non-commutative spaces - pennsylvania state universityalgebra c(x) of continuous complex-valued...

Non-Commutative Spaces

John RoePenn State [email protected]

June 19, 2000

AbstractIn this lecture we will describe some examples

of noncommutative spaces in the sense ofConnes.

Noncommutative Spaces and Groupoids

Commutative geometry

Theorem 1. [R. Descartes] Euclidean geometry isthe study of three functions x, y, z on R3.

In other words, one can recover the geometry of thespace R3 by studying functions on it (the coordinates).For another similar example, consider the followingsimple theorem: a compact space X is connected i! thealgebra C(X) of continuous complex-valued functionson it has no non-trivial idempotent (an idempotent inan algebra A is an element a = a2 ! A.)

These results are special cases of a general theorem:the commutative algebra C(X) completely determinesthe compact Hausdor! space X . (The points of X arejust the homomorphisms C(X) " C.) If X is locallycompact we replace C(X) by C0(X), the algebra ofcontinuous complex-valued functions that vanish atinfinity.

Noncommutative Spaces and Groupoids 1

What is noncommutative geometry?

Basic idea of NCG: we want to consider‘spaces’ whose ‘coordinate functions’ do not commute.Motivation comes from the observables of quantumtheory.

Key features of Connes’ NCG are:

• Replace points by functions,

• Matrices,

• Positivity and Hilbert space.

For instance, we might want to consider the‘quotient space’ S1/Z, where Z acts by an irrationalrotation R. Classically this quotient space isstructureless because the Z action is ergodic. Butthe quotient space is naturally represented bya noncommutative C!-algebra, which has a richstructure.


Noncommutative Quotient Spaces

Consider a simple example, of two points identifiedby an equivalence relation.

This corresponds to the matrix algebra M2(C).

More generally consider an etale equivalencerelation E on a compact space X . For example in thecase of the irrational rotation algebra, E = S1#Z andr, s: E " S1 are given by r(x, n) = x, s(x, n) = Rnx.It now makes sense to multiply ‘continuous matricescompactly supported on E ’, getting a noncommutativealgebra CcE .


The irrational rotation algebra

Keep thinking about the example above. Clearlythe algebra is generated by functions on S1 togetherwith an operator V corresponding to the rotation.

But functions on S1 can be expanded in Fourierseries. Using U to denote the generator (i.e. thefunction z) we get

Example The irrational rotation algebra A! isgenerated by two unitaries U and V subject to therelation

UV = e2"i!V U.

This is the algebra corresponding to the transformationgroupoid (see later) S1 ! Z, where Z acts on S1 viathe irrational rotation !.

Note the word ‘unitary’ — We have implicitlyrepresented the algebra on an appropriate Hilbertspace. This allows us to remove the asymmetrybetween U and V ; we complete to a C!-algebra.More about this later.


Quotients and Morita equivalence

Let E be an etale equivalence relation on the locallycompact Hausdor! space X . In order that X/E be a‘good’ (i.e. Hausdor!) space, it is su"cient that E beproper (that is, r and s are proper maps).

Theorem 2. (Rie!el) Let E be a proper etaleequivalence relation on X as above. Then C!(E)is Morita equivalent to the algebra C0(X/E).

(Two C!-algebras are Morita equivalent if they are‘the same up to finite matrices’, for instance C andM2(C) are Morita equivalent. It is a theorem that Aand B are Morita equivalent i! A $ K and B $ K areisomorphic.)

Summary: C!(E) is a noncommutativegeneralization of C0(X/E). But C!(E) can haveinteresting structure even when C0(X/E) is highlydegenerate.


The Powers-Rie!el idempotent

To show the non-triviality of the irrational rotationalgebra A! we will construct an interesting idempotent.Write elements of A! as

!fnV n, where the fn are

functions on the circle R/2"Z (regard U as the functionei#).

Choose [a, b] % [0, 2"] disjoint from its 2"!-translate. Then let f and g", g+ be functions whosegraphs are illustrated below, with g2

" + g2+ = f2 & f ,

V g"V ! = g+, and put

e = g"V ! + f + g+V.

This is an idempotent (exercise!).


Measuring the size of idempotents

The linear functional

# :"

fnV n '" 12"

#f0

is a trace on A!, that is, #(aa#) = #(a#a). We use itto measure the size of idempotents.

Clearly #(1) = 1, #(e) = ! for the Powers-Rie!elidempotent e.

Theorem 3. For any idempotent x ! A! (or in amatrix algebra Mn(A!)) we have #(x) ! Z + !Z.

Why? A geometric perspective comes throughfoliation theory.


Groupoids 101

Definition 1. A groupoid is like a group, except thatcomposition is not defined everywhere.

More precisely: a groupoid is comprised of a set Gof arrows, a set G0 of objects, two maps s, r:G " G0,and an associative composition law which allows one toform g1g2 ! G provided that s(g1) = r(g2). One alsorequires that each object should have an associatedidentity arrow in G0, and that each arrow should havea two-sided inverse.

We will usually consider locally compact topologicalgroupoids (obvious definition). Such a groupoid isetale if r and s are local homeomorphisms, andit is smooth if everything is C$ and r and s aresubimmersions.


Examples of Groupoids

• A topological space X can be considered as agroupoid (with only the identity arrows, i.e. G =G0 = X).

• An equivalence relation on X defines a groupoidwith objects X (there is exactly one arrow betweentwo points of X if they are equivalent, otherwisenone).

• A topological group can be considered as a groupoid(with only one object).

• Suppose that a group ! acts on a space X (onthe right). The transformation groupoid X ! !associated to this action has objects X , and arrowspairs (x, $) with s(x, $) = x$, r(x, $) = x, and

(x, $#) · (x$#, $) = (x, $#$).

• A foliation gives rise to a smooth groupoid (insimple cases, just the groupoid of the associatedequivalence relation).


Groupoids and algebras

Let G be an etale groupoid. The groupoid algebraCc(G) consists of all continuous, compactly supportedfunctions f :G " C, with composition defined by

(f1 ( f2)(g) ="

g1g2=g

f1(g1)f2(g2).

We regard this as an algebra of functions on the‘noncommutative space’ defined by the groupoid.Examples: If G comes from a topological space X ,we get the algebra Cc(X). If G comes from a discretegroup ! we get the usual group algebra.

If G is a smooth groupoid there is also a version ofthis construction — one needs to replace summationby integration, but this can be done canonically usingsmooth densities, cf. the theory of integration on Liegroups.


The Kronecker foliation of a torus

The foliation algebra for this foliation is Moritaequivalent to the irrational rotation algebra.

Each transversal to the foliation gives rise to aprojection in the foliation algebra. The trace of sucha projection turns out to be the transverse measure ofthe transversal.

Thus, the range Z+!Z of dimensions for projectivemodules is revealed as the image of the integration mapH2(T2; Z) " R given by the transverse measure.


Levels of smoothness

We have talked about ‘algebras of functions’. Butwhat kinds of functions?

