisfy the integrable Yang-

Baxter equation [25].

FROM XY TO ADE 89

@r

+A3

r r r

r r r

r r

r r

Figure 1. Ising model: Dynkin diagram A3 and cong-

uration space

The centre of SU(n), the abelian Z=n, acts on A(n;k), e.g. Z=2 on the

Dynkin diagram Ak+1 by a ip i! k i which may or may not have a

xed point depending on the parity of k. The interesting case is when

there is a xed point. In any case, the Boltzmann weights are preserved

under the symmetry, and yield new Boltzmann weights on the orbifold

graphs A(n;k)=(Z=p), whenever p divides n, satisfying the integrable

Yang Baxter equation [28] [35]. For example, when n = 2; k = 2m

we blow up the xed point m to a pair (a copy of Z=2) and replace

each distinct pair i; k i (i 6= m) interchanged by the symmetry with

a singleton yielding the graph Dm+2. The case A3 is self dual in that

A3=(Z=2) = A3. Nevertheless, the situation here is not entirely trivial.

This is Kramers-Wannier high temperaturelow temperature duality.

This duality replaces the Boltzmann weights at a temperature t with

ones at dual temperature t. Again the xed point of the symmetry

t ! t is what provides the interesting structure | at the critical

temperature tc of Onsager.

We have mentioned the phenomena of AF algebras with non-AF xed

point algebras under symmetries. Such examples were rst found using

similar orbifold constructions. As a continuous version of the ip on

a Dynkin diagram which yields symmetries on AF algebras, consider

instead the ip on the interval around its midpoint or a ip on a circle

around an axis in its plane through its centre. The orbifold space is

best described by taking the cross product. For a pair of points inter-

changed by the symmetry, the local crossed product is simply a two

by two matrix algebra. The diagonal elements represent the contin-

uous functions on the pair, and the o-diagonal elements come from

the transition between the two points. Each xed point is replaced

by a pair arising from the transitions only generating a copy of C 2 as

the continuous functions on the group (dual). Thus gluing together, we

90 DAVID E. EVANS

represent the action of the ip on the interval [0; 1] by the C-algebra of

continuous functions on a half-interval [0; 1=2], which on the half-open

half-interval [0; 1=2) (the part of the interval which has a non-trivial

orbit) are two by two matrix valued but at other end point 1=2 are

diagonal. So the spectrum of this algebra is a continuous version of

the D-Dynkin diagram. Topologically it is an interval with two non

Hausdor points at one end. The analogous action of the ip on the

two torus which conjugates each variable has four xed points, yielding

a sphere with four singular points. The corresponding cross product is

the space of two by two matrix valued functions on the sphere which

are diagonal at four distinguished points. Replacing the two torus by

a non-commutative torus generated by two unitaries U and V satis-

fying the commutation relation V U = qUV where q = exp2i we

obtain a non commutative toroidal orbifold when taking the symme-

try which inverts the generators U and V . It is Morita equivalent to

the algebra of a singular ow on a sphere obtained as the quotient

of the Kronecker ow on the torus as illustrated in [36], page 137 or

http://www.cf.ac.uk/maths/opalg/ncto/. Remarkably these algebras

are AF (when is irrational)([20][19] or [32],[65]) although the corre-

sponding irrational rotation algebras and algebras of the Kronecker ow

are not. The non-commutative torus has a representation on L2(T2)

where U and V are represented as multiplication and translation oper-

ators. In this representation, or at least if one takes the Fourier trans-

form, the Hamiltonian H = U + U1 + (V + V 1) are the Mathieu

or discrete Schrodinger operators with almost periodic potentials. The

natural home to study these operators is the xed point algebra under

our ip because when is irrational then U+U1 and V +V 1 generate

the xed point algebra. It is still a tantalizing mystery as to whether

there is a relation between the AF property of the xed point algebras,

a strong form of non-commutative disconnectedness, and the Cantor

spectra of such almost Mathieu operators - which are at least known

to be Cantor for generic coupling constant and rotation number .

Symmetries on such algebras, where there are underlying xed points

can produce algebras with totally dierent properties. Similarly, sym-

metries on subfactors, statistical mechanical models, conformal eld

theories can produce totally dierent subfactors etc from what one

started with.

FROM XY TO ADE 91

From a lattice model one may obtain a eld theory by taking a con-

tinuum or scaling limit, letting the lattice spacing go to zero whilst

simultaneously approaching the critical temperature. As the scale or

correlation length becomes innite, one obtains a scale invariant or con-

formal invariant theory. Belavin et al [4] suggested that the scale in-

variance at a critical point is enhanced to conformal invariance. One of

the invariants of the conformal eld theory is the central charge a mul-

tiplier in projective representations Lm of the vector elds zm+1d=dz

on the circle, the Virasoro algebra. However the central charge can

already raise its head in the statistical mechanical model. Going back

to the partition function Z of Eq. (1) the free energy f = logZ=NM

is independent of boundary conditions as N;M ! 1. However the

asymptotics depend on boundary conditions; if 1 << N << M , then

Z exp(NMf +Mc=N6), where c is the central charge. (See e.g.

[26] (or [36], Chapter 8) for explicit computation in the case of the Ising

model).

Let us however proceed to the conformal eld theory at criticality.

It is argued on physical grounds that the partition function Z() in

a conformal eld theory on the torus should be invariant under re-

parametrization of the torus by SL(2;Z): Z() = Z((a + b)=(c + d))

[23]. In the string theory formulation, modular invariance is essentially

built into the denition of the partition function (although Nahm [53]

has argued the case for modular invariance in terms of the chiral algebra

and its representations rather than a functional integral setting). In

the transfer matrix picture of the statistical mechanical picture, we

wrote the partition function as an average over eH , where H is the

Hamiltonian. The Hamiltonian is now L0+ L0 c=12 where L0; L0 are

commuting generators of rotation groups, c the central charge and the

shift by c=24 comes from mapping the Virasoro algebra on the plane to

a cylinder. We also have a momentum P = L0 L0 describing evolution

along the closed string, so taking both evolutions into account we rst

compute

Z() = tr eHeiP = tr e2i(L0c=24) e2i(L0c=24):

Here 2i = + i parametrizes the metric of the torus, and we

then have to average over . If we choose one from each orbit under

the action of SL(2;Z) and integrate we implicitly assume that Z()

92 DAVID E. EVANS

is modular invariant. In our SU(n) setting the Hilbert space decom-

poses according to the associated loop group representations. The loop

group LG is the group of smooth maps from S1 into a compact Lie

group G under pointwise multiplication. We are interested in projec-

tive representations of LG o Rot(S1) where Rot(S1) is the rotation

group, which are highest weight representations in that the generator

L0 of the rotation group is bounded below. Such representations are

called positive energy representations and are classied by irreducible

representations of G (by restriction to the constant loops) and a level k

describing the multiplier in the projective representation. For unitary

irreducible positive energy representations, the possibilities are severely

restricted. Indeed k must be integral and for a given value of the level,

there are only a nite number of admissible (vacuum vector) irreducible

representations of G: For G = SU(n) the admissible ones are precisely

the vertices of A(n;k), the same labeling set as used to construct our

statistical mechanical model.

The partition function then decomposes as

Z() =X

Z;()()

where

Z = 0; 1; 2; :::; Z00 = 1(2)

and characters () = tre2i(L0c=24), Im > 0.

Here the label 0 refers to the vacuum representation, and the condi-

tion Z00 = 1 refers to the physical concept of uniqueness of the vacuum

state. The matrix Z arising in this way is called a modular invari-

ant mass matrix. (More precisely, for current algebras the characters

depend also on variables other than , corresponding to Cartan sub-

algebra generators which are omitted here for simplicity. These extra

variables mean we are dealing with SL(2;Z) rather than PSL(2;Z)).

From the canonical generators

S =

0 1

1 0

, and T =

1 1

0 1

of SL(2;Z) we obtain the unitary Kac-Petersen matrices S = [S]; T =

[T] transforming characters, where S is symmetric as well as S0

FROM XY TO ADE 93

S00 > 0 and T is diagonal:

(1=) =X

S(); ( + 1) =X

T():

Then the classication of modular invariant partition functions can

be reformulated as a matrix problem. Find all matrices Z subject to

Eq. (2) commuting with S and T . This is a rather concrete problem.

For SU(2) at level k, SU(2)k, the admissible weights are the spins

= 0; 1; ::; k and the Kac-Peterson matrices are given explicitly as

S =

r2

k + 2sin

(+ 1)(+ 1)

k + 2

T = exp

i(+ 1)2

2k + 4

i

4

with ; = 0; 1; :::; k, and the characters as

(q) =q(+1)2=4(k+2)

(q)3

Xn2Z

(2n(k + 2) + + 1)qn(n(k + 2) + + 1)

if q = e2i , and the Dedekind function (q) = q1=24Q

1

n=1(1 qn):

For the Ising model, the characters are (in the notation of Fig. 1),

= [#2=2]3=2; = ([#3=]

3=2 [#4=n]3=2)=2

with the #-functions:

p#3= = q1=48

1Yn=0

1 + qn+1=2

p#4= = q1=48

1Yn=0

1 qn+1=2

p#2= = q1=24

1Yn=1

(1 + qn):

Here the Kac-Petersen matrices are simply

S = 12

0@ 1

p2 1p

2 0 p2

1 p2 1

1A ; T = ei=24

0@ 1 0 0

0 ei3=8 0

0 0 ei

1A

94 DAVID E. EVANS

so that there is only one modular invariant the diagonal mass matrix

or:

Z =j#2j3 + j#3j3 + j#4j3

=2jj3:

Whilst the mass matrix is trivial, the partition function itself has

some structure. The following is also a modular invariant particular

function

Z = (j#2j+ j#3j+ j#4j)=2jjthis time for the coset model su(2)1

Lsu(2)1=su(2)2, which also ex-

hibits Ising fusion rules (as does (E8)1 (E8)1=(E8)2) and so(5)1).

At rst sight, it might appear that generally there may be an innite

number of solutions to this modular invariant problem. However, there

is a following estimate [12]: Z dd which is a strengthening of the

inequality of Gannon [42]:P

Z 1=S200 if d = S0=S00: Thus since

Z is positive and integral there are at most nitely many solutions,

for a xed representation of SL(2;Z). In the case of SU(2), there are at

most three solutions for a xed level k. This is the ADE classication

of Capelli, Itzykson and Zuber [22]. A Dynkin diagram is associated

to each invariant through the identication of diagonal terms of the

invariant f : Z 6= 0g = I with eigenvalues fSf=S00 : 2 Ig of thecorresponding Dynkin diagram if f = 1 the fundamental representa-

tion of SU(2). The A refers to the diagonal invariant, D to orbifold

invariants and E to the three E6; E7; E8 exceptional invariants. For

SU(3) there is an anologous ADE classication due to Gannon [43]; di

Francesco and Zuber [28] sought to show systematically the existence

of graphs with spectra matching the modular invariant, give a meaning

to these graphs themselves and compute them in a number of examples.

As we have said there are at most nitely many solutions to the

modular invariant conditions. There is always one solution the trivial

diagonal invariant: X2A

jj2

where the corresponding mass matrix is diagonal Z = . In some

sense, [52] [29] [11] 'every' modular invariant is diagonal if looked at

properly. If we can extend the A system to a B system so that the

characters decompose

=X2A

b ; 2 B

FROM XY TO ADE 95

according to some branching rules, then the diagonal B-modular in-variant will give an A-modular invariantX

2B

j j2 =X2B

jX2A

bj2:

In some sense, every modular invariant should look like this or with a

possible twist w(); for a permutation w of the extended fusion

rules, preserving the vacuum. The problem in general is then to nd

such extensions. When there is no twist present we have what are

sometimes called type I invariants:

Z =X

b;b:

These are automatically symmetric: Z = Z. In the presence of a

non-trivial twist, we have the type II invariants

Z =X

bbw():

These are not necessarily symmetric, but at least there is symmetric

vacuum coupling Z0 = Z0. Not every modular invariant is even

symmetric in this sense, (e.g. for SO(16n)1) but every known SU(n)

modular invariant is even symmetric in the usual sense.

Our aim is to study or even construct modular invariants from sub-

factors. The framework can be summarised as follows. We have a

hypernite III1 factor N on which there is a system of endomorphisms

f 2 Ag labelled by our positive energy representations or our originalstates in the original statistical mechanical setting. We induce these

endomorphisms to endomorphisms on a larger ambient factor M|

there will be two natural ways to do this labelled .The modular invariant will then be constructed or recovered as

Z = h+ ;

i

where the right hand side will be computed as dimensions of intertwiner

spaces or the number of common sectors when we decompose into

irreducibles. The original endomorphism 2 A will be irreducible but

may not be. The factor N will carry the modular data for S and T

matrices, varying the inclusion may change the modular invariant but

somehow the inclusion will have to be related to the original A-system.The system A on the factor N can be constructed via the method

of Jones-Wassermann. First for any positive energy representation

96 DAVID E. EVANS

2 A, the objects LISU(n) and LI0SU(n), if LI ; LI0 denote loops

on SU(n) concentrated on complementary non-trivial intervals I and I0

in the circle. We can thus form the inclusion

LISU(n)00 LI0SU(n)

0:(3)

For = 0, the vacuum representation, there is Haag duality and this in-

clusion is not proper but gives us a single hypernite III1 factorN (more

precisely a net N(I) of such factors). The inclusion Eq. (3) then deter-

mines a system of endomorphisms 2 A, so that the inclusion Eq. (3)is isomorphic to N N , with index [N , N ] = d2. Wassermann [66]

has shown that the fusion rules of such endomorphisms are precisely

the same as that of SU(n) at a root of unityq = e2i=(k+n)

. Moreover,

rotation through 1800 on the circle, interchanges the role of I and I0.

This has the eect that the system A is naturally braided, i.e. not only

is the system commutative = as sectors if ; 2 A End(N),

but there is a choice "(; ) of unitaries taking to satisfying the

Yang Baxter equation, braiding fusion equation etc.

Thus we have commutative matrices N = [N];; 2 A, deter-

mining the fusion

=X

N

with composition of endomorphisms, or rather sectors, their unitary

equivalence classes and a natural notion of addition. Fusion by the

endomorphism of the conjugate of is given by N = N tr , the trans-

pose. Thus fN : 2 Ag is a family of commuting normal matricesand so simultaneously diagonalisable. By the Verlinde theory the uni-

tary matrix which performs this diagonalisation is the S matrix itself.

Inverting the consistency condition or the regular representation

NN =X

NN(4)

we obtain

N =

X

S

S0

S S

or

N =X

S

S0

jSihSj:

The modular invariants will provide representations other than the

regular representation Eq. (4), and pick out subsets fS=S0 : 2 Ig

FROM XY TO ADE 97

where I = f : Z 6= 0g; the diagonal part of the modular invariant.At the same time these representations will replace N by other families

G of graphs | to be associated with the modular invariant or at least

the subfactor N M which yields that particular invariant.

As we have said the inclusion, which is meant to duplicate the mod-

ular invariant, should be related to the original A-system. This is

achieved as follows. There is a conjugation on endomorphisms of N ,

(extending for groups the notion of inverting automorphisms or conju-

gating a representation in the dual) compatible with the conjugation

on A. Similarly one can conjugate endomorphisms or sectors of M , or

those between N and M , M and N . In particular, we can take the

inclusion = N ! M , its conjugate = M ! N and compose to get

endomorphisms = on M and = jN = on N called the canoni-

cal and dual canonical endomorphisms respectively. What we need is

lies in the system generated by A, i.e. decomposes as a sum of sectors

from A. Note that we do not need to specify M when we ask whether

a particular endomorphism of N is a dual canonical endomorphism.

It may not be particularly clear in a given situation whether a certain

endomorphism is a dual canonical endomorphism or what M may be.

When Z is a modular invariant typical candidates for dual canonical

endomorphisms will beP

Z0;P

Z0 on N andP

ZN

opp

on NN

Nopp where Nopp is the opposite algebra, etc.

The rst non trivial (i.e. exceptional) invariant for SU(2) occurs at

level 10:

ZE6= j0 + 6j2 + j4 + 10j2 + j3 + 7j2:(5)

The diagonal part of the invariant I = f : Z 6= 0g matches the spec-trum of the Dynkin diagram E6, namely fS1=S0 = 2 cos(+1)=12 :

= 0; 6; 4; 10; 3; 7g: For this reason Capelli, Itzykson and Zuber la-

belled the invariant by the graph E6. In the subfactor setting we can

derive this graph as follows. First, we turn to the conformal embedding

description of this invariant due to Bouwknegt and Nahm [17] which

provides the extended system B which diagonalises the invariant. The

embedding SU(2)10 SO(5)1 means there is a mapping of SU(2) in

SO(5) such that the three level 1 representations B of SO(5) decom-

pose into level 10 representations of SU(2) with nite multiplicity. The

98 DAVID E. EVANS

system SO(5)1 has three inequivalent representations, b,v,s basic, vec-

tor and spinor which reproduce the Ising fusion rules. They decompose

(cf Eq. (5)) as

b = 0 + 6; v = 4 + 10; s = 3 + 7(6)

so that the E6 modular invariant for SU(2)10 arises from the diagonal

invariant for SO(5)1:

ZE6= jbj2 + jvj2 + jsj2:

Moving now to the loop group factors the conformal embedding gives

us an inclusion of factors:

LSU(2) LSO(5)

using the vacuum representation on LSO(5), a net of subfactorsN(I) M(I): Fixing I, we have subfactor N M on which there are systems

A = SU(2)10 and B = SO(5)1 of endomorphisms acting respectively.

These two systems can be related by a form of Mackey induction-

restriction which in the subfactor setting goes back to Longo-Rehren

[51]. Using the braiding "+ or its opposite braiding ", we can lift en-

domorphisms in A to those of M: = 1Ad"(; ) . The maps

[]! [ ] preserve all the operations of conjugation, addition and mul-

tiplication of sectors [67][8][9][10]. However, they are not injective, and

may be reducible. We nd that f+ : 2 Ag decomposes into six

irreducible sectors such that the graph E6 is multiplication by +1 [67]

[9]. In fact [+1 ] = G; is part of a system of matrices with non-negative

entries fG : 2 Ag which represents the original A-fusion rules.

This had been noticed empirically in e.g. [28] [55] which now gets a

subfactor explanation.

To bring the B system into the game we use -restriction, =

to take M -sectors to N -sectors. This map is not multiplicative, but

in the type I situation there is a reciprocity: h ; i = h; i (withinequality on the type II setting) as long as say is a subsector of

the induced system E6 respectively. Since restriction takes the E

6

systems into the A-system by Eq. (6), namely

b = = 0 + 6; v = 4 + 10; s = 3 + 7

the reciprocity means that the B system, coming from the three level

1 representations of SO(5) must lie in the induced systems E6 , i.e.

FROM XY TO ADE 99

A3 = B E+6 \ E

6 . In fact we can identify as sectors b = 0; v =

10; s = 3 9; and E+6 \ E

6 = A3 precisely. The dual canonical

endomorphism lies inA, but a priori we do not have much informationabout its Fourier transform . In fact as sectors: = idM + +

1 1 so

that 2 E+6 _ E

6 , the full induced system. Indeed A = E+6 _ E

6 ,

the system generated by E6 is precisely all subsectors of f : 2 Ag

in fact the latter has global dimension w =P

A d2, whilst if w =P

2A d2 denotes the global dimension of the induced system then [10]:

w=w =X

Z0d

with the sum over only the degenerate sectors inA|which have trivial

monodromy with all other sectors. In this case the A-system, as farSU(n)k is non-degenerate, the vacuum is the only degenerate sector.

Moreover we can recover the modular invariant as

Z = h+ ;

i; ; 2 A:

In this case the E6 systems are commutative (but not braided) as is the

E+6 _E

6 system | but this is not always the case. The neutral system

A0 = A+ \ A, if A are the induced chiral systems, is braided, with

the braiding non-degenerate if that of A is. Complexifying the nite

dimensional algebrasA we can decompose them in the non-degenerate

case as [15]:

A =M2A0

M2A

Mat(b):(7)

Here b; are the chiral branching coecients h; i; 2 A; 2 A.(In the case of chiral locality where the extended net M(I), is local,

i.e. observables associated with disjoint intervals commute, then b; =

h; i = h ; i; 2 A0; 2 A:) In particular the extended systemsare commutative only when b; 1; 2 A0; 2 A Thus the informal

inclusions SU(n)n SU(n21)1 give non-commutative chiral systemswhen n 4; and it explains the computations of Feng Xu [67] who

found non-commutativity in case n = 4 by a direct computation.

Thus we can decompose the modular invariant as

Z = h+ ;

i =

X2A0

b;b;

100 DAVID E. EVANS

and from Eq. (7) by counting dimension we see that

jAj =X

(b;)2 = trbtb:

In the case of chiral locality where b+ = b, so that the invariant is

type I, we see that jAj = trZtZ, more generally jAj only sees the

type I parent of a type II invariant. Thus with

ZD10= j0 + 16j2 + j4 + 12j2 + j6 + 10j2 + 2j8j2 + j2 + 14j2

ZE7= j0+16j2+j4+12j2+j6+10j2+j2j2+(2+14)

8 +8(2+14)

then in either case trbb = 10 so that multiplication by [1 ] gives the

graph D10 in either case so we do not get the graph E7 for ZE7(where

we can use the dual canonical endomorphisms 0 + 16, 0 + 16 + 8respectively).

In general (and this will work when either chiral locality holds or

fails) we look at the action of A on the M -N sectors MAN which are

the irreducible sectors of = (which can be identied with A

when chiral locality holds). This action decomposes as:L

Mat(Z),

with

=M

S

S0

1Z:

Thus we get the desired representation with spectrum matching the

diagonal part of the modular invariant, and counting dimension then

jMAN j = trZ; e.g. trZE7= 7 so that we do indeed now recover the

correct graph.

The subfactor framework is rich enough to produce a Moore-Seiberg

type decomposition of modular invariants as well as handle possibly

non-symmetric modular invariants. As we have already observed, in the

case of chiral locality, b+; = b;(= h; i) for 2 A; 2 A0: So the

question arises as to how far we can identify b+ and b, say b

; = b+w();

for a permutation w of the extended neutral system B or if we need

dierent labellings B+ or B to handle possibly non-symmetric modular

invariants. Now locality holds if and only if =P

Z0 =P

Z0:

In general we dene + =P

Z0; =P

Z0: Using the theory of

intermediate subfactors of [45], we can show [16] that both are dual

canonical endomorphisms for inclusions N M which satisfy chiral

locality and M M . This means we can use -induction on both

FROM XY TO ADE 101

inclusions N M, to obtain type I modular invariants Z, such

that

Z+0 = Z+

0 = Z0; Z

0 = Z

0 = Z0;

where we can identify both neutral systems M

A0M

with MA0M . If

M+ =M; then we can write

Z

=X2A0

bb

in particular (Z+ = Z) and using the identications M

A0M

with

MA0M to produce an automorphism ! of the neutral elements MA0

M we

have:

Z =X

bbw():

In the case of E7 invariant we have N M M where M+ =M

and the dual canonical endomorphism for N M; N M are

0 + 16, 0 + 16 + 8 as we have said before.

It may happen that M+ 6= M and this does occur for SO(16n)1where there are non-symmetric modular invariants where we must use

dierent labelling M+A0

M+;M

A0M

on the left and right to decompose

Zext+;

as ;#(+), where # = ##

1+ is the identication. The situation

is summarised [10] using recent work of Rehren [62] on canonical tensor

product subfactors as a pair of inclusions:

NO

Nopp M+

OM

opp B

where the dual canonical endomorphisms for NN

Nopp B and

M+

NM

opp B as

PZ

Nopp;

P2A0 #+()

N#()

opp respec-

tively.

There is a connection between the two chiral inductions and the pic-

ture of left- and right-chiral algebras in conformal eld theory. Suppose

that our factor N is obtained as a local factor N = N(I) of a quantum

eld theoretical net of factors fN(I)g indexed by proper intervals I R

on the real line, and that the system NXN is obtained as restrictions of

DHR-morphisms (cf. [44]) to N . This is in fact the case in our examples

arising from conformal eld theory where the net is dened in terms of

local loop groups in the vacuum representation. Taking two copies of

such a net and placing the real axes on the light cone, then this denes

a local net fA(O)g, indexed by double cones O on two-dimensional

Minkowski space (cf. [61] for such constructions). Given a subfactor

N M , determining in turn two subfactors N M obeying chiral

102 DAVID E. EVANS

locality, will provide two local nets of subfactors fN(I) M(I)g as alocal subfactor basically encodes the entire information about the net

of subfactors [51]. Arranging M+(I) and M(J) on the two light cone

axes denes a local net of subfactors fA(O) Aext(O)g in Minkowski

space. The embedding M+ Mopp B gives rise to another net of

subfactors fAext(O) B(O)g, where the net fB(O)g obeys local com-mutation relations. The existence of the local net was already proven

in [62], and now the decomposition of [ext] tells us that the chiral

extensions N(I) M+(I) and N(I) M(I) for left and right chiral

nets are indeed maximal (in the sense of [61]), following from the fact

that the coupling matrix for fAext(O) B(O)g is a bijection. This

shows that the inclusions N M should in fact be regarded as the

subfactor version of left- and right maximal extensions of the chiral

algebra.

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FROM XY TO ADE 105

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School of Mathematics, University of Wales Cardiff, PO Box 926,

Senghennydd Road, Cardiff CF24 4YH,Wales, U.K., [email protected]

THE HEAT-FLOW METHOD IN CONTACTGEOMETRY

ROBERT GULLIVER

Contact geometry treats such questions as the existence and clas-sication of contact structures on manifolds of odd dimension andspecied topological structure. See inequality (1) below. The geo-metric/analytic approach treated in this report introduces parabolicsystems of partial dierential equations (PDEs) in a way which comple-ments the more algebraic methods, which until now are better knownin contact geometry.This is a report on joint work in progress with Hansjorg Geiges of the

University of Leiden, Netherlands and Matthias Schwarz of the Univer-sity of Leipzig, Germany. Many of the specic results reported on hereappeared rst in a paper [1] by Steve Altschuler, which introduced theheat- ow method to study contact structures, and in a recent preprint[2] of Altschuler and Lani Wu.

1. Introduction to Contact Geometry

Many of the participants in this conference apply analytical meth-ods to geometrically motivated problems, or use geometric methods tostrengthen their analysis. However, it cannot be assumed that every-one is familiar with all of the most modern concepts and techniques ofdierential geometry. For that reason, this section will be devoted toan introduction to contact geometry appropriate for analysts, amongothers, and may be skipped by those with a good knowledge of thearea. I was until rather recently a complete novice in this area of ge-ometry, and the reader should not expect a polished nor absolutelyconcise presentation. See [4], [5] and [6] for more complete referencesto the literature. I expect that analysts will be interested to see thisnovel application of parabolic operators.

Date: September 29, 2000; November 24, 2000.This work was initiated during my stay at the Max Planck Institute for Math-

ematics in the Sciences, Leipzig, and completed while I was visiting Monash Uni-versity and the University of Melbourne. I would like to thank MPI, Monash andMelbourne for their generous hospitality.

106

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY 107

A hyperplane distribution in an open setM of R2n+1 ; or in a smooth(2n+ 1)-dimensional manifold M; species a subspace x of dimension2n in R2n+1 ; or rather in the tangent space to M at each point x 2M;which depends smoothly on the point x:

1.1. Example: a foliation. A familiar example of a hyperplane dis-tribution would be the two-dimensional distribution in R3 spanned bythe vector elds e1(x) = (1; 0; 2x1) and e2(x) = (0; 1; 2x2): Here wehave written x = (x1; x2; x3): This distribution is especially easy tovisualize, since e1(x) and e2(x) are a basis for tangent vectors to thefamily of paraboloids of revolution x3 x 2

1 x 22 = C; for various real

constants C: This family of surfaces is a foliation of R3 ; which meansthat every point of R3 lies on one of the surfaces, the surfaces and thefamily are smooth, and in some neighborhood of any point, the familylooks like the family of coordinate planes x3 = const :; up to a localdieomorphism. In this situation, we say that the distribution is inte-grable, meaning in this case, where the rst and second components ofe1 and e2 are (1; 0) and (0; 1); that their third components 2x1 and 2x2are simultaneously the partial derivatives of a scalar function, locally.(The scalar function is x 2

1 + x 22 +C; of course.) Integrability is equiva-

lent to saying that for any two vector elds V;W in ; the Lie bracket[V;W ] also lies in : Alternatively, we may describe a hyperplane dis-tribution as the kernel of a nowhere vanishing dierential 1-form :(A 1-form is the dual of a vector eld, so that for any vector eld V;

(V ) denes a scalar function and depends linearly and pointwise onV:) Given ; the 1-form is determined up to a nonvanishing scalarfactor by the requirement that (e1) = (e2) = 0; where e1; e2 forma local basis for the distribution : (Computationally, has the samecomponents as the cross product of e1 and e2.) The integrability con-dition for the distribution may be written in terms of the 1-form asan identity between 3-forms: ^d = 0: (The exterior derivative d ofa 1-form is the 2-form dened by the alternating part of the matrixof rst partial derivatives; the wedge product of dierential forms isthe alternating part of their tensor product.)A contact structure is a hyperplane distribution which is maximally

non-integrable. In terms of Lie brackets, we may write !(V;W ) for thetransversal component ([V;W ]) of the Lie bracket of two vector eldsV;W in : This makes ! a 2-form. The integrability condition requiresthat ! 0; for to be a contact structure, we require not merely that! 6= 0 but far more: that the 2n-form !n = ! ^! ^ ^! be nowherezero on . Via the appropriate Riemannian metric, this is equivalent tosaying that ! denes an almost-complex structure on the hyperplane

108 ROBERT GULLIVER

distribution : Restricted to = ker; ! is the same as d: Thus,the contact criterion may be written entirely in terms of the 1-form :

^ dn 6= 0:(1)

Note that inequality (1) depends on but is independent of the choiceof 1-form ; since if e = f for some nonvanishing scalar function f;then e ^ den fn+1 ^ dn: Note also that since d is a two-form,the left-hand side of (1) is a dierential form of degree 2n + 1; so onR2n+1 or M2n+1; it has only one real component. In this sense, contact

structures and contact forms only have their full meaning in domainsand manifolds of odd dimension. A 1-form on a (2n + 1)-manifoldwhich satises inequality (1) is called a contact form.Inequality (1) is unusual, in the context of geometric analysis, for

two reasons: it is a strict partial dierential inequality, and it is anunderdetermined \system" consisting of one real, rst-order, fully non-linear partial dierential inequality for the 2n+1 real components ai(x)of the 1-form : Specically, in the 5-dimensional case n = 2, we maywrite in local coordinates (x0; : : : ; x4) as

=

4Xi=0

ai(x) dxi:

Then the contact inequality (1) is equivalent to the inequality

X

sgn() a(0)@a(1)

@x(3)

@a(2)

@x(4)6= 0;

where the sum is over all permutations of f0; 1; 2; 3; 4g: Systems ofpartial dierential equations of this general form are rather poorly un-derstood at present. In the case of contact geometry, however, we shallsee that there is a parabolic method available to attack inequality (1);see Section 2 below.

1.2. Example: the standard contact structure. A familiar exam-ple of a contact structure would be the two-plane distribution in R3

with the subspace x at the point x = (x1; x2; x3) having basis vec-tor elds e1(x) = (x1; x2; 0) and e2(x) = (x2; x1; r

2); where we havewritten r2 = x 2

1 + x 22 : In order to visualize ; we note that e1 is the

horizontal vector eld pointing away from the x3-axis, and that e2 is avector orthogonal to e1 and with slope r; as measured from the (x1; x2)-plane. Then the distribution is not a foliation, which may be seenas follows. Suppose (x1(t); x2(t)); 0 t T; describes a closed curvein the (x1; x2)-plane. Since x is never vertical, there is a unique wayto lift this curve to a curve x(t) = (x1(t); x2(t); x3(t)) in R

3 ; so that


the tangent vector x0(t) is always in the distribution x(t): If wereintegrable, then the space curve would stay on the same surface of thefoliation, and would therefore be a closed curve. However, to be spe-cic, suppose that the plane curve (x1(t); x2(t)) describes the boundary@; in the positive sense, of = one-fourth of an annulus: in polarcoordinates (r; ); is given by a < r < b; 0 < < =2: Then alongeach of the two straight sides 0; =2; the tangent vector lifts toa multiple of e1 = (x1; x2; 0), so x3(t) remains constant. But along thequarter-circle r b; 0 < < =2; the tangent vector lifts to a multipleof e2 = (x2; x1; r

2), so x3(t) increases by the slope times the lengthin the plane = b2=2: Returning along the quarter-circle r a; as decreases from =2 to 0; x3(t) decreases by a

2=2: Thus, the change inx3(t) as t increases from 0 to T is (b2 a2)=2; which is exactly twicethe area of the quarter-annulus :In fact, for any domain in the (x1; x2)-plane, the change in x3(t)

as (x1(t); x2(t)) describes @ equals twice the area of : This may beseen by computing a form 0 so that = ker0:

0 = x2 dx1 x1 dx2 + dx3:

Since x0(t) is in the distribution x(t); we get 0(x0(t)) = 0; which means

that x03(t) = x2(t)x0

1(t)+x1(t)x0

2(t); hence the change x3(T )x3(0) inthe height as (x1(t); x2(t)) goes around @ equals the integral around@ of x2 dx1 + x1 dx2; which is twice the area of :The 1-form 0 is the standard contact form on R3 ; and is the stan-

dard contact structure. More precisely, this is the rotationally symmet-ric version; the contact form x2 dx1+dx3 is translationally invariant intwo coordinate directions, and is also known as \the" standard contactform. In higher dimensions, the standard contact form on R2n+1 is

0 = dx0 +

nXk=1

(x2k dx2k1 x2k1 dx2k);(2)

which is invariant under the (n + 1)-dimensional group generated byrotation in the(x2k1; x2k)-plane, 1 k n; plus translation along the x0-axis.

A natural question is: when are two contact forms equivalent? Thelocal version of this question has a surprisingly simple answer:

Theorem 1.1. (Darboux): Let be a contact form on a neighbor-hood of x in R

2n+1 : Then on a smaller neighborhood of x, there is adieomorphism into R2n+1 such that is mapped to the standard con-tact form 0:

110 ROBERT GULLIVER

Darboux' Theorem may be interpreted as saying that the contactcondition (1) is a very \soft" condition, as compared to the familiar par-tial dierential equations traditionally treated by geometric analysts.This softness is apparent from the recent work of Gromov, Eliashbergand others on noncompact manifolds, which showed, for example, thatany noncompact, odd-dimensional manifold which has a hyperplanedistribution with an almost-complex structure also carries a contactstructure (see [8] and references therein.)

1.3. Global non-uniqueness: the Lutz Twist. Since contact struc-tures are locally unique, it might seem reasonable to think that a topo-logically simple space like R3 has only one contact structure up to achange of coordinates. However, there are subtle criteria which distin-guish other contact structures on R3 from the standard 0:Recall the description in subsection 1.2 of basis vector elds e1; e2

for the standard contact structure on R3 : e2 is orthogonal to the radialvector e1; and has slope r; which means that it makes an angle ' =arctan r with the (x1; x2)-plane. As r ! 1; e2 becomes vertical, so' ! =2: Instead, suppose that ' = '(r) increases beyond =2 tomake one or more revolutions before slowly approaching arctan r+2m(m 2 Z) as r ! R < 1: Outside the cylinder r < R; the contactstructure may be continued smoothly, to join up with the standardcontact structure. This construction is known as the Lutz twist (see[9].)In terms of the contact form, in cylindrical coordinates (r; ; x3);

0 = dx3 r2 d is replaced by = h0(r)dx3 h1(r) d for somefunctions h0; h1 : [0;1) ! R with h01h0 h1h

0

0 > 0; and with h0(r) =1; h1(r) = r2 for all r R: Then h1 and h0 are related to the angle 'by rh0(r) tan'(r) = h1(r):This new contact structure is overtwisted, that is, there is a topolog-

ical disk D R3 with jD nowhere zero along @D and j@D 0: In

fact, let r0 be the rst value of r with '(r0) = : Then we may chooseD = f(r; ; x3) : x3 = r20r

2; 0 2; 0 r r0g: It may be shownthat no such disk exists in R3 with the standard contact structure.

1.4. Compact Manifolds. What about compactmanifolds? The onlyknown obstruction to the existence of an orientable contact structure onan oriented, odd-dimensional manifold M2n+1 is the requirement thatsome hyperplane distribution on M should have an almost-complexstructure; this can be written as a topological condition onM , that theeven-dimensional Stiefel-Whitney classes w2i (certain natural cohomol-ogy classes with Z=2Z coecients) are in the image of cohomology withinteger coecients. However, there are many manifolds which satisfy


this condition but have not been shown to carry a contact structure.Specically, one would like to know whether there is a contact structureon the odd-dimensional torus T 2n+1.We shall assume for the rest of this paper that manifolds are com-

pact, oriented and have no boundary. A readily visualized exam-ple is the interesting case of the torus T 2n+1, which is just the cube[; ]2n+1 R

2n+1 after opposite faces have been identied.A contact form may be found on the three-torus T 3 as the rst case

of a classical construction. Begin on the two-dimensional torus T 2;introduce local coordinates (q0; q1) on T 2 and then extend these coor-dinates to the 4-dimensional phase space, or cotangent bundle, T (T 2):Then a cotangent vector x at the point x = (q0; q1) has components(p0; p1), meaning that x = p0 dq0 + p1 dq1: (In certain applications,(q0; q1) are coordinates of position and (p0; p1) are components of themomentum vector.) Then ! = dp0 ^ dq0 + dp1 ^ dq1 is the naturalsymplectic form on phase space T (T 2): One notes that ! is the exte-rior derivative d; where is the canonical 1-form p0 dq0 + p1 dq1 onphase space. When is restricted to the unit-sphere bundle M3 :=f(x; p) : x 2 T 2; p 2 T

x (T2); jpj2 = 1g, it satises the contact condition

(1). Here jpj2 = p20 + p21: The verication of inequality (1) reduces toshowing that p0 @jpj

2=@p0+ p1 @jpj2=@p1 6= 0 on M . Meanwhile, on T 2;

there is a global basis of tangent vector elds, which implies that theunit sphere bundle M3 of T 2 is T 2 S1 = T 3: In coordinates (q0; q1; )for T 3 = (R=2Z)3; we have = cos dq0 + sin dq1: This is the mostnatural construction for a contact structure on T 3:The construction above generalizes to higher dimensions. Let N be

an oriented (n+1)-dimensional manifold, equipped with a Riemannianmetric, and introduce local coordinates (q0; : : : ; qn; p0; : : : ; pn) for thecotangent bundle T N of N; where (q0; : : : ; qn) are local coordinates onN and a cotangent vector is represented as

Pn

i=0 pi dqi: Let be thecanonical 1-form

Pn

i=0 pi dqi: When is restricted to the unit-sphere

bundleM2n+1; dened as(q; p) : q 2 N; p 2 T

qN; jpj2 = 1

; it satises

the contact condition (1). That is, the unit sphere of the cotangentbundle of any manifold carries a natural contact structure. This ishow contact structures arise naturally, on suitable energy surfaces inHamiltonian systems.When one applies the same construction to N = T 3; n = 2; one nds

a contact 1-form 1 on the 5-dimensional unit sphere bundle of T N:But the unit sphere bundle M5 is now T 3 S2; not T 5: However, T 5

can still be given a contact structure, as was rst shown by Lutz [9].Another way to nd a contact structure on T 5 is to apply the followingresult of Gromov (see [8] and [5], p. 456):

112 ROBERT GULLIVER

Theorem 1.2. : If M2 is a branched covering of M1; branched alonga codimension-2 submanifold of M1; and if M1 has a contact form1 whose restriction to also makes into a contact manifold, thenM2 has a contact form close to the pullback of 1:

In our case, T 2 may be written as a branched double cover F : T 2 !

S2 of the sphere, branched simply over four points of S2; which we mayassume are the four equidistant points (1; 0; 0); (0;1; 0) along theequator fp2 = 0g of S2 R

3 :

We construct a branched covering eF : M2 ! M1 from M2 = T 5 =T 3T 2 toM1 = T 3S2; by twisting F; as follows. Let q = (q0; q1; q2) becoordinates for T 3 = (R=2Z)3; and let p = (p0; p1; p2) be coordinatesfor S2; where p 20+p

21+p

22 = 1: For each q2 2 R=2Z; let (q2) : S

2 ! S2

be the rotation in the (p0; p1)-plane by angle q2; leaving p2 xed. TheneF : M2 ! M1 is dened by eF (q0; q1; q2; z) := (q0; q1; q2;(q2)(F (z))) :eF : M2 !M1 is a branched covering, with branch locus

= f(q; p) 2 T 3 S2 : p0 = cos(q2 + k=2);

p1 = sin(q2 + k=2); p2 = 0; k 2 Zg:

has four connected components k; k = 0; 1; 2; 3; each of whichprojects dieomorphically onto the the T 3 factor of M1:

Write 1 for the canonical contact 1-formP2

i=0 pi dqi on M1; viewedas the unit cotangent bundle of T 3: On each component k of ; wehave 1 jk

= cos(q2 + k=2) dq0 + sin(q2 + k=2) dq1: We compute(1 ^ d1) jk

= dq0 ^ dq1 ^ dq2; k = 0; 1; 2; 3; which shows that is a (disconnected) contact 3-manifold with contact form 1 j :We may now apply Theorem 1.2 to nd a contact form onM2 = T 5

which is close to the pullback of 1: Thus, the 5-torus T5 has a contact

structure.The existence of a contact structure on the 7-torus, and on numerous

higher-dimensional manifolds, was unclear until now.

2. The Heat Flow

Recall that we are assuming that manifolds are compact, connected,oriented and have no boundary. In addition, we will assume that aRiemannian metric has been chosen.A property of parabolic PDEs familiar to analysts is the strong

maximum principle: if the solution f(t; x) satises at the initial timef(0; x) 0 but f(0; x) 6 0; then at time t > 0; f(t; x) will be positiveeverywhere. That is, heat ows instantaneously to warm a connecteddomain. This property makes parabolic methods ideal for the studyof strict inequalities such as the contact inequality (1). The idea is to


use a hands-on construction to make f(0; x) > 0 for x in an appropri-ate, possibly quite small, open set, while f(0; ) 0 everywhere, andthen to replace f(0; x) with the strictly positive solution f(t; x) at somesmall positive time t: Since f(t; x) is close to f(0; x) in certain strongnorms, other relevant conditions will be maintained.Altschuler in [1] considers any orientable compact 3-manifold M :

he combines the Lutz twist with the strong maximum principle to con-struct a contact form onM: (The result was proved using entirely dier-ent methods in [10]; see also [4].) Altschuler's technique is to start witha foliation, or equivalently with a 1-form 1 satisfying 1 ^ d1 0;and then to use the Lutz twist to construct a 1-form 2 satisfying2 ^ d2 > 0 on a certain open set U; with 2 = 1 near @U: The re-sulting 1-form 2 on all of M satises 2 ^ d2 0; such 1-forms havebeen called confoliations by Eliashberg and Thurston [4]. Altschulerthen denes a degenerate parabolic system of equations for a 1-form(t; x); 0 < t < "; x 2 M; and uses 2 as the initial condition attime t = 0: The system of PDEs is chosen so that the scalar quan-tity f(t; x) := (^d); which is initially nonnegative everywhere andstrictly positive on U; becomes everywhere positive for small t > 0:One diculty is that the system of PDEs is degenerate parabolic, sothat \heat" will ow reliably only in certain directions. Altschuler de-nes the system of equations so that heat ows in directions tangent toker2; which is the original foliation ker1 on the more troublesomeset MnU; and ensures that the Lutz twist was carried out so that theopen set U meets each leaf of ker1:A nonlinear version of the system of equations Altschuler uses on a

3-manifold is

@

@t= ( ^ df) ;where f(t; x) = ( ^ d):(3)

Here, for a p-form on an oriented Riemannian (2n+1)-manifold, isa (2n+1p)-form, the Hodge star of ; which depends linearly on andis dened at each point so that for any oriented orthonormal coframe0; : : : ; 2n of 1-forms, (p ^ : : : ^ 2n) = 0 ^ : : : ^ p1: The system(3) appears quite complicated, but it may be dealt with successfully bythe following trick. The real-valued function f(t; x) satises a singledegenerate parabolic PDE:

@f

@t= ( ^ d ( ^ df)) + h ^ df; di:(4)

Thus, the system (3) uncouples weakly, in the sense that appearsin the PDE (4) only as a coecient. Once f(t; x) is determined, theequation (3) for (t; x) becomes a parameterized system of ODEs. Of

114 ROBERT GULLIVER

course, the unknown 1-form also appears in the coecients of (4), sothis version of Altschuler's method succeeds by requiring t to remainsmall, implying that (t; ) is close to the initial 1-form 2:More generally, on a (2n + 1)-manifold, choose a (2n 1)-form ,

and consider the system of equations

@

@t= ( ^ df) ;where f(t; x) := ( ^ d):(5)

The PDE satised by f is now

@f

@t= ( ^ d ( ^ df)) +

@

@t^ d

:(6)

Again, the system (5) uncouples weakly. Further, we have

Proposition 2.1. : Equation (6) is a weakly parabolic PDE, a degen-erate heat equation, where the right-hand side denes a second-orderpartial dierential operator, which is strongly elliptic when restrictedto the distribution H TM given by

H = (ker())? :

Recall that for a 2-form on M; ker x := f v 2 TxM j (v; ) =0 on TxM g. The coecients of the principal part of the PDE (6) at(t; x) are ATA; where the skew-symmetric matrix A represents x incoordinates which are orthonormal at x; the subspace Hx is spannedby the columns of ATA:In the nonlinear version of Altschuler's heat ow on a 3-manifold,

as we have seen, one chooses = , the evolving 1-form itself. Wewould like to apply Proposition 2.1 in this case. At a given small timet > 0; the 1-form (t; ) never vanishes, so we may complete to a localorthonormal basis (1; 2; 3) with a nonvanishing scalar multiple of1: We compute ker() = ker(1) = ker(2^3) = Re1 ; and thereforeH = (ker())? = Re2 + Re3 = ker: In particular, for small positivet; the distribution H is close to ker2: Thus, if 2 is a contact form onan open set U which is a neighborhood of some point on each leaf ofthe original foliation ker1, then heat will ow out of U to warm eachpoint of M:In general, one may show that

Lemma 2.2. : If is locally decomposable as a product of 1-forms,then

H := (ker())? = ker :


3. The Higher Lutz Twist

For higher dimensions 2n+ 1, in the recent paper of Altschuler andWu [2], the degenerate parabolic system (5) is studied with the choice = ^ (d)n1: They show the existence of a smooth solution forall positive time, via a parabolic regularization. (More precisely, in [2]Altschuler and Wu consider a partly linearized degenerate parabolicsystem which is easier to analyze than equation (5), and slightly morecomplicated, but has consequences equivalent to those of equation (5).)The PDE (6) now becomes

@f

@t= n ( ^ d ( ^ df)) + h ^ df; (d)ni:(7)

Thus, the system of equations uncouples in the same sense as in the3-dimensional case n = 1 (compare equations (3) and (4).)Another part of their paper carries out a higher analogue of the

Lutz twist for the ve-dimensional product case M5 = N3 F 2; usinga contact structure on the 3-dimensional manifold N and its paralleliz-ability. They are thereby able to prove that every product 5-manifoldof this form carries a contact structure. Incidentally, this gives anotherconstruction of a contact form on the 5-torus T 5:Let us proceed in an analogous, but in applications rather dierent,

fashion. Consider a (2n+1)-manifoldM2n+1 = N2n1F 2 which is theproduct of a contact (2n1)-manifold (N;N) and an oriented surfaceF . We shall write 1 for the 1-form on M = N F pulled back fromN : For simplicity, assume that (N;N) has a closed Reeb orbit . Thismeans that 0(s) 6= 0 and that d( 0(s); v) = 0 for all parameter valuess along the curve and for all vectors v 2 T (s)M: Then, accordingto an extension of Darboux' Theorem 1.1, in some neighborhood W

of in N , there are multipolar coordinates (z; r1; 1; : : : ; rn1; n1);r 21 + : : : + r 2n1 < R2; k 2 R mod 2; so that N is the standardcontact form (2), which in these coordinates means that

N = dz +

n1Xk=1

r 2k dk:

In a small ball B F 2; let polar coordinates (rn; n) be chosen, 0 rn < R; n 2 R mod 2: For some choice of real-valued functionshk(r1; : : : ; rn), 0 k n; dene

2 = h0(r1; : : : ; rn) dz +

nXk=1

hk(r1; : : : ; rn) dk :(8)

116 ROBERT GULLIVER

Then 2 will satisfy the contact inequality (1) in U :=W B provided

1

r1 : : : rn

h0 h1 : : : hn@1h0 @1h1 : : : @1hn...

......

@nh0 @nh1 : : : @nhn

> 0; for ri 0 :(9)

Here the operator @k denotes @=@rk: Inequality (9) is equivalent tothe orientation preserving local dieomorphism property for the cen-tral projection of (h0; : : : ; hn) 2 R

n+1 to the sphere Sn: Observe thatinequality (9) continues to hold when 2 is multiplied by a positivescalar function.Recall that we wish to carry out this higher Lutz twist on the open

set U , but we need to construct 2 on all of M . Therefore it will benecessary for the coecients hk(r1; : : : ; rn) to satisfy boundary con-ditions on @U , so that the extension of 2 to all of M by dening2 = 1 on MnU will be smooth. However, only the oriented con-tact structure is important to us, which means that 2 only needsto be dened modulo a (nonconstant) positive multiple. Specically,there needs to hold on the boundary hk(r1; : : : ; rn) = r 2k h0(r1; : : : ; rn);1 k n 1; and hn(r1; : : : ; rn) = 0; as well as inequality (9) inthe interior of U: This requires us to nd a mapping from the sectorV := f(r1; : : : ; rn) 2 (0;1)n : r 21 + : : : + r 2n < R2g to the sphere Sn

which is a dieomorphism of V with an open subset of Sn; having thefollowing boundary values on @V . On the curved part of the boundaryfr 21 + : : :+r 2n = R2g; we require hk = r 2k h0; 1 k n1; and hn = 0:For 1 k n; on the face frk = 0g; we require hk = 0; 1 k n: Inthe ve-dimensional case n = 2; this may be done using a conformalmapping from the quarter-disk V to the hemisphere of S2 with a slitfrom an interior point to the equator removed. The boundary of thequarter-disk covers the slit twice and the equator once. For the generalcase n 2, another more hands-on construction of the map from Vinto the hemisphere of Sn may be carried out.Closed Reeb orbits may be rare for a given contact manifold (N;N),

but the above procedure may be modied appropriately.A covering argument may then be used to arrange disjoint open sets

of M of the above form so that their projections from M = N Fto N cover all of N: For small time t > 0; the solution (t; ) willbe close to the initial value 2: On the complement of the union ofthe sets U where the higher Lutz twist has been carried out, we have2 = 1: Write 1 = 1 ^ (d1)

n1: Since 1 is the pullback of thecontact form N ; we see that 1 is the pullback of a volume form onN; and thus is decomposable as a product of 1-forms. It follows from


Lemma 2.2 that on the complement of the sets U; the distribution H1

of elliptic directions for the system (5), with replaced by 1; equalsker 1; which is the tangent plane TF to fygF 2 in TM = TN TF:Therefore, for small time t > 0; the distribution H is close to thefoliation TF: It follows by Proposition 2.1, for small time t > 0; thatheat ows out of the union of open sets U along directions arbitrarilyclose to TF to warm all ofM = NF: Numerous points omitted here,in part rather technical, will be treated in [7] to prove

Theorem 3.1. : If N2n1 is a compact contact manifold and F 2 is acompact oriented surface, then M2n+1 = N F has a contact 1-form,which is C2-close to a 1-form 2 obtained from the contact form of Nby means of the higher Lutz twist.

By induction on n = 2; 3; : : : ; with N = T 2n1 and F = T 2; wededuce

Corollary 3.2. : Any odd-dimensional torus T 2n+1 carries a contactstructure.

References

[1] S.J. Altschuler, A geometric heat ow for one-forms on three dimensional man-

ifolds, Illinois J. Math. 39 (1995), 98118.[2] S.J. Altschuler and L.F. Wu, On deforming confoliations, preprint 1999.[3] S.-S. Chern, Pseudo-groupes continus innis in Geometrie Dierentielle, Col-

loques Internat. CNRS, Strasbourg 1953.[4] Y. Eliashberg and W. Thurston, Confoliations, Univ. Lecture Ser. 13, Amer.

Math. Soc., Providence, 1998.[5] H. Geiges, Constructions of contact manifolds, Proc. Camb. Phil. Soc. 121

(1997), 455464.[6] H. Geiges, Contact topology in dimension greater than three, to appear in Proc.

Eur. Congr. Math. (Barcelona, 2000). Progress in Math., Birkhauser, Basel.[7] H. Geiges, R. Gulliver and M. Schwarz, Heat ow and contact structures on

product manifolds, in preparation.[8] M. Gromov, Partial Dierential Relations, Ergebnisse vol. 9, Springer, Berlin

1986.[9] R. Lutz, Sur la geometrie des structures de contact invariants, Ann. Inst.

Fourier (Grenoble) 29 (1979), 283306.[10] J. Martinet, Formes de contact sur les varietes de dimension 3, Lecture Notes

in Math. 209 (1971), 142163: Springer, Berlin.

School of Mathematics, University of Minnesota


MANIPULATING THE ELECTRON CURRENT

THROUGH A SPLITTING

M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV

Abstract. The description of electron current through a splitting

is a mathematical problem of electron transport in quantum net-

works [5, 1]. For quantum networks constructed on the interface

of narrow-gap semiconductors [29, 2] the relevant scattering prob-

lem for the multi-dimensional Schodinger equation may be substi-

tuted by the corresponding problem on a one-dimensional linear

graph with proper selfadjoint boundary conditions at the nodes

[11, 10, 25, 24, 16, 19, 4, 28, 20, 18, 6, 5, 1]. However, realistic

boundary conditions for splittings have not yet been derived.

Here we consider some compact domain attached to a few semi-

innite lines as a model for a quantum network. An asymptotic

formula for the scattering matrix for this object is derived in terms

of the properties of the compact domain. This allows us to pro-

pose designs for devices for manipulating quantum current through

a splitting [3, 15, 22, 9, 21].

Introduction: current manipulation in the resonance

case

In this paper we discuss the scattering problem on a compact do-main with a few semi-innite wires attached. This is motivated by thedesign of quantum electronic devices for triadic logic. In the papers[3, 15] a special design of the one-dimensional graph which permitsmanipulation of the current through an elementary ring-like splittingis suggested. This permits, in principle, manipulation of quantum cur-rent in the resonance case to form a quantum switch. Another devicefor manipulating quantum current through splittings is discussed in[22, 9]. In [21] the special design of the splitting formed as a circulardomain with four one-dimensional wires attached is used to produce atriadic relay.

Date: 20 September 2000

Revised 20 January 2001.

118

MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 119

In order to illustrate the basic principle of operation consider the self-adjoint Schrodinger operator

L + q(x)@

@n

@

= 0:

on some compact domain . In this paper we will only consider thecase R

3 [22] (for other cases see also [3, 15, 21]). Roughly speakingthe solution of the Cauchy problem

@i@t

= L(x; 0) = 0(x)

(1)

is given in terms of eigenfunctions 'n

(x; t) =Xn

neint'n(x):

Picking a specic mode '0 with energy 0 we suppose that '0 disap-pears on some subset l0 . Connecting `thin channels' at various

Ω 0λ

0λ

l0

gure 1. Resonance switch

points on the boundary of and introducing an excitation of energy0 along the channels we can hope to create a switching eect. Essen-tialy this is achieved by varying q(x) so that l0 \ @ coincides with theconnection point of a `thin channel'.Implicit in our construction is the assumption that the energy of theelectrons in the device is equal to some resonance eigenvalue of theSchrodinger operator on . We refer to this as the resonance case1.

1This has interesting implications when we consider the eect of decreasing the

length scale|or equivalently scaling up the energy|viz. the eect of non-zero

temperature becomes negligable for suciently small length scales, see [14].

120 M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV

Another assumption which we have made above is that 0 is a simple

eigenvalue of L. We will show that the case of multiple eigenvalues isa simple generalisation of the case for simple eigenvalues, see [3, 15].In the rst section we give a brief description of the connection of thethin channels (here they are modelled by one-dimensional semi-lines)to the compact domain, for more details see [22]. In the second sectionwe derive an asymptotic formula for the scattering matrix in termsof the eigenfunctions on the compact domain. In the last section webrie y discuss some simple models of a quantum switch constructed onthe basis of this asymptotic formula.

1. Connection of compact domain to thin channels

As we mentioned above the thin channels are modelled by one-dimensional semi-lines. This is justied by an appropriate choice of ma-terials (narrow-band semiconductors) and energies, see [29, 2, 21]. Weassume that these channels are attached at the points fa1; a2; :::; aNg @ (perturbation of the operator L at inner points faN+1; aN+2; : : :; aN+Mgmay be considered using the same techniques as for fa1; a2; :::; aNg[2, 3, 21] although we do not consider this here).We refer to L, dened above, as the unperturbed Schrodinger operator.L is restricted to the symmetric operator L0 dened on the class D0

of smooth functions with Neumann boundary conditions which vanishnear the points a1; a2; : : : ; aN . The deciency subspaces, Ni, of therestricted symmetric operator L0,

[L?0 i] esi = 0

for complex values of the spectral parameter coincide with Greensfunctions G(x; as) of L which are elements of L2() but do not be-long to the Sobolev class W 1

2(). In the case when is a compact

one-dimensional manifold (a compact graph) these Greens functionsare continuous and can be written in terms of a convergent spectralseries [3]. However, when R

2 ;R3 the deciency elements will havesingularities and we must use an iterated Hilbert identity to regularisethe values of the Greens function at the poles.It is well known, for R

3 , that the Greens function admitts therepresentation inside

G0

(x; y) =eipjxyj

4jx yj+ g(x; y; )(2)


where g(x; y; ) is continuous. The potential-theory approach gives theasymptotics of the Green function near the boundary point as 2 @

G0

(x; as) 1

2jx asj+ Ls(x) +Bs(x; );(3)

where Ls is a logarithmic term depending only on @ and Bs is abounded term containing spectral information [8].In order to choose regularised boundary values we use the followinglemma [22] (here we assume L > 1 is semi-bounded from below):

Lemma 1. For any regular point from the complement of the spec-

trum (L) of L and any a 2 fasgN+M

s=1 the following representation is

true:

G(x; a) = G1(x; a) + (+ 1)G1 G(x; a);

where the second addend is a continuous function of x and the spectral

series of it in terms of eigenfunctions 'l of the nonperturbed operator

L

(+ 1)G1 G(x; a) = (+ 1)Xl

'l(x)'l(a)

(l + 1)(l )

is absolutely and uniformly convergent in .

The proof of this lemma is based on the classical Mercer theoremalong with the Hilbert identity [22].It is well known that the domain of L?

0can be written as the direct

sum

D?0= D0 +Ni +Ni(4)

so for any u 2 D?0

u = u0 +Xs

A+

s Gi(x; as) +Xs

As Gi(x; as):

We dene u 2 D?0in terms of the coordinates

As A+

s + As ;

Bs limx!as

"u(x)

Xt

At<Gi(x; at)

#;

the singular and regular amplitudes respectively since it is clear fromthe above lemma that As is the coecient of the singular part and Bs

the coecient of the regular part of u 2 D?0. The boundary form of L?

0

may be written in terms of As Bs as a hermitian symplectic form

hL?0u; vi hu;L?

0vi =

XBusAvs Au

sBvs :(5)


1.1. Self-adjoint extensions. We recall that to each boundary pointas, s = 1; : : : ; N , there is attached a semi-innite ray. On the s-th raywe dene the symmetric operator

ls;0 = d2

dxs+ qs(xs);

on functions which vanish at xs = 0 (which is identied with as 2 ).Let us consider the symmetric operator L0 l1;0 l2;0 ::: lN;0. Theconnection between the compact domain and the rays is given by (aparticular) self-adjoint extension of this operator. The boundary formof the adjoint L?

0 l?

1;0 l?2;0 ::: l?N;0 is easily seen to be

NXs

BusA

vs Au

sBvs

+

NXs

u0s(0)vs(0) us(0)v0s(0)

:(6)

It is well known that the self-adjoint extensions of L0 l1;0 l2;0 :::lN;0 correspond to Lagrange planes in the Hermitian symplectic spaceof boundary values equipped with the above boundary form [26]. Ingeneral, if A, B are (vectors of) boundary values for some symmetricoperator then any self-adjoint extension can be described by

i

2(U I)A+

1

2(U + I)B = 0

for some unitary matrix U [14, 13].We choose the particular family of self-adjoint extensions which corre-spond to the following boundary conditions at the N points of contactof the rays

As

us(0)

=

0 0

Bs

u0s(0)

;(7)

for s = 1; : : : ; N and > 0. The resulting self-adjoint extension wedenote by L. The parameter is a measure of the strength of theconnection between the rays and the compact domain|in the limit ! 0 the resolvent of L converges uniformly to the resolvent of L oneach compact subset of the resolvent set of L [22].

2. Asymptotics of the scattering matrix

For the remainder we assume that the potential on the rays qs(xs) 0 is zero.We use the ansatz

us = fs(xs;k)s1 + fs(xs; k)Ss1;(8)


for the scattered wave generated by the incoming wave from the rayattached to the point a1. Here fs(xs;k) are the Jost solutions [27], inthis case (qs(xs) 0) just the exponentials

fs(xs;k) = eikxs;

and = k2 is the spectral parameter.From the boundary conditions (7) we get N equations

As = f 0s(0;ik)s1 + f 0s(0; ik)Ss1Bs = fs(0;ik)s1 + fs(0; ik)Ss1:(9)

Inside the eigenfunction u(x; k) may be written as a sum of Greensfunctions at the spectral parameter = k2

u(x; k) =

NXs

CsG(x; as):

Using the Cayley transform between the spectral points i and onegets a relationship between these Greens functions and the deciencyelements (as dened above) so that [22]

limx!as

[G(x; as; ) <G(x; as; i)] =

I+ L

L IGi(as); Gi(as)

gs():

Consequenttly we can show that u has the following asymptotics asx! as

u Cs<Gi(x; as) + Csgs() +

Xt6=s

CtG(as; at) + o(1):(10)

It follows that for the scattering wave the symplectic variables are re-lated by

As = Cs

Bs = gs()Cs +Xt6=s

CtG(as; at);

that is B = QA where

Q() =

0BB@

g1() G(a1; a2) G(a1; aN)G(a2; a1) g2() G(a2; aN)

.... . .

...G(aN ; a1) gN()

1CCA :(11)

Putting this into (9) we can solve for the scattering matrix to get

S = I+ ik2Q

I ik2Q:(12)


Let us choose an eigenvalue 0 of the unperturbed operator L on .We suppose that 0 has a p-dimensional eigenspace, which we denoteR0, with orthonormal basis f'0;ig

p. The following important technicalstatement close to Lemma 1 above is true [22]:

Theorem 1. The elements of the Q-matrix have the following asymp-

totics at the spectral point 0:

Qst()

pXi=1

'0;i(as)'0;i(at)

0 +Q0(as; at; );

where Q0(as; at; ) is a continuous function at the point = 0.

We will use this result to prove an asymptotic formula for the scat-tering matrix in the limit of weak connection between the compactdomain and the rays.Consider the mapping P : L2() ! C

N which gives the vector of val-ues of a function in L2() at the nodes of each of the N rays. Todistinguish between functions and elements of C N we use the notation

P( ) = j i 2 CN ;

and we denote

R0 P(R0):

Proposition 1. It is possible to choose an orthonormal basis f0;igp

for R0 which forms an orthogonal, but not necessarily normalised, basis

for R0 under P.

Proof: Given some orthonormal basis f'0;igp for R0 we see that

0;i =

pXj=1

Uij'0;j

is also an orthonormal basis where U 2 U(p).The inner product of the image under P

h0;ij0;ji =

pXr;s=1

Uirh'0;rj'0;siUjs

shows that nding an orthogonal basis for R0 amounts to nding theunitary matrix U which diagonalises Ars = h'0;rj'0;si. 2


This allows us to write Q in `diagonal' form

Q =1

0 [j0;1ih0;1j+ + j0;mih0;mj]

+Q0()

=Dl

0 +Q0()(13)

where m p is the dimension of R0.

Theorem 2. If 0 is an eigenvalue of L then for vanishing coupling

0 the scattering matrix of L has the form

S(0) = I+ 2P0 2Xs=1

(ik02P?

0Q0P

?0)s

= I+ 2P0 +O(2)(14)

where P0 is the orthogonal projection onto R0.