Theory Type offunction

Noncommutativeversion

Measure Theory Borel Von Neumannalgebra

Topology Continuous C!-algebraDi!erentialTopology

C$ Holomorphicallyclosed sub-algebra of aC!-algebra

AlgebraicGeometry

Rational CG

We obtain the first three by completing the lastwith respect to a suitable topology.


Groupoid C!-algebras

To get C(S1) from the trigonometric polynomials,let the algebra of polynomials act on L2(S1) bymultiplication (= on %2(Z) by convolution) and takethe closure in the operator norm.

Similarly one obtains a C!-algebra from CGby forming the closure in a suitable operatorrepresentation on Hilbert space.

Problem. What is suitable?

The regular representation of Z on %2(Z) enjoysthe following property: if fn is a sequence oftrigonometric polynomials, and )fn) " 0 in theregular representation, then )fn) " 0 in anyother representation (exercise!). Thus the regularrepresentation is universal (and thus uniquely suitable).

Not all groupoids enjoy this property. This causestrouble.


Lecture 2Noncommutative Measure

Theory

Nigel Higson and John RoePenn State University

June 20, 2000

Clay Symposium: Lecture 2

Spectral Theory

T =bounded selfadjoint operator on aHilbert space H (thus !Tu, v" = !u, Tv").

Spectrum(T) = { ! # C : T ! !I is singular } $ R.

Some rather amazing facts:

• The mapn!

i=1

aixi %"

n!

i=1

aiTi

from polynomials to operators determines anisometric &-algebra homomorphism

C(Spectrum(T)) " B(H).

• The map further determines an isometric &-algebra homomorphism from bounded Borelfunctions to operators. Pointwise convergence offunctions corresponds to pointwise convegenceof operators.

Clay Symposium: Lecture 2 1

Measure classes

The image of the Borel functional calculushomomorphism is isomorphic to the algebraL#(X, µ) for some measure µ on X = Spectrum(T)(assuming the Hilbert space H is separable).

The operator T determines the measure class[µ] (two measures belong to the same class if theyhave the same nullsets).

Theorem. The measure class [µ] and a multiplicityfunction (discussed later) determine T up to unitaryequivalence (that is, up to change of orthonormalbasis in the Hilbert space).

Example. Associated to T = i ddx

is theLebesgue measure class on X = R. (Thisoperator is unbounded, and requires somehatcareful treatment).


Operator Algebras

Definition. Operator algebra = von Neumannalgebra = W!-algebra = unital &-algebra ofoperators on H which is closed under the topologyof pointwise convergence.

This definition is not optimal . . . but it is simplest.

Example L#(X, µ) $ B(L2(X, µ)).

Example: If " is a unitary representation of a groupon a Hilbert space then its commutant, the algebraof all intertwining operators, is a von Neumannalgebra.

Double Commutant Theorem. Every vonNeumann algebra is the commutant of its commutant.

Example, continued: Observe that a subspace ofH is a subrepresentation of " if and only if theorthogonal projection onto it is an operator in thecommutant of ".

Projections in M $ Subrepresentations of "


Decomposition of Representations

Algebraic Background:

G = finite groupK = field" = finite-dimensional representation over K

Theorem. Commutant of " = Direct sum of centralsimple algebras.

Decomposition of Hilbert Space:

(X, µ) = measure space (only [µ] is relevant){Hx}x"X = family of Hilbert spacesS = measurable family of sections

% We can form H = '&

XHx dµ(x).

Theorem. If M is a von Neumann algebraon H (separable) then there are canonicaldecompositions

H = '&

XHx dµ(x) M = '

&XMx dµ(x)

where the Mx have trivial center.


Factors

Definition. A factor is a von Neumann algebra withtrivial center.

Finite dimensional case: M != Mn(C) = (central)simple algebra.

Message: Algebraically, factors play the roleof central simple algebras over C, in infinitedimensions (more on this later).

Definition. A factor M is of type I if M != B(H)(algebraically) for some H.

Are there non-type I factors? . . . Yes . . . Let Gbe a group and let M be the commutant of the left-regular representation on #2(G).

Theorem. The commutant of M is the commutantof the right-regular representation on #2(G).

Lemma. T %" !T$e, $e" is a faithful trace on M.

Corollary. If G has no finite conjugacy classes thenM is not of type I.


Comparison of ProjectionsProjections in M $ Subrepresentations of "

P1 ! P2 $ "1 and "2 areunitarily equivalent

[P1] ( [P2] $ "1 is unitarily eq. to asubrepresentation of "2

Theorem. This is a partial order on equivalenceclasses. If M is a factor then this is a linear order.

P1 is minimal $ "1 is irreducible

P1 is infinite $ "1 is equiv. to a propersubrepresentation of itself

P1 is finite $ "1 is not equiv. to a propersubrepresentation of itself

Proposition. M is a factor if and only if " is isotypic:every two subrepresentations of " have equivalentsubrepresentations.

Proposition. A factor has a minimal projection ifand only if it is algebraically isomorphic to B(H) forsome H (i.e. iff it is of type I).


Decomposition of Representations

Theorem. If M is the commutant of a commutativevon Neumann algebra then

M = '&

XMx dµ(x),

where each Mx is algebraically isomorphic to thefull algebra of bounded operators on some Hilbertspace

If M is the commutant of a selfadjoint operatorT then X above is the spectrum of T , µ is themeasure discussed earlier, and x %"

!dim(Mx)

is the multiplicity function which, together with [µ],serves to characterize T .

Theorem. If the commutant of " decomposes asabove then " decomposes as a direct integral ofmultiples of irreducible representations.

Remark. Among discrete groups, type I impliesvirtually abelian.


Types I, II and III

Relative dimension of projections in factors:

For finite projections P0, P1 # M (with P1 )= 0) form

dim(P0)/dim(P1) # [0,#)

using a ‘Euclidean algorithm’.

Absolute dimension function:

dim(P) =

'(()

((*

0 if P is zero# if P is infinite

dim(P)/ dim(Pref) if P is finite

Proposition. There is a maximum dimension andrange (dim) is arithmetically closed within [0, dmax].

Types of Factors:

Type I {0, 1, 2, . . . , n} or {0, 1, 2, . . . ,#}

Type II [0, 1] or [0,#]

Type III {0, #}


Representations of &-Algebras

GNS construction:

A = complex &-algebra

% : A " C, %(a!a) * 0

H& = GNS Hilbert space, !a, b" = %(b!a)

"% = GNS representation, "% : A " B(H%)(assuming %(a!b!ba) ( Cb%(a!a))

Example:

A = C(X)

%(f) =&

Xf(x)dµ(x) (Riesz Theorem)

H% = L2(X, µ)


Infinite Tensor Products

Powers factors:

A ="#

j=1 M2(C), CAR algebra

% ="#

j=1 %!, product state

%!

#a bc d

$=

1

1 + !(!a + d)

R! = double commutant of "%[A]

Examples:

! = 0 % Type I factor

! = 1 % Type II factor

Theorem. The R! are type III factors.

Theorem. (R. Powers) For ! # (0, 1) the R! arepairwise nonisomorphic type III factors.


What Type?

Does R! have any non-zero, finite projections?

Is there a faithful, normal, semifinite trace on R!?

T * 0 # R!