Proof: Using equation (13),

S() =

I+

ik2D0

0 + ik2Q0

I

ik2D0

0 ik2Q0

1:

Since D0 = D?0, the matrix E0 = I

ik2D0

0 is invertable. Consequently

the denominator can be writtenI

ik2D0

0 ik2Q0

1=

[I ik2Q0E

10]E0

1= E1

0

I ik2Q0E

10

1:

Again the matrix I ik2Q0E10

has an inverse for 0 since Q0 =Q?0. This gives the following expression for the scattering matrix

S() = E?0E10

+ ik2Q0E10

I ik2Q0E

10

1=

E?0E10

+ ik2Q0E10

Xs=0

(ik2Q0E10)s:

Denoting i ph0;ij0;ii and diagonalising we can write,

E10

= diag

1

ik221

0 ; : : : ; 1

ik22m0

; 1; : : : ; 1

1

= diag

0

0 ik221

; : : : ;0

0 ik22m; 1; : : : ; 1

:

Therefore

lim!0

E10

= P?0= I P0:(15)


Furthermore

E?0E10

= diag

0 + ik22

1

0 ik221

; : : : ;0 + ik22m0 ik22m

; 1; : : : ; 1

which gives us the limit

lim!0

E?0E10

= P?0 P0 = I 2P0:(16)

From these limits we get

S(0) = I 2P0 + ik0

2Q0P

?0

Xs=0

(ik02Q0P

?0)s

= I+ 2P0 2Xs=1

(ik02P?

0Q0P

?0)s

= I+ 2P0 +O(2): 2

This formula appears to imply that there may be non-zero trans-mission in the case of zero connection between the rays. Actually thetransmission coecients are not continuous with respect to uniformlyin [3, 22]. The physically signicant parameters of the system areobtained by averaging as functions of over the Fermi distribution sothat there is no transmission for = 0.

Corollary 1. If 0 is an eigenvalue of L such that P0 = I then the

above formula is independent of , ie.

S(0) = I

Consequently, when we have pure re ection at an eigenvalue of theunperturbed operator, we have pure re ection regardless of the strengthof the interaction between the rays and the compact domain.

3. Simple models

In [21] the authors discuss the case where is the unit disc in R2

and there are four one-dimensional wires attached at the points ' =0; ;=3. The dynamics on is given, using polar coordinates (r; ),by the dimensionless Schrodinger equation

+ [V0 + "rcos()] = (17)

on the domain with Neumann boundary conditions at the boundary:

@

@n

r=1

= 0:

The dimensionless magnitude " of the governing eld is choosen so thatthe eigenfunction corresponding to the second smallest eigenvalue hasonly two zeroes on the boundary of the unit circle which divide the


circumference in the ratio 2 : 1. It is then easy to see that by rotatingthe potential V one may redirect the quantum current from the wireattached to the point ' = 0 to any other wire with all of the otherwires blocked [21].The analysis in this case is similar to the analysis given above exceptthere is now only a logarithmic singularity in the Greens function andthe Krein formula for innite deciency indices [17, 23] and innite-dimensional Rouchet theorem [12] play a central role. A large amountof the calculation was done using Mathematica.

In [15], using the above asymptotic formula for the scattering matrixto choose appropriate parameters, the author discusses the case where is simply a one-dimensional ring and there is an angle of =2 betweenthe rays|see gure 2. a). By applying a uniform eld to the ring, q = 0for the open state and q = 3 for the closed state, it is easy to see thata switching eect is produced where the Fermi energy is assumed tocorrespond to the smallest eigenvalue of the unperturbed operator onthe ring, ie. 0 = 1. See also [7] where a similar construction isconsidered.For the purposes of comparison we also consider a device|see gure

q = 0q =-3

oc

q = 0q = 3

oc

a) Interference b) Potential barrierswitch switch

gure 2.

2.b), the angle between the rays is now |with similar parameterswhere now we switch the current by raising a potential barrier, q = 0for the open state and q = 3 for the closed state, instead of usinginterference eects. It is easy to see that, unlike the rst case, theeciency of such a switch will be limited by tunneling.Using Maple we numericaly integrate over the Fermi distribution to


produce plots of the averaged conductance c o in the closed and openstates respectively|see gure 3. a) and b) which show the plots forthe `interference switch' and the `potential barrier switch' respectively.Here is a temperature parameter in the model.

1.21.41.61.8

22.22.42.62.8

33.23.43.63.8

4

0 0.2 0.4 0.6 0.8 1|β|

= 0:0225

= 0:045

= 0:09

= 0:18

gure 3. a) log10(c=o) versus

3.5

4

4.5

5

5.5

6

6.5

0 0.2 0.4 0.6 0.8 1|β|

= 0:0225

= 0:18

gure 3. b) log10(c=o) versus


It appears that the open state|possibly due to tunneling eects|is more dicult to achieve. This also appears to explain why, in thelimit of small , the properties of the switches improve: weak couplingbetween the ring and rays improves the open state of the switches. Onthe other hand, in the limit ! 1, the ratio c=o for the secondexample rapidly decreases to a bound due to tunneling which may becalculated from the transmission coecient

lim!0

c

o

=1

4:57 103;

see gure 3 b). The rst switch does not have this bound and conse-quently for suciently low temperature or small radius (see rst foot-note) we conjecture that it will have better properties.

References

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1996. Springer.

[3] V. Bogevolnov, A. Mikhailova, B. S. Pavlov, and A. Yafyasov. About scattering

on the ring. Department of Maths Report series 413 ISSN 1173-0889, Univer-

sity of Auckland, New Zealand, 1999. To be published in `Operator Theory:

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[7] P. Exner, P. Seba, and P. Stovicek. Quantum interference on graphs controlled

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cation to simple nanoelectronic devices. J. Math. Phys, 36:28132825, 1995.

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ment of Maths Report series 439, University of Auckland, New Zealand, 2000.

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tering on a compact domain with few semi-innite wires attached: resonance

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Arnol'dfest, 1998. Proceedings of the Fields Institute Conference in Honour

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Communications Series.

[25] S. P. Novikov and I. A. Dynnikov. Discrete spectral symmetries of low-

dimensional dierential operators and dierence operators on regular lattices

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Mathematics Department, Auckland University, Private Bag 92019,

Auckland, New Zealand



Laboratory of the Theory of Complex Systems-Institute of Physics,

St Petersburg University, Ulyanovskaja 1, Petrodvorets 198504, St

Petersburg, Russia

E-mail address : anna [email protected] [email protected]

Mathematics Department, Auckland University, Private Bag 92019,

Auckland, New Zealand


WHAT'S NEW FOR THE BELTRAMI EQUATION ?

TADEUSZ IWANIEC AND GAVEN MARTIN

Abstract. The existence theorem for quasiconformal mappings

has found a central role in a diverse variety of areas such as holo-

morphic dynamics, Teichmuller theory, low dimensional topology

and geometry, and the planar theory of PDEs. Anticipating the

needs of future researchers we give an account of the \state of the

art" as it pertains to this theorem, that is to the existence and

uniqueness theory of the planar Beltrami equation, and various

properties of the solutions to this equation.

This paper surveys the recent work of the authors' paper [14] and

parts of our monograph [15]. Readers interested in more details, and inparticular rather greater discussion of related work by others, shouldconsult those works. In what follows we use fairly standard notation,in particular B denotes a disk, usually the unit disk, in the complex

plane C and C = C [ f1g is the Riemann sphere.

The Beltrami equation has a long history. Gauss rst studied theequation, with smooth coecients, in the 1820's while investigating theproblem of existence of isothermal coordinates on a given surface. The

complex Beltrami equation was intensively studied by Morrey in thelate 1930's, and he established the existence of homeomorphic solutionsfor measurable [18]. It took another 20 years before Bers recognisedthat homeomorphic solutions are quasiconformal mappings. Since thenthese ideas have found diverse applications in a variety of areas such as

holomorphic dynamics, Teichmuller theory, low dimensional topologyand geometry, and the planar theory of PDEs.

1. PDEs

There is a strong interaction between linear and non-linear ellipticsystems in the plane and quasiconformal mappings. The most generalrst order linear (over R) elliptic system takes the form

@ f = 1 @f + 2 @f

Research supported in parts by grants from the U.S. National Science Foundation

and the N.Z. Marsden Fund.

132

WHAT'S NEW FOR THE BELTRAMI EQUATION ? 133

where 1 and 2 are complex valued measurable functions such that

j1(z)j + j2(z)j K 1

K + 1< 1 a:e:

The complex Beltrami equation is simply that equation which is linearover the complex numbers.

@f(z) = (z) @f(z)(1)

Classically one assumes ellipticity bounds:

kk1 =K 1

K + 1< 1(2)

When = 0 we have the CauchyRiemann system.These sets of equations are particular cases of the genuine non-linear

rst ord er system

@f = H(z; @f)M(3)

where H : C ! C is Lipschitz in the second variable,

jH(z; )H(z; )j K 1

K + 1j j; H(z; 0) 0

A feature of (3) is that the dierence of two solutions need not solvethe same equation but it is Kquasiregular (the term used to describenon-injective quasiconformal functions). Thus quasiconformal mapsare a central tool used to establish a priori estimates needed for the

existence and uniqueness.

2. Classical Regularity Theory

Typically one seeks solutions to the Beltrami equation in the Sobolev

space W1;2loc (). However, the solutions to this equation have the fol-

lowing striking regularity result nally established in complete form byK. Astala (the Area Distortion Theorem) [2].

Theorem 2.1. Let be a measurable function dened in with kk1 =

k < 1. Let f be any solution to the Beltrami equation with f 2 W1;qloc (),

q > 1 + k. Then f 2 W1;ploc () for all p < 1 + 1

k.

Moreover, there may be solutions in W1;1+kloc () not in any higher

Sobolev space, and there may be solutions inW1;2loc () not inW

1+1=kloc ().

Notice the indices p and q in the above result form a Holder conjugatepair.

The Neumann iteration procedure based on invertibility of the Bel-trami operator I S, (S the complex Hilbert transform) yields a

representation formula (rst found by Bojarski [5]) and existence.

134 TADEUSZ IWANIEC AND GAVEN MARTIN

Theorem 2.2. Let be a measurable function in C and suppose

kk1 < 1. Then there is a homeomorphic solution g : ! C to theBeltrami equation.

Moreover every W1;2loc (; C ) solution f is of the form

f(z) = (g(z))

where : g()! C is a holomorphic function.

This last fact is known as \factorisation".

3. A Fundamental Example

This example re ects what is possible in the degenerate elliptic set-ting, and shows why it is necessary to use the Orlicz-Sobolev spaces in

order to discuss the ne properties of solutions.

Theorem 3.1. Let A : [1;1) ! [1;1) be a smooth increasing func-tion with A(1) = 1 and such thatZ

1

1

A(s)

s2ds <1:(4)

Then there is a Beltrami coecient compactly supported in the unitdisk, j(z)j < 1, with the following properties:

1. The ellipticity bound K(z) =1+j(z)j

1j(z)jsatisesZ

B

eA(K(z))dz <1(5)

2. Every W1;1loc (B)solution to the Beltrami equation

fz = fz a:e: B(6)

continuous at the origin is constant.

3. There is a bounded solution w = f(z) to the Beltrami equation inthe space weak-W 1;2(B)

T1q<2W

1;q(B) which homeomorphi-

cally maps the punctured disk onto the annulus 1 < jwj < R.

A few remarks.

First, W1;1loc (B) is really the weakest space in which one can begin to

discuss what it means to be a solution.

Secondly, the integrability condition (4) implies that A is sublinear.As examples the function

A() =

(log )1+(7)


satises (4) for all > 0, but not for = 0. More generally, if we put

log1 = log ; logn+1 = log(logn )

we have the iterated logarithm functions. Then for each n > 0 thefunction

log1(1 + ) log2(e+ ) log3(ee + ) (logn(e

e

e

+ ))1+

again satises (4) for all > 0, but not for = 0.

Finally, (3) is no accident. We can prove that if eA(K) 2 Lploc(B) for

some p 0 and certain types of A (and in particular the log log : : :examples), with Z

B

eA(K(z))dz =1

then there is a homeomorphic (and hence continuous) solution inW 1;p(B)for all p < 2. In fact the solution lies in an OrliczSobolev class just

below W1;2loc

4. Mappings of Finite Distortion

We next give a general denition of the mappings which mostly oc-cur. Roughly, solutions to a Beltrami equation with ellipticity boundswhich are pointwise nite will be mappings of nite distortion as soonas they are ACLabsolutely continuous on lines.

Denition A mapping f : ! C is said to have nite distortion if:

1. f 2 W1;1loc (),

2. The Jacobian determinant, J(z; f) = detDf(z), of f is locallyintegrable and does not change sign in

3. There is a measurable function K = K(z) 1, nite almost ev-erywhere, such that f satises the distortion inequality

jDf(z)j2 K(z) jJ(z; f)j a:e: (8)

Notice that the hypotheses are not sucient to guarantee that f 2

W1;2loc () unless the distortion function K is bounded. Nor do they

imply that the Jacobian does not vanish on a set of positive measure.

The motivational philosophy behind the condition that the distortionfunction is exponentially integrable is now clear. We wish to exploit the

BMOH1 duality (and even more rened versions of this) to achieveuniform estimates on approximating sequences of solutions.


5. Maximum Principle and Continuity

One of the rst tasks is to establish a maximum type principle andmodulus of continuity estimates.A continuous function u : ! R dened in a domain is monotone

if

oscB u osc@B u

for every ball B . This denition in fact goes back to Lebesgue in1907 where he rst showed the relevance of the notion of monotonicity

to elliptic PDEs in the plane. In order to handle very weak solutionsof dierential inequalities, such as the distortion inequality, we need toextend this concept, dropping the assumption of continuity, and to thesetting of OrliczSobolev spaces.

Denition. A real valued function u 2 W 1;P () is said to be weaklymonotone if for every ball B and all constants m M such that

jM uj jumj+ 2umM 2 W1;P0 (B)(9)

we have

m u(x) M(10)

for almost every x 2 B.

For continuous functions (9) holds if and only if m u(x) M

on @B. Then (10) says we want the same condition in B, that is the

maximum and minimum principles.Here, and in what follows we assume, unless otherwise stated, that

the Orlicz function P satisesZ1

1

P (t)dt

t3=1(11)

and that the function t 7! t5

8 is convex. For example Orlicz functionsof the form

P () = 2

log1(1 + ) log2(e+ ) log3(ee + ) (logn e

e

e

+ )

are of this form.

The OrliczSobolev space W1;Ploc () consists of functions which, to-

gether with their rst derivatives, lie in the space LP (). Thus in theexample given above we are looking in Zygmund type spaces just below

W1;2loc ().


Lemma 5.1. Let be a bounded domain and suppose that u 2 W 1;P ()\

C() is weakly monotone. Then

min@

u u(x) max@

u(12)

for every x 2 .

The paper [17] by Manfredi should be mentioned as the beginning

of the systematic study of weakly monotone functions.We now recall a fundamental monotonicity result in the Orlicz

Sobolev classes.

Theorem 5.1. The coordinate functions of mappings with nite dis-

tortion in W 1;P () are weakly monotone.

There is a particularly elegant geometric approach to the continuityestimates of monotone functions. The idea goes back to Gehring inhis study of the Liouville theorem in space where he developed theOscillation Lemma.

We need the Pmodulus of continuity P () dened for 0 < 1as follows. For > 0 the value t of P at is uniquely determined bythe equation Z 1=

1

P (st)ds

s3= P (1):(13)

Certainly P is a non-decreasing function with

lim!0

P () = 0:(14)

Given the transcendental nature of the equation one must solve, itis impossible in all but the most elementary situations, to calculate. Here are a few explicit formulas for () which exhibit the correctasymptotics for near 0.

P (t) = t2; () = j log j1

2

More generally for all > 0 we have,

P (t) = t2 log1(e+ t); > 0; () j log j2

P (t) =t2

log(e + t); () [log j log j]

1

2 ;

and nally

P (t) =t2

log(e+ t) log log(3 + t); () [log log j log j]

1

2 ;

We now have the fundamental modulus of continuity estimate.


Theorem 5.2. Let u 2 W 1;P (B) be weakly monotone inB = B(z0; 2R)

. Then for all Lebesgue points a; b 2 B(z0; R) we have

ju(a) u(b)j 16RkrukB;P

ja bj

2R

:(15)

In particular, u has a continuous representative for which (15) holds

for all a and b in the disk B(z0; R).

In the statement above we have used

krukB;P = inf

1

:1

jBj

ZB

P (jruj) P (1)

(16)

to denote the P -average of ru over the ball B.

Theorem 5.3. Every mapping with nite distortion in the Orlicz

Sobolev class W1;Ploc (), is continuous.

6. Liouville Type Theorem

Here is a rst taste of the power of Theorem 5.1.

Theorem 6.1. Let f : C ! C be a mapping of nite distortion whosedierential belongs to LP (C ). Then f is constant.

The proof consists in showing that R krukB;P ! 0 as R ! 1 in(15) using the Dominated Convergence Theorem.

7. Solutions

A principal solution is a homeomorphism h : C ! C with

1. a discrete set E (the singular set) such that h 2 W1;1loc (C nE),

2. the Beltrami equation hz(z) = (z)hz(z) holds for a.e. z 2 C , and

3. we have the normalisation h(z) = z + o(1) at 1

It will become clear that the key to understanding the Beltrami equa-tion and its local solutions is in the existence and uniqueness propertiesof the principal solutions.

A function f , not necessarily a homeomorphism, is a very weak so-lution if it satises:

there is a discrete set Ef (the singular set) such that h 2

W1;1loc ( n Ef ).

the Beltrami equation

fz(z) = (z)fz(z)

holds for almost every z 2


The point is that one expects a solution f to be the composition

of the principal solution h with a meromorphic function ' dened onh(),

f = ' h;

(the Stoilow factorisation theorem). Thus away from the poles, the

solution f is locally as good as the principal solution.

7.1. Uniqueness of Principal Solutions. Here is the most generaluniqueness result that we are aware of.

Theorem 7.1. Every elliptic equation

hz = H(z; hz)

admits at most one principal solution in the Sobolev-Orlicz class z +W 1;P (C )

We use the term z +W 1;P (C ) to denote the mappings h with jhzj+jhz 1j 2 LP (C ). As far as the ellipticity is concerned, we assume that

there is a measurable compactly supported function k : C ! B suchthat for almost every z 2 C and all ; 2 C

jH(z; )H(z; )j k(z)j j

Proof. Let h be a solution to the equation. Thus hz(z) = 0 forz suciently large. The point is that given two principal solutions h1

and h2, the mapping f = h1h2 has nite distortion and its dierentialDf = Dh1 Dh2 belongs to LP (C ). To see this note

j(h1 h2)zj = jh1z h2zj

= jH(z; h1z)H(z; h2z)j

k(z)j(h1 h2)zj

whence J(z; h1h2) 0. It follows that f is constant from the Liouville

theorem. The normalisation at 1 implies that this constant is 0.

8. Stoilow Factorisation

We now state that if a Beltrami equation admits a homeomorphicsolution, then all other solutions in the same class are obtained fromthis solution via composition with a holomorphic mapping.

Theorem 8.1. Suppose we are given a homeomorphic solution h 2

W1;Ploc () to the Beltrami equation

hz = (z)hz a:e: (17)


Then every solution f 2 W1;Ploc () takes the form

f(z) = (h(z)); z 2

where : h()! C is holomorphic.

9. Failure of Factorisation

Now an example which shows that for fairly nice solutions one cannotexpect a factorisation theorem even in the case of bounded distortion.

Theorem 9.1. Let K > 1 and q0 <2KK+1

. Then there is a Beltrami

coecient supported in the unit disk with the following properties.

kk1 = K1K+1

,

The Beltrami equation hz = hz admits a Holder continuous so-

lution h 2 z +W 1;q0(C ) which fails to be in W1;2loc (C ),

The solution h is not quasiregular, and therefore not the principalsolution, nor obtained from the principal solution by factorisation.

10. Distortion in the Exponential Class

Theorem 10.1. There exists a number p0 > 1 such that every Bel-

trami equation

hz(z) = (z) hz(z) a:e: C

with Beltrami coecient such that

j(z)j K(z) 1

K(z) + 1B(z)

and

eK 2 Lp(B)(18)

with p p0, admits a unique principal solution h 2 z +W 1;2(C ).

There are examples to show that in order for there to be a principalsolution in the natural Sobolev space z +W 1;2(C ) it is necessary thatthe exponent p at (18) is large, at least p 1.

As a matter of fact, somewhat more is true in Theorem 10.1. The

higher the exponent of integrability of epK the better the regularity ofthe solution. That is even beyond L2, such as L2 logL with any 0,see [16].

The situation is dierent if the integrability exponent of eK is smaller

than the critical exponent p0. Here the principal solution need not bein z +W 1;2(C ), but we still obtain a satisfactory class of solutions.


Theorem 10.2. Suppose the distortion function K = K(z) for the

Beltrami equation is such that eK 2 Lp(B) for some positive p. Thenthe equation admits a unique principal solution h

h 2 z +W 1;Q(C ); Q(t) = t2 log1(e+ t)(19)

Moreover, every W1;Qloc () solution is factorisable.

11. Distortion in the Subexponential Class

We assume here that the Beltrami coecient is supported in the unit

disk B.

Theorem 11.1. There is a number p 1 such that every Beltramiequation whose distortion function has

exp

K(z)

1 + logK(z)

2 Lp(B)

for p > p, admits a unique principal solution h 2 z +W 1;Q(C ) with

Orlicz function Q(t) = t2 log1(e+ t). Moreover we have

Modulus of Continuity;

jh(a) h(b)j2 CK

log log(1 + 1jabj

)(20)

for all a; b 2 2B. Inverse; The inverse map g = h1(w) has nite distortion K =

K(w) and

logK 2 L1(C )

Factorization; each solution g 2 W1;Qloc () to the equation

gz = (z)gz; a:e:

admits a Stoilow factorisation

g(z) = h(z)(21)

where is holomorphic in h1(). In particular, all non-constant

solutions in W1;Qloc () are open and discrete.

12. Existence Theory.

Various reductions show the important case to be when the Beltramicoecient is compactly supported in the unit disk B. Then any

solution is analytic outside the unit disk.


12.1. Results from Harmonic Analysis. The existence proof pre-

sented here exploits a number of substantial results in harmonic analy-sis. The arguments clearly illustrate the important role that the higherintegrability properties of the Jacobians have to play. The critical ex-ponent p0 in Theorem 10.1 depends only on the constants in three

inequalities which we now state.The rst is a direct consequence of [8].

Theorem 12.1. (Coifman, Lions, Meyer, Semmes) The Jacobian de-

terminant J(x; ) of a mapping 2 W 1;2(C ) belongs to the Hardy spaceH1(C ) and we have the estimate

kJ(x; )kH1(C) C1

ZC

jDj2(22)

Next we have from [9]

Theorem 12.2. (Coifman, Rochberg) Let be a Borel measure inC such that its HardyLittlewood maximal function M(x; ) is niteat a single point (and therefore at every point). Then logM(x; ) 2

BMO(C ) and its norm is bounded by an absolute constant,

k logM(x; )kBMO C2:(23)

Finally we shall need the constant C3 which appears in the H1-BMO

duality theorem of Feerman, [12].

Theorem 12.3. (Feerman) For K 2 BMO(C ) and J 2 H1(C ) wehave

ZK(x) J(x) dx

C3kKkBMOkJkH1(24)

Having these prerequisites we can reveal that the exponent in The-orem 10.1 is

p0 = 8C1C2C3:(25)

13. Sketch of Proof for Theorem 10.1

We again refer the reader to [14] for more details, but the basic ideascan be found here.

Let K be the distortion function. Set

Ap =

ZC

epK(z) ep

dz <1


We approximate by smooth functions via mollication. We have

K K and the uniform boundZC

epK(z) ep

dz

ZC

epK(z) ep

dz = Ap(26)

Put = p

2to see

e K(z) e

2 L2(C ).

Next, the maximal function of e K(z) is nite everywhere,

M(z; e K ) = e +M(z; e K e ))(27)

and this last term is a constant plus a function in L2(C ).Now consider the BMO functions

K(z) =1

logM(z; e K )(28)

By Theorem 12.2, the BMO norm of this function does not depend on,

kKkBMO 2C2

p(29)

Moreover this function pointwise majorises the distortion function. Weneed a uniform L2 bound for K . Clearly

1

j2Bj

Z2B

K(z)2 dz

=1

4 2j2Bj

Z2B

log2 [M(z; e K )]2

1

4 2log2

1

j2Bj

Z2B

[M(z; e K e ) + e ]2

1

4 2log2

2e2 +

2C

j2Bj

ZC

(e K e )2

Here we have used the L2 inequality for the maximal operator. Thisgives Z

2B

jK(z)j2 dz C4 log

2(1 + Ap)(30)

where C4 is an absolute constant.Let us now return to the mollied Beltrami equation,

f z = (z)fz(31)

We look for a C1solution of (31) in the form

f z (z) = e(z); f z (z) = (z)e(z)(32)


where 2 W 1;p(C ; C ), for some p > 2, is compactly supported. The

necessary and sucient condition for is that

(e)z = (e)z

or equivalently

z = z + ()z(33)

This equation is uniquely solved using the BeurlingAhlfors transform,that is the singular integral operator dened by

(Sg)(z) = 1

2i

ZC

g()jd ^ d

(z )2

Note that z = Sz and so equation (33) reduces to

(I S)z = ()z(34)

As kSk2 = 1 and as kk1 < 1, there is p > 2 such that

kk1kSkp < 1

In this case the operator I S has a continuous inverse. Thus

z = (I S)1()z 2 Lp (C )(35)

and also

z = Sz 2 Lp (C )(36)

Note that z vanishes outside the support of which is containedin B(0; 2). Also z = Sz = O(z2) as z ! 1. Thus (z) C

z

asymptotically, for a suitable constant C. In fact

(z) = (Tz)(z) =1

2i

ZC

z()

zd ^ d(37)

where T is the complex Riesz Potential. Hence is Holder continuous

with exponent 1 2p, by the Sobolev Imbedding Theorem.

Now the solution f of equation (32) is unique up to a constant asf z = 0 outside 2B and as f z 1 2 Lp (C ). That is, f is a principalsolution to the Beltrami equation (31). It is important to realise here

that the Jacobian of f is strictly positive,

J(z; f ) = jf z j2 jf z j

2 = (1 jj2)e2 > 0(38)

The Implicit Function Theorem tells us that f is locally one-to-one.Another observation to make is that limz!1 f (z) = 1. It is an el-ementary topological exercise to show that f : C ! C is a global

homeomorphism of C . It's inverse is C1smooth of course.


We now digress for a second to outline the existence proof in the clas-

sical setting where K(z) K < 1. As the sequence K is uniformlybounded we nd there is an exponent p = p(K) > 2 such that

kf z kp + kf z 1kp CK(39)

where CK is a constant independent of . Hence the Sobolev ImbeddingTheorem yields the uniform bound

jf (a) f (b)j CKja bj12

p + ja bj(40)

The same inequality holds for the inverse map and hence

jf (a) f (b)j ja bj

p

p2

CK + ja bj2

p2

(41)

We may assume that f (0) = 0. As the pnorms of f z and f z 1are uniformly bounded, we may assume that each converges weakly

in Lp(C ) after possibly passing to a subsequence. From the uniformcontinuity estimates and Ascoli's Theorem, we may further assumef ! f locally uniformly in C . Obviously f satises the same modulusof continuity estimates and is therefore a homeomorphism. Moreover,

it follows that the weak limits of f z and f z 1 must in fact be equal tofz and fz1 respectively. Hence f is a homeomorphism in the Sobolevclass z +W 1;p(C ), that is fz and fz 1 in Lp(C ). Finally observe that ! pointwise almost everywhere, and hence in Lq(C ), where q isthe Holder conjugate of p. The weak convergence of the derivatives

shows that f is a solution to the Beltrami equation.

Back to the more general setting. If we followed the above argumentwe nd the Lp bounds are useless as we cannot keep them uniform.We therefore seek an alternative route via a Sobolev-Orlicz class whereuniform bounds might be available. We note the elementary inequality

(juj+ jvj)2 2K(juj2 jvj2) + 4K2jv wj2(42)

whenever u; v; w are complex numbers such that jwj K1K+1

juj and

K 1. We apply this inequality pointwise with

u = z ; v = z ; w = z

and K = K(z) as dened at (28), where

(z) = f (z) z 2 W 1;2(C ); K = K(z)(43)

and use equations (31), (29) we can write

(jz j+ jz j)2 2K(j

z j2 jz j

2) + 4(K)2jj

2


and hence

jD(z)j2 2KJ(z; ) + 4jKj

2(44)

Next we integrate this and use Theorems 12.1 and 12.3 to obtain

ZC

jDj2 2C3kKkBMOkJ(z; )kH1 + 4

Z2B

jKj2

4C1C2C3

p

ZC

jDj2 + 4C4 log2(1 + Ap)

where in the latter step we have used the uniform bounds at (29) and

(30).It is clear at this point why we have chosen p0 = 8C1C2C3 at (25).

The termRCjDj2 in the right hand side can be absorbed in the left

hand side. After doing this we obtain the uniform bounds in L2

ZC

jDj2 8C4 log2(1 + Ap)(45)

which read as

kDkL2(C) C5 log

ZB

epK

(46)

and in turn leaves us with the local estimate for the mapping f (z) =

(z) + z, namely

kDf kL2(BR) C5

R + log

ZB

epK

(47)

where BR = B(0; R). As f is monotone (being a C1 homeomorphism)we can apply the modulus of continuity estimate of Theorem 5.2,

jf (a) f (b)j C6

R + logRBepK

log

1

2

e+ R

jabj

(48)

for all a; b 2 BR.

Now consider the inverse map to f . Let us denote it by h =

(f )1 : C ! C . As both f and h are smooth dieomorphisms we


nd ZBR

jDh(w)j2 dw =

ZBR

K(w; h)J(w; h) dw

=

Zh(BR)

K(z; f ) dz

ZC

(K(z) 1) dz + jh(BR)j

CpR2 +

ZB

(K(z) 1) dz

In this last inequality we have put in the uniform bound jh(BR)j CpR

2. One interesting way to see this estimate (though perhaps notthe easiest) is via the Koebe distortion theorem. Anyway, we haveZ

B(0;R)

jDhj2 C (R2 +

ZB

K)(49)

and consequently we have the continuity estimate at (5.2) for h ,

jh(x) h(y)j2 CR2

RBK

loge + R

jxyj

(50)

For f this reads as

jf (a) f (b)j R exp

CR2

RBK

ja bj2

(51)

whenever a; b 2 B(0; R) and R 1. The uniform W 1;2 bounds, and

the continuity estimates from above and below now enable us to passto the limit. We nd f ! f and h ! h = f1 locally uniformly inC and Df and Dh converging weakly in L2

loc(C ). As in the classicalsetting this implies that f is a homeomorphic solution to the Beltrami

equation. Moreover fz; fz 1 2 L2(C ) and the same is true of theinverse function

References

[1] L.V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton

1966; Reprinted by Wadsworth Inc. Belmont, 1987.

[2] K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173,

(1994), 3760.

[3] K. Astala, T. Iwaniec, P. Koskela and G.J. Martin, Mappings with BMO

bounded Distortion, Math. Annalen, 317, (2000), 703726.

[4] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity,

Arch. Rat. Mech. Anal. 63, (1977), 337-403.


[5] B. Bojarski, Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk.

SSSR, 102, (1955), 661664.

[6] M.A. Brakalova and J.A. Jenkins, On solutions of the Beltrami equation, J.

Anal. Math., 76, (1998), 6792.

[7] R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certain

Espaces Homogenes, Lecture Notes in Math., 242, Springer-Verlag, 1971.

[8] R.R. Coifman, P.L Lions, Y. Meyer and S. Semmes, Compensated compactness

and Hardy spaces, J. Math. Pures Appl., 72, (1993), 247286.

[9] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc.

Amer. Math. Soc., 79, (1980), 249254.

[10] R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy

spaces in several variables, Ann. Math., 103, (1978), 569645.

[11] G. David, Solutions de l'equation de Beltrami avec kk1 = 1, Ann. Acad. Sci.

Fenn. Ser. AI Math., 13, (1988), 2570.

[12] C. Feerman, Characterisations of bounded mean oscillation, Bull. Amer.

Math. Soc., 77, (1971), 587588.

[13] F.W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer.

Math. Soc., 103, (1962), 353393.

[14] T. Iwaniec and G.J. Martin The Beltrami Equation, Memoirs of the Amer.

Math. Soc., to appear.

[15] T. Iwaniec and G.J. Martin Geometric Function Theory and Non-linear Anal-

ysis, to appear, Oxford University Press.

[16] T. Iwaniec , P. Koskela, G.J. Martin and C. Sbordone Mappings of exponen-

tially integrable distortion, to appear,

[17] J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 3, (1994),

393-402.

[18] C.B. Morrey, On the solutions of quasi-linear elliptic partial dierential qua-

tions, (1938), 126166.

[19] S. Muller, A surprising higher integrability property of mappings with positive

determinant, Bull.Amer. Math. Soc. 21 (1989), 245248.

[20] Rochberg and Weiss Analytic families of Banach spaces and some of their uses,

Recent progress in Fourier analysis (El Escorial, 1983), 173201, North-Holland

Math. Stud., 111, North-Holland, Amsterdam-New York, 1985.