X = Spectrum(T)

+

%

'()

(*

µT(E) = dim('E(T))

Trace(T) =

,

X

x dµT(x)

Obstacles:

• the trace, should it exist, need not be finite on anyelement of +jM2(C)

• +jM2(C)= (the norm closure) is independent of !

• algebraic structure need not pass to weakclosure: for the tracial product states it happensthat [+jM2(C)] ## != [+jM3(C)] ##


Geometric Examples of Factors

Data:

(X, µ), measure spaceG, discrete groupG , X " X, measurable action

Assumptions:

action preserves class of µaction is free

Example:

PSL(2, Z) , P1(R) " P1(R).

Group-measure space construction:

{#2(Gx)}x"X = measurable field of Hilbert spaces

M =-measurable families {Tx # B(#2(Gx))}x"X/G

.

Theorem. M is a factor iff the action is ergodic.


Type III Factors

Theorem. Suppose that the action is ergodic.

• M is type I iff there are atoms.

• M is type II iff there are no atoms but thereis an invariant measure in the class of µ.

• M is type III iff there are no atoms and noinvariant measures in the class of µ.

This applies to the factors R!:

X =/

j Z/2Z

µ =/

j µ!

G = 'jZ/2Z

Theorem. If ! # (0, 1) then there is no invariantmeasure in the class of µ.


Dynamics

Prelude: Let G act on X, preserving the class ofµ. There is an associated unitary representation onL2(X, µ):

"(g)f =%

dg!µdµ

&12g!f

In fact there is a whole ‘principal series’ of unitaryrepresentations:

"t(g)f =%

dg!µdµ

&it

"(g)f (t # R)

Groupoid perspective: Note that

'&

X#2(Gx)dx = L2(X , G)

On L2(X , G) define

Uth =%

dg!µdµ

&it

h

Then UtMU!t = M, where M is the von Neumann

algebra associated to the action of G on X.


Tomita’s Theorem

Summary: The dynamics (t : M " M can beobtained directly from the weight on M associatedto µ by an algebraic construction.

Theorem. Let % be a faithful, normal, semifiniteweight on M and let H% be the associated GNSHilbert space. The unbounded operator S : x %" x!

on H% has a polar decomposition

S = J)12, ) = S!S,

and JMJ = M # while )itM)!it = M.

The Powers examples: On R! we have

(t = AdUt : R! " R!

Ut = Ait! + Ait

! + . . .

A! =

#!

1+! 0

0 11+!

$(note: %!(T) = trace2(A!T))


Connes’ Theorem

What does Tomita’s theory do?

Weights " Automorphism groups

Compare this to:

T = T! " U = exp(iT)

Heisenberg relations " Weyl relations

In measure theory, equivalent measures are relatedby Radon-Nikodym derivatives:

&Xf(x) dµ =

&Xf(x)h(x)d*

%(T) = +(A!TA)

Connes’ R-N Theorem. Let M be a von Neumannalgebra. For any % and + there is a unitary ‘cocycle’{Ut} $ M such that

(%t (T) = (+

t (U!t TUt),

for all T # M and t # R.


Invariants in the Type III situation

Definition. T(M) = { t # R : (%t is inner }.

Lemma. T(M) is independent of % — it is aninvariant of M alone.

Recall that for R!,

(t(T1 + T2 + . . . ) = Ait! T1A

!it! + Ait

! T2A!it! + . . .

where A! =

#!

1+!0

0 11+!

$.

Lemma. This is inner for a given t # R iff Ait! = I.

Hence T(R!) = 2"log!Z.

Hence the invariant T(M) distinguishes the Powersfactors from one another.


Non-Commutative Topology


June 21, 2000

AbstractC∗-algebras are the noncommutative

counterpart of topological spaces. Theiralgebraic topology can be studied by means ofK-theory.

Clay Symposium Lecture 3: Noncommutative Topology

The big picture

Theory Type offunction

Noncommutativeversion

Measure Theory Borel Von Neumannalgebra

Topology Continuous C∗-algebraDifferentialTopology

C∞ Holomorphicallyclosed sub-algebra of aC∗-algebra

AlgebraicGeometry

Rational CG

Today’s Reading Assignment: A. Connes, Ananalogue of the Thom isomorphism for crossedproducts. . . , Advances in Math 39(1981), 31–55.

Clay Symposium Lecture 3: Noncommutative Topology 1

C∗-algebras

Definition 1. A C∗-algebra is a Banach ∗-algebrawhich is isomorphic to a subalgebra of B(H), for someHilbert space H.

Thus a C∗-algebra is a norm-closed subalgebra ofB(H); contrast the ‘pointwise’ topology considered inlecture 2. (Since the pointwise topology is weaker thanthe norm topology, every von Neumann algebra is inparticular a C∗-algebra; but it is not usually helpful tothink like this.)

Example Let X be a compact metrizable space.Let µ be a suitable measure on X . The representationρ: C(X) → B(L2(X,µ)) by multiplication operatorsshows that C(X) is a C∗-algebra.

Note that in this example we can recover themeasure by

∫

f dµ = 〈ρ(f)ξ, ξ〉

where ξ ∈ L2(X,µ) is the constant function 1. Thispoints the way to an abstract characterization of C∗-algebras.


More about C∗-algebras

Definition 2. A C∗-algebra is a Banach ∗-algebra Ain which the norm and involution are related by theC∗-identity ‖a∗a‖ = ‖a‖2.

Theory of positive linear functionals (theGNS construction) connexts this abstract definitionwith the concrete one; Hahn-Banach argumentsproduce sufficiently many states to give a concreterepresentation of every abstract C∗-algebra. Theabstract definition is useful e.g. to show that A/Iis a C∗-algebra when A is a C∗-algebra and I is aclosed ideal.

Commutative C∗-algebras are precisely of the formC0(X) (Gelfand-Naimark). Consequently: If a ∈ Ais selfadjoint, and f is continuous on the spectrum ofa, then f(a) ∈ A also — the (continuous) functionalcalculus. If f is only (bounded) Borel then f(a) ∈ A′′

but usually f(a) /∈ A.


Group (and Groupoid) C∗-algebras

Let Γ be a discrete group. The group ring CΓis a complex involutive algebra. We may obtain C∗-algebras by completing CΓ in various norms.

• Define τ : CΓ → C by x =∑

cgg 7→ c1. Then τis a positive linear functional (in fact a trace) andthe associated GNS representation is the regularrepresentation of CΓ on ℓ2Γ. Completing inthis norm gives the reduced C∗-algebra, C∗

red(Γ).Notice that by construction, τ extends to a traceon C∗

red(Γ).

• Alternatively, for every state σ, let ‖x‖σ be thenorm in the GNS representation associated toσ. The maximal norm on CΓ is ‖x‖max =supσ ‖x‖σ, and completing in this norm givesthe maximal C∗-algebra C∗

max(Γ). Note thatthe trivial representation gives a homomorphismC∗

max(Γ) → C.

Theorem 1. (Hulanicki) C∗max(Γ) = C∗

red(Γ) if andonly if Γ is amenable.


Projections in C∗-algebras

C∗-algebras are usually not stuffed full ofprojections.