[21] V. Ryazanov, U. Srebro and E. Yacubov, BMOquasiregular mappings, To

appear, J. D'Analyse Math.

[22] E.M. Stein, Note on the class L logL, Studia Math., 32, (1969), 305310.

T. Iwaniec, Department of Mathematics, Syracuse University, Syra-

cuse NY, USA


G.J. Martin, Department of Mathematics, The University of Auck-

land, Auckland, NZ


SOME SECOND-ORDER PARTIAL DIFFERENTIAL

EQUATIONS ASSOCIATED WITH LIE GROUPS

PALLE E. T. JORGENSEN

To Derek Robinson on the occasion of his 65th birthday

Abstract. In this note we survey results in recent research paperson the use of Lie groups in the study of partial dierential equa-tions. The focus will be on parabolic equations, and we will showhow the problems at hand have solutions that seem natural in thecontext of Lie groups. The research is joint with D.W. Robinson,as well as other researchers who are listed in the references.

1. Introduction

When the Hamiltonian of a quantum-mechanical system is relatedto a Lie algebra, it is often possible to use the representation structure

of the Lie algebra to decompose the Hilbert space of the quantum-mechanical system into simpler (irreducible) pieces. For example, ifa Hamiltonian commutes with the generators of a Lie algebra, theHilbert space of the system can be decomposed into irreducibles of the

Lie algebra, and the Lie algebra elements themselves can be used aselements in a set of commuting observables.We have aimed at making the present paper accessible to a wide

audience of non-specialists, stressing the general ideas and motivatingexamples, as opposed to technical details.

2000 Mathematics Subject Classication. Primary 35B10; Secondary 22E25,22E45, 31C25, 35B27, 35B45, 35C99, 35H10, 35H20, 35K10, 41A35, 43A65, 47F05,53C30.

Key words and phrases. approximating variable coecient partial dierentialequation with constant coecients, t ! 1 asymptotics, boundary value problem,Gaussian estimates, heat equation, Hilbert space, homogenization, nilmanifold, par-abolic, partial dierential equations, scaling and approximation of solution, spec-trum, stratied group.

This research was partially supported by two grants from the U.S. National Sci-ence Foundation, and by the Centre for Mathematics and its Applications (CMA)at The Australian National University (ANU).

This paper is an expanded version of a lecture given by the author at the NationalResearch Symposium on Geometric Analysis and Applications at the ANU in Juneof 2000.

149

150 PALLE E. T. JORGENSEN

The class of such Hamiltonians is quite large: see [JoKl85] and[Jor88]. In this introduction we will review those Hamiltonians Hwhose interaction terms are polynomial in the position variables. Such

Hamiltonians are directly and naturally related to nilpotent Lie alge-bras. The nilpotent case is studied in Section 2.The spectrum of H is obtained by decomposing the physical space

on which the HamiltonianH acts into irreducible representations of the

underlying nilpotent group. Sometimes this decomposition is decisive,as is the case with a particle in a constant magnetic eld, where thedecomposition leads to a harmonic-oscillator Hamiltonian. Sometimesthe decomposition leads to a new Hamiltonian that requires further

analysis, as is the case with a particle in a curved magnetic eld.The time evolution of the system is obtained by solving the heat

equation of the underlying nilpotent Lie group. By writing the Hamil-tonian as a quadratic sum of Lie-algebra elements and then using therepresentation of these Lie-algebra elements arising from the regular

representation, it is possible to write etH as the convolution of a ker-nel (which is a solution of the heat equation) with a representationacting on the physical Hilbert space; see [Jor88].The simplest case of this spectral picture is as follows: Consider a

nonrelativistic spinless particle of mass m in an external magnetic eldB (x). The Hamiltonian for such a system is given by

H =1

2m

p

e

cA

2;(1.1)

where p = hir and A is the vector potential satisfying B = r A.

Consider the commutatorshpi

e

cai; pj

e

caj

i=

h

i

e

c"ijkbk;h

pi e

cai; bj

i=h

i

@bj

@xih

ibij;(1.2) h

pi e

cai; bjk

i=h

i

@bjk

@xih

ibijk;

...... ;

where A = (a1; a2; a3), B = (b1; b2; b3), x = (x1; x2; x3). If B is apolynomial in x, eventually the derivatives of B will give zero, so that

the set of commutators closes. The resulting Lie algebra formed by reallinear combinations of the elements

pi e

cai; bi; bij; : : :(1.3)

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS 151

is therefore a nilpotent Lie algebra, and the Hamiltonian (1.1) is qua-

dratic in the rst three Lie algebra elements Xi :=pi

ecai, i =

1; 2; 3, from the list (1.3). By general theory, e.g., [Rob91], this Liealgebra is the Lie algebra g of some Lie group G, which we may taketo be simply connected.We show further in [JoKl85] and [Jor88] that there is a unitary rep-

resentation U of G on L2 (R3) such that

2mH = dU

3X

i=1

pi

e

cai

2!:

If there is a constant of motion for the Lie-algebra elements pi ecai,

then U is a direct integral over a corresponding spectral parameter

. We then get H =R

d H() where H has absolutely continuous

spectrum, while each H() has purely discrete spectrum. If 0 ()

1 () is the spectrum of H(), then each 7! i () is realanalytic, and we get the following typical spectral picture.

0()

1()2()

In this paper we will focus attention on a more restricted case whereinthe coecients are periodic. As shown in Section 3, this case shares the

spectral band structure with the polynomial-magnetic-eld case. Weshow that in the periodic case the regularity of the coecients may berelaxed, and in fact, our spectral-theoretic results will be valid whenthe operator has L1-coecients.

2. Periodic operators

We begin by recalling some elementary denitions and facts aboutstratied Lie groups from [FoSt82]. A real Lie algebra g is called


stratied if it has a vector-space decomposition

g =

rMk=1

g(k);(2.1)

for some r, which we shall take nite here, all but a nite number ofthe subspaces g(k) are nonzero,

g(k); g(l)

g

(k+l)(2.2)

for all k; l 2 N , and g(1) generates g as a Lie algebra. Thus a stratied

Lie algebra is automatically nilpotent, and if r is the largest integersuch that g(r) 6= 0, then g is said to be nilpotent of step r. A Lie groupis dened to be stratied if it is connected and simply connected and

its Lie algebra g is stratied.Let G be a stratied Lie group and exp : g ! G the exponential

map. The CampbellBakerHausdor formula establishes that

exp (X) exp (Y ) = exp (H (X; Y )) ;

where H (X; Y ) = X + Y + [X; Y ] =2 + a nite linear combination ofhigher-order commutators in X and Y . Thus X; Y ! H (X; Y ) denes

a group multiplication law on the underlying vector space V of g whichmakes V a Lie group whose Lie algebra is g and the exponential mapexp : g ! V is simply the identity. Then V with the group law isdieomorphic to G. Next let dk denote the dimension of g(k) and dthe dimension of g and for each k choose a vector-space basis X(k) =X

(k)1 ; : : : ; X

(k)

dk

of g(k) such thatX1; : : : ; Xd = X

(1)

1 ; : : : ; X(r)

dris a basis

of g. If 1; : : : ; d is the dual basis for g, i.e., if k (Xl) = k;l, dene

k = k exp1. Then 1; : : : ; d are a system of global coordinates for

G, and the product rule on G becomes

k (xy) = k (x) + k (y) + Pk (x; y) ; x; y 2 G;

where Pk (x; y) is a nite sum of monomials in i (x), i (y) for i < kwith degree between 2 and m. It follows that both left and right Haarmeasure on G can be identied with Lebesgue measure d1 dd.

If Xi denotes one of the (abstract) Lie generators, we denote by Ai

the corresponding right-invariant vector eld on G, i.e., Ai on a test

function on G is given by A(l)i = dL (Xi), or more precisely,

A(l)i (g) =

d

dt (exp (tXi) g) jt=0; g 2 G;(2.3)

and similarly A(r)i = dR (Xi) given byA(r)i

(g) =

d

dt (g exp (tXi)) jt=0:(2.4)


Since we can pass from left to right with the adjoint representation,

the formulas may be written in one alone, and we will work with A(l)i ,

and denote it simply Ai.If 1 j d1 we will need the functions yj on G dened by

yj

exp

dX

k=1

kXk

!!= j:(2.5)

These functions satisfy the following system of dierential equations:

A(l)i yj = A

(r)i yj = i;j:(2.6)

It follows by the standard ODE existence theorem that the functionsyi on G are determined uniquely by (2.6) and the \initial" conditionsyi (e) = 0. Also note that (2.6) is consistent only for the dierentialequations dened from a sub-basis A1; : : : ; Ad1 , and that they wouldbe overdetermined had we instead used a basis: hence the distinction

between subelliptic and elliptic.In addition, we have given a discrete subgroup in G such that

M = G= is compact. It is well-known that it then has a unique (upto normalization) [Jor88, Rob91] invariant measure . The correspond-

ing Hilbert space is L2 (M;), and the invariant operators on G passnaturally to invariant operators on M ; see [BBJR95]. Let X1; : : : ; Xd1

be the generating Lie-algebra elements. Then the corresponding in-variant vector elds on G will be denoted A1; : : : ; Ad1 , and those on M

will be denoted B1; : : : ; Bd1. Functions ci;j 2 L1 (G) are given, and we

form the quadratic form

h (f) =

d1Xi;j=1

hAif j ci;jAjfi :(2.7)

If further

ci;j (g ) = ci;j (g) for g 2 G; 2 ;(2.8)

then we have a corresponding form hM on M = G=.

Introducing

c"i;j (x) = ci;j"1x

; " > 0;(2.9)

we get for each " a periodic problem corresponding to the period lattice". To speak about " for " 2 R+ , we must have an action of R+ onG which generalizes the familiar one

" : (x1; : : : ; xd) 7! ("x1; : : : ; "xd)

of Rd . It turns out that this can only be done if G is stratied, and so inparticular nilpotent; see [FoSt82], [Jor88]. In that case it is possible to


construct a group of automorphisms f"g"2R+ of G which is determined

by the dierentiated action d" on the Lie algebra g. If g is specied

as in (2.1)(2.2), then

d"X(1)

= "X(1); X(1)

2 g(1); " 2 R+ :(2.10)

Let H, respectively H", be the selfadjoint operators associated to theperiod lattices and " (see [BBJR95] or [Rob91]), and let St = etH ,S"t = etH" .We now turn to the homogenization analysis of the limit " ! 0

which leads to our comparison of the variable-coecient case to theconstant-coecient one. It should be stressed that in the Lie case, eventhe \constant-coecient" operator

Pi;j Aici;jAj is not really constant-

coecient, as the vector elds Ai are variable-coecient.Take even the simplest example where G is the three-dimensional

Heisenberg group of upper triangular matrices of the form

g =

0@1 x z0 1 y0 0 1

1A ; x; y; z 2 R:(2.11)

In this case, dimg(1) = 2, and dim g

(2) = 1, with g(2) spanned by the

central element in the Lie algebra. Dierentiating matrix multiplication(2.11) on the left as in (2.3), we get the following three identities:

A1 =@

@x+ y

@

@z= dL (X1) ;

A2 =@

@y= dL (X2) ;

A3 =@

@z= dL (X3) ;

where the rst vector eld is of course variable coecients.

We will use standard tools [ZKO94] (see also [Dau92], [Tho73],[Wil78]) on homogenization.

Theorem 2.1. [BBJR95] Suppose the system ci;j 2 L1 is given and

assumed strongly elliptic. Then there is a C0-semigroup St on L2 (G; dx)

with constant coecients, where dx is left Haar measure, such that

lim"!0

S"t St

f 2= 0

for all f 2 L2 (G; dx) and t > 0.

The constant coecients of the limit operator ci;j may be determinedas follows: We show in [BBJR95] that if

ci;j (g) := h (gi yi; gj yj)(2.12)


and if C (g) is the corresponding quadratic form, then the problem

infgC (g) =: C(2.13)

has a unique solution, i.e., the inmum is attained at f1; : : : ; fd1 suchthat

C (f) = C:(2.14)

The order relation which is used in the inmum consideration (2.13)

is the usual order on hermitian matrices: For every g, the matrix

C (g) := (ci;j (g))d1i;j=1

is hermitian, and the matrix inequality C (g) C

may thus be spelled out as follows:Xi;j

zici;j (g) zj Xi;j

zici;jzj for all z1; : : : ; zd1 2 C :

Solvability of this variational problem is part of the conclusion of ouranalysis in [BBJR95], i.e., the existence of the minimizing functions

f1; : : : ; fd1 .Then the coecients of the homogenized operator can also be com-

puted with the aid of the coordinates yi, i = 1; : : : ; d1, introduced in(2.5) and (2.6). One has the representation

ci;j =

ZY

dy

d1Xk;l=1

(Ak (fi (y) yi)) ck;l (y) (Al (fj (y) yj))(2.15)

= hY (fi yi; fj yj) ;

where h denotes the sesquilinear form associated with H, and the sub-script Y refers to the region of integration. Specically, Y is a funda-

mental domain for the given lattice in G. For example, we may takeY to be dened by

Y =\ 2

x 2 G ; jxj

x 1 ;(2.16)

and j j dened relative to a geodesic distance d, jxj := d (x; e), x 2 G.Then

(i)S

2 Y = G, and

(ii) meas (Y 1 \ Y 2) = 0 whenever 1 6= 2 in .

(These are the axioms for fundamental domains of given lattices, but

we stress that (2.16) is just one choice in a vast variety of possiblechoices.)The simplest case of the construction is G = R, and it was rst

considered in [Dav93, Dav97] by Brian Davies. This is the simplest


possible heat equation, and we then have the conductivity representedby a periodic function c, say

c (x+ p0) = c (x) ; x 2 R;

where p0 is the period. Then H = ddxc (x) d

dx, and it can be checked

that

c =

1

p0

Z p0

0

dx

c (x)

1:

Theorem 2.2. [BBJR95] Adopt the assumptions of Theorem 2.1. Then

limt!1

tD=2 ess supjxj2+jyj2at

Kt (x ; y) Kt (x ; y) = 0

for each a > 0 where jxj = dc (x ; e), and where

dc (x ; y) = sup

( (x) (y) ; 2 C1c (G) ;

d1Xi;j=1

ci;j (Ai ) (Aj ) 1 pointwise

)

and Ai refers to the Lie action of the vector eld Ai on from (2.3).

It is our aim here only to sketch the ideas, and the reader is referredto our papers for details, but we stress that the proof is based onhomogenization, see, e.g., [BLP78], [ZKON79], [Koz80], and [AvLi91]The number D is the homogeneous degree dened from the given

grading, or stratication, g(i) of the nilpotent Lie algebra g. As spelled

out in [Jor88] and [FoSt82], there are numbers i depending on theLie-structure coecients such that

D =Xi

i dim g(i):

To be specic, the numbers i are determined in such a way that weget a group of scaling automorphisms f"g"2R+ of g, and therefore on

G, and it is this group which is fundamental in the homogenizationanalysis. Specically, extending (2.10), " : g! g is dened by

"X(i)

= "iX(i); X(i)

2 g(i);(2.17)

and then extended to g by linearity via (2.1), in such a way that

" ([X; Y ]) = [" (X) ; " (Y )] ; X; Y 2 g; " 2 R+ :(2.18)

Hence if (2.2) holds, then it follows from (2.17) and (2.18) that i = ifor i = 1; 2; : : : . In the case of the Heisenberg Lie algebra g, we have


[X; Y ] = Z as the relation on the basis elements; Z is central. Then

g(1) = span (X; Y ), g(2) = RZ, 1 = 1, 2 = 2, so D = 4.

Let Kt and Kt be the respective integral kernels for the semigroups

St and St, and set

jjjKjjjp = essx2G

ZG

dy jK (x; y)jp

1=p

and

jjjKjjj1= ess

x;y2GjK (x; y)j :

Then

Theorem 2.3. [BBJR95] Adopt the assumptions of Theorem 2.1. Then

limt!1

tD=2Kt Kt

1= 0; lim

t!1

Kt Kt

1= 0:

3. G = Rd

The case G = Rd was considered in [BJR99], where we further

showed that the limit S"t ! St then holds also in the spectral sense.

In that case, we scale by " = 1=n, n!1, and then identify the limitoperator as having absolutely continuous spectral type, and we prove

spectral asymptotics. (A general and classical reference for periodicoperators is [Eas73].)Starting with an equation which is invariant under the Zd-transla-

tions, we then use the Zak transform [Dau92] to write St = etH as a

direct integral over Td (= Rd=Zd), viz.,

St =

Z

Td

S(z)t ;(3.1)

and we establish continuity of z 7! S(z)t in the strong topology [BJR99,

Lemma 2.2]. Pick a positive C1-function on Rd of integral one, andset

c(n)i;j (x) = nd

ZRd

dy (ny) ci;j (x y) ;

and form the corresponding C0-semigroup

S(n)t = etH

(n)

;

where H(n) is dened from c(n)i;j . We then show in [BJR99] that S

(n)t

approximates St, not only in the strong topology, but also in a spectral-theoretic sense. Using this, we establish the following connection be-

tween St = etH and S(z)t = etH

(z)

in (3.1). Setting z =ei1 ; : : : ; eid

,

we get


Theorem 3.1. If n (z) denotes the eigenvalues of Hz then

(3.2) limN!1

N2n (w) ; w

N = z; n = 0; 1; : : :

=nD

(n ) j C (n )E; n 2 Z

do;

where the limit is in the sense of pointwise convergence of the ordered

sets, and where C = (ci;j) is the constant-coecient homogenized case.

The rate of convergence of the eigenvalues in (3.2) can be estimatedfurther by a trace norm estimate.We refer the reader to [BJR99] for details of proof, but the argu-

ments in [BJR99] are based in part on the references [Aus96], [DaTr82],[Eas73], and [ZKON79]. In addition, we mention the papers [Aus96],[AMT98], and [TERo99], which contain results which are related, butwith a dierent focus.

Finally, we mention that our result from [BJR99], Theorem 3.1, hassince been extended in several other directions: see, e.g., [Sob99] and[She00].

Acknowledgements. We are grateful to Brian Treadway for excellenttypesetting and graphics production, and to the participants in theNational Research Symposium at The Australian National University

for fruitful discussions, especially A.F.M. ter Elst.

References

[Aus96] P. Auscher, Regularity theorems and heat kernel for elliptic operators,J. London Math. Soc. (2) 54 (1996), 284296.

[AMT98] P. Auscher, A. McIntosh, and P. Tchamitchian, Heat kernels of secondorder complex elliptic operators and applications, J. Funct. Anal. 152(1998), 2273.

[AvLi91] M. Avellaneda and F.-H. Lin, Lp bounds on singular integrals in homog-

enization, Comm. Pure Appl. Math. 44 (1991), 897910.[BBJR95] C.J.K. Batty, O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson,

Asymptotics of periodic subelliptic operators, J. Geom. Anal. 5 (1995),427443.

[BLP78] A. Bensoussan, J.-L. Lions, and G.C. Papanicolaou, Asymptotic Analy-sis for Periodic Structures, North-Holland, Amsterdam, 1978.

[BJR99] O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson, Spectral asymptoticsof periodic elliptic operators, Math. Z. 232 (1999), 621650.

[DaTr82] B.E.J. Dahlberg and E. Trubowitz, A remark on two-dimensional peri-

odic potentials, Comment. Math. Helv. 57 (1982), 130134.[Dau92] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf.

Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathe-matics, Philadelphia, 1992.

[Dav93] E.B. Davies, Heat kernels in one dimension, Quart. J. Math. OxfordSer. (2) 44 (1993), 283299.


[Dav97] E.B. Davies, Limits on Lp regularity of self-adjoint elliptic operators, J.Dierential Equations 135 (1997), 83102.

[Eas73] M.S.P. Eastham, The Spectral Theory of Periodic Dierential Equa-

tions, Scottish Academic Press, Edinburgh, Chatto & Windus, London,1973.

[FoSt82] G.B. Folland and E.M. Stein, Hardy Spaces on Homogeneous Groups,Princeton University Press, Princeton, 1982.

[Jor88] P.E.T. Jorgensen, Operators and Representation Theory: Canonical

Models for Algebras of Operators Arising in Quantum Mechanics,North-Holland Mathematics Studies, vol. 147, Notas de Matematica,vol. 120, North-Holland, AmsterdamNew York, 1988.

[JoKl85] P.E.T. Jorgensen and W.H. Klink, Quantum mechanics and nilpotent

groups, I: The curved magnetic eld, Publ. Res. Inst. Math. Sci. 21(1985), 969999.

[Koz80] S.M. Kozlov, Asymptotics of fundamental solutions of second-order

divergence dierential equations, Mat. Sb. (N.S.) 113(155) (1980),no. 2(10), 302323, 351 (Russian), English translation: Math. USSR-Sb. 41 (1982), no. 2, 249267.

[Rob91] D.W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathemat-ical Monographs, The Clarendon Press, Oxford University Press, NewYork, 1991.

[She00] Z. Shen, The periodic Schrodinger operators with potentials in the

MorreyCampanato class, preprint, University of Kentucky, 2000.[Sob99] A.V. Sobolev, Absolute continuity of the periodic magnetic Schrodinger

operator, Invent. Math. 137 (1999), 85112.[TERo99] A.F.M. ter Elst and D.W. Robinson, Second-order subelliptic operators

on Lie groups, I: Complex uniformly continuous principal coecients,Acta Appl. Math. 59 (1999), 299331.

[Tho73] L.E. Thomas, Time dependent approach to scattering from impurities

in a crystal, Comm. Math. Phys. 33 (1973), 335343.[Wil78] C.H. Wilcox, Theory of Bloch waves, J. Analyse Math. 33 (1978), 146

167.[ZKO94] V.V. Zikov, S.M. Kozlov, and O.A. Olenik, Homogenization of Dieren-

tial Operators and Integral Functionals, Springer-Verlag, Berlin, 1994,translated by G.A. Yosian from the Russian Usrednenie differ-

encial~nyh operatorov, \Nauka", Moscow, 1993.[ZKON79] V.V. Zikov, S.M. Kozlov, O.A. Olenik, and H.T. Ngoan, Averaging and

G-convergence of dierential operators, Uspekhi Mat. Nauk 34 (1979),no. 5(209), 65133, 256, Russian Math. Surveys 34 (1979), no. 5, 69148.

Department of Mathematics, The University of Iowa, 14 MacLean

Hall, Iowa City, IA 52242-1419, U.S.A.

E-mail address : [email protected]: http://www.math.uiowa.edu/~jorgen/

PRINCIPAL SERIES AND WAVELETS

CHRISTOPHER MEANEY

Abstract. Recently Antoine and Vandergheynst [1, 2] have pro-

duced continuous wavelet transforms on the n-sphere based on a

principal series representation of SO(n; 1). We present some of

their calculations in a more general setting, from the point of view

of Fourier analysis on compact groups and spherical function ex-

pansions.

1. Coherent States

We begin with Antoine and Vandergheynst's denition of a coherent

state, as presented in [1, 2]. Here G is a locally compact group.

Suppose thatX is a homogeneous space ofG,X = G=H, equipped

with a G-invariant measure.

Let (U; L2(Y )) be a unitary representation of G on some Lebesgue

space L2(Y ). Assume there is a Borel cross section

: X ! G; (x)H = x; 8x 2 X:

Say that 2 L2 (Y ) is admissible mod(H; ) whenZX

jhU ( (x)) j'ij2dx <1; 8' 2 L2 (Y ) :

The orbit of an admissible vector under (X),

fU ( (x)) : x 2 Xg

is called a coherent state.

Note that there are other variations on the theme of \restricted

square integrability", such as the case described in [3].

1991 Mathematics Subject Classication. 43A90,22E46,43A75,42C40.Key words and phrases. Semisimple Lie group, coherent state, continuous

wavelet transforms, principal series, Plancherel formula, admissible vectors.This is the content of my lecture at the National Research Symposium on Geo-

metric Analysis and Applications in Canberra, June 2000. In the past year my

research was partially supported by the ARC Small Research Grants Scheme.

160

PRINCIPAL SERIES AND WAVELETS 161

2. Frames

Suppose now that is an admissible vector in L2(Y ). Dene a linearoperator

A; : L2 (Y ) ! L2 (Y )

by

hA;'1j'2i =

ZX

h'1jU ( (x)) i hU ( (x)) j'2i dx; 8'1; '2 2 L2(Y ):

When this has a bounded inverse, say that the coherent state is a frame.

When the orbit of under (X) is a frame of L2 (Y ) there is the

continuous wavelet transform,

W : L2 (Y ) ! L2 (X)

dened by

W' (x) = h'jU ( (x)) i ; 8' 2 L2(Y ):

This operator is one-to-one and its range H is complete with respect

to the inner-product:

hW'jW iH =W'jWA

1;

L2(X)

; ; ' 2 L2(Y ):

Hence there is a unitary isomorphism W : L2 (Y ) ! H.

3. The setting

For the calculations which we will describe here, the ingredients are:

G is a noncompact connected semisimple Lie group with nite

centre and Cartan involution . K is the corresponding maximal compact subgroup.

G = KAN is an Iwasawa decomposition.

M is the centralizer of A in K.

X = G=N .

Y = K=M .

U is a certain principal series action of G on L2(K=M), to be

dened below.

Assume that (K;M) is a Gel'fand pair.

See Knapp's book for details [5, page 119].

162 CHRISTOPHER MEANEY

4. Decompositions

There are Iwasawa projections K : G! K, A : G! A, N : G! N ,

for which

g = K(g)A(g)N(g); 8g 2 G:

The Haar measure on G is given in terms of that of K and right Haar

measure of AN , [5, page 139] with

dg = dk dr(an):

The measure on K is normalized so thatZK

dk = 1:

There is a mapping log : A! a with

exp(log(a)) = a; 8a 2 A:

For each 2 a let

a = e(log(a)); 8a 2 A:

5. Invariant Integration

There is the special functional 2 a determined by the structure of

the group G. For f 2 Cc(G) the integral formula for Haar measure on

G is ZG

f(x) dx =

ZK

ZA

ZN

f(kan) a2 dndadk:

See [6, Prop. 7.6.4] for details.

We can use KA to parametrize G=N and the G-invariant integral onG=N is given byZ

G=N

F (y)dy =

ZK

ZA

F (kaN)a2 dadk

for F 2 Cc(G=N). Hence, we take the Borel section : G=N ! G to

be

(kaN) = ka; 8a 2 A; k 2 K:


6. Induced Representations

Consider the space of continuous covariant functions:

I(G) =

8<:f :

f : G! C continuous

f(gman) = af(g);8g 2 G;m 2M; a 2 A; n 2 N

9=; :

Left translation by elements of G preserves the property of covariance:

(U(g)f) (x) = fg1x

; 8g; x 2 G; f 2 I(G):

U(g) : I(G) ! I(G); 8g 2 G:

For a covariant function f 2 I(G),

f(x) = f(K(x)A(x)N(x)) = A(x)f(K(x)); 8x 2 G:

Equip I(G) with the inner product

hf1jf2i =

ZK

f1(k)f2(k) dk

and norm

kfk =

ZK

jf(k)j2dk

1=2

:

The completion of I(G) is

HU= L2(K=M):

The action of G on HU is an example of a principal series represen-

tation, see section 8.3 of Wallach's book [6]. For our purposes, the

essential fact is that U jK is the regular representation of K on a sub-

space of L2(K). If f 2 L2(K=M), extend it to be an element of HU by

assigning

f(kan) = af(k):

Notice that if f 2 L2(K=M),

U(g)f(k) = A(g1k)f(K(g1)k); k 2 K; g 2 G:

For each g 2 G the action of U(g) extends to a continuous linear

operator on HU . It is a unitary representation:

hU(g)f1jU(g)f2i =

ZK

(U(g)f1)(k)(U(g)f2)(k) dk

=

ZK

A(g1k)2f1(K(g1k))f2(K(g1k)) dk: = hf1jf2i

Lemma 1. The representation (U;HU) is unitary. When restricted to

K, it is the action of K by left translation on L2(K=M).


7. Fourier analysis on the compact group K

We review some basic facts about analysis on compact groups. LetbK be the dual object of K, consisting of a maximal set of inequivalent

irreducible unitary representations ( ; V ) of K.

For each integrable function f on K there is the Fourier series:

f(x) =X 2 bK

d f (x):

Convolution with a character is

f (x) =

ZK

f(y) tr( (y1) (x)) dy = tr( bf( ) (x))where the Fourier coecient is

bf( ) = ZK

f(x) (x1) dx =

ZK

f(x) (x) dx:

The Fourier coecients are linear transformationsbf( ) 2 HomC (V ; V ):

Fourier coecients of convolutions are products of Fourier coecients:

(f g)^( ) =

ZK

ZK

f(x)g(x1y) (y1) dxdy

=

ZK

ZK

f(x)g(x1y) (y1xx1) dxdy

= bg( ) bf( ):Dene left translation on K by

xf(y) = f(x1y); 8x; y 2 K;

and the composition with inversion

f_(x) = f(x1); 8x 2 K:

Fourier coecients of left translates satisfy

(xf)^( ) =

ZK

f(x1y) (y1xx1) dy = bf( ) (x1)

Fourier coecients of adjoints satisfy

(g_)^( ) = bg( ):The L2(K) inner product can be viewed as a convolution:Z

K

f(x)g(x) dx =

ZK

f(x)g_(x1) dx = f g_(1):


For f; g 2 L2(K), the Fourier series of their convolution is absolutely

convergent, see [4],

f g(x) =X 2 bK

d f g (x)

f and g in L2(K):

f g(x) =X 2 bK

d trbg( ) bf( ) (x) ;

ZK

f(x)g(x) dx =X 2 bK

d tr( bf( )bg( ));kfk

2

2 =X 2 bK

d

bf( ) 22

:

In particular, for each 2 bK,

k bf( )k22 = d kf k22:

See Appendix D of Hewitt and Ross [4] for details about the norms

k kp; 1 p 1:

If h 2 L1(K) then

f 7! f h; L2(K) ! L2(K);

is a bounded linear operator which commutes with left translation.

Similarly,

f 7! h f; L2(K) ! L2

(K);

is a bounded linear operator which commutes with right translation.

The norm of both of these operators is

sup 2 bK

bh( ) 1

:

8. Homogeneous Spaces

Now we return to dealing with functions on K=M , which we identify

with right-M -invariant functions on K.

For each 2 bK, let

V M = fv 2 V : (m)v = v; 8m 2M g

and P : V ! V M , the orthogonal projection on to this subspace.

Let be the normalized Haar measure onM . Its Fourier coecients

are b( ) = P ; 8 2 bK:


If f 2 L1(K=M) then

f = f ; =) bf( ) = P bf( ); 8 2 bK:We are restricting our attention to the case where (K;M) is aGel'fand

pair, which means that

dimV M

1; 8 2 bK:

Lemma 2. If (K;M) is a Gel'fand pair and f 2 L1(K=M), then for

all 2 bK,

rank( bf( )) 1 and (V M )? ker( bf( )):

Lemma 3. If (K;M) is a Gel'fand pair and f 2 L1(K=M), then for

all 2 bK, bf( ) bf( ) = k bf( )k22P :Lemma 4. If (K;M) is a Gel'fand pair and f 2 L1(K=M), then for

all 2 bK,

k bf( )kp = k bf( )k2 ; 1 p 1:

Lemma 5. If (K;M) is a Gel'fand pair and h 2 L1(K=M), then the

norm of the operator

f 7! f h; L2(K) ! L2(K=M);

is

sup

nkbh( )k2 : 2 bK o

= sup

npd kh k2 : 2 bK o

:

In this lemma, if dimV M

= 0 then bh( ) = 0 and so we need only

take the supremum over those for which dimV M

= 1.

9. Admissible Vectors

In [2] the unitary representation (U;HU) of G is said to be square-

integrable modulo N if there is a non-zero vector for whichZK

ZA

jhU(ka)jij2a2 dadk <1

for all 2 HU . Such an is called admissible.

Notice that this can be rearranged to sayZK

ZA

U(a)jU(k1)2 a2 dadk <1

for all 2 HU . Recall that U jK is left translation.


We then nd thatZK

jhU(ka) j i j2dk =

ZK

ZK

(U(a)) (x) (kx) dx

2

dk

=

ZK

(U(a)) _(k)2 dk

= (U(a)) _

22

Using the Plancherel formula for this, (U(a)) _ 22=X

d tr(U(a))^( )b( )b( )(U(a))^( )

=X

d k(U(a))^( )k22k

b( )k22We arrive at the general version of Antoine and Vandergheynst's crite-

rion for admissibility.

Theorem 1. If 2 HU = L2(K=M) has the property that

sup 2 bK

ZA

k(U(a))^( )k22 a2 da <1

then is admissible.

Since the functions here are right-M -invariant, the only non-zero

parts of the Fourier series correspond to those for which P 6= 0.