Kaplansky Conjecture Let Γ be a torsionfreediscrete group. Then C∗

red(Γ) has no nontrivialprojections. Consequently, every selfadjoint operatorin C∗

red(Γ) has connected spectrum.

Note that, in physical terms, to study the spectrumof a selfadjoint operator is to study the range of valuesthat an observable can assume.

Atiyah Conjecture (simplified version) Let Γ be asabove, let x ∈ CΓ. Then the operator λ(x) ∈ B(ℓ2Γ)is either zero or injective. (This is really a conjectureabout the L2 Betti numbers.)

The example of Z shows that the Kaplanskyconjecture belongs to (noncommutative) topology,whereas the Atiyah conjecture belongs to(noncommutative) (semi?)-algebraic geometry.


Projections + Matrices = K-Theory

Definition 3. Let A be a C∗-algebra with unit.K0(A) is the abelian group with one generator foreach isomorphism class of finitely generated projectiveA-modules, and with the relations

[M ] + [N ] = [M ⊕ N ].

Lemma 1. If e, f are projections in a unital C∗-algebra A and ‖e − f‖ < 1, then there is a unitaryu ∈ A with f = u∗eu.

Proof The operator v = ef + (1 − e)(1 − f) hasev = vf and satisfies ‖1 − v‖ < 1, so it is invertible.Use polar decomposition to replace v by its unitarypart u. 2

Thus one can translate the definition of K-theoryas follows: the generators are homotopy classes ofprojections e = e2 = e∗ in matrix algebras Mn(A).This makes the link with topology. Note that K-theory is Morita invariant.


Where do elements of K0(A) come from?

• Spectral projections.

• Indices of elliptic operators ‘over A’ (for instance,leafwise elliptic operators on a foliation).

Remark Even though the kernel and cokernelprojections of an elliptic operator T over A belongto the von Neumann algebra A′′, and not necessarilyto A itself, their formal difference does define a K-theory class for A. This is another consequence of theideas underlying the ‘heat equation’ method.

For example, the integrality of the L2 Bettinumbers individually (Atiyah conjecture) is a muchdeeper statement than the integrality of the L2 Eulercharacteristic (Atiyah L2-index theorem).


Traces and K-theory

A trace τ on a C∗-algebra A is a state thatis invariant under inner automorphisms: τ : A → C

satisfies τ(u∗au) = τ(a) for unitary u. It willthen extend to a trace on Mn(A) by the formula[aij] 7→

∑

i τ(aii).

A trace τ gives rise to a dimension functionτ∗:K0(A) → R.

Remark In practice we often need to consider‘unbounded traces’, e.g. on K(H). Then delicatequestions of analysis intervene.


Smooth subalgebras

Definition Let A be a C∗-algebra, A ⊆ A a densesubalgebra. We say that A is smooth in A if thefollowing is true: for every a ∈ A, and every functionf holomorphic on a neighborhood of the spectrum (inA) of a, the element f(a) ∈ A in fact belongs to A.

Examples (i) C∞(M), M a compact smoothmanifold, is a smooth subalgebra of C(M); (ii) Thesubalgebra of Aα consisting of those formal sums∑

amnUmV n, with {amn} of rapid decay, is smooth;(iii) Let τ be a (densely defined) unbounded trace ona C∗-algebra A, which is semicontinuous; then thedomain of τ is a smooth subalgebra. For the standardtrace on K(H) one obtains in this way the subalgebraL

1(H) of trace class operators.

(A trace τ on A is semicontinuous if for everyα ∈ R, the set {a ∈ A : a ≥ 0, τ(a) ≤ α} is closed inA; compare Fatou’s Lemma.)

Theorem 2. Let A be a smooth subalgebra of A;then the inclusion A → A induces an isomorphism onK0.


Spectral topology

Using K-theory to investigate the spectrum of aselfadjoin operator T :

1. Embed T in a C∗-algebra A with a faithful trace τ ,normalized so that τ(1) = 1.

2. Compute K0(A) and the range of τ : K0(A) → R.

3. Each connected component of spectrum(T ) givesrise to a projection in A. Use the information aboutthe traces of such projections.

Elementary example A selfadjoint n × n matrix hasno more than n eigenvalues.


K1 and Bott periodicity

The group K0(A) has an ‘odd’ counterpart K1(A),generated by homotopy classes of unitary elementsin matrix algebras Mn(A). There is an elementaryisomorphism K1(A) ≡ K0(SA), where the suspensionSA is defined to be C0(R) ⊗ A.

Bott Periodicity Theorem There is also anisomorphism K0(A) ≡ K1(SA).

It follows that K0 and K1 together comprise ageneralized homology theory for C∗-algebras. Forinstance a short exact sequence

0 → J → A → A/J → 0

of C∗-algebras gives rise to a cyclic six-term exactsequence in K-theory.

Remark For a space X we have Ki(C0(X)) =K−i(X), so that this is a true generalization of Atiyah-Hirzebruch K-theory.


K-Theory for Aα

We ask: What are the K-theory groups for theirrational rotation algebra Aα?

Answer: (Pimsner-Voiculescu) K0(Aα) = Z ⊕ Z,K1(Aα) = Z ⊕ Z. Moreover, the dimension functiongiven by the canonical trace on Aα maps K0(Aα) tothe lattice Z ⊕ αZ ⊆ R.

The Pimsner-Voiculescu calculation proceeds via anexact sequence for the K-theory of a crossed productby Z. More geometrically, work with the (Moritaequivalent) foliation algebra for the Kronecker foliationon T

2. This is a transformation groupoid algebra for aflow (R-action). Thus it suffices to compute K-theoryfor the crossed product C∗-algebra A ⋊ R.

Theorem 3. (Connes) The leafwise index of id/dxgives a ‘Thom isomorphism’ Ki(A) → Ki±1(A ⋊ R).

Note that if the R-action is trivial this is just theBott periodicity theorem.


The Thom isomorphism

Idea of proof — reduce to the case of a trivialR-action.

• If A = C we always have trivial action.

• Every K0-class for A arises from a homomorphismC → A (up to finite matrices).

• Can change the R-action to an ‘exterior equivalent’one which preserves a given smooth projection e.

Discussion Let the derivation δ generate the R-action. The extent to which the action fails to preservee is measured by the bounded operator [e, δ(e)]. Usethis to make a ‘small’ perturbation of the unboundedinfinitesimal generator of the R-action, so as to makeit commute with e.

Cannot do this for R2-actions — curvatureintervenes.


The range of the trace

Suppose given a flow on a compact space X , andlet µ be an invariant measure. One obtains from µ atrace τ on A = C(X) ⋊ R.

Theorem 4. The range of dimτ : K0(A) → R is equalto the range of the map H1(X ;Z) → R given by firstpairing with the vector field which generates the flow,and then integrating with respect to µ. (exercise —this is well defined on cohomology.)

Proof proceeds via the formula

dimτ(φ(u)) =1

2πi

∫

u−1δ(u) dµ

for u unitary over C(X). (φ is the Thom isomorphism.)

Corollaries: (i) Computation of the range of thetrace on Aα, above; (ii) A simple C∗ algebra withno nonzero idempotent — take X = S3 and a minimaldiffeomorphism.


The tangent groupoidReference: Noncommutative Geometry pp102–111

Let M be a compact smooth manifold. ThenM × M is (in a trivial way) a smooth groupoid.Its associated algebra C[M × M ] is the algebra ofsmoothing operators on M . The tangent bundle TMis a smooth groupoid too: it is a bundle over M whosefibers are groups. Let us glue these together.

Definition 4. The tangent groupoid to M , G(M),has objects M × [0, 1], and it is equal to TM overM × {0} and to M × M over M × {t} for t 6= 0.

This gives a smooth groupoid — local coordinatesnear (x, 0) may be given by

(x, v, t) 7→

{

(x, expx(tv), t) (t > 0)(x, v) (t = 0)

Notice that the decomposition of the objects M ×[0, 1] = M × {0} ∪ M × (0, 1] allows us to writeG(M) = G0(M) ∪ G+(M), the disjoint union of aclosed and an open subgroupoid.


The tangent deformation

A decomposition like this gives an exact sequence

0 → C∗(G+) → C∗(G) → C∗(G0) → 0.

Here C∗(G+) has trivial K-theory and soK∗(C

∗(G)) ∼= K∗(C∗(G0)) ∼= K∗(T ∗M). Evaluation

at some finite value of t (say t = 1) gives ahomomorphism C∗(G) → K, and so on the level ofK-theory we obtain a homomorphism

K0(T ∗M) → Z.

Theorem 5. The above homomorphism is theAtiyah-Singer analytical index map.

To prove this, construct a linear section ofC∗(G) → C∗(G0), using the symbol calculus forpseudodifferential operators.

kt(x, y) = (2π)−n

∫

eiξ·(x−y)/t σ(x, ξ) dξ.


A noncommutative view of the index

theorem

One can regard the index theorem as a reduction ofthe above exact sequence of noncommutative spacesto the level of commutative geometry. Suppose Membedded in Euclidean space R

N . From this weobtain a free action of the groupoid G on the spaceR

N . The transformation groupoid G ⋊ RN is then aproper equivalence relation, as are the subgroupoidsG0 ⋊ RN and G+ ⋊ RN .

By Rieffel’s Theorem (lecture 1) these groupoids areMorita equivalent to ordinary ‘commutative’ spaces.

The index theorem is thus reduced to a version ofBott periodicity relating the K-theory groups of thealgebras associated to the groupoids G and G ⋊ RN

— that is, to the Thom isomorphism theorem.


Introduction to the Baum-Connes

conjecture

It is natural to enquire whether every groupoidG has the property that its C∗-algebra C∗(G) is ‘K-theoretically equivalent’ to an ordinary commutativespace. In a more precise form, this is the Baum-Connes conjecture: a certain assembly map

µ:Ktop∗ (G) → K∗(C

∗

red(G))

is conjectured to be an isomorphism.

The assembly map is a generalization of the Atiyah-Singer index map.

The BC conjecture has been verified in a widevariety of examples. Recent constructions lead togroupoids G for which it fails: these exploit the tensionbetween C∗

max and C∗

red, together with the fact thata decomposition like G = G0 ∪ G+ above in generalgives an exact sequence of C∗-algebras only on theC∗

max level.


Hike

For those who will be here on Friday June 23rd, ahike is being planned. Would those interested pleasesign below. Consult Guoliang Yu for details.


Lecture 4Noncommutative Differential

Topology


June 22, 2000


Characteristic Numbers

, smooth manifold

, projection-valued functionvector bundle

, oriented closed submanifold

Definition. trace .

Proposition. Fixing , the scalar onlydepends on .

trace is a closed form.

Given ,

tracetrace

Remark: If dim is odd then (in facttrace ).


Characteristic Numbers inNoncommutative Geometry

, noncommutative algebra

, idempotent over

Example:

)

Powers-Rieffel projection

Problem: Define characteristic numbers asbefore.

If trace then

What is ?

What is ?

What is ?


CyclesDefinition. An -cycle over an algebra is

a differential graded algebra with an algebramap from into , and

a closed, graded trace on .

The trace properties:deg deg .

.

Warning: It is not necesarily true that northat , nor that .

Proposition. If is an -cycle then thecharacteristic number

trace

depends only on .

As before, if is odd then is zero.


Traces

-cycles over traces on

If is a path of projections and ˙ then

˙ ˙ ˙ ˙

and therefore

˙ ˙ ˙ .

Commutative case:

, = any distribution.

However, constant dim .

Noncommutative examples:

,

,


Topological Invariance

= Banach or -algebra

dense subalgebra of

Not every cycle over induces a ‘topologicallyinvariant’ map,

.

Example: Let . Then for, no corresponding map

is known, in general. (Note that is even a ‘state’:.)

Sufficient condition for traces: Semicontinuity.

Sufficient condition for -cycles:

(These are called -traces.)


Cycles for the Irrational Torus

For with one can take

.

Lemma. The same formulas define a -cycle on ,for any .

Computation: For the Powers-Rieffel projection,

(See Connes, A survey of foliations and operatoralgebras.)


Flows and Multiflows

, smooth action

lim , derivation

, invariant trace

, ,

Given commuting flows and , we can form

and (supposing is invariant for both flows)


CurvatureLemma. The characteristic number

depends only on the outer equivalence class of theaction of on .

Proof. Given , form the combined action

Aut ,

Construct from it a -cycle over . Since

we get

Example: Powers-Rieffel projection (compare Thomisomorphism theorem of Connes).


Cyclic -Cocycles

cyclicity:

Proposition. Cyclic -cocyles are precisely thefunctionals associated to -cycles on theuniversal differential graded algebra over .


Cyclic Cohomology

Lemma. Let be a cyclic -multlinearfunctional on .

is cyclic too.

.

Definition. = cyclic -cocyles modulocyclic coboundaries.

Pairings:

The latter is

trace

trace

assuming that .


Cyclic Cohomology and Manifolds(First Look)

For oriented we define

(de Rham differential)

We obtain:geometric-cycles

closed de Rhamcurrents cyclic -cocyles

However:

(manifold -chain) =manifold -cycle

(cyclic -cochain) = cyclic -cocycle

Note the mismatch — there are some surprises!


Products and Suspension

Lemma.evenodd

Product on cycles:

This induces

Lemma. is a polynomial algebra withdegree two generator .

(Compare with .)

Suspension:

from generator of .

Proposition.


Bordism

Lemma. If bounds then for all .

Proof. Stokes’ Theorem.

Definition. An -cycle bounds if thereexists a d.g.a.–trace pair , and asurjection , such that

“Stokes’ Theorem”

Lemma. If bounds then , forall projections .

Remark: This gives a natural context for showingthat

trace

is well defined (depends only on ).


Algebraic Structure of Bordism andSuspension

= Hochschild cohomology (drop cyclicitycondition on -linear functionals; keep thesame differential ).

Theorem. A cycle bounds iff its cyclic cohomologyclass is in the image of the map

cyclic symmetrization

Theorem. A cycle bounds iff its cyclic class is in thekernel of

Definition.

stable bordism class of cycles

lim


Example: Bass Conjecture

,

,

If is torsion free, is

?