Theorem 2. If 2 HU = L2(K=M) is admissible and there are con-

stants 0 < c1 c2 for which

c1

ZA

k(U(a))^( )k22 a2 da c2

for all 2 bK with P 6= 0, then the corresponding coherent state is a

frame.

We can reword this to see that the criterion for to give rise to a

frame for L2(K=M) is that there are constants 0 < c1 c2 for which

c1 d

ZA

k(U(a)) k2

2a2 da c2;

for all 2 bK with P 6= 0.


10. Spherical Functions

Let bKM denote the set of those 2 bK with P 6= 0. For each 2 bKM

dene the spherical function

' = = :

If f 2 L1(K=M) its Fourier series isX 2 bKM

d f ' :

When K=M = Sn, this is the usual spherical harmonic expansion.

To use the criterion for a frame, we need estimates on

d

ZA

k(U(a)) ' k2

2a2 da;

uniformly in 2 bKM .

11. Zonal Functions

A special case occurs when is bi-M -invariant, since it is then ex-

panded in a series

=X 2 bKM

d c ' with c = hj' i :

But U(a) is also bi-M -invariant and its expansion is

U(a) =X 2 bKM

d c (a)'

with

c (a) = hU(a)j' i =jU(a1

)' :

Since the spherical functions ' are matrix entries of irreducible rep-

resentations,

' ' 0 =

(' =d if = 0

0 if 6= 0;

and k' k22 = 1=d . Hence, Theorem 2 says that a bi-M -invariant func-

tion produces a frame for L2(K=M) when there are positive constants

c1 c2 for which

0 < c1

ZA

jc (a)j2a2 da c2

for all 2 bKM .


12. Antoine and Vandergheynst

The results in [2] are concerned with the case where:

G = SOe(1; 3), K = SO(3), M = SO(2), and K=M = S2.

A = (0;1) with multiplication, X = SO(3) A, = 1.

bKM = f0; 1; 2; 3; : : :g, dn = 2n+1, and the spherical functions 'nare normalized ultraspherical polynomials.

Suppose we use spherical coordinates (; ) to parametrize S2. Propo-

sition 3.4 of [2] states that if 2 L2(S2) is admissible andZ 2

0

(; ) d 6= 0

then gives rise to a frame. This is achieved using the spherical har-

monic expansion of U(a) and the asymptotics of the zonal spherical

functions, to get the inequality in Theorem 2 above.

In [2] there is presented a sucient condition on a function 2 L2(S2)

so that it satises the hypotheses of Theorem 1. These are similar to

the moment conditions in the Euclidean space setting, see Proposition 7

in [3]. Proposition 3.6 [2] states that if 2 L2(S2) satisesZ

0

Z 2

0

(; )

1 + cos()sin() dd = 0

then it is admissible.

References

1. J.-P. Antoine and P. Vandergheynst, Wavelets on the n-sphere and related man-

ifolds, J. Math. Phys. 39 (1998), no. 8, 39874008.

2. , Wavelets on the 2-sphere: a group-theoretical approach, Appl. Comput.

Harmon. Anal. 7 (1999), no. 3, 262291.

3. J. E. Gilbert, R. A. Kunze, and C. Meaney, On derived intertwining norms for

the Lorentz group, Representation theory and harmonic analysis (Cincinnati,

OH, 1994), Amer. Math. Soc., Providence, RI, 1995, pp. 5773.

4. Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Struc-

ture and analysis for compact groups. Analysis on locally compact Abelian groups,

Springer-Verlag, New York, 1970, Die Grundlehren der mathematischen Wis-

senschaften, Band 152.

5. Anthony W. Knapp, Representation theory of semisimple groups, Princeton Uni-

versity Press, Princeton, N.J., 1986, An overview based on examples.

6. Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker

Inc., New York, 1973, Pure and Applied Mathematics, No. 19.

Dept. Mathematics, Macquarie University, North Ryde, NSW 2109,

Australia


SINGULARITIES AND THE WAVE EQUATION ON

CONIC SPACES

RICHARD B. MELROSE AND JARED WUNSCH

Introducing polar coordinates around a point in Euclidian space re-duces the Euclidian metric to the degenerate form

dr2 + r2 d!2(1)

where r is the distance from the point and d!2 is the round metric onthe sphere. If X is an arbitrary manifold with boundary, the class ofconic metrics on X is modeled on this special case. Namely, a conicmetric is a Riemannian metric on the interior of X such that for somechoice of the dening function x of the boundary (x 2 C1(X) with@X = fx = 0g, x 0, dx 6= 0 on @X), the metric takes the form

g = dx2 + x2h on X = Xn@X; near @X:Here h is a smooth symmetric 2-cotensor on X such that h0 = hj@X isa metric on @X:In fact a general conic metric can be reduced to a form even closer

to (1) in terms of an appropriately chosen product decomposition of Xnear @X; that is, by choice of a smooth dieomorphism

[0; )x @XF! O X; O an open neighborhood of @X :(2)

The normal variable in x 2 [0; ) is then a boundary dening function,at least locally near @X; and the slices F x=x0 have given dieomor-phisms to @X: Now such a product decomposition can be chosen sothat

F g = dx2 + x2hx; in x < ;(3)

where hx is a family of metrics on @X.This reduced form is closely related to the behavior of geodesics near

the boundary. Up to orientation and parameterization there is a uniquegeodesic reaching the boundary at a given point p: In particular thenormal bration of X near @X given by the segments F ([0; ) fpg),p 2 @X, consists of geodesics which hit the boundary, each at thecorresponding point p:We shall discuss here the behavior of solutions to the wave equation

(D2

t )u = 0 on R X(4)

170

SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES 171

when X is endowed with a conic metric, is the associated (positive)Laplacian on functions, and Dt = i@=@t. For simplicity we take X tobe compact. It is only really important that @X be compact.Our primary concern is to describe the phenomenon of the propa-

gation of singularities for solutions to (4). To do so it is necessaryto understand the behavior of solutions in a way related to the func-tional analytic domain of : For the moment we simply say that we aredealing with `admissible' solutions. This condition is explained furtherbelow.In the interior of X the propagation of singularities, described pre-

cisely in terms of the notion of wavefront set, was treated in detail byHormander ([4]). We paraphrase Hormander's result here as

\Singularities travel along null bicharacteristics, which inthe case of the wave equation project to time-parameter-ized geodesics."

Thus, in the microlocal sense of singularities described by the wavefront set, a bicharacteristic segment, which covers a light ray, eitherconsists completely of singularities for a given solution or the solutionhas no singularity along it.This quite adequately describes the propagation of singularities ex-

cept where a light ray hits the boundary at some point and at sometime. Here a `splitting' of singularities will usually take place. Thisis generally called a diractive eect. The contrapositive of this eectcan be succinctly stated as follows:

\If no singularity reaches the boundary at time t then nosingularity leaves at time t:"

The point here is that the regularity along any one of the `radial'rays leaving the boundary at a given time is related, in general, to thesingularities on all the incoming rays (although there are two separatecomponents, as described below) arriving at the boundary at that time.Thus, even if singularities arrive at the boundary at time t along justone ray, they will in general depart along all rays leaving the boundaryat time t:There are, however, some important exceptions to this general spread-

ing of singularities. For instance, if X is a conic manifold with `trivial'conic metric dened by the blowup of a point in a smooth Riemann-ian manifold. In this case, admissible solutions are just the lifts ofsolutions in the usual sense and, because of Hormander's theorem oninterior singularities, the singularities are carried outward only on theone ray continuing the incoming ray in the original manifold.

172 RICHARD B. MELROSE AND JARED WUNSCH

For a general conic metric there is a similar notion of the `geometriccontinuation' of an incoming geodesic which hits the boundary. For atrivial conic metric obtained from a blowup, the boundary metric h0 isthe standard metric on the sphere. The geometrically related incomingand outgoing rays hit this sphere at antipodal points; these can alsobe thought of as the points separated by geodesics of length on theunit sphere. In the case of a general conic metric we may mimic thisby dening the relation

(5) G(p) =

fq 2 @X; 9 a geodesic in @X for h0 of length with end points p; qg:In general of course, G(p) is not smooth. Generically it is a hyper-surface with Lagrangian singularities; it is always the projection of asmooth Lagrangian relation.A geometric renement of the diraction result is obtained by con-

sidering the order of singularities with respect to Sobolev spaces andan additional `second microlocal' regularity condition. For simplicitysuppose that the (admissible) solution u is singular only near @X andonly near a single incoming ray hitting the boundary at time t and atthe point p: In the past (for t < t) we may suppose that the solution islocally in some Sobolev space Hs: Suppose further that the singularitiesof the solution are not too strongly focused on @X insofar as tangentialsmoothing raises the overall regularity, that is, for some k; ` > 0,

(@X + 1)ku 2 Hs+`loc

in t < t near @X;(6)

where @X is the Laplacian on @X with respect to the metric h0, ex-tended to act on a neighborhood of X using the product decomposition(2). Under these two assumptions and the additional requirement that

0 < ` <n

2;(7)

we obtain the following `geometric theorem'.

\If an admissible solution is singular only near an incom-ing ray arriving at @X at time t and (6) and (7) hold,then on outgoing rays with initial point in the comple-ment of G(p),

u 2 Hs+` 1

2

loc8 > 0 in t > t near @X:"(8)

When slightly generalized and rened, as described below, this resultapplies to the fundamental solution

sin tpp


p incoming singularity

outgoing regularityG(p)

Figure 1. Given the tangential smoothing property,there is greater regularity on outgoing rays from pointsnot -related to incoming rays carrying singularities.

with pole close to @X and with ` < n12. The diractive theorem

merely tells us that if the pole is specied at (x; p) at t = 0, thensingularities cannot emanate from @X except at time t = x. On theother hand, while `strong' singularities can emanate from all points inG(p); this geometric theorem tells us that the solution is microlocallymore regular on rays starting from @X at t = x but with initial pointoutside G(p): We can in fact use the conormality of the fundamentalsolution to obtain a sharper result than (8): the analogue of (8) for thefundamental solution yields Hs+l regularity, i.e. we obtain one-halfderivative of improvement over the general case.In the special case in which the metric g takes precisely the `product'

form

g = dx2 + x2h(y; dy);(9)

near the boundary, Cheeger and Taylor [1, 2] have given an explicitanalysis of the fundamental solution constructed by separation of vari-ables. (See also the discussion by Kalka-Meniko [6].) They Sobolevregularity they obtain is the same, and is therefore optimal in general.They also show that the outgoing solution is conormal and computethe precise order. A version of the results of Cheeger-Taylor has beenestablished in the analytic category by Rouleux [14]. Lebeau [7, 8]has also obtained a diractive theorem in the setting of manifolds withcorners in the analytic category.Detailed proofs of the results in this paper will appear in [11].The rst author acknowledges partial support from NSF grant DMS9625714.

The second author was partially supported by an NSF VIGRE instructorship at

Columbia University and by an NSF postdoctoral fellowship, and is grateful to


Alberto Parmeggiani and Cesare Parenti for invaluable conversations. The authors

wish to thank the referee for several helpful suggestions.

1. Friedrichs extension

To describe the admissibility condition, near the boundary, for solu-tions to (4) we rst describe the domain of the Laplacian for a conicmetric. We take the Friedrichs extension of . By denition, isassociated to the Dirichlet form

F (u; v) =

ZX

hdu; dvig dg ; u; v 2 C1c (X)(10)

Hence, dg is the metric volume form; in this case

dg = 'xn1 dx dh0 near @X; n = dimX ; ' 2 C1; ' > 0 :

The inner product in (10) is that induced, by duality, by the metric onT X: Following Friedrichs we dene,

D(1=2) = closnC1c(X) w.r.t. F (u; u) + kuk2L2

g

o;

whenever X is a compact conic manifold with boundary of dimensionn 2:This is a Hilbert space with dense injection D(1=2) ,! L2

g(X) so

there is a dual injection L2

g(x) ,! (D(1=2))0: The natural operator

: D(1=2)! (D(1=2))0 is determined by

(u; ')L2g= F (u; ') 8 u; ' 2 D(1=2):

Then the Friedrichs extension of is the unbounded operator withdomain

D() =u 2 D(1=2); u 2 L2

g(x):

It is a self-adjoint, non-negative operator and in this case has discretespectrum of nite multiplicity. This allows its complex powers to bedened by reference to an eigenbasis. The real powers are isomorphismso the null space, which consists precisely of the constants. Each ofthe powers is therefore a Fredholm map

s : D(s)! L2

g(X) 8 s 2 R:

with null space the constants and range the orthocomplement of theconstants. The domains form a scale of Hilbert spaces, and

D(s) ,! D(t) is dense 8 s t

with D(0) = L2

g(X).


For our purposes it is also important to note when the domains

consist of extendible distributions, i.e. those dual to _C1(X). This isthe case only for s > n

4and more precisely

_C1(X) ,! D(s) ,! C1(X)(11)

are dense inclusions forn4< s < n

4: The limits of this range correspond

to the occurrence of formal solutions of u = 0:

2. Wave group

The Cauchy problem for the wave equation

(D2

t )u = 0; on R X

ujt=0 = u0; Dtujt=0 = u1(12)

has a unique solution

u 2 C0(R;D(1=2)) \ C1(R;L2

g (x))

8 (u0; u1) 2 E = E1 = D(1=2) L2

g(X) :

These `nite energy solutions' are the main object of study here. Moregenerally, with the equation interpreted in C1(R;D(

s21)) the Cauchy

problem has a unique solution

u 2 C0(R;D(s2 )) \ C1(R;D(

s2

1

2 ))

8 (u0; u1) 2 Es = D(s2 )D(

s2

1

2 ) :(13)

The regularity hypothesis on the solution can be weakened to

u 2 L2

loc(R;D(

s2 )) \H1

loc(R;D(

s2

1

2 ))

without changing the unique solvability.Notice that these calculations are consistent under decrease of s:

Furthermore, partial hypoellipticity in t shows that the solution to(12) satises

u 2 Hkloc(R;D(

s2+

k2 )) 8 k 2 R :(14)

An admissible solution to the wave equation is one that satises

u 2 Hploc(R;D(

q

2 )) for some q; p 2 R(15)

with (4) holding in Hp2loc

(R;D(q21)). Such a solution automatically

satises (14) for some s:These statements can be reinterpreted in terms of the wave group

U(t) :

u0u1

7!

u(t)

Dtu(t)

; U(t) : Es ! Es 8 s:(16)


3. Hormander's theorem

Let M be a manifold without boundary. The wave front set of adistribution u 2 C1(M) is a closed subset of the cosphere bundle

WF(u) SM:

It may be dened by decay properties of the localized Fourier transform,or the FBI (FourierBrosIagonitzer) transform, or by testing withpseudodierential operators. The projection (WF(u)) M is exactlythe C1 singular support, the complement of the largest open subset ofM to which u restricts to be C1.A rened notion of wavefront set is the Sobolev-based wavefront set,

denoted WFs; this is a closed subset of SM , where now the projectionis the complement of the largest open subset of M to which u restrictsto be Hs.If u satises a linear dierential equation, Pu = 0; then

WF(u) (P ) SM

when (P ) is the characteristic variety of P , the set on which its (ho-mogeneous) principal symbol, p; vanishes.If p is real then the symplectic structure on T M , or the contact

structure on SM; denes a `bicharacteristic' direction eld VP on SM ,tangent to (P ): The integral curves of VP are called bicharacteristics;those lying in (P ) are called null bicharacteristics.

Theorem 1 (Hormander). Let P be a (pseudo)-dierential operator

with real principal symbol. If Pu = 0 then WF (u) (P ) is a unionof maximally extended null bicharacteristics. The same result also holdswith WF replaced by WFs for any s:

In our case, M = R X so T M = T R T X. The principal

symbol of the d'Alembertian is 2 j j2g, where is the dual variable

to t and j jg is the (dual) metric on T X. Then

(P ) = +(P ) [ (P ) SM

when I(P ) = R SX are the disjoint parts of (P ) in > 0 and < 0. In this representation of (P ) the null bicharacteristics aregeodesics on X, lifted canonically to SX, with t as ane parameter.Thus, for the wave equation overX, Hormander's theorem does indeedreduce to the informal propagation statement described above.Combined with standard results relating the singularities of the solu-

tion to singularities of the initial data, Hormander's theorem applied tothe wave equation on a conic manifold yields complete information on


the behavior of singularities except along bicharacteristics lying abovegeodesics which hit the boundary.

4. Diffractive theorem

On parametrized geodesic segments with an end point on the bound-ary, the dening function x is either strictly increasing or strictly de-creasing near the boundary. For each sign of and for each t 2 R

the bicharacteristics covering such geodesics which hit the boundary att = t and along which t is increasing (resp. decreasing) as x decreases,form a smooth submanifold of T (ft < tgX) (resp. T (ft > tgX)).We denote these `radial' surfaces (near @X) by

R;I(t) and R;O(t) (P )

where is the sign of and I; O refers to whether these are `incoming'or `outgoing' and hence, equivalently, whether they lie in t < t or t > t:

Theorem 2 (Diractive theorem). If u is an admissible solution to(4) then for any t 2 R, s 2 R, = ,

R;I(t) \WFs(u) = ; ) R;O(t) \WFs(u) = ;:Here, WFs(u) is the wave front set computed relative to the Sobolev

space Hs; locally in the interior.This is a precise form of the diractive result described informally

above. Notice that the singularities for dierent signs of are com-pletely decoupled. This does not, however, represent any renementin terms of propagation along the underlying geometric rays, since allgeodesics are covered by bicharacteristics with xed and of eithersign.The proof of this result is discussed brie y below in x7.

5. Geometric theorem

Consider a geodesic onX which hits the boundary at a point p 2 @X:An open set of perturbations of the geodesic, meaning geodesics start-ing near some interior point on the geodesic and with initial tangentclose to the tangent to the geodesic, will miss the boundary. A limitof such curves as the perturbation vanishes consists of three segments.The rst is the incoming geodesic segment. The second is a geodesicsegment in the boundary, of length . The third is the outgoing geo-desic from the end point of the boundary segment, which is thereforea point in G(p) as dened in (5) (see Figure 2). Thus it is reason-able to suppose that, amongst the outgoing bicharacteristics leavingthe boundary at time t; those with initial points in G(p) will be more


Figure 2. A sequence of geodesics nearly missing theboundary, and the three segments which they approach.

closely related to an incoming bicharacteristic with end point p arrivingat time t: We call these the geometrically-related bicharacteristics (orgeodesics).For instance, if there are incoming singularities on a single ray the

singularities on the `non-geometrically-related' outgoing bicharacteris-tics might be expected to be weaker than the incoming singularity.However, this is not in general the case. To obtain such a geometricrenement of the diraction result we need to impose an extra `nonfo-cusing' assumption.

Theorem 3 (Geometric theorem). Let u be an admissible solution to(4) and let = . Suppose that R;I(t)\WFs u = ; near @X. Supposeadditionally that for some k and 0 < ` < n

2

WFs+`(1 +@X)k u \ R;I(t) = ;:(17)

For any 0 < r < ` 1=2; if no incoming bicharacteristic hitting the

boundary at time t at a point in G(p) with sgn = is in WFs+r u, thenthe outgoing bicharacteristic with initial point p 2 @X and sgn = is not in WFs+r u for any > 0.

If in addition to (17) we have

WFs+`(xDx + (t t)Dt)(1 +@X)k u \R;I(t) = ;:(18)

then the same conclusion follows for all 0 < r < l.

Thus the additional assumption (17) allows regularity on outgoingrays to be deduced from regularity in the incoming geometrically-related rays up to the corresponding level above `background' regu-larity.


As already noted, this result may be applied to the fundamentalsolution with initial point near the boundary. If the initial pole of thefundamental solution is suciently close to the boundary then there isa unique short geodesic segment from it to the boundary, arriving ata point p: If t is the length of the segment then, provided t is smallenough, (17) and (18) hold with s < n

2+ 1 for any ` < n1

2. It

follows that on R;O(t); the outgoing set, the fundamental solution is

in H1=2, for all > 0, microlocally near the non-geometrically relatedrays, those with end point not in G(p), whereas the general regularityis H

n2+1 for all > 0. This is a gain of `nearly' n1

2derivatives over

the background regularity.In this way we extend part of the result of Cheeger and Taylor [1,

2] in the product case (9) to the general conic case. Inspection ofthe fundamental solution constructed in [1] reveals the `nearly' n1

2

dierence in smoothness between geometric and non-geometric rays tobe sharp.

6. Spherical conormal waves

Around a given point q in a compact Riemann manifold there are`spherical' conormal waves which are singular only on the sphericalsurfaces r = t, for small t of both signs. These just correspond toconormal data at t = 0 at the (ctive) cone point q. An important ex-ample is the fundamental solution, in which case the result follows fromHadamard's construction. In the more general case of a conic mani-fold with boundary there are similar contracting, and then expanding,conormal waves.

Theorem 4. If u is an admissible solution near @X and t = 0 whichis conormal to t = x for t < 0 then it is conormal to t = x, near the

boundary, for small t > 0.

These conormal solutions to the wave equation in the general coniccase are at the opposite extreme to those considered in the GeometricTheorem above. Namely, they are already smooth in the tangentialvariables, so no tangential smoothing in the sense of (6) is possible.Further analysis of the structure of these waves shows that the principalsymbols undergo a transition at x = 0, the boundary, given by thescattering matrix for the model cone with the same boundary metric.Since this scattering matrix should have full support in general, thisprovides counterexamples to any extension of the geometric theoremin which the tangential smoothing condition is dropped.


7. Methods

The basic method we use is microlocal, but non-constructive. It isa direct extension of one of the proofs by Hormander of the interiorpropagation theorem. This `positive' commutator method is itself amicrolocalization of the energy method for hyperbolic equations. In ita `test' pseudodierential operator, A, is applied to the equation andthe essential positivity of the symbol of the commutator 1

i[P;A] gives

a local regularity estimate on the solution.To extend this method to cover behavior of solutions near the bound-

ary we replace the ordinary notions of wavefront set, pseudodierentialoperators and microlocalization with versions appropriately adapted tothe geometry. When considering the Laplacian itself on the manifoldwith boundary with conic metric, the appropriate notion is that of aweighted b-pseudodierential operator (see [13]). This for instance al-lows the precise description of the domains of the powers of whichis used at various points in the argument.However, for the wave operators for the conic Laplacian the appro-

priate notion corresponds to the `edge' calculus of pseudodierentialoperators discussed originally by Mazzeo [9], arising from a ltrationof the boundary (see also Schulze [15]). In this case, the manifold withboundary is X R and the bers of the boundary are the surfacest = const. Thus t is the base variable of the bration.To the edge calculus of pseudodierential operators, given by mi-

crolocalization from the dierential operators generated by xDx, Dy

(where the y's are tangential variables) and xDt, we associate a notionof wavefront set. We can prove the propagation theorem analogousto that of Hormander for this `edge' wavefront set. However, in thisnew sense, D2

t is not globally of principal type but rather has tworadial surfaces. These correspond to the end points of bicharacteristicsarriving at, and leaving from, the boundary. At these surfaces thereare restrictions on the propagation results, very closely related to thosefor scattering Laplacians in [10]. The construction of the test operatorA; which away from the radial surfaces is essentially given by owoutalong the geodesic spray on @X; becomes more delicate at the radialsurfaces. Positivity relies on the precise form of the singularity of theHamilton vector eld there.These propagation estimates form the basis of both the diractive

and geometric theorems. In the former we combine the estimates witha variant of the one-dimensional FBI transform, scaled with respect tothe normal variable x. This reduces the diractive result to an iterative


application of a uniqueness theorem for the Laplacian on the model,non-compact cone.To obtain the geometric theorem, showing that the outgoing singu-

larities on non-geometrically related rays are weaker than the incomingones, we use a division theorem. The additional hypothesis of microlo-cal tangential smoothing is shown to imply that the solution actuallylies in a weighted Sobolev space with a higher x weight (hence more`divisible' by x) than is given, a priori, by energy conservation. Thisallows the microlocal propagation results indicated above to be pushedfurther at the outgoing radial surface and so yields the extra regularity.

8. Applications and extension

The propagation of singularities results of the type discussed aboveshould allow estimates of the spectral counting function as shown orig-inally by Ivrii ([5], see also [12] and [3]).We expect these methods to extend to more complicated geometries,

including manifolds with corners and iterated conic spaces.

References

[1] Je Cheeger and Michael Taylor, On the diraction of waves by conical singu-

larities. I, Comm. Pure Appl. Math. 35 (1982), no. 3, 275331, MR84h:35091a.

[2] , On the diraction of waves by conical singularities. II, Comm. Pure

Appl. Math. 35 (1982), no. 4, 487529, MR84h:35091b.

[3] L. Hormander, The analysis of linear partial dierential operators, vol. 3,

Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

[4] Lars Hormander, On the existence and the regularity of solutions of linear

pseudo-dierential equations, Enseignement Math. (2) 17 (1971), 99163. MR

48 #9458

[5] V. Ivrii, On the second term in the spectral asymptotics for the Laplace-

Beltrami operator on a manifold with boundary, Funct. Anal. Appl. 14 (1980),

98106.

[6] M. Kalka and A. Meniko, The wave equation on a cone, Comm. Partial Dif-

ferential Equations 7 (1982), no. 3, 223278, MR83j:58110.

[7] G. Lebeau, Propagation des ondes dans les varietes a coins, Seminaire sur

les Equations aux Derivees Partielles, 19951996, Ecole Polytech., Palaiseau,

1996, MR98m:58137, pp. Exp. No. XVI, 20.

[8] Gilles Lebeau, Propagation des ondes dans les varietes a coins, Ann. Sci. Ecole

Norm. Sup. (4) 30 (1997), no. 4, 429497, MR98d:58183.

[9] R. Mazzeo, Elliptic theory of dierential edge operators I, Comm. in P.D.E. 16

(1991), 16151664.

[10] R.B. Melrose, Spectral and scattering theory for the Laplacian on asymptoti-

cally Euclidian spaces, Spectral and scattering theory (Sanda, 1992) (M. Ikawa,

ed.), Marcel Dekker, 1994, pp. 85130.

[11] R.B. Melrose and J. Wunsch, Propagation of singularities for the wave equation

on conic manifolds, In preparation.


[12] Richard Melrose, The trace of the wave group, Microlocal analysis (Boulder,

Colo., 1983), Amer. Math. Soc., Providence, R.I., 1984, pp. 127167. MR

86f:35144

[13] Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, A K Peters Ltd.,

Wellesley, MA, 1993. MR 96g:58180

[14] Michel Rouleux, Diraction analytique sur une variete a singularite conique,

Comm. Partial Dierential Equations 11 (1986), no. 9, 947988. MR

88g:58182

[15] B.-W. Schulze, Boundary value problems and edge pseudo-dierential oper-

ators, Microlocal analysis and spectral theory (Lucca, 1996), Kluwer Acad.

Publ., Dordrecht, 1997, pp. 165226. MR 98d:47112

Department of Mathematics, MIT, Cambridge MA 02139

Department of Mathematics, SUNY at Stony Brook, Stony Brook

NY 11794

SOME REMARKS ON OSCILLATORY INTEGRALS

GERD MOCKENHAUPT

1. Introduction

The purpose of this note is to describe some results about oscillatoryintegral operators. Specically we are interested in bounds in Lebesguespaces of operators given by

Tf(x) =

ZRk

ei'(x;) f() d;

with '(x; ) a real-valued smooth function on Rn Rk; k n. Obvi-ously T is bounded as maps from Lqcomp to L

p

loc. What is of interesthere is the dependence of the norm for increasing . This will of coursedepend on the conditions we put on the phase function '. To guaranteethat ' lives on an open subset of Rn Rk it is natural to start withthe condition

rank ddx' = k; x 2 Rn and 2 Rk:(1)

We will assume this condition throughout this note. For work relatedto weaker assumptions see, e.g. [21] and [18]. One of the questions wewill ask is: What is the optimal (q; p)-range for which the operator Thas norm of order n=p? In particular we would like to understandhow this range will depend on k.To put things in perspective let us begin by describing what is known

for the case k = n: A model phase function here is '(x; ) = x , forx; 2 Rn. Then T is a localized version of the Fourier transform andthe (Lqcomp; L

p

loc)-boundedness properties are covered by the Hausdor-

Young inequality. For general phase functions satisfying (1) the L2-theory of Fourier integral operators gives

kjTkjLqcomp!Lp

loc

C n=p;

with p = q0 2 the dual exponent of q, i.e. 1=q0 + 1=q = 1.Next we consider the case k = n 1: A basic result was obtained by

E. M. Stein in the sixties. He discovered, for n 2, that the Fourier

1991 Mathematics Subject Classication. 42B15.Key words and phrases. oscillatory integrals, restriction theorems.The support of the Australian Research Council is greatfully acknowledged.

183

184 GERD MOCKENHAUPT

transform has the following restriction property: For the unit sphereSn1 in Rn and d a rotationally invariant measureZ

Sn1

j bf()j2 d() C kfk2Lp

0

(Rn);(2)

for some p0 > 1. By localizing to a ball of radius inRn, the dual of thisinequality states that the operator T with phase function '(x; ) =x (), where : U ! Sn1 parameterizes a coordinate neighborhood

of the unit sphere in Rn, has norm of order n=p as a map from L2(U)to Lp

locfor some p <1. Improvements on the range of exponents p were

made by P. Tomas [26] and E.M. Stein. Moreover, it was shown by E.M.Stein (see [24]) that for nonlinear phase functions ' the norm of T has

order n=p as an operator from L2comp to L

p

locfor p 2(n+1)=(n 1),

provided ' satises the following curvature condition: for each x 2 Rn

the hypersurface parameterized by

7! rx'(x; ) has nonvanishing Gaussian curvature.(3)

This (L2; Lp)-result is sharp in the sense that p = 2(n + 1)=(n 1)is critical. Moreover, due to an example of J. Bourgain [2, 4]under the conditions (3) and (1) Stein's result can not be im-proved in case n is odd if we require q = 1 (see also [15]).

QQQQQQQQ

1qq = 2

1p

p = 2nk

p = 4n2kk

Figure: (p; q)-range for k = n and k = n 1.

However under some furtherconditions, remarkable im-provements have been madeby J. Bourgain [1]. Hismethod, which led to fur-ther improvements in [27, 5]and [25], showed in particu-lar for the situation of theunit sphere described above,that for certain exponentsp less then the critical L2-exponent 2(n+1)=(n1) that

kTkL1(U)!Lp

loc

C n

p : It is

expected that the (q; p)-range for which this inequality holds is deter-mined by: p > 2n=k and p (2n k)=kq0 (see Figure). For n = 2the norm of T is essentially well understood due to work of L. Car-leson and P. Sjolin [6] provided the curvature condition (1) and (3) aresatised. We note that for the expected bounds the crucial point is tounderstand for the operators T the (L1; Lp)-bounds.Our main concern here are the cases k < n 1. There have been

some results in the past addressing the problem of (Lqcomp; Lp

loc)-bounds

SOME REMARKS ON OSCILLATORY INTEGRALS 185

for oscillatory integral operators in these cases. However, these resultsmainly discussed the cases k = 1; n 2 or n=2 for n even (see e.g. [7],[8], [12], [11], [19], [20]). For dierent k some results are obtained in[10] and [17]. A natural question which we ask here is the following:Suppose (1) holds. Under which conditions on the phase function '

does T map Lqcomp(Rk) to Lp

loc(Rn) with norm of order

n

p in the full

range

p 2n k

kq0 and p >

2n

kfor k < n?(4)

One of our results will be that we can expect these optimal boundsonly when k n=2. We will also see that in some situations wherethe phase function is linear in the x-variables an analogue of the Stein-Tomas result holds, i.e. optimal (L2; Lp)-bounds hold, but the (L1; Lp)-bounds fail to hold in the range given in (4). This apparently appearsonly if k < n 2.We should mention that one of the main diculties which distin-

guishes the case k < n1 from k = n1 lies in the fact that, althougha stationary phase argument shows that for 2 C1(Rk) and most

x 2 Rn the decay of T (x) is of order k=2, in general isotropicbounds for Tf(x) decay slower (see e.g. [9]).

2. A necessary curvature condition

Here we derive a necessary condition on the phase function ' suchthat T is bounded in the full range described in the above gure (for

k n). First we observe that if T has norm of order n=p, then foreach x0 the operator with phase function 'R(x; ) = R ('(x0+x=R; )'(x0; )) satises the same bounds uniformly in R > 0. Hence, the op-erator with the linearized phase function (x; ) 7! x rx'(x0; ) hasthe same bounds. By reparameterizing the k-dimensional submani-fold 7! rx'(x0; ) over the tangent plane at a given point usingtranslation invariance we may assume that x rx'(x0; ) has the form(x1 ; x2 ()), with (0) and d (0) both vanishing. A further scalingargument replacing x1 ! Rx1; x2 ! R2x2 and ! =R and letting

R ! 1 shows that the phase function x1 + x2 ~ (), here ~ isthe second order part of the Taylor expansion of , gives rise to anoperator

~Tf(x) =

ZRk

ei(x1+x2~ () ) f() ejj

2=2 d;

which is bounded from Lq(Rk) to Lp(Rn) for (q; p) on the line p =

(2n k)=kq0 provided that T has norm of order n=p on this line.


Write

x2 ~ () =1

2 Q(x2);

with Q(x2) =Pnk

j=1 x2;j Bj and Bj 2 Sym(k), where Sym(k) denotes

the space of symmetric matrices on Rk. To emphasize the dependence

on Q the operator ~T will be denoted by TQ in the following and werefer to the submanifolds parameterized by

H : Rk 3 7! (; B1; : : : ; Bnk) 2 Rn

as the associated quadratic submanifold MQ.

If TQ maps L1(Rk) to Lp(Rn), then in particular for the constant

function 1 we have G = ~TQ1 2 Lp. A computation gives

kGkpp = C

ZRnk

j det(E + iQ(x2))jp=2+1 dx2;(5)

here E denotes the unit matrix in Sym(k). To ensure that the aboveintegral is nite for some p <1 we need that the symmetric matricesBj; 1 j n k, are linearly independent which requires that n k(k + 3)=2. To nd a further restriction we show

Proposition 2.1. If the function G above is in Lp(Rn), for all p >2n=k, thenZ

Snk1

j detQ(x)j d(x) <1 for all <n k

k;(6)

with the uniform measure on the unit sphere Snk1.

Proof. To see this we use polar coordinates in (5) and write x2 = ry,and r = jxj. Then

j det(E + iQ(x))j2 = det(E +Q(x)2

= 1 + r2c21 + : : : c2k1r2k2 + detQ(y)2r2k:

Suppose that supj;y jcj(y)j c and let L(y) = maxf1; c=j detQ(y)jg.Then we get the following lower bound on kGkpp for p = 2n=k + 2, > 0: Z

Snk1

Z 1

L(y)

rnk1

jrk detQ(y)j(nk)=k+drd(y);

which evaluates to (6) by integrating the inner integral.

As a consequence we show:

Corollary 2.2. Suppose the function G dened above is in Lp(Rn) forall p > 2n=k. Then the following hold:

If k is odd, then k n

2.


If k is even and the subspace fQ(x)jx 2 Rnkg intersects the cone

of positive denite matrices in Sym(k), then k n=2.

Proof. The idea here is to nd a hypersurface on the unit sphere inRnk where the function detQ vanishes at least of order 1. Assumingk < n=2, i.e. (n k)=k > 1, Proposition (2.1) implies that the inte-gral

RSnk1

j detQj1 d is nite. Since the polynomial detQ(x) =det(x1B1 + + xnkBnk), with Bi 2 Sym(k), is homogenous of de-gree k for some power > 0 the function jxjj detQ(x)j1 must be in-tegrable over the unit ball in Rnk. We can assume that detQ(x) doesnot vanish identically and that B1 is a diagonal matrix with entries 1.If k is odd we can write locally detQ(x) = (x1 '(x2; : : : ; xnk)) (x)where '; are real continuous functions and '(0) = 0. Hence , forall > 0, j detQj1 is not locally integrable on the unit ball in Rnk

and therefore k n=2. To show the second part we may assume thatB1 = E. Then x1 ! Q(x1; x2; : : : ; xnk) is the characteristic polyno-mial of the symmetric matrix Q(0; x2; : : : ; xnk) and therefore has onlyreal zeros. So again detQ(x) = (x1 '(x2; : : : ; xnk)) (x). As beforewe nd that k has to be n=2.

The condition in the proposition above may be phrased in an in-variant way. Consider the submanifold M parameterized by 7!rx'(x0; ) and x a point P = rx'(x0; 0). We assume that M car-ries the induced Euclidean metric. Let NP (M) be the normal planeat P 2 M , TP (M) be the tangent plane at P , v 2 NP (M) and letGP (v) be the Gaussian curvature at P of the orthogonal projection ofM (along v) into Rv TP (M). Then (6) states thatZ

Snk1Np(M)

jGP (v)j d(v) < 1 for all <

n k

k;(7)

where denotes a nontrivial rotationally invariant measure on the unitsphere in NP (M).

3. Restriction to quadratic submanifolds

In the following we show some positive results for the operators TQ.

We write TQf =\f dQ, where dQ is the measure on Rn with supporton MQ dened by

Q(f) =

ZRk

f(;H()) ejj2=2 d:

Theorem 3.3. IfRSnk1

j detQ(x)j d(x) <1 for = nkk. Then

TQ is bounded from L2(Rk) to Lp(Rn) for p 22nkk

.


Proof. It is enough to show (and in fact equivalent) that the compo-

sition TQTQf = cd f maps the dual space Lp

0

(Rn) into Lp(Rn) for

p 22nkk

. Our strategy is now to dene an analytic family Tz which

evaluates at z = 0 to TQ and is bounded from L1 to L1 on the line

<z = 1=2 and from L2 to L2 for <z = nkk. A complex interpolation

argument will then give the theorem. This is analogous to Stein's proofof the Tomas-Stein theorem. The main point here is to nd a suitableanalytic family. To dene this analytic family we split variables andwrite as in the previous section x = (x1; x2) 2 Rk Rnk. For z 2 C

we put

Kz(x) =(1 + j detQ(x2)j)

z

(n k + kz)ddQ(x1; x2):

A computation shows that the latter expression is a constant multipleof

(1 + j detQ(x2)j)z

(n k + kz)det(E + iQ(x2))

1=2 ex1(E+iQ(x2))1x1=2(8)

We dene Tzf = Kzf . Note that T0f = ccdf; c 6= 0; and the familyTz is analytic in the whole complex plane. For (L1; L1)-bounds for Tzwe have to get uniform bounds for Kz on <z = 1=2. This follows easilyfrom (1+ j detQ(x2)j)

2 det(E+Q(x2)2). For the L2-boundedness we

have to bound the Fourier transform of Kz. To compute the Fouriertransform of Kz we rst evaluate the Fourier transform with respect tothe x1-variable. This gives

cKz(1; 2) = C

ZRnk

eix22(1 + j detQ(x2)j)

z

(n k + kz)e1(E+iQ(x2))1=2 dx2

Hence, to bound cKz it is enough to get bounds on the Fourier transformof (1 + j detQj)z. Now, for <z = nk

k+ "; " > 0; we nd, using polar

coordinates x2 = ry; r = jx2j the following bound for kcKzk1:

C sup2Rnk

ZSnk1

Z 1

0

(1 + rkj detQ(y)j)z

(n k + kz)eiry rnk1 dr dy

Since we are assuming

RSnk1

j detQ(y)jnk

k d(y) <1; we see thatthe above integral is bounded by a constant times

supx2R

j1

(n k + kz)

Z 1

0

rnk1(1 + rk)z eixr dr j:

On the line <z = nkk, the function rnk1(1 + rk)z is essentially

r1+kz+nk, i.e., homogeneous of degree 1 + is. Its Fourier transformis homogeneous of degree is and produces a pole at z = nk

kwhich


cancels with the Gamma function in front of the last integral. Hence

jcKzj is bounded.

We note that if we would have been working with the analytic family

~Kz(x1; x2) =1

(n k + kz)det(E +Q(x2)

2)z=2 cd(x1; x2)then the above method gives the following

Corollary 3.4. IfRSnk1

j detQ(x)j d(x) < 1 for all < nkk

.

Then TQ is bounded from L2(Rk) to Lp(Rn) for p > 22nkk

.

Arguing similarly one can show that if for a suitable polynomial p(z)the -distributions

z(f) = p(z)

ZRnk

j detQ(y)jz f(y) dy;

has a bounded Fourier transform on the line <(z) = nkk

then TQis bounded from L2(Rk) to Lp(Rn) (p has only to annihilate nitely

many poles of bz). Using this observation one can show that in certaincases one has optimal (L2; Lp)-bounds for TQ, although the (L1; Lp)-bounds fail to hold for some p > 2n=k. We provide a few examples inthe following.

First we dene for (x;X) 2 Rk Sym(k) = Rn with n =k(k+3)

2

Tf(x;X) =

ZRk

ei(x+X) f() ejj2=2 d:(9)

Then we have the following theorem, whose rst part was indepen-dently shown in [10] and for the special case k = 2 in [7].

Theorem 3.5. The operator T has the following properties:

(1) T is bounded from L2(Rk) to Lp(Rn) i p 22nkk

.

(2) T is unbounded as an operator from L1(Rk) to Lp(Rn) for p 2(k + 1) (= 2n

k+ k 1).

For the proof we will need the Fourier transform of j detXjz; X 2Sym(k); z 2 C. This has been computed rst by T. Shitani and morerecently by Faraut and Satake [13] using the theory of Jordan algebras.To state the result, we note rst that Sym(k) \GL(k;R) decomposesunder the operation (g;X)! gXgt into k+1 GL(k;R)-orbits, j; j =0; : : : ; k, where j is the cone of symmetric matrices of signature (k j; j). Let 0 be the orbit of the unit matrix E 2 Sym(k). Associated


to 0 is the Gamma function

0(s) =

Z0

etrace(X) (detX)sk+12 dX

= (2)k(k1)

4

Y0jk

(sj 1

2):

For 0 i k we dene Zeta distributions

i(f; s) =

Zi

f(X) j detXjs dX:

The poles here lie on the arithmetic progression 12Z \ (1;1]. We

have

i( bf; s k + 1

2) = (2)k(k+1)=2 eiks=2 0

(s)X0ik

ui;j(s) j(f;s);

where ui;j is a polynomial of degree k in eis. Putting s = k+12(1 z),

then it easily follows that the Fourier transform of

1

0(k+1

2(1 z))

1

j detXjzk+12

is a bounded function inX 2 Sym(k) on the imaginary line<z = 1 withbounds growing at most exponentially along this line. Hence part (i)follows. For the second part we will show that kT1kp <1 if and only if

p > 2(k+1). In fact, since kT1kpp = CRSym(k)

j det(E+iX)jp=2+1 dX,

we nd using generalized polar coordinates

ZRk

Y1jk

j1 + ijjp=2+1

Y1i<jk

ji jj d1 : : : dk <1

Now, the worst decay of the integrant is along the coordinate axes.Checking exponents it follows that the last integral is nite if and onlyif p > 2(k + 1).

We remark that one can show that the operator (9) is a bounded

operator from L1(Rk) to L2k+2(BR) with norm of order (logR)1

2k+2 ,where BR is a ball of radius R in Rn (note that 2k + 2 is an eveninteger).As a second example we consider for m > 1 the set M(m;C) of

complex m m-matrices which we might consider as a real subspace


of Sym(4m) via the following real linear map

Q :M(m;C) 3 Z = X + iY !

0BB@

0 0 X Y0 0 Y XtX tY 0 0tY tX 0 0

1CCA 2 Sym(4m);

where X; Y denote the real resp. imaginary part of Z. Note that for 2 C we have det(E+iQ(Z)) = det(2E+ZZ)2. It has been shownby E.M. Stein [23] that the Fourier transform of the Zeta distribution

(f; s) =

ZM(m;C)

j detZjs f(Z)dZ

is given by the 1 (s)

j detZjs2m, where (s) = (s) (s 2) : : : (s

2m + 2), (s) = ( s

2)=(2+s

2). Let k = 4m and dene for (x; Z) 2

Rk M(m;C) = Rn, with n = 2m(m + 2), the oscillatory integraloperator

Tf(x; Z) =

ZRk

ei(x+Q(Z))) f() ejj2=2 d:

Then we have

Theorem 3.6. The operator T has the following properties:

(1) T is bounded from L2(Rk) to Lp(Rn) i p 22nkk

.

(2) T is unbounded as an operator from L1(Rk) to Lp(Rn) for p 2m+ 1 (= 2n

k+ k

4 1).

Using polar coordinates associated to the Cartan decomposition cor-responding to the symmetric space SU(n; n)=S(U(n) U(n)) it is nothard to check that we have T1 2 Lp iZ

Rm

h1 : : : hmQ

1i<jm(h2i h2j)

2Q1im(1 + h2i )

p2dh1 : : : dhm

is nite, i.e. p > 2m+1. This conrms the second part of the theorem.These examples suggest that sharp L2-restriction estimates should

hold for most quadratic submanifolds. It would be interesting to ndout for which sets inside Sym(k)nk for which sharp L2-restriction fails(so far we have only some insight in the case n k = 2; 3).In the above examples k was always < n=2. However, there even in

case k = n3 examples for which we have optimal (L2; Lp)-bounds butthe (L1; Lp)-bounds fail for some p > 2n=k: An example is providedby

Q(x) = (x1 + x3)2n3 + x1(

21 + 23 + + 2n5)x3(

22 + 24 + : : : 2n4)

+ 2x2(12 + 34 + + n5n4)


It can be shown that the Fourier transform of the corresponding -distribution is essentially a cone multiplier of order 3=(2n 6) hencebounded function. But we do not have (L1; Lp)-boundedness forall p > 2n=(n 3). To see this one has to check when j det(E +

iQ(x))jp=2+1 is integrable, where Q is the Hessian of the quadraticform ! q(x; ). Now, E + iQ(x1 x3; x2; x1 + x3) has eigenvalues

1+2ix1; 1+i(x1px22 + x23) and we nd using polar coordinates in the

x2; x3-variables that the L1-norm of j det(E+ iQ(x))jp=2+1 is bounded

from below by a multiple ofZx1>0

1

(1 + jx1j)p

21

Zjx1rj1

r dr(1 + jx1 + rj)

n42 (1 + jx1 rj)

n42

p

21dx1:

The last integral is nite i p

2 1 + n4

2(p2 1) > 2, i.e. p > 2n+2

n2.

For more details and a description of how this examples arises in thecontext of nonregular orbits under certain Lie group actions we refer to[16]. Finally we mention the following theorem for the case k = n 2(see [16] and [7]).

Theorem 3.7. If B1; B2 2 Sym(n2) are linear independent then for

the operator TQ corresponding to Q(x1; x2) = x1B1+x2B2 the following

statements imply each other

(1) TQ is bounded from L2(Rk) to Lp(Rn) for p 2n+2n2

.

(2) TQ1 2 Lp(Rn) for p > 2n

n2.

(3)RS1

j detQ(x)j <1 for < 2n2

.

References

[1] J. Bourgain. Besicovitch type maximal operators and applications to Fourieranalysis. Geom. and Func. Anal, 1:147187, 1991.

[2] J. Bourgain. Lp estimates for oscillatory integrals in several variables. Geom.

and Func. Anal, 1:321374, 1991.[3] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets

and applications to nonlinear evolution equations. Geom. and Func. Anal.,

3(2):107262, 1993.[4] J. Bourgain. Some new estimates on oscillatory integrals. Essays on Fourier

analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 83112, PrincetonMath. Ser., 42, Princeton Univ. Press.

[5] J. Bourgain. On the dimension of Kakeya sets and related maximal inequalitiesGeom. and Func. Anal., 9(2):256282, 1999.

[6] L. Carleson and P. Sjolin. Oscillatory integrals and a multiplier problem forthe disc Studia Math., 44:287299, 1972.

[7] M. Christ. Restriction of the Fourier transform to submanifolds of low codi-mension. Thesis, University of Chicago, 1982.

[8] M. Christ. On the restriction of the Fourier transform to curves: endpointresults and degenerate curves. Trans. Amer. Math. Soc., 287:223238, 1985.


[9] J.J. Duistermaat. Oscillatory integrals, Lagrange immersions and unfoldingsof singularities. Comm. Pure and Appl. Math., 27:207281, 1974.

[10] L. de Carli and A. Iosevich. Some sharp restriction theorems for homogeneousmanifolds preprint

[11] S.W. Drury. Restriction of Fourier transforms to curves. Ann. Inst. Fourier,Grenoble, 35(1):117123, 1985.

[12] S.W. Drury and K. Guo. Some remarks on the restriction of the Fourier trans-form to surfaces. Math. Proc. Cambridge Philos. Soc., 113(1):153159, 1993.

[13] J. Faraut and I. Satake. The functional equation of Zeta distributions asso-ciated with formally real Jordan algebras. Tohoku Math. Jour., 36:469482,1984.

[14] A. Greenleaf and A. Seeger. Fourier integral operators with fold singularities.J. Reine Angew. Math., 455:3556, 1994.

[15] W. Minicozzi and C. Sogge. Negative results for Nikodym maximal functionsand related oscillatory integrals in curved spaces. Math. Res. Lett., 4(2-3):221-237, 1998.

[16] G. Mockenhaupt, Bounds in Lebesgue spaces of oscillatory integrals, Habilita-tionschrift, Universitat Siegen, 1996.

[17] G. Mockenhaupt. A restriction theorem for the Fourier transform. Bull. Amer.

Math. Soc., 25(1):3136, 1991.[18] D.H. Phong and E.M. Stein. The Newton polyhedron and oscillatory integral

operators. Acta Math., 179(1):105152, 1997.[19] E. Prestini. Restriction theorems for the Fourier transform to some manifolds

in Rn . Proc. of Symp. in Pure Math., 35:101109, 1979.

[20] A. Ruiz. On the restriction of Fourier transforms to curves. Conference onharmonic analysis in honor of Antoni Zygmund, Vol. I, II, 186212, WadsworthMath. Ser., Wadsworth, Belmont, Calif., 1983.

[21] A. Seeger. Degenerate Fourier integral operators in the plane. Duke Math. J.,71(3):685745, 1993.

[22] A. Seeger and C.D. Sogge. Bounds for eigenfunctions of dierential operators.Indiana Univ. Math. J., 38(3):669682, 1989.

[23] E.M. Stein. Analysis on matrix spaces and some new representations ofSL(n; C ). Ann. Math., 86:461490, 1967

[24] E.M. Stein. Harmonic Analysis: Real-variable methode, orthogonality and os-

cillatory integrals. Princeton Math. Ser., 43, Princeton Univ. Press, 1993.[25] T. Tao, A. Vargas and L. Vega. A bilinear approach to the restriction and

Kakeya conjectures. J. Amer. Math. Soc., 11:9671000, 1998.[26] P.A. Tomas. A restriction theorem for the Fourier transform. Bull. Amer.

Math. Soc., 81:477478, 1975.[27] T.H. Wol. Recent work connected with the Kakeya problem. Prospects in

Mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999.

School of Mathematics, Georgia Institute of Technology, Atlanta,

GA 30332, USA


YET ANOTHER CONSTRUCTION OF THE CENTRAL

EXTENSION OF THE LOOP GROUP

MICHAEL K. MURRAY AND DANIEL STEVENSON

Abstract. We give a characterisation of central extensions of aLie group G by C in terms of a dierential two-form on G and adierential one-form on G G. This is applied to the case of thecentral extension of the loop group.

1. Introduction

Let G and A be groups. A central extension of G by A is another

group G and a homomorphism : G ! G whose kernel is isomorphic

to A and in the center of G. Note that because A is in the center of Git is necessarily abelian. We will be interested ultimately in the case

that G = (K) the loop group of all smooth maps from the circle S1 toa Lie group K with pointwise multiplication but the theory developedapplies to any Lie group G.

2. Central extension of groups

Consider rst the case when G is just a group and ignore questionsof continuity or dierentiability. In this case we can choose a section of

the map . That is a map s : G ! G such that (s(g)) = g for everyg 2 G. From this section we can construct a bijection

: AG! G

by (g; a) = (a)s(g) where : A ! G is the identication of A with

the kernel of . So we know that as a set G is just the product AG.

However as a group G is not generally a product. To see what it is notethat (s(g)s(h)) = (s(g))(s(h)) = gh = (s(gh)) so that s(g)s(h) =

c(g; h)s(gh) where c : GG! A. The bijection : AG! G inducesa product on AG for which is a homomorphism. To calculate this

The rst author acknowledges the support of the Australian Research Council.The second author acknowledges the support of the Australian Research Council.

194

CENTRAL EXTENSION 195

product we note that

(a; g)(b; h) = (a)s(g)(b)s(h)

= (ab)s(g)s(h)

= (ab)c(gh)s(gh):

Hence the product on A G is given by (a; g) ? (b; h) = (abc(g; h)gh)

and the map is a group isomorphism between G and AG with thisproduct.Notice that if we choose a dierent section ~s then ~s = sh were

h : G! A.It is straightforward to check that if we pick any c : GG! A and

dene a product on AG by (a; g) ? (b; h) = (abc(g; h)gh) then this isan associative product if and only if c satises the cocycle condition

c(g; h)c(gh; k) = c(g; hk)c(h; k)

for all g, h and k in G.If we choose a dierent section ~s then we must have ~s = sd for some

d : G! A. If ~c is the cocycle determined by ~s then a calculation showsthat

c(g; h) = ~c(g; h)d(gh)d(g)1d(h)1:(1)

We have now essentially shown that all central extensions of G by A

are determined by cocycles c modulo identifying two that satisfy thecondition (1). Let us recast this result in a form that we will see againin this talk.Dene maps di : G

p+1 ! Gp by

di(g1; : : : ; gp+1) =

8><>:(g2; : : : ; gp+1); i = 0;

(g1; : : : ; gi1gi; gi+1; : : : ; gp+1); 1 i p 1;

(g1; : : : ; gp); i = p:

(2)

IfMp(G;A) = Map(Gp; A) then we dene : Mp(G;A)!Mp1(G;A)by (c) = (c d1)(c d2)

1(c d3) : : : . This satises 2 = 1 and denes

a complex

M0(G;A)!M1(G;A)

!M2(G;A)

! : : :

The cocycle condition can be rewritten as (c) = 1 and the condition

that two cocycles give rise to the same central extension is that c =~c(d). If we dene

Hp(G;A) =kernel : Mp(G;A)!Mp1(G;A)

image : Mp+1(G;A)!Mp(G;A)

196 MICHAEL K. MURRAY AND DANIEL STEVENSON

then we have shown that central extensions of G by A are classied byH2(G;A).

3. Central extensions of Lie groups

In the case that G is a topological or Lie group it is well-known

that there are interesting central extensions for which no continuous ordierentiable section exists. For example consider the central extension

Z2 ! SU(2) = Spin(3)! SO(3)

of the three dimensional orthogonal group SO(3) by its double coverSpin(3). Here SU(2) is known to be the three sphere but if a sectionexisted then we would have SU(2) homeomorphic to Z2 SO(3) and

hence disconnected.From now on we will concentrate on the case when A = C

. ThenG! G can be thought of as a C principal bundle and a section willonly exist if this bundle is trivial. The structure of the central extension

as a C bundle is important in what follows so we digress to discuss

them in more detail.

3.1. C bundles. Let P ! X be a C bundle over a manifold X.

We denote the bre of P over x 2 X by Px. Recall [1] that if P isa C

bundle over a manifold X we can dene the dual bundle P asthe same space P but with the action pg = (pg1) and, that if Qis another such bundle, we can dene the product bundle P Q by

(P Q)x = (Px Qx)=C where C

acts by (p; q)w = (pw; qw1).We denote an element of P Q by p q with the understanding that(pw) q = p (qw) = (p q)w for w 2 C

. It is straightforward tocheck that P P is canonically trivialised by the section x 7! p p

where p is any point in Px:

If P and Q are C bundles on X with connections P and Q thenPQ has an induced connection we denote by PQ. The curvatureof this connection is RP +RQ where RP and RQ are the curvatures ofP and Q respectively. The bundle P has an induced connection

whose curvature is RP .Recall the maps di : G

p ! Gp1 dened by (2). If P ! Gp is a C

bundle then we can dene a C bundle over Gp+1 denoted (P ) by

(P ) = 11 (P ) 12 (P ) 13 (P ) : : : :

If s is a section of P then it denes (s) a section of (P ) and if isa connection on P with curvature R it denes a connection on (P )

which we denote by (). To dene the curvature of () let us denote


by q(Gp) the space of all dierentiable q forms on Gp. Then we denea map

: q(Gp)! q(Gp+1)(3)

by =Pp

i=0 d

i , the alternating sum of pull-backs by the various mapsdi : G

p+1 ! Gp. Then the curvature of () is (R). If we consider

((P )) it is a product of factors and every factor occurs with its dualso ((P )) is canonically trivial. If s is a section of P then under thisidentication (s) = 1 and moreover if is a connection on P then() is the at connection on (P ) with respect to ((s)).

4. Central extensions

Let G be a Lie group and consider a central extension

C ! G

! G:

Following Brylinski and McLaughlin [2] we think of this as a C bundle

G ! G with a product M : G G ! G covering the product m =d1 : GG! G.

Because this is a central extension we must have that M(pz; qw) =M(p; q)zw for any p; q 2 P and z; w 2 C

. This means we have asection s of (P ) given by

s(g; h) = pM(p; q) q

for any p 2 Pg and q 2 Ph. This is well-dened as pwM(pw; qz)qz =pwM(p; q)(wz)1qz = pM(p; q)q. Conversely any such sectiongives rise to an M .Of course we need an associative product and it can be shown that

M being associative is equivalent to (s) = 1. To actually make G into

a group we need more than multiplication we need an identity e 2 Gand an inverse map. It is straightforward to check that if e 2 G is the

identity then, becauseM : GeGe ! Ge, there is a unique e 2 Ge suchthat M(e; e) = e. It is also straightforward to deduce the existence of

a unique inverse.Hence we have the result from [2] that a central extension of G is a

C bundle P ! G together with a section s of (P ) ! G G such

that (s) = 1. In [2] this is phrased in terms of simplicial line bundles.

Note that this is a kind of cohomology result analogous to that in therst section. We have an object (in this case a C

bundle) and ofthe object is `zero' i.e. trivial as a C bundle.For our purposes we need to phrase this result in terms of dier-

ential forms. We call a connection for G ! G, thought of as a C

bundle, a connection for the central extension. An isomorphism of


central extensions with connection is an isomorphism of bundles with

connection which is a group isomorphism on the total space G. Denoteby C(G) the set of all isomorphism classes of central extensions of G

with connection.Let 2 1(G) be a connection on the bundle G ! G and consider

the tensor product connection (). Let = s(()). We then havethat

() = ((s))(())

= (1)(2())

= 0

as 2() is the at connection on 2(P ). Also d = s(d()) = (R).Let (G) denote the set of all pairs (;R) where R is a closed, 2i

integral, two form on G and is a one-form on GG with (R) = dand () = 0.

We have constructed a map C(G) ! (G). In the next section weconstruct an inverse to this map by showing how to dene a centralextension from a pair (;R). For now notice that isomorphic centralextensions with connection clearly give rise to the same (;R) and that

if we vary the connection, which is only possible by adding on the pull-back of a one-form fromG, then we change (;R) to (+(); R+d).

4.1. Constructing the central extension. Recall that given R wecan nd a principal C bundle P ! G with connection and curvatureR which is unique up to isomorphism. It is a standard result in the

theory of bundles that if P ! X is a bundle with connection whichis at and 1(X) = 0 then P has a section s : X ! P such thats() = 0. Such a section is not unique of course it can be multiplied bya (constant) element of C . As our interest is in the loop groupG which

satises 1(G) = 0 we shall assume, from now on, that 1(G) = 0.Consider now our pair (R; ) and the bundle P . As (R) = d wehave that the connection (w) () on (P ) ! G G is at andhence (as 1(GG) = 0) we can nd a section s such that s((w)) = .The section s denes a multiplication by

s(p; q) = pM(p; q) q:

Consider now (s) this satises (s)(((w))) = (s((w)) = () =0. On the other hand the canonical section 1 of ((P )) also satises

this so they dier by a constant element of the group. This means thatthere is a w 2 C

such that for any p, q and r we must have

M(M(p; q); r) = wM(p;M(q; r)):


Choose p 2 Ge where e is the identity in G. Then M(p; p) 2 Ge andhence M(p; p) = pz for some z 2 C

. Now let p = q = r and it is clearthat we must have w = 1.So from (;R) we have constructed P and a section s of (P ) with

(s) = 1. However s is not unique but this is not a problem. If we

change s to s0 = sz for some constant z 2 C then we have changed

M to M 0 = Mz. As C is central multiplying by z is an isomorphismof central extensions with connection. So the ambiguity in s does notchange the isomorphism class of the central extension with connection.

Hence we have constructed a map

(G)! C(G)

as required. That it is the inverse of the earlier map follows from the

denition of as s(()) and the fact that the connection on P ischosen so its curvature is R.

5. Conclusion: Loop groups

In the case where G = L(K) there is a well known expression for the

curvature R of a left invariant connection on ^L(K) | see [5]. We can

also write down a 1-form on L(K) L(K) such that (R) = d and() = 0. We have:

R(g)(gX; gY ) =1

42

ZS1

hX; @Y id

(g1; g2)(g1X1; g2X2) =1

42

ZS1

hX1; (@g2)@g1

2 id:

Here h ; i is the Killing form on k normalised so the longest root haslength squared equal to 2 and @ denotes dierentiation with respectto 2 S1. Note that R is left invariant and that is left invariant in

the rst factor of GG. It can be shown that these are the R and arising in [3].In [4] we apply the methods of this talk to give an explicit construc-

tion of the `string class' of a loop group bundle and relate it to earlier

work of Murray on calorons.

References

[1] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization,Progr. Math., 107, Birkhauser Boston, Boston, MA, 1993.

[2] J.-L. Brylinski and D. A. McLaughlin, The geometry of degree-four character-

istic classes and of line bundles on loop spaces. I, Duke Math. J. 75 (1994),no. 3, 603638;


[3] M. K. Murray, Another construction of the central extension of the loop group,Comm. Math. Phys. 116 (1988), 7380

[4] M.K. Murray and D. Stevenson, Higgs elds, bundle gerbes and string struc-

tures. In preparation.[5] A. Pressley and G. Segal, Loop groups. Oxford, Clarendon Press, 1986.

Department of Pure Mathematics, University of Adelaide, Ade-

laide, SA 5005, Australia

E-mail address, Michael K. Murray: [email protected]

E-mail address, Daniel Stevenson: [email protected]

SPECTRUM OF THE RUELLE OPERATOR AND ZETA

FUNCTIONS FOR BROKEN GEODESIC FLOWS

LUCHEZAR STOYANOV

1. Introduction

Let K be an obstacle in IRn, where n 3 is odd, i.e. K is a compact

subset of IRn with C1 boundary @K such that

= IRnnK

is connected. One of the main objects of study in scattering theory (byan obstacle) is the so called scattering matrix S(z) related to the wave

equation in IR with Dirichlet boundary condition on IR . Thisis (cf. [LP], [M] or [Z]) a meromorphic operator-valued function

S(z) : L2(Sn1) ! L2(Sn1)

with poles fjg1j=1 in the half-plane Im(z) > 0.

A variety of problems in scattering theory deal with nding geometricinformation about K from the distribution of the poles fjg. In whatfollows we describe one particular problem of this type.The obstacle K is called trapping if there exists an innitely long

bounded broken geodesic (in the sense of Melrose and Sjostrand [MS])

in the exterior domain . For example, if contains a periodic brokengeodesic (this is always the case when K has more than one connectedcomponent), then K is trapping.It follows from results of Lax-Phillips (1971) and Melrose (1982) that

if K is non-trapping, then fz : 0 < Im(z) < g contains nitely manypoles j for any > 0 (cf. the Epilogue in [LP] for more preciseinformation).In the rst edition of their monograph Scattering Theory published

in 1967, Lax and Phillips conjectured that for trapping obstacles thereshould exist a sequence fjg of scattering poles such that Imj ! 0as j ! 1. However M. Ikawa [I1] showed that this is not the casewhen K is a disjoint union of two strictly convex compact domainswith smooth boundaries. It turned out that in this particular case

the scattering matrix has poles approximately at the pointsk

d+ i,

k = 0;1;2; : : : , where d is the distance between the two connected201

202 LUCHEZAR STOYANOV

components K1 and K2 of K and > 0 is a constant depending onlyon the curvatures of @K at the ends of the shortest segment connecting

K1 and K2. Substantial new information concerning the distributionof poles in this case was later given by C. Gerard [G].Ikawa modied the initial conjecture of Lax and Phillips in the fol-

lowing way.

Lax-Phillips Conjecture (LPC) (in the form given by M. Ikawa):If K is trapping, then there exists > 0 such that the strip fz : 0 <Im(z) < g contains innitely many poles j.

So far results concerning this conjecture are only known in the casewhen K has the form

K = K1 [K2 [ : : : [Ks ;(1)

where Ki are strictly convex disjoint compact domains in IRn (with C1

boundaries) satisfying the following condition introduced by M.Ikawa:

(H) Km\ convex hull(Ki [Kj) = ; for all m 6= i 6= j 6= m.