Stupid question: is a cyclic coboundary? No— there are no coboundaries in degree 0.

But we can try to solve

Example: If then take

Exercise: What to do when ?

Other important examples arise in index theory


Homological Algebra

From the map associatedto the inclusion of the cyclic complex into theHochschild complex we get

Consider . One has.

cohomology of -complex (there is aspectral sequence )

and

fits into the ordinary framework ofhomological algebra


Smooth Manifolds

Example: (topological algebra)

de Rham currents.

de Rham boundary

ev/oddev/odd


Groups

Group -cocycles:

Cyclic cocycles from group -cocycles:

ifif

Example:

, group homomorphism

, group cocycle

ifif


Group Cocycles and Novikov

Example, continued:

, where generates theautomorphism group exp .

is a -trace (it pairs with )

from , a product of group -cocycles, we get a -trace and a map on

.

Index formula of Connes and Moscovici: forand for an elliptic operator on

,

constant

the Novikov conjecture for products of onedimensional cohomology classes


Elliptic operators, cyclic theory,and zeta functions


June 25, 2000

AbstractIn this lecture we will show how abstract

elliptic operators give rise to cyclic cocycles whichhave strong integrality properties.

Clay Institute Symposium Lecture 5

A quick review

• Noncommutative spaces — group and groupoidalgebras.

• Noncommutative measure theory — von Neumannalgebras — dynamic.

• Noncommutative topology — K-theory — spectralprojections, indices.

• Di!erential topology — characteristic class (Chern-Weil) theory leads to the formalism of cycliccohomology.

Key examples — C(X), C!, the irrationalrotation algebra

A! =!U, V |UV = e2"i!V U

".

Clay Institute Symposium Lecture 5 1

Cyclic cohomology

For an algebra A one defines an n-cycle over A tobe a package

A !! "0d

!! . . .d

!! "n

!

!! C

where (", d) is a di!erential graded algebra (notnecessarily commutative) and

#: "n ! C is a closed

graded trace.

The essential information is contained in themultilinear functional !(a0, . . . , an) =

#a0da1 · · · dan,

which is a cyclic cocycle, that is

! " " = (#1)#!, b! = 0,

where " can be any cyclic permutation of a0, . . . , an,and b is the ‘Hochschild coboundary’.

It turns out that b preserves the cyclic symmetrycondition (Surprise!) and therefore n-cycles are in factthe (co)cycles for cyclic cohomology theory HC!(A).


Structure on cyclic cohomology

HC!(A) is a module over HC!(C), which isa polynomial ring with one 2-dimensional generator.The corresponding operation on HC!(A) is denotedS:HCn(A) ! HCn+2(A).

One has the Connes exact sequence

HCn"1(A)S

!! HCn+1(A)I

!! HHn+1(A)B

!! HCn(A) . .

where HH!(A) is the Hochschild cohomology of Awith coe"cients in A!.

There are pairings of HCn(A) with K[n](A), [n]being n modulo 2, and these pairings are compatiblewith the S operation. The pairings are given by‘integration of the Chern character’.


Elliptic operators

Let D be a first order elliptic operator on a compactn-dimensional manifold M . Example: M = Tn and Dis the operator

#1$

$x1+ · · · + #n

$

$xn

where the Pauli matrices #1, . . . , #n are generators ofa Cli!ord algebra.

Key properties:

1. D is an unbounded selfadjoint operator on L2.

2. D has discrete eigenvalues tending to infinity, withfinite-dimensional eigenspaces; which is to say theresolvent operators (D ± i)"1 are compact.

3. [D, f ] is bounded for a dense set of functions f onM (for instance the smooth functions).

Note. Z/2-grading in the even-dimensional case.


Index theory and K-homology

Properties 1 and 2 imply that kerD is a finite-dimensional space. When n is even, kerD is gradedand the index IndexD = dimkerD+ # dimkerD" isan important invariant.

Property 3 allows us to ‘couple’ the index to avector bundle V — there are many choices for sucha coupled operator DV but (3) implies that theyall di!er by bounded operators and hence have thesame index. The assignment V $! IndexDV gives ahomomorphism K0(M) ! Z.

Atiyah, Brown-Douglas-Fillmore, and Kasparovdeveloped this idea into analytic K-homology —‘abstract’ elliptic operators (satisfying appropriateversions of 1–3 above) are the cycles for this theorywhich pairs integrally with K-theory.

The definition of K-homology works even fornoncommutative algebras.


Eigenvalues and zeta functions

Let µ1, µ2, . . . be the eigenvalues of |D| arrangedin increasing order. Property (2) of our list says thatµj ! % as j ! %. But one can obtain a more precisequantitative form of this statement.

Theorem 1. (Weyl) One has µj & j1/n, where werecall that n is the dimension of M .

It follows that the series defining the zeta function

%D(s) =#$

j=1

""sj = Tr |D|"s

is convergent for 's > n. (One must make a specialcase of the zero eigenvalues, if any.) Note that thisis the Riemann zeta function when D = id/dx on thecircle S1.

Theorem 2. The function %D(s) extends to ameromorphic function on C with poles at n, n# 1, . . ..

The proofs involve studying the heat equation.


Normalized elliptic operators

Let D be as above. To simplify assume that D isinvertible, and let F = D|D|"1 be the phase of D. Itis a bounded operator and the results above translateto:

1. F is selfadjoint and F 2 = 1,

2. For all continuous functions f on M , thecommutator [F, f ] is a compact operator onH = L2. In fact for a dense set of functionsf on M (for instance the smooth functions) thiscommutator belongs to the Schatten ideal Lp(H),for every p > n.

Discussion. The Schatten ideal Lp is generatedby those positive operators whose p’th powers areof trace class — compare the Lebesgue space Lp.Most familiar inequalities between Lebesgue spaceshave noncommutative counterparts — see B. Simon,Trace ideals and their applications.


Summable Fredholm modules and cycles

A (normalized) Fredholm module over a C!-algebraA is given by a representation & of A on a Hilbert spaceH, together with a selfadjoint operator F ( B(H) withF 2 = 1 and [F, &(a)] ( K(H) for all a ( A.

(There is a graded variant of the above definition:H is a graded Hilbert space, the representation of A isby even operators, and F itself is odd.)

The Fredholm module (&, H, F ) is p-summable ifthere is a dense subalgebra A ) A such that [F, &(a)]belongs to the Schatten ideal Lp for all a ( A.

Theorem 3. (Connes) Let (&,H, F ) be a p-summable Fredholm module. Then the equalities

d' = [F,'],%

' = Tr((')

define a p-cycle over A, called the character of theFredholm module. (( is the grading operator.)

One should take p * 0, 1 mod 2 according tograding.


Small technical improvement

Observe that the mapping

Tr$(') = 12 Tr((F [F,'])

agrees with Tr(') whenever ' ( L1, but is defined forall ' such that [F,'] = d' is in L1. This allows us torelax slightly the order of summability required in thedefinition of the character. For instance, a 1-summableFredholm module has a character in HC0(A) definedby

a $!% $

a = 12 Tr((F [F, &(a)]).

Exercise. Check the trace property.


Pairing theorem

Let (&,H, F ) be a Fredholm module over A andlet e ( A be a smooth projection. Then &(e)F&(e) isa graded Fredholm operator on the space &(e)H andwe can form its index. This gives a map

K0(A) ! Z, e $! Index &(e)F&(e)

corresponding to V $! IndexDV in the commutativecase.

Now suppose that F is p-summable. Then wemay pair the cycle corresponding to F with theprojection e according to the recipe of lecture 4, thatis

#edede · · · de, and we have

Theorem 4. Index(eFe) = ck

#e dede · · · de& '( )

k, even.

Proof If P is Fredholm, Q a parametrix modulo Lp,then

Index(P ) = Tr(1 # QP )p # Tr(1 # PQ)p.


Stability theorem

There is an ambiguity in the dimension p of thecharacter — any integer, of appropriate parity, andgreater than the degree of summability of the module,will do.

Theorem 5. Let (&, H, F ) be a p-summableFredholm module and let !p, !p+2 be its p-dimensionaland (p + 2)-dimensional characters. Then [!p+2] =cp S[!p] ( HCp+2(A), where cp is a universal constant.

Exercise. Do the computation by hand for p = 0.

Thus the character is unambiguously defined in theperiodic theory HCP !(A).

Remark. Cycles coming from summable Fredholmmodules automatically satsify the summabilityconditions to be higher traces — that is, they givelinear functionals on the K-theory of the full C!-algebra A.


Example I: ordinary manifolds

Let F be the Fredholm module defined by the Diracoperator D over a spin manifold Mn. It is p-summablefor all p > n (as we saw above), so it has a characterin HCn(C#(M)).

Recall from lecture 4 that

HCn(C#(M)) = Zn(M)+Hn"2(M)+Hn"4(M)+· · · .

Theorem 6. The character of the Dirac modulehas top-dimensional component equal to the current(f0, . . . , fn) =

#M f0df1 · · · dfn, and the homology

components are those of the Poincare dual of the*A-genus.

The proof uses the heat equation.


Example II: the free group

Let ! = F2, A = C!r (!), A = C!, and define a

Fredholm module as follows:

e

l(v)v

Graded Hilbert space H = L2(V ) + L2(E).

F =+

0 U!

U 0

,,

where U sends vertex v to edge )(v). (1-dimensionalcorrection gets F 2 = 1 on the nose.)

This Fredholm module is 1-summable. Similarfinitely summable ‘dual Dirac’ constructions arepossible on noncompact symmetric spaces G/K ofrank one.


Kaplansky-Kadison for the free group

Lemma 1. The character in HC0(C!) of theFredholm module defined above is equal to the vonNeumann trace ! : C! ! C.

It follows that the map dim$ : K0(A) ! R is givenby pairing with a Fredholm module, and so is integervalued. In particular the trace of a projection in Amust be either 0 or 1. Since ! is faithful we get

Theorem 7. The C!-algebra A = C!r (F2) has no

non trivial projections.

Problem/Conjecture Is it true that for anytorsionfree group, Sn! is the character of a Fredholmmodule for su"ciently large n? This would implyKadison-Kaplansky conjecture by arguments as above.


Atiyah conjecture for the free group

Lemma 2. Let U , T be operators on a Hilbert spaceH, and suppose that U commutes modulo finite rankoperators with T . Then U also commutes modulofinite rank operators with the orthogonal projectionoperator onto the kernel of T .

(Connes’ terminology: the orthogonal projection isquasicontinuous.) Proof uses the fact that the set ofoperators of rank , r is closed in the weak (pointwise)topology. No analog for e.g. trace class operators.Thus this is special to the free group.

Hence find that F gives a Fredholm module evenover the algebra A obtained by adjoining kernelprojections to A = C!. (A is a subalgebra of thevon Neumann algebra A$$, but it contains elements notbelonging to the C!-algebra A.) Hence we get:

Theorem 8. (Linnell) Any nonzero element of A =C! is injective as an operator on )2(!).


Example III: the irrational rotationalgebra

As described in lecture 4, the algebra

A! =!U, V |UV = e2"i!V U

"

admits a canonical 2-trace

!2(a0, a1, a2) = !0-a0(*1(a1)*2(a2) # *2(a1)*1(a2))

..

We observed the integrality of -e, !2. for the Powers-Rie!el projection e. Why is this?

Theorem 9. !2 is the character of a Fredholmmodule.

Extension of this to quasicontinuous elementsrelated to the quantum Hall e!ect, seeNoncommutative Geometry pp355–367.


Seeking a local index formula

The index formulae above all depend ultimately onthe formula

Index(P ) = Tr(1 # QP )p # Tr(1 # PQ)p

which is not local.

A truly local formula would neglect trace classoperators (because, for example, pseudodi!erentialoperators are trace class o! the diagonal). Can suchan index formula be found in the abstract case?

Connes-Moscovici showed that the answer isyes under a natural hypothesis. Recall thatthe p-summability hypothesis is equivalent to theconvergence of the Dirichlet series for a certain zetafunction in a right-hand half plane. It turns out that afully local index formula can be achieved if we assumein addition that %(s) has an analytic continuation.

See lecture 9.


Lecture 6Classification of Factors


June 25, 2000


Von Neumann Algebras

B(H) = Bounded operators on aseparable Hilbert space

L1(H) = Trace class operators on H

Proposition. L1(H)! != B(H) via the pairing!T1, T2" = trace(T1T2).

Definition. The ultraweak (u.w.) topology on B(H)is the associated weak! topology.

Double Commutant Theorem. A unital #-algebraM $ B(H) is u.w.-closed iff M "" = M.

Definitions.

A #-subalgebra M $ B(H) is a von Neumannalgebra iff M = M "".

M is a factor iff M % M " = CI.


Classification into TypesSuppose M is the commutant of a unitaryrepresentation ! (of some object, e.g. a group).Then recall that:

Projections in M ! Subrepresentations of !

Moreover there are further correspondences:

Equivalence ofprojections ! Equivalence of

subrepresentations

Order on equivalence classes ! Inclusion of sub-

representations

Theorem (Murray and von Neumann). If M is afactor then the ordering on equivalence classes ofprojections is one of:

• {0, 1, . . . , n}{0, 1, . . . ,"}

Type I

• [0, 1][0,"]

Type II

• {0, "} Type III


Algebraic versus Spatial Isomporphism

Obvious: Each von Neumann algebra is a dualspace.

Not so obvious: The space M! $ M! is unique.

# The u.w. topology is intrinsic to M.

Now suppose given!1 : M $ B(H1)

!2 : M $ B(H2)

%

u.w. continuous

• The commutant of !1 & !2 in B(H1 & H2) is afactor (if M is).

• Subreps of !1&!2 (like !1 and !2) are classifiedby dimension of projections:

! '$ dim(P!) (

&''(

'')

{0, . . . , dmax}

[0, dmax]

{0,"}

• Hence the representation theory of a given Mis essentially trivial. (Especially in the type III case!)