As we mentioned above, the LPC holds in the case s = 2. There arepartial results in the case s 3 also due to Ikawa (cf. [I3], [I4]). Belowwe brie y describe Ikawa's approach in dealing with this case.The starting point is the distribution

u(t) =Xj

eijtjj ; t 6= 0 ;

where as above fjg is the set of all poles of the scattering matrix

S(z). Guillemin and Melrose [GM] showed that the sum of the principalsingularities of u(t) on IR+ isX

(1)m T j det(I P )j1=2(t d ) ;(2)

where runs over the set of periodic broken geodesics in K , d is theperiod (return time) of , T the primitive period (length) of , andP the linear Poincare map associated to .Applying the Laplace transform to (2), Ikawa [I3] introduced the

following zeta function (which could be called the scattering zeta

function)

Z(s) =X 2

(1)m T jI P j1=2esd ; s 2 C :

SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 203

He then showed that the existence of analytic singularities of Z(s)implies the existence of a band 0 < Im(z) < containing an innite

number of scattering poles j (i.e. LPC holds).Clearly Z(s) is a Dirichlet series. Let s0 be its abscissa of absolute

convergence. Assuming n = 3, Ikawa showed that there exists > 0such that in the region s0 < Re(s) s0 the analytic singularitiesof Z(s) coincide with these of the zeta function

Z0(s) =

1Xm=0

X

(1)mr T em(sT + ) ;

where runs over the set of primitive periodic broken geodesics in ,r = 0 if has an even number of re ection points and r = 1 other-wise, and 2 IR is determined by the spectrum of the linear Poincare

map related to . The function Z0(s) is rather similar to a dynami-cally dened zeta function (cf. the survey of Baladi [Ba] for generalinformation on this topic). One of the main tools to study this sort ofzeta function is the so called Ruelle operator (well known e.g. from the

study of Gibbs measures in statistical mechanichs and ergodic theory,cf. [R1] and [Si]). Ikawa [I4] succeeded to implement results of Pollicott(1986) and Haydn (1990) concerning the spectrum of the Ruelle oper-ator and derived sucient conditions for Z0(s) (and therefore Z(s)) tohave a pole in a small neighbourhood of s0 in C. From this he derivedthe following.

Theorem 1. ([I4]) If K is a nite union of disjoint balls in IR3 with

the same radius > 0 and > 0 is suciently small, then LPC holds.

The study of the scattering zeta function itself seems to be rather dif-cult and very few results concerning it are known. Petkov [P] showedthat Z(s) admits an analytic continuation in the domain Re(s) s0.Moreover, under some additional assumptions about the geometry ofthe obstacle K and assuming rational independence of the primitiveperiods of periodic orbits, Petkov proved that supt2IR jZ(s0 + it)j =1

(cf. [P] for some further results on the properties of Z(s)). Assumingthat the broken geodesic ow has two periodic orbits 1 and 2 suchthat T 1=T 2 is a diophantine number, Naud [N] proved that Z(s) hasan analytic continuation to a domain of the form

s0 C

jtj< Re(s) s0 ; jtj = jIm(s)j 1 ;

for some constants C > 0 and > 0.


2. Spectrum of the Ruelle operator and dynamical zeta

function

In the case when the obstacle K has the form (1), it is sometimesmore convenient to work with the billiard ball map B (i.e. the shiftalong the broken geodesic ow from boundary to boundary) on the non-wandering set M0. The latter is the set of all points 2 S@K(IR

n) that

generate trapped broken geodesics in both directions. As a dynamicalsystem B : M0 ! M0 is naturally isomorphic to the Bernoulli shifton the symbolic space

=

1Y1

f1; 2; : : : ; sg ;

the isomorphism being given by the natural coding of the geodesicsby sequences fijg, where ij = 1; : : : ; s is the number of the connectedcomponent Kij containing the jth re ection point. Ikawa [I4] used theclassical interpretation of the Ruelle operator as an operator actingon the space of Lipschitz functions on the symbol space . However,having in mind some signicant recent developments, it seems more

convenient to use a dierent model which is more closely related to thedynamics of the ow.

In this section we describe some recent results of the author con-cerning the spectrum of the Ruelle operator and the dynamical zetafunction related to the broken geodesic ow in the exterior of an ob-

stacle of the form (1) in the IR2. The results are similar to these of

Dolgopyat [D] in the case of Anosov ows on compact manifolds. Onewould expect that similar results could be obtained (by using similartechniques) for obstacles in IR

n, n 3. Such results would very likelylead to partial solutions of the LPC for cases much more general then

the one considered in Theorem 1 above.From now on we assume that K is an obstacle of the form (1) in

IR2. Let be the non-wandering set of the broken geodesic ow in .

Clearly is the union of all orbits generated by elements of the set M0

dened above. For x 2 n S@K() and a suciently small > 0 let

W s (x) = fy 2 S() : (t(x); t(y)) for all t 0 ;

(t(x); t(y))!t!1 0 g ;

W u (x) = fy 2 S() : (t(x); t(y)) for all t 0 ;

(t(x); t(y))!t!1 0 g

be the (strong) stable and unstable sets of size . It is easy to show that

for every x 2 nS@K() and every suciently small > 0, W s (x) and


W u (x) are 1-dimensional submanifolds of S

(). It is worth mentioningthat both W s

(x) \ and W u (x) \ are Cantor sets.

Given > 0, set

S () = fx = (q; v) 2 S() : dist(q; @K) > g ; = \ S () :

In what follows in order to avoid ambiguity and unnecessary complica-tions we will consider stable and unstable manifolds only for points xin S () or ; this will be enough for our purposes.It follows from the genaral theory of horocycle foliations (cf. [S] or

[KH]) that if > 0 is suciently small, there exists > 0 such that ifx; y 2 and (x; y) < , thenW s

(x)\ and [;](Wu (y)\) intersect

at exactly one point [x; y]. That is, there exists a unique t 2 [; ]such that t([x; y]) 2 W u

(x). Setting (x; y) = t, denes the so calledtemporary distance function ([Ch1]). If z 2 and U W u

(z)and S W s

(z) are closed (i.e. containing their end points) curvescontaining z and such that U \ and S \ have no isolated points,then

= [U \ ; S \ ] = f[x; y] : x 2 U \ ; y 2 S \ g

is called a rectangle in . Notice that is "foliated" by leaves [x; S\]of stable manifolds.Let R = fRig

ki=1 be a family of rectangles such that each Ri is

contained in a C1 cross-section Di S () to the ow t. Thus, foreach i, Ri = [Ui\; Si\], where Ui and Si are closed curves inW

u (zi)

and W s (zi), respectively, for some zi 2 . Set

R =

k[i=1

Ri :

The family R is called complete if there exists T > 0 such that forevery x 2 there exist t1 2 [T; 0) and t2 2 (0; T ] with t1(x) 2 Rand t2(x) 2 R. Thus, (x) = t2(x) > 0 is the smallest positive timewith P (x) = (x)(x) 2 R, and P : R ! R is the Poincare map

related to the family R.Following [B] and [Ra] we will say that a complete family R =

fRigki=1 of rectangles in S () is a Markov family of size 2 (0; =2)

for the billiard ow t if:(a) Ri \ Rj = ; for i 6= j and for each i the sets Ui \ and Si \

are contained in the interior of the curves Ui and Si, respectively;(b) diam(Ri) ;(c) For any i 6= j and any x 2 Ri\P

1(Rj) we have P ([x; Si\]) [P (x); Sj] and P ([Ui \ ; x]) [Uj; P (x)];


(d) For any i = 1; : : : ; k and x 2 Ri the function is constant onthe set [x; Si \ ].

The existence of a Markov family R of an arbitrarily small size > 0for t follows from the construction of Bowen [B] (cf. also Ratner [Ra]).It follows from a result of C. Robinson [Ro] that there exists an open

neighbourhood V of in S() and C1 transverse foliations F u andF s on V such that W u

(x) \ V = F u(x) and W s (x) \ V = F s(x) for

any x 2 \ V . Fix a neighbourhood V and C1 foliations F u and F s

with these properties.

Choosing > 0 suciently small, we may assume that our Markovfamily R satises the following additional condition:

(e) For each i the cross-section Di (and therefore the rectangle Ri)

is contained in V .

In what follows we assume that R = fRigki=1 is a xed Markov family

for t satisfying the extra conditions (e) and (f). Then

U =

k[i=1

Ui ;

is a nite disjoint union of compact curves in V .Using the foliations F u and F s in V , assuming again that V is su-

ciently small, we can extend the product [x; y] over the whole Ui Sifor any i as follows. Given x 2 Ui W u

(zi) = F u(zi) \ V andy 2 Si W s

(zi) = F s(xi) \ V , the sets F s(x) and [;](Fu(y))

intersect at exactly one point [x; y].The image of the C1 map UiSi 3 (x; y) 7! [x; y] is a 2-dimensional

sumanifold of S () which will be denoted by ~Ri = [Ui; Si]. The pro-

jection p : ~R = [ki=1

~Ri ! U along the leaves of F s is C1. Then the

Poincare map P : ~R ! [ki=1Di and the corresponding root function

: ~R ! [0;1) are well-dened and C1. Thus, can be extended to aC1 map : U ! U by the same formula: = p P . In the same way

one observes that (x; y) is well-dened and C1 for (x; y) 2 [ki=1

~Ri~Ri.

Let C(U) be the space of bounded continuous functions on U . Forg 2 C(U) denote kgk = kgk0 = supx2U jg(x)j. Given g, the Ruelle

operator Lg : C(U) ! C(U) is dened in the usual way:

(Lgh)(u) =X

(v)=u

eg(v)h(v):


If g 2 C1(U), then clearly Lg preserves the space C1(U). Here C1(U)

denotes the space of dierentiable function with bounded derivatives

on U .A conjugacy between \ R and the Bernoulli shift on a symbolic

space A is now dened in the usual way. Let A = (Aij)ki;j=1 be the

matrix given by Aij = 1 if P (Ri)\Rj 6= ; and Aij = 0 otherwise. Then

A = f(ij)1j=1 : 1 ij k; Aij ij+1 = 1 for all j g;

and the Bernoulli shift ~ : A ! A is given by ~((ij)) = ((i0j)),

where i0j = ij+1 for all j. Dene : R ! A by (x) = (ij)1j=1,

where P j(x) 2 Rij for all j 2 Z. Notice that P : R ! R is invertible,

so P j is well-dened for all integers j. The map is a homeomorphismwhen A is considered with the product topology, and ~ = P .The projection r of on A is given by r = .The subset \ U of R can be naturally identied with

+A = f (ij)

1j=0 : 1 ij k ; Aij ij+1 = 1 for all j 0 g:

Namely, if : A ! +A is the natural projection, then + =

j\U : \ U ! +A is a bijection with ~ + = + , where ~

denotes the corresponding Bernoulli shift on +A.

The dynamical zeta function of the broken geodesic ow t is denedby

(s) =Y

(1 es`( ))1 ;

where runs over the set of closed orbits of t and `( ) is the leastperiod of . One can easily see that

(s) = exp

0B@ 1X

n=1

1

n

Xn(x)=xx2U\

esn(x)

1CA ;

where

n(x) = (x) + ((x)) + : : :+ (n(x))

is the period (length) of the closed orbit generated by x. That is, wehave

(s) = exp

1Xn=1

1

nZn(s)

!;

where

Zn(s) =X

n(x)=xx2U\

esn(x) :


The following lemma1 of Ruelle ([R2], cf. also [PoS]) partially ex-plains the relationship between the Ruelle operator and the behaviour

of the dynamical zeta function (s).

Lemma 1. (Ruelle) Let xj be an arbitrary point in Uj \ for every

j = 1; : : : ; k. There exist constants C > 0, > 0, 2 (0; 1) such thatZn(s)

kXj=1

LnsUj

(xj)

CjIm(s)jnn

for all n 1 and all s 2 C with Re(s) hT, hT being the topological

entropy of the ow t, and Uj is the characteristic function of Uj in U .

The suspended ow ~r over rA = f(; t) : 2 A; 0 t r() g

is dened by ~rs(; t) = (; t + s), where we use the identication

(; r()) = ((); 0) in A IR. Then r;+A = f(; t) 2 r

A : 2 +Ag is

a closed subset of rA which is invariant under the semi ow ~rt , t 0

(cf. [PP2] for details). A conjugacy between ~rt and t on is dened

by r(s(x)) = ((x); s) for x 2 R and 0 s (x).Given 2 (0; 1), consider the metric d on A given by d(; ) = 0 if

= and d(; ) = n if i = i for jij n and n is maximal with thisproperty. In a similar way one denes a metric d on +

A. Using it, we

get a metric on r;+A by setting dr((; t); (; s)) = d(; )+ jtsj. The

spaces of Lipschitz functions on +A and

r;+A with respect to the metrics

just dened will be denoted by F(+A) and F(

r;+A ), respectively.

In the present setting the well-known Perron-Ruelle-Frobenius theo-rem reads as follows.

Perron-Ruelle-Frobenius Theorem. Let f 2 F(+A) be a real-

valued function.

(a) The Ruelle operator Lf : F(+A) ! F(

+A) has a simple eigen-

value = ePr(f), where Pr(f) is the topological pressure of f , and a

strictly positive eigenfunction h 2 F(+A).

(b) spec(Lf) n fg is contained in a disk of radius strictly less than

.(c) There exists a unique probability measure = f on +

A such thatZLf (g) d =

Zg d

for all g 2 F(+A).

1Its statement is slightly modied to suit the present context.


The measure f is the so calledGibbs measure related to the potentialf ([R1], [Si]).

The complex case is more complicated. It was estabslished by Polli-cott [Po] that for any f = u+ iv 2 F(

+A) the spectral radius of Lf as

an operator on F(+A) is not greater than ePr(u). Moreover, if Lf has

an eigenvalue with jj = ePr(u), then is simple and unique and therest of the spectrum is contained in a disk of strictly smaller radius. If

Lf has no eigenvalues with jj = ePr(u), then the whole spectrum of

Lf is contained in a disk of radius less than ePr(u).The following result was obtained by using a modication of the

technique developed by Dolgopyat [D] in order to prove a similar resultin the case of Anosov ows on compact manifolds2, in particluar forgeodesic ows on surfaces of negative curvature.

Theorem 2. ([St2]) There exist constant c0 < hT and 2 (0; 1)such that for Re(s) c0 and jIm(s)j >> 0 the spectral radius of the

Ruelle operator Ls does not exceed .

Using the above theorem and applying the argument of Pollicottand Sharp [PoS] in the case of geodesic ows on compact surfaces ofnegative curvature, one derives that the zeta function

(s) =Y

(1 es`( ))1

of the billiard ow t has an analytic continuation in a half-planeRe(s) > c0 for some c0 < hT except for a simple pole at s = hT . More-over, folowing [PoS] again, one derives that there exists c 2 (0; hT ) suchthat

() = #f : `( ) g = li(ehT ) +O(ec)

as ! 1, where li(x) =R x

2du= logu. The latter is a much stronger

result than the standard Prime Orbit Theorem for open planar billiards

in [Mor] (cf. [St1] for the higher dimensional case) derived by meansof a result of Parry and Pollicott [PP1].

3. Exponential decay of correlations

In this section we continue to consider the case when K is an obstacleof the form (1) in IR

2, and we also use most of the notation from Sect.2.

2In fact the primary aim of Dolgopyat was to establish exponential decay of

correlations for such ows. See Sect.3 for more information.


Let F 2 F() for some > 0. Denote by ~F the function on rA

such that ~F r = r F . There exists = () 2 (0; 1) such that~F 2 F(

rA). Let ~ be the Gibbs measure related to ~F with respect to

the suspended ow ~r, and let P = Pr~r( ~F ) be the topological pressure

of ~F with respect to ~r. A function ~f 2 F(+A) related to ~F is dened

by

~f() =

Z r()

0

~F (; s) ds :

Let ~ be the Gibbs measure on A determined by the function ~f Pr.Then (cf. [PP2])

d~(; s) =1

(r)d~()ds ; where (~g) =

ZA

~g() d~() :

Moreover, we have Pr~( ~f Pr) = 0: The Gibbs measures ~ and ~give rise to measures and on R and \ U , respectively, via the

conjugacies r and . If f is the function on \ U such that ~f

= f , then f(x) =

Z (x)

0

F (s(x)) ds, so f 2 F( \ U). The

measures and are called the Gibbs measures related to F and fP ,respectively. It follows from above that Prt(F ) = P and Pr(fP) =0.

Given a a Holder continuous potential F on and arbitrary A;B 2

F(), the correlation function of A and B is dened by

A;B(t) =

Z

A(x)B(t(x)) dF (x)

Z

A(x) dF (x)

Z

B(x) dF (x)

:

It is an important problem in smooth ergodic theory (and also in var-

ious areas in physics) to know whether such a function decyas expo-nentially fast as t!1.Using Theorem 2 above and the technique developed by Dolgopyat

[D], one immediately gets the following.

Theorem 3. Let F be a Holder continuous function on in S()and let F be the Gibbs measure determined by F on . For every

> 0 there exist constants C = C() > 0 and c = c() > 0 such that

jA;B(t)j CectkAk kBk

for any two functions A;B 2 F().

Here khk denotes the Holder constant of h 2 F().


We refer the reader to the recent survey article of Baladi [Ba] forgeneral information and historical remarks on decay of correlations.

Amongst the most recent achievements one should mention the impor-tant articles of Liverani [L], Young [Y], Chernov [Ch1] and Dolgopyat[D] answering long standing questions. It apppears that for billiardsthe only results of this type that have been known so far concern the

corresponding discrete dynamical system (generated by the billiard ballmap from boundary to boundary). To my knowledge, these results are:the subexponential decay of correlations established for a very largeclass of dispersing billiards by Bunimovich, Sinai and Chernov [BSC]

and the exponential decay of correlations for some classes of dispersingbilliards in the plane and on the two-dimensional torus established byYoung [Y] and Chernov [Ch2] as consequences of their more generalarguments. Theorem 3 above describes a non-trivial class of billiard(broken geodesic) ows with exponential decay of correlations for any

Holder continuous potential.

References

[Ba] V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys.

18 (1998), 255-292.

[B] R. Bowen, Symbolic dynamics for hyperbolic ows, Amer. J. Math. 95 (1973),

429-460.

[BSC] L. Bunimovich, Ya. Sinai,and N. Chernov, Statistical properties of two-

dimensional hyperbolic billiards, Russ. Math. Surveys 46 (1991), 47-106.

[Ch1] N. Chernov, Markov approximations and decay of correlations for Anosov

ows, Ann. of Math. 147 (1998), 269-324.

[Ch2] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94

(1999),

[D] D. Dolgopyat, On decay of correlations in Anosov ows, Ann. of Math. 147

(1998), 357-390.

[G] Gerard C.: Asymptotique des poles de la matrice de scattering pour deux

obstacles strictement convexes. Bull. de S.M.F., Memoire n 31, 116 (1988).

[GM] V.Guillemin and R.Melrose, The Poisson sumation formula for manifolds

with boundary, Adv. in Math. 32 (1979), 204-232.

[I1] M.Ikawa, On the poles of the scattering matrix for two strictly convex obstacles,

J. math. Koyto Univ. 23 (1983), 127-194.

[I2] M. Ikawa, Decay of solutions of the wave equation in the exterior of several

strictly convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146.

[I3] M. Ikawa, On the distribution of poles of the scattering matrix for several

convex bodies, pp. 210-225 in Lecture Notes in Math., vol. 1450, Springer,

Berlin, 1990.

[I4] M. Ikawa, Singular perturbations of symbolic ows and poles of the zeta func-

tions, Osaka J. Math. 27 (1990), 281-300; Addendum: Osaka J. Math. 29

(1992), 161-174.

[KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical

Systems, Cambridge Univ. Press, Cambridge 1995.


[LP] P. Lax, R. Phillips, Scattering Theory, Revised Ed., Academic Press, London,

1989.

[L] C. Liverani, Decay of correlations, Ann. of Math. 142 (1995), 239-301.

[M] R. Melrose: Geometric Scattering Theory, Cambridge Univ. Press, 1994

[MS] R. Melrose, J. Sjostrand: Singularities in boundary value problems. Comm.

Pure Appl. Math. 31 (1978), 593-617; 35 (1982),129-168.

[Mor] T. Morita, The symbolic representation of billiards without boundary con-

dition, Trans. Amer. Math. Soc. 325 (1991), 819-828.

[N] F. Naud, Analytic continuation of the dynamical zeta function under a dio-

phantine condition, Preprint 2000.

[PP1] W. Parry and M. Pollicott, An analogue of the prime number theorem and

closed orbits of Axiom A ows, Ann. Math. 118 (1983), 573-591.

[PP2] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure

of hyperbolic dynamics, Asterisque 187-188, 1990.

[P] V. Petkov, Analytic singularities of the dynamical zeta function, Nonlinearlity

12 (1999), 1663-1681.

[Po] M. Pollicott, On the rate of mixing of Axiom A ows, Invent. Math. 81 (1985),

413-426.

[PoS] M. Pollicott and R. Sharp, Exponential error terms for growth functions of

negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042.

[Ra] M. Ratner, Markov partitions for Anosov ows on n-dimensional manifolds,

Israel J. Math. 15 (1973), 92-114.

[Ro] C. Robinson, Structural stability of C1 ows, Lect. Notes in Math. 468 (1975),

262-277

[R1] D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading, Mass., 1978.

[R2] D. Ruelle, Resonances for Axiom A ows, Commun. Math. Phys. 125 (1989),

239-262.

[Si] Ya. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 27 (1972),

21-69.

[S] S. Smale, Dierentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967),

747-817.

[St1] L. Stoyanov, Exponential instability and entropy for a class of dispersing bil-

liards, Ergod. Th. & Dynam. Sys. 19 (1999), 201-226.

[St2] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of corre-

lations for open billiard ows, Amer. J. Math., to appear.

[Y] L.-S. Yang, Statistical properties of systems with some hyperbolicity including

certain billiards, Ann. Math. 147 (1998), 585-650.

[Z] M. Zworski, Counting scattering poles, In: Spectral and Scattering Theory, M.

Ikawa Ed., Marcel Dekker, New York 199 ; pp. 301-331.

Luchezar Stoyanov, Department of Mathematics and Statistics, Uni-

versity of Western Australia, Perth WA 6709, Australia


ON THE BANACH-ISOMORPHIC CLASSIFICATION

OF Lp SPACES OF HYPERFINITE SEMIFINITE VON

NEUMANN ALGEBRAS

F. A. SUKOCHEV

Abstract. We present a survey of recent results in the Banachspace classication of non-commutative Lp-spaces.

An important role in Banach space theory has always been playedby the problem of classifying Banach spaces. This problem has manyfacets. In our present setting we address this problem by looking at aBanach space as a linear topological space. The natural maps then arecontinuous linear operators and we look for invariants under isomor-phism (=bicontinuous one-to-one linear operator).

In general, the development of Banach space theory has clearly shownthat there is no hope left for a complete structural theory of Banachspaces, although one can still hope to have such a theory in somespecial cases. Our objective in the present talk is to describe recentresults in this direction in the special case of non-commutative Lp-spaces associated with seminite von Neumann algebras.

To place this work in its proper context we brie y review its origins,beginning with the work of both the mathematicians mentioned so far:Banach and von Neumann.

Many fruitful directions in Banach space theory emerged from thefamous book [B] by Banach, and the date of appearance of the Frenchedition of this book (1932) is usually regarded as the date of birth ofthe theory itself. The nal chapter (XII) of this book discusses in depththe problems of comparison between the elements of the two familiesof Banach spaces (perhaps, the most important families of classicalBanach spaces): the spaces lp and Lp = Lp(0; 1), 1 p < 1. Recallthat lp is the space of all innite complex-valued sequences a = (an)1

n=1,

such that

kaklp := (

1Xn=1

janjp)1=p <1;

Research supported by the Australian Research Council.213

214 F. A. SUKOCHEV

and Lp = Lp(0; 1) is the space of all (equivalence classes of) Lebesguemeasurable functions f on (0; 1) such that

kfkLp:= (

Z1

0

jf(t)jpdt)1=p <1:

The result which is of direct relevance to our present theme can beformulated as follows.

Theorem 1 [B], (1932) There exists an isomorphic embedding ofthe space Lp into lq if and only if p = q = 2.

In other words, these two families are pairwise non-isomorphic.In contrast to Banach's book, the paper of J. von Neumann [N] is

almost completely unknown, even to experts. It appeared in 1937 (veyears after Banach's book), in an obscure Russian journal, which ceasedto exist almost immediately after its rst volume was printed. From thepresent point of view, the theory of non-commutative Lp-spaces beganfrom this paper. Let me brie y describe (a version of) von Neumann'sconstruction of Lp-spaces associated with the von Neumann algebraMn of all nn complex matrices. As a linear space it is identied withMn. Given the matrix A = (aij)

ni;j=1

2 Mn, let jAj = (AA)1=2. Fixing

the standard trace Tr on Mn, we set

kAkp := Tr(jAjp)1=p; 1 p <1:

It is established in [N] that k kp is a norm on Mn and it is customaryto denote the space (Mn; k kp) by Cn

p . In the modern terminology, thespace Cn

p is a non-commutative Lp-space associated with von Neumann

algebra (Mn; T r).It seems clear from [N] that von Neumann was well aquainted with

Banach's book and after having constructed the n2-dimensional spacematrix space Cn

p , he remarks that another natural way to metrize the

n2-dimensional linear space Mn is to identify standard matrix unitseij; 1 i; j n with the rst n2 coordinate vectors of the space

lp, in other words to convert Mn into the n2-dimensional space ln2

p .

This leads to an immediate problem: whether these two n2-dimensional

spaces, Cnp (non-commutative) and ln

2

p (commutative), coincide. In [N],

von Neumann easily established that the spaces Cnp and ln

2

p are non-isometric.

From the viewpoint adopted in our present setting, the natural ques-

tion would be whether the Banach-Mazur distance between Cnp and ln

2

p

ON THE BANACH-ISOMORPHIC CLASSIFICATION OF Lp SPACES ... 215

is uniformly bounded. Recall that the Banach-Mazur distance d(X; Y )between Banach spaces X and Y is

d(X; Y ) := inffkTk kT1k j T : X ! Y isomorphismg;

where we adopt the convention inf ; = +1. This question was an-swered only 30 years later, in the negative, by McCarthy (see [M] andalso comments and additional references in Pisier's paper [P]). Beforeformulating McCarthy's result, recall rst that the innite-dimensionalanalogues of the spaces Cn

p are Schatten-von Neumann ideals Cp; wemay also refer to them as to non-commutative Lp-spaces associatedwith von Neumann algebra B(H) of all bounded linear operators onthe Hilbert space H equipped with the standard trace Tr. Recall thata compact operator A 2 B(H) belongs to Cp if and only if

kAkCp:= Tr(jAjp)1=p <1:

Theorem 2 [M], (1967) There exists a constant C > 0 such thatfor any n 2 N and n2-dimensional subspace X of Lp we have

d(Cnp ; X) > Cn

1

3 j 1p 1

2 j:

The following consequence is straightforward.

Corollary 3 [M], (1967) There exists an isomorphic embedding of

the space Cp into Lp if and only if p = 2.

The converse to the result of Corollary 3 was obtained by Arazy andLindenstrauss [AL].

Theorem 4 [AL], (1975) There exists an isomorphic embeddingof the space Lp into Cp if and only if p = 2.

216 F. A. SUKOCHEV

One by-product of the Arazy-Lindenstrauss arguments was an iden-tication of yet another member of the Lp-family, the spaces Sp; 1 p <1. The denition is very simple

Sp := (1n=1

Cnp )p;

i.e. each element x 2 Sp is represented by an innite sequence (xn)1n=1

with xn 2 Cnp and

kxkSp:= (

1Xn=1

kxnkCnp)1=p:

This space can be easily viewed as a subspace of Cp, in our terminologywe may say that Sp is the Lp-space associated with the von Neumannalgebra (1n=1Mn) (von Neumann subalgebra of B(H)) equipped withthe standard trace Tr.

Theorem 5 [AL], (1975) There exists an isomorphic embedding

of the space Cp into Sp if and only if p = 2.

Thus development from 1932 till 1975 has clearly shown that thefollowing four families of innite-dimensional separable Lp-spaces arenon-isomorphic:

(a) The Lp-spaces associated with the von Neumann algebra l1 =L1(N) with the trace given by counting measure on N , i.e. the spaceslp;

(b) The Lp-spaces associated with the von Neumann algebra L1 =L1(0; 1) with trace given by Lebesgue measure on (0; 1), i.e. the spacesLp;

(c) The Lp-spaces associated with the von Neumann algebra (1n=1Mn)with trace Tr, i.e. the spaces Sp;

(d) The Lp-spaces associated with the von Neumann algebra B(H)with trace Tr, i.e. the spaces Cp.


Let us now overview the situation. Let M be an innite dimensionalseminite von Neumann algebra acting on a separable Hilbert space,let be a normal faithful seminite trace on M, and let Lp(M; ); 1 p <1 be the Banach space of all -measuarble operators A aliatedwith M such that (jAjp) < 1 with the norm kAkp := ((jAjp))1=p,

where jAj = (AA)1=2 (see e.g. [FK]). It is quite natural to ask thefollowing question:What is the linear-topological classication of the spaces Lp(M; );

p 6= 2?It is natural to subdivide the above question to further subcategories

accordingly to various classication schemes for von Neumann algebras.Tne following two results (obtained jointly with V. Chilin) are a simpleapplication of Pe lczynski's decomposition method.

Proposition 6 [SC1], (1988) Let M be a commutative von Neu-

mann algebra with a normal faithful seminite trace . Then Lp(M; ),p 6= 2 is Banach isomorphic to one of the spaces lp or Lp.

Further, recall that a von Neumann algebra M is called atomic if ev-ery nonzero projection in M majorizes a nonzero minimal projection.

Proposition 7 [SC1], (1988) Let M be an atomic von Neumannalgebra with a normal faithful seminite trace . Then Lp(M; ); p 6= 2is Banach isomorphic to one of the spaces lp, Sp or Cp.

The next logical step is the description of Lp-spaces associated withvon Neumann algebras of type I with separable predual. It is well-known that such an algebra can be represented as a countable l1-directsum of von Neumann algebras of the typeAn Mn andAB(H), whereAn, A are commutative von Neumann algebras with separable predu-als. One can easily see that the following Banach spaces are actuallyLp-spaces associated with von Neumann algebras of type I: the directsums Lp Sp and Lp Cp, the Lebesgue-Bochner spaces Lp(Sp) andLp(Cp), as well as the space Cp Lp(Sp). The following result (an-nounced in [SC2]) actually shows that these examples actually exhaustthe list of such spaces.

218 F. A. SUKOCHEV

Theorem 8 [SC2], (1990) LetM be a type I von Neumann algebrawith a normal faithful seminite trace . Then Lp(M; ); p 6= 2 is

Banach isomorphic to one of the following spaces

lp; Lp; Sp; Cp; Sp Lp; Lp(Sp); Cp Lp; Lp(Cp); Cp Lp(Sp):

Moreover, the spaces

lp; Lp; Sp; Cp; Sp Lp

are pairwise Banach non-isomorphic and non-isomorphic to any of thefour remaining spaces

Lp(Sp); Cp Lp; Lp(Cp); Cp Lp(Sp):

Thus, the number of distinct Lp-families has been raised from 4 to5. However the question whether the remaining 4 spaces are pairwisedistinct proved to be very hard. The following result proved to be oneof the necessary ingredients.

Theorem 9 [S1], (1996) Let N be a nite von Neumann algebrawith nite, normal, faithful trace 1, let M be an innite von Neu-mann algebra with seminite, normal, faithful trace 2. Then for p > 2there is no Banach isomorphic embedding of Cp into Lp(N ; 1), whenceLp(N ; 1) and Lp(M; 2) are non-isomorphic for all p 2 (1;1); p 6= 2.

This result was subsequently used (together with other methods) inthe rst part of the following theorem, which raises the number of dis-tinct families of re exive Lp-families to 8. The second part of Theorem10 below yields a complete linear-topological classication of the pred-uals to von Neumann algebras of type I. The proof of the second partis based on the study of the Dunford-Pettis property in von Neumannalgebras and its preduals.

Theorem 10 [S2], (2000) Let M be an innite-dimensional vonNeumann algebra of type I acting in a separable Hilbert space H,let be a normal faithful seminite trace on M, let Lp(M; ); p 2

[1;1); p 6= 2 be the Lp-space associated with M. Then


(a) the space Lp(M; ) is isomorphic to one of the following ninespaces:

lp; Lp; Sp; Cp; Sp Lp; Lp(Sp); Cp Lp; Lp(Cp); Cp Lp(Sp); (L)

and if (E; F ) is a pair of distinct spaces from (L), which does notcoincide with the pair (Lp(Cp); CpLp(Sp)), then E is not isomorphicto F ;(b) all nine spaces from (L) are pairwise non-isomorphic, provided

p = 1.

However the question whether the Lp(Cp) and Cp Lp(Sp) are Ba-nach distinct for 1 < p < 1 remained unresolved. The technique of[S1] for dealing with embeddings of Cp for p > 2 has not been su-cient. The breakthrough has come with the following joint result of theauthor with U. Haagerup and H. Rosenthal.

Theorem 11 [HRS1], [HRS2] (2000) Let N be a nite von Neu-mann algebra with nite, normal, faithful trace 1, letM be an innitevon Neumann algebra with seminite, normal, faithful trace 2. Thenfor 1 p < 2 there is no Banach isomorphic embedding of Cp intoLp(N ; 1), whence Lp(N ; 1) and Lp(M; 2) are non-isomorphic for all

p 2 [1;1); p 6= 2.

Theorem 11 is crucial in the proof of the following theorem whichcompletes the isomorphic classication of separable Lp-spaces associ-ated with von Neumann algebras of type I for p > 1; it yields morethan just the non-isomorphism of Lp(Cp) and CpLp(Sp) and strength-ens the second part of Theorem 10.

Theorem 12 [HRS1], [HRS2] (2000) Let N be a nite vonNeumann algebra with a xed faithful normal tracial state on Nand 1 p < 2. Then Lp(Cp) is not isomorphic to a subspace of

Cp Lp(N ; ).

Thus, all nine spaces listed in L (see Theorem 10) are pairwise non-isomorphic also for 1 p 6= 2 <1.

Much more follows via application of (a strengthened version of)Theorem 11. Let M be a hypernite (i.e., M is a weak closure ofa union of an increasing sequence of nite dimensional von Neumann

220 F. A. SUKOCHEV

algebras) seminite von Neumann algebra. In this setting, M can bedecomposed as MI MII1 MII1, where MI , MII1, MII1 are hy-pernite von Neumann algebras of types I, II1, and II1 respectively.Further, using disintegration and deep results of A. Connes [C], thealgebras MII1 (respectively, MII1) can be realized as a countable l1-direct sum of von Neumann algebras of the form AB, where A is asabove and B the unique hypernite factor R of type II1 (respectively,the unique hypernite factor R0;1 = R B(H) of type II1). Againvia Pe lczynski's decomposition method (and results of A. Connes) wearrive at the following classication result.

Proposition 13 [HRS1], [HRS2] (2000) If M is a hyperniteseminite von Neumann algebra with a normal faithful seminite trace

and 1 p <1, then Lp(M; ) is isomorphic to one of the followingthirteen spaces:

lp; Lp; Sp; Cp; Sp Lp; Lp(Sp); Cp Lp; Lp(Cp); Cp Lp(Sp) ;

Lp(R); Cp Lp(R); Lp(R) Lp(Cp); Lp(R0;1):

However, the question whether all spaces listed above are pairwisenon-isomorphic is much harder. It required the full strength of Theo-rem 11 combined with very recent results of M. Junge [J].

Theorem 14 [HRS1], [HRS2] (2000) Let N be a nite von Neu-mann algebra with a xed faithful normal tracial state and 1 p < 2.Then Lp(R0;1) is not isomorphic to a subspace of Lp(N ) Lp(Cp).

Theorem 15 [HRS1], [HRS2] (2000) IfM is a hypernite semi-

nite von Neumann algebra with a normal faithful seminite trace and1 p < 1, then Lp(M; ) is Banach isomorphic to precisely one ofthe spaces listed in Proposition 13.


References

[AL] J. Arazy and J. Lindenstrauss, Some linear topological properties of the

spaces Cp of operators on Hilbert spaces, Compositio Math. 30 (1975),81-111.

[B] S. Banach, Theorie des operations lineaires, Warszawa (1932).[C] A Connes, Classication of injective factors, Ann. of Math. (2) 104

(1976), 73-115.[FK] T. Fack and H. Kosaki,Generalized s-numbers of -measurable operators,

Pacic J. Math. 123 (1986), 269-300.[HRS1] U. Haagerup, H.P. Rosenthal and F.A. Sukochev, On the Banach-

isomorphic classication of Lp spaces of hypernite von Neumann al-

gebras, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), 691-695.[HRS2] U. Haagerup, H.P. Rosenthal and F.A. Sukochev, Banach embedding

properties of non-commutative Lp spaces, to appear.[J] M. Junge, Non-commutative Poisson process, to appear.[M] C. McCarthy, Cp, Israel J. Math. 5 (1967), 249-271.[N] J. von Neumann, Some matrix-inequalities and metrization of matric-

space, Rev. Tomsk Univ. 1 (1937), 286-300.[P] G. Pisier, Some results on Banach spaces without local unconditional

structure, Compositio Math. 37 (1978), 3-19.[SC1] F. Sukochev and V. Chilin, Isomorphic classication of separable non-

commutative Lp-spaces on atomic von Neumann algebras, Dokl. Akad.Nauk. UzSSR (1) (1988), 9-11 (Russian).

[SC2] F. Sukochev and V. Chilin, Symmetric spaces on seminite von Neu-

mann algebras, Soviet Math. Dokl. 42 (1991), 97-101.[S1] F. Sukochev, Non-isomorphism of Lp-spaces associated with nite and

innite von Neumann algebras, Proc. Amer. Math. Soc. 124 (1996),1517-1527.

[S2] F. Sukochev, Linear topological classication of separable Lp-spaces asso-

ciated with von Neumann algebras of type I, Israel J. Math. 115 (2000),137-156.

Department of Mathematics and Statistics, School of Informatics

and Engineering, The Flinders University of South Australia, GPO

Box 2100, Adelaide, SA 5001, Australia


INTRODUCING QUATERNIONIC GERBES.

FINLAY THOMPSON

Abstract. The notion of a quaternionic gerbe is presented as

a new way of bundling algebraic structures over a four manifold.

The structure groupoid of this bration is described in some detail.

The Euclidean conformal group R+SO(4) appears naturally as a

(non-commutative) monoidal structure on this groupoid. Using

this monoidal structure we indicate the existence of a canonical

quaternionic gerbe associated to a conformal structure on a four

manifold.

It is natural to think that quaternionic algebra and four dimensionalgeometry should be closely linked. Certainly complex algebra and anal-ysis provide indispensable tools for exploring two dimensional Riema-niann geometry.However, despite many attempts, quaternionic algebra has not been

usefully applied to the dierential geometry of four manifolds.1 Themost commonly held view is that quaternionic algebra is too rigid tobe useful in studying four manifolds. It is generally assumed that thenatural setting for quaternionic dierential geometry is hyperKahler orhypercomplex. [10]The purpose of this talk/article is to present the notion of a quater-

nionic gerbe, and to demonstrate that they appear naturally as aquaternionic algebraic structure on four manifolds. This work appearsas part of an eort to realize the goal of \doing four dimensional ge-ometry and topology with quaternionic algebra."Although quaternionic structures are dened [8] for all 4n dimen-

sional manifolds, the basic structures and diculties are already presentin only four dimensions. The notion of \quaternionic curve" has beenequated with that of a \self dual conformal" structure.[2] Note thateven this class of manifolds is strictly larger than the hyperKahler man-ifolds. Here we restrict our attention to smooth oriented four manifolds,including hyperKahler and self dual conformal manifolds.It is proposed that a \quaternionic structure" on a four manifold is

essentially a Euclidean conformal structure. This compares favourably

1Except perhaps Atiyah's notes on solutions to the Yang-Mills equations on the

four sphere, [1]

222

QUATERNIONIC GERBES 223

with the two dimensional case where xing a complex structure is equiv-alent to xing a conformal structure.

1. The Problem.

The most obvious denition of a quaternionic structure on a fourmanifold M requires the existence of three integrable complex struc-tures, I; J;K 2 End(TM), such that,

I2 = J2 = K2 = IJK = 1:

In terms of holonomy, this implies a reduction of the frame bundle'sstructure group to H , the group of unit quaternions. Note that H =GL(1; H ), which generalises the complex case in an obvious way.The problem comes when we consider Berger's list [2] of holonomy

groups for Riemannian manifolds:

real O(n); SO(n);

complex U(n); SU(n);

quaternionic Sp(n) Sp(1); Sp(n)

exceptional G2; Spin(7)

The quaternionic-Kahler series Sp(n) Sp(1) is clearly relatedto quaternionic algebraic structures, however it is not contained inGL(n; H ). Does this mean that quaternionic-Kahler manifolds are notquaternionic? In reaction to this apparent contradiction, S. Salamondened quaternionic manifolds (see [8]) as having a holonomy reductionto GL(n; H ) Sp(1). Then quaternionic-Kahler implies quaternionic,as you might expect.There are two interesting low dimensional coincidences in Berger's

list. The rst U(1) = SO(2) tells us the complex Kahler curves are sim-ply Riemannian surfaces. Moreover, because xing a conformal struc-ture on a Riemannian surface corresponds to a holonomy reduction toR+SO(2), and R

+SO(2) = C = GL(1; C ), geometrically speaking,

xing a conformal structure is equivalent to xing a complex structureon two dimensional manifolds.The second coincidence Sp(1) Sp(1) = SO(4) seems similar, with

\complex" replaced by \quaternionic". We also have,

GL(1; H ) Sp(1) = H Sp(1) = R

+Sp(1) Sp(1) = R+SO(4):

The implication is that xing a quaternionic structure is equivalent toxing a conformal structure on four manifolds. But what exactly dowe mean by a \quaternionic structure"?

224 FINLAY THOMPSON

1.1. The Impasse. The algebra of quaternions appears naturally asthe generator of the Brauer group of the reals, Br(R) = fR; H g. Thegroup structure is given by the tensor product, moduli \matrix" alge-bra. It is not necessary to go into the details of the Brauer group here,instead we simply note that H generates Br(R) because of the followingalgebra isomorphism,

: H R H ' EndR(H );

where (p q) : v 7! p v q for any p; q; v 2 H . Note that we haveused both the left and the right module structures in dening .The Euclidean conformal group R+SO(4) has a natural quaternionic

presentation using the isomorphism . Let i : H H ! H H bethe canonical map associated to the tensor product. The image ofthe multiplicative group H

H under these maps is precisely the

conformal group. We have the following exact sequence of groups,

1 ! R ! H

H

i! R+SO(4) ! 1

where R ! H H

acts as r 7! (r; r1).

Proposition . The Euclidean conformal group in four dimensions ap-pears in a natural and quaternionic way as,

R+SO(4) = fp q = i(p; q) j p; q 2 H

g:

Proof: The Euclidean norm of x 2 H is jxj2 = x x. Let p q =i(p; q). Then,

jp q(x)j = jp x qj =pp x q q x p

= jpjjqjjxj = jxj

The above presentation of the conformal group, using the isomor-phism : H H ! End(H ), places equal emphasis on the left andright module structures of H on itself. Indeed, the isomorphism isthe H -bimodule structure on H ! It is then natural to consider the fullbimodule structure as the important structure that we want to inte-grate over four manifolds. However this way is blocked.

Proposition . The automorphisms of H considered as a H -bimoduleare all scale multiples of the identity,

AutH-bimodule(H ) = R

+ Id :

Proof: This is simply a consequence of Shur's lemma applied tothe representations of M4(R).


Thus a four manifold with an integrable H -bimodule structure de-ned on the tangent bundle has holonomy contained in R+ Id, whichforces the manifold to be ane.So we reach an impasse:

The Euclidean conformal group has a natural quaternionic pre-sentation using the H -bimodule structure on H .

The automorphisms of H as a H -bimodule are simply scale mul-tiples of the identity.

We show that quaternionic gerbes provide a way of going past thisimpasse.

1.2. The Suggested Solution. The central idea is to use a moresophisticated way of \gluing" local data together.Although H has very few automorphisms when considered as an H -

bimodule, it does have an interesting group of automorphisms as anR-algebra,

Aut(H ) = Inn(H ) = SO(3):

Note that all the automorphisms are inner. The suggestion is to con-sider the set of linear maps in End(H ) that commute with the H -bimodule structure, up to inner automorphisms. Such a map: f : H !H is required to satisfy the equation,

f(p v q) = (p)f(v)(q);

where ; are inner automorphisms associated to f . It turns out thatthe set of all such generalised automorphisms is precisely the Euclideanconformal group, i(H H

) = R+SO(4).

The idea of allowing the two actions to commute up to an automor-phism is natural in category theory. A gerbe is a special kind of sheafof categories and provides a rich enough language to handle the innerautomorphisms coherently.An excellent presentation of the theory of Abelian gerbes has been

presented by Jean-Luc Brylinski in \Loop Spaces, Characteristic Classesand Geometric Quantisation." [5].Nigel Hitchen, studying special Lagrangian sub-manifolds in dimen-

sion three, has also made use of Abelian gerbes. Hitchen's approach[6] stresses the idea that Abelian gerbes certain cohomology classes.Michael Murray has presented [7] Abelian gerbes in a dierent light

as bundle gerbes.However the theory we are looking for is non-Abelian. L. Breen has

dened non-Abelian gerbes [3, 4] for arbitrary Lie groups and hasdeveloped the theory of 2-gerbes. Breen's work applies quite well toour present situation.

226 FINLAY THOMPSON

2. The Structure Groupoid

Just as there is a structure group associated to a principal bundle, agerbe has an associated groupoid. In this section we will describe thestructure groupoid associated to a quaternionic gerbe.Following Breen [3], we can associate to any crossed module a groupoid.

We start from the crossed module dened by,

: H ! SO(3):

where is the natural map onto the inner automorphisms, (p) =p p1, and SO(3) acts on H

as automorphisms.Recently R. Brown and collaborators have been relating groupoids

and crossed modules to algebraic topology. (see [12])

Denition . The quaternionic structure groupoid H is dened:

objects of H are elements of SO(3), any element p 2 H

is considered a morphism p 2 H(; )

p : !

whenever (p) = . For any two morphisms p : ! and q : ! , the compositionis given by the map,

q p = qp : ! = (qp) = (q)(p):

It is easy to check that all of the axioms of a small category aresatised. In addition, because H is a division algebra, all the morphismsare invertible so that H is a groupoid.Note that the set of all morphisms inH consists of pairs (p; ) 2 H

SO(3). We will abuse notation a little and say that H = H

SO(3)as sets.Although we have used the left SO(3)-action on itself, we have not

used the group structure on SO(3).

2.1. Tensor product on H. The small category H has a monoidalstructure coming from the group structure on SO(3),

: HH ! H:

We use the tensor product symbol because the central example of amonoidal structure on a category is that of the tensor product in vec-tor spaces. This tensor product is not commutative, however it isassociative.For any ; 2 SO(3),

=


For any two maps p : ! (p) and q : ! (q) we dene p qas,

p q = p[q] : ! (p[q]):

To see that is well dened we should check that tensor product ofthe ranges is the range of the tensor product of the maps,

(p[q]) = (p)([q]) = (p)(q)1

= ((p)) ((q))

Note that is simply the semi-direct group structure coming fromthe action of SO(3) on H via inner automorphisms,

(H;) = Ho SO(3):

where (p; ) (q; ) = (p[q]; ).Moreover, this semi-direct product is isomorphic to the Euclidean

conformal group,

Ho SO(3) ' R

+SO(4):

2.2. H -bimodules. We can also represent the groupoid H as a cate-gory of quaternionic bimodules in such a way that the tensor product isreally a tensor product. In order to do this we need to dene carefullywhat we mean by an H -bimodule.

Denition . An H -bimodule is a vector space V with two commutingactions of the quaternions. Or equivalently, a bilinear map,

: H H ! End(V ):

Given an H -bimodule (V; ) we can present the action as,

: H H ! End(V );

by using the universal property of the tensor product. In this formwe see that the H -bimodules are simply the modules over End(H ) =M4(R), the algebra of four by four matrices. Therefore the only sim-ple module is R4 . For our purposes we restrict ourselves to real fourdimensional H -bimodules. The objects of H will be identied with thefour dimensional H -bimodules.Although all such objects are structurally identical, we will distin-

guish between dierent quaternionic structures on the same underlyingvector space.

Proposition . Let V be a H -bimodule in H. Then there is an H -bimodule isomorphism : H ! V .

228 FINLAY THOMPSON

Proof: V is a simple module over H , in two dierent ways. Com-paring these actions we dene for each v 2 V , v 6= 0, a map H ! H :p 7! pv by the rule,

p v = v pv:

The map p 7! pv is an R-algebra automorphism for all v,

v (pq)v = (pq) v = p (q v) = p (v qv)= (p v) qv = v (pvqv)

So we have dened a map V ! Aut(H ) = SO(3). To show surjectivitywe start by xing some v and dene [p] = pv. For any in Aut(H )the tansitivity of SO(3) implies that = for some 2 SO(3). Allautomorphisms are inner, so there is some r 2 H such that = (r) =r r1. Then we observe,

p (v r1) = (v pv) r1 = (v r1) (rpvr1)

= (v r1) [pv] = (v r1) [p]= (v r1) [p]

and we see that v r1 maps onto . It is easy to see that the mapV ! SO(3) bres through the projection V ! P (V ), where P (V ) isthe real projective space of one dimensional subspaces in V . Let e 2 Vbe chosen in the preimage of the identity of SO(3). Then it is clearthat p e = e p for all p 2 H , and,

: H ! V

p! p e

is the isomorphism of H -bimodules.

We see from the above proof that each H -bimodule structure createsan identication of P (V ) with SO(3). Recall that as a smooth manifoldSO(3) = RP

3 = P (V ).We can go in the other direction as well. Let V be a right H -module

and : V ! SO(3) be a H equivariant map,

(xp) = (p)1(x):

Then we dene a left H action on V as,

px = x(x)[p]


The left action commutes with the right action and so V is an H -bimodule,

p(xq) = xq(xq)[p] = xq(q)1(x)[p]

= x(x)[p]q = (px)q

We distinguish the dierent objects in H by using the dierent iden-tications P (V )! SO(3) associated H -bimodule structures. Two ob-jects dier by an element of SO(3).Now the tensor product can actually be represented as a tensor prod-

uct. If V and W are the H -bimodules associated to objects and inH, then the H -bimodule associated to = is V H W .A quaternionic gerbe consists of the structure groupoid bred over

a four manifold. To see how we do that, we need a closer look at thetheory of sheaves of categories.

3. Sheaves of Categories or Stacks.

A gerbe is a special kind of sheaf of categories. Our objective inthis section is to present enough of the general theory so that we canunderstand what is the nature of gerbes, and how they can be useful.We will not present a self contained account here, instead we refer thereader to [5].A presheaf of categories involves the interplay of locally dened \ob-

jects" and \morphisms". A stack2 requires that the objects satisfy adescent property, up to an isomorphism. The concept is is quite ex-ible, but still very precise. The isomorphisms that glue together theobject data must satisfy additional coherence identities.

3.1. Local Homeomorphisms. Instead of working with the categoryof open sets on a manifold X, we work with local homeomorphisms:continuous map f : Y ! X such that,

any y 2 Y has an open neighbourhood U whose image f(U) isopen in X, and,

the restriction of f to U gives a homeomorphism between U andf(U).

Denition . The category of spaces over X, CX, has,

objects are local homeomorphisms to X, f : Y ! X; a morphism g : (f : Y ! X) ! (h : Z ! X) is a local homeo-morphisms, g : Y ! Z such that f = h g.

2For us the terms \stack" and \sheaf of categories" refer to the same concept.

230 FINLAY THOMPSON

An important example to keep in mind is associated to an opencover fUig of X. The canonical projection f : Y =

`i Ui ! X from

the disjoint union onto X is a local homeomorphism.

3.2. Presheaves of Categories. In the same way that a presheaf ofsets is simply a functor from CX to the category of sets, a presheafof categories over X is a functor C from the category of spaces overX, CX , to the (bi)-category of small categories, functors and naturaltransformations. Or, more explicitly,

to every local homeomorphism f : Y ! X we associate a smallcategory,

C(f : Y ! X)

to every arrow of local homeomorphisms k : (Z; g) ! (Y; f) weassociate a functor,

C(k) = k1 : C(f : Y ! X)! C(g : Z ! X);

to every composition k l : (W;h)! (Z; g)! (Y; f) we associatean invertible natural transformation,

k;l : l1k1 ) (kl)1

This data must satisfy the following coherence condition,

m1l1k1k;l! m1(lk)1

??yl;m

??ylk;m

(lm)1k1k;lm! (lkm)1

It would be possible to dene a presheaf of categories with the re-quirement that l1k1 is strictly identical to (kl)1. However that doesnot take advantage of the extra exibility provided. We will see laterhow quaternionic gerbes make use of this exibility.

3.3. Descent for Morphisms. Let C be a presheaf of categories. Wesay that the morphisms satisfy descent if for any two objects A;Bin C(f : Y ! X), the presheaf of sets on Y dened by,

Hom(A;B)(k : Z ! Y ) = Hom(k1(A); k1(B))

is actually a sheaf on Y .We can explain this in terms of objects and maps more directly.

Let V X be a open neighbourhood, and let A;B be objects inC(V ) = C(V ,! X). Now let fUig be a open cover of V . Take acollection of morphisms i : AjUi ! BjUi , where i 2 C(Ui). Themorphisms satisfy descent if,

ijUij = jjUij 8i; j;


implies the existence a unique morphism : A! B in C(V ) such thati = jUi for all i.In the above we have denoted AjUi for the \restriction" of A to Ui.

Of course the restriction is really a functor C(V )! C(V \Ui), and thatfunctor is not necessarily trivial or obvious. However the presentationbecomes much easier to if we make use of these small abuses of thenotation.

3.4. Descent for Objects. The objects satisfy a much more com-plicated descent property, making use of the natural transformationsappearing in the denition.Let C be a presheaf of categories. Let V be any open set in X and

f : Y ! V be any surjective local homeomorphism. The descent datafor any A 2 C(Y ) consists of an isomorphism : p1

2 (A) ! p11 (A) in

C(Y X Y ) such that,

p112 () p1

23 () p131 () = Idp1

1(A)

in H(Y X Y X Y ).3

We say that the objects satisfy descent if every pair (A; ) asabove implies the existence of an object A0 2 C(V ) and an isomorphism : f1(A0)! A in C(Y ) such that the following diagram in C(Y X Y )commutes,

p11 f1(A0)

1

f;p2f;p1! p1

2 f1(A0)

??y

??y

p11 (A)

! p12 (A)

This rather complicated prescription can also be understood in termsof open sets in the normal sense.Let fUig be an open cover of V X. The descent data is equivalent

to a set of local objects Ai 2 C(Ui), and isomorphisms ij : AijUij !AjjUij in C(Ui \ Uj). The isomorphisms are required to satisfy,

ikjUijk = ijjUijk jkjUijk ;in the category over the triple intersection, C(Ui\Uj \Uk). Again notethat we have implicitly used the natural transformations by glossingover the restrictions.

Denition . A stack (or sheaf of categories) on X is a presheaf ofcategories where objects and morphisms satisfy the descent conditionsabove.

3The natural projections Y X Y ! Y are denoted p1 and p2, and the three

natural projections Y X Y X Y ! Y X Y are denoted by p12,p23 and p13.

232 FINLAY THOMPSON

4. Quaternionic Gerbes

Now we rene the notion of a stack to that of a gerbe by imposingthree more conditions:

1. gerbes take values in groupoids, the full sub-category of smallcategories whose morphisms are invertible.

2. gerbes are locally non-empty. This means that there exists asurjective local homeomorphism f : Y ! X such that C(Y ) isnon-empty. We could also state this by saying that there existsan covering fUig of X such that the C(Ui) are all non-empty.

3. gerbes are locally connected. This means that for any twoobjects A;B in C(f : Y ! X), there exists an surjective localhomeomorphism g : Z ! Y such that g1(A) and g1(B) areisomorphic. In terms of covers: if A;B are objects in C(U) forsome U X, then there exists an open covering fUig of U suchthat A jUi is isomorphic to B jUi for all i.

Denition . A gerbe on X is a locally non-empty and locally con-nected sheaf of groupoids on X.

For any group G let GX be the sheaf of G-valued functions on X. Agerbe is said to have band inG if for any object A 2 C(f : Y ! X), thesheaf Aut(A) of automorphisms of A on Y is isomorphic to GY , and theisomorphism : Aut(A)! GY is unique up to an inner automorphismof G.

Denition . Quaternionic Gerbe is a gerbe with band in H .

4.1. Neutral Gerbes. A gerbe G is said to be neutral if there existsa global object, A 2 G(X). Because the automorphism sheaf of Aut(A)is isomorphic to the sheaf of H -valued functions, we can identify thegroupoid G(X) with the groupoid of principal H -bundles,

: G(X)! Tor(Aut(A))

B 7! Isom(B ! A)

Quaternionic gerbes are locally non-empty so we can always ndan object AU 2 G(U) over the open set U . Using that local objectwe can identify G(U) with the groupoid of H -bundles over U . Localnon-emptiness implies that the gerbe is locally neutral.In order to understand this local neutrality, it is helpful to consider

an analogy with the relation between a principal G-bundle and anassociated vector bundle. To any vector bundle we can associate theprincipal bundle of frames. The local neutralisation associated to alocal object AU is sort of \frame" for G over U . The set of all framesfor G forms a local groupoid.


Assuming that U is contractable, all the H -bundles dier by an

automorphisms valued function : U ! SO(3). We dene a H -

bundle associated to AU and by letting AU = AU as a bre bundle.The action of H however is twisted by . Let a 2 AU and let a bethe same element considered in AU . Then for any quaternion p 2 H

,

a p = (a [p]):

If U is not contractable there can be topologically inequivalent H -bundles. Then we can replace the function above with an H -bimoduleM ! U . If AU and BU are two dierent objects in G(U), then there isan H -bimodule M such that,

AH M = B:

The local groupoid H(U) consists of the \frames" of G(U). Notethat because of the local neutrality axiom, all quaternionic gerbes lookthe same locally.

4.2. The Local Groupoid. We describe here the local structure ofH.An object of the local groupoid H(U) is a diagram of the form,

A! Aut(H )

??y

U

where : A ! U is a principle H -bundle and is an H -equivariant

map (xp) = (p)1(x).As we have seen, this data can also be presented in terms of H -

bitorsors.An H -bitorsor is a principle right H -bundle that is also a principle

left H -bundle for a commuting action of H . For any (A; ) 2 H(U),the left H action on A is, px = x(x)[p].The morphisms of H -bitorsors are simply bundle maps that commute

with both the left and right actions.

4.3. Tensor Product on H(U). In terms of bitorsors we can presentthe product structure on H(U) by using the quaternionic tensor prod-uct.For any A;B 2 H(U),

AH B = AR B=

where xp y x py.

234 FINLAY THOMPSON

Assuming U is contractable and by xing a coordinate basis, we geta canonical trivialisation of the tangent bundle, TU = U H . In thisway TU can be considered as an object in H(U).Relative to this xed object, all the others are given by SO(3) valued

functions on U , the morphisms are given by H valued functions.Over U H the local groupoid consists of sections C1(U;H). How-

ever the strength of this approach is in terms of the global structure.A global quaternionic gerbe is given in terms of \transition functions".

4.4. Transition Functions or Bitorsor Cocyle. The transition func-tions for a quaternionic gerbe are given in terms of H -bitorsors. Maybewe should say \transition bitorsors".Let G be a quaternionic gerbe on X and let fUig be a good cover.4

Choose Ai 2 G(Ui). Then Aut(Ai) is isomorphic to the sheaf of H -valued functions. Using Ai we have the following local neutralisation,

i : G(Ui)! Tor(Aut(Ai))

B 7! Isom(B ! Ai)

On any intersection Uij = Ui \ Uj we can dene,

Eij = Isom(Aj jUij ; Ai jUij );

The Eij are H -bitorsors and are the transition functions. The twoH -actions are given by the composition of an isomorphism with auto-morphisms ofAijUij and AjjU ij, which are each isomorphic to H -valuedfunctions. Note that the isomorphisms H ' Aut(Ai) are unique up toan automorphism. To be really careful we should take care of thoseautomorphisms as well, however that will work will be presented in acomprehensive way later.The H -bitorsors Eij need to be compared over triple intersections.

The natural transformations in the denition of a stack give us thefollowing morphisms as extra data:

ijk : Eij H Ejk ! Eik;

These morphisms live in H(Uijk) and must satisfy the following co-herence condition on four intersections,

Eij H Ejk H Ekl ijkId! Eik H Ekl

Id jkl

??y

??y ikl

Eij H Ejl ijl! Eil

4All intersections Ui \ Uj are contractable.


The pair (Eij; ijk) is called a quaternionic bitorsor cocycle onX.Of course the above description of a particular quaternionic gerbe

depends on the choice of Ai 2 G(Ui). We can measure the dependenceon those choices with a coboundary.

4.5. Coboundary. Let Bi be a dierent choice of local objects and(Fij; ijk) be the associated bitorsor cocyle.Let Mi 2 H(Ui) be dened by,

Bi = Ai H Mi

The pair (Mi; ij) is a coboundary relating (Fij; ijk) to (Eij; ijk)if ij is a map in H(Uij),

ij : Fij !M

i H Eij H Mj

such that as morphisms in H(Uijk),

ik ijk = ijk (ij jk)

We can present this equation with a commutative diagram,

Fij H Fjkijjk! M

i H Eij H Ejk H Mk

ijk

??y

??yId ijkId

Fik !ik

M

i H Eik H Mk

It was demonstrated in [11] that coboundaries dene an equivalencerelation on the set of quaternionic bitorsor cocycles. Moreover, it ispossible to construct a quaternionic gerbe from a given cocyle, andthat gerbe will be isomorphic to any gerbe constructed from a cocylefrom the same equivalence class.Although we have used the terminology of cohomology at present

there is no actual theory of H -valued cohomology. We use the termi-nology because it is convenient, and perhaps to be a little optimistic.

5. Conformal Four Manifolds

A conformal structure on a four manifold is a reduction of the framebundle to R+SO(4). As we saw at the beginning, the Euclidean con-formal group can be presented using the groupoid H with its tensorproduct acting as the group structure.We will indicate here brie y how to construct a quaternionic bitorsor

cocycle from a given a conformal structure on a four manifold X. Thepresentation here is very sketchy and a more detailed presentation isbeing prepared.

236 FINLAY THOMPSON

We can choose charts f i : Ui ! H g that are compatible with theconformal structure, i.e.,

@( i 1j ) 2 i(H H

);

where @(f) is the Jacobian matrix of f considered as an element ofH H . Therefore there are H -valued functions xij and yij on Uij suchthat,

@( i 1j ) = xij yji:

Over each chart Ui the tangent bundle has a canonical H -bitorsorstructure coming from the coordinate . The tangent gerbe cocycleallows us to relate these various H -bitorsor structures.The xij yij can be used to dene Eij by twisting the left and right

H -actions by (xij) and (yij) respectively. In terms of an SO(3) valuedfunction, we can dene Eij relative to TUi with the function (yijxij).Over the triple intersections Uijk it is possible to construct isomor-

phisms ijk : Eij Ejk ! Eik.It can also be shown that the i are coordinate charts compatible

with the conformal structure if and only if the (Eij; ijk) form a quater-nionic gerbe cocyle.

References

[1] M. Atiyah, Geometry on Yang-Mills Fields., Scuola Normale Superiore Pisa,

Pisa, 1979.

[2] Arthur L Besse, Einstein manifolds., Springer-Verlag, Berlin-New York, 1987.

[3] L. Breen, On the classication of 2-gerbes and 2-stacks., Asterisque, (1994),

no. 225.

[4] L. Breen, Tannakian Categories., Proceedings of Symposia in Pure Mathemat-

ics, Volume 55(1994), part 1.

[5] Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quan-

tization., Progress in Mathematics 107, Birkhauser, 1993.

[6] N. Hitchin, Lectures on Special Lagrangian Submanifolds., School on Dieren-

tial Geometry (1999), the Abdus Salam International Centre for Theoretical

Physics.

[7] M. K. Murray, Bundle gerbes., J. London Math. Soc. (2) 54 (1996), no. 2,

403416.

[8] S. Salamon, Dierential geometry of quaternionic manifolds., Annales Scien-

tiques de l'Ecole Normale Superieure. Quatrieme Serie, 19 (1986), no. 1, 31

55.

[9] S. Donaldson and P. Kronheimer, The Geometry of Four Manifolds., Oxford

Mathematical Monographs, Oxford 1990.

[10] D. Joyce, Hypercomplex Algebraic Geometry., The Quarterly Journal of Math-

ematics, Vol 49, no. 194, p129 (1998), Oxford Second Series.

[11] F. Thompson, New Approachs to Quaternionic Algebra and Geometry., PhD.

Thesis, 1999, SISSA, Italia.


[12] R. Brown, Groupoids and Crossed Objects in Algebraic Topology., Ho-

mology, Homotopy and Applications, Vol 1, no. 1, 1-78 (1999),

http://www.emis.de/journals/HHA/.

Finlay Thompson, Victoria University of Wellington, Wellington,

New Zealand


geometric analysis and applications, proc. nat. res. symposium, canberra

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