Brauer TheoryK = Field

M = finite-dim’l central simple algebra over K

Theorem. M != Mj(D), for some central divisionalgebra D.

Definition. Br(K) = isomorphism classes of centraldivision algebras over K.

Theorem. Br(K) is an abelian group with theoperation

[D1] + [D2] = [D], D1 )K D2 = Mj(D).

For K = C the group Br(K) is trivial, but e.g. for Qp

there is an isomorphism Br(K) != Q/Z.

Example: For a, b ( K# set

i2 = a, j2 = b, ij = !ji = k.

If char(K) *= 2 then

D !=

*M2(K) if aX2 + bY2 = 1 has a sol’n

Div. Ring if aX2 + bY2 = 1 has no sol’n


AFD FactorsDefinition. A von Neumann algebra M isapproximately finite-dimensional if it is generated byan increasing family of f.d. #-subalgebras.

Example: R" = Powers factor from Lecture 2

= Double commutant of )M2(C)in a GNS representation

Theorem (Murray and von Neumann). There is aunique AFD factor of type II1.

Compare: Group von Neumann algebras. For non-amenable G, classification is largely unknown.

Proof of Theorem. Uses the trace&'(

')

µt(E) = dim(#E(T))

$(T) =

+

Spectrum(T)

"dµT(")

and associated norm +S+22 = $(S!S) to compute

estimates—does not generalize even to II" (sincepossibly +S+2 = " for all S in approximating family).


ITPFI Factors

Sj =

!

"#"

(j)1 . . .

"(j)nj

$

%& %j(T) = trace(SjT)

& =,"

(j)k : 1 , k , nj, j = 1, 2, . . .

-

R& =Double commutant of )Mnj

(C)in GNS representation for )%j

Recall:

Theorem (Powers). For Sj -' 1

1+""

1+"

(and for the

values 0 < " < 1 the ITPFI factors obtained arepairwise non-isomorphic.

A negative indication for the classification problem:

Theorem (Woods). The moduli space of ITPFIfactors is not countably separated.


Tomita’s Modular Theory• % : M $ C, faithful normal state (or weight):

%(T!T) . 0

%(T!T) = 0 . T = 0

% u.w. continuous

• H%, GNS Hilbert space.

• S : H% $ H%, Sx = x!.

Theorem. There is a polar decomposition S = J'12

(isometry/positive operator) and

JMJ = M "

'itM'!it = M.

Recall: This is a difficult theorem in analysiswhich serves to reduce unbounded operator theoryto bounded operator theory (c.f. Weyl’s form ofHeisenberg’s relations).


Connes’ Theorem

Theorem. Any two modular flows (% and () arerelated by a unitary cocyle in M:

()t (T) = (%

t (UtTU!t ).

In other words one has

( : R / M2(M) $ M2(M)

(t

)a

d

*=

)(%

t (a)

()t (d)

*

It follows that associated to M there is a canonicalhomomorphism

R 0$ Out(M) = Aut(M)/Inn(M)

We are eventually going to distill from this the basicclassification invariant for AFD factors.


KMS ConditionKMS = Kubo, Martin and Schwinger

Example:

M = Mn(C)

%(T) = trace(e!HT)

H%!= M

/''0

''1

&''(

'')

J : x '$ e!12Hx!e

12H

' : x '$ e!HxeH

(z : T '$ eizHTe!izH

Computation:

%((z(T1)T2) = trace(ei(z+i)H)T1e!izHT2)

= trace(e!HT2ei(z+i)HT1e

!i(z+i)H)

= %(T2(z+i(T1))

KMS Condition:

For every T1 and T2 there is a continuous,bounded function F on the strip 0 , Im(z) , 1,holomorphic in the interior, with

*F(t) = %(T2(t(T1))

F(t + i) = %((t(T1)T2)


KMS Condition, Continued

Theorem. If ( is Tomita’s modular flow associatedto a normal state % then the KMS condition issatisfied.

Theorem. Given a normal state % there is aunique flow ( (namely Tomita’s) for which the KMScondition holds.

Example: For the Powers factors, one can checkthat if

S =

)1

1+""

1+"

*

then

(t(T1 ) T2 ) · · · ) = SitT1S!it ) SitT2S

!it ) · · ·

satisifies the KMS condition.


The Invariant S(M)

G = locally compact abelian group.

Definition. The spectrum of an action * : G/M $M is the complement of the largest open set U in Gsuch that

Supp(f) $ U #2

Gf(g)*(g) dg = 0

(here f ( L1(G)).

Definition. The Connes spectrum of an action* : G / M $ M is

S(*) = %p Spectrum(* : G / pMp $ pMp),

where the p ( M are nonzero *-fixed projections.

Proposition. The Connes spectrum is a closedsubgroup of G, and is invariant under exteriorequivalence of actions.

Definition. The closed subgroup S(M) $ R+ is theConnes spectrum of the modular flow (we identifyR+ and R via !x, y" = xiy).


A Finer Classification of Factors• " = 0 M is Type III0 if S(M) = { 1 }.

• 0 < " < 1 M is Type III" if S(M) = "Z.

• " = 1 M is Type III1 if S(M) = R+.

Connes’ discrete decomposition:

Let 0 < " < 1. The analysis of III" factors isreduced to II" factors and their automorphisms bya crossed product decomposition

III" = II" /* Z

associated to an automorphism * of a II" factorsuch that

trace(*(T)) = " trace(T)

for all T . 0. These ideas are developed to theirfullest in Takesaki’s theory of crossed products andduality, to be discussed next . . .


Crossed Products* : G/M $ M, action of a locally compact abeliangroup on a von Neumann algebra M $ B(H).

Definition. The crossed product M /* G $B(L2(G,H)) is the von Neumann algebra generatedby

*(Tf)(h) = *h(T)f(h), T ( M

(Ugh)(h) = f(g ! h), g ( G,

or equivalently by L1(G,M) with the twistedmultiplication

F1 ! F2(g) =2

GF1(h)*h(F2(g ! h)) dh.

Definition. The dual action of G on M /* G is:*

+(T) = T, T ( M

+(Ug) = !+, g" · Ug, g ( G,

Takesaki’s Duality Theorem.

•'M / *G

(G != M

• M /* G /* G != M ) B(L2(G))


Duality and Modular Theory• ( : R / M $ M, modular group

• N = M /( R, crossed product

• ( : R+ / N $ N, dual action

Proposition. The kernel of the restricted action (on the center of N is precisely S(M).

Proposition. The von Neumann algebra N has afaithful and semifinite trace $, and moreover

$(("(T)) = "$(T),

for all T . 0 and all " ( R+.

Proof: Recall that N is generated by L1-functionsF : R $ M. If F has compactly supported Fouriertransform (so that F extends to an analytic functionon C) then define

$(F) = %(F(i)),

where % is the weight generating the modular flow(. By the KMS condition, this is a trace.


The Flow of Weights Mod(M)

The story so far . . .

M ! (M,%) ! (M,%,() ! (N, $, ()

From M, by choosing a faithful normal semifiniteweight % we have ultimately obtained a semifiniteN, with a flow ( scaling the trace $.

Crucial point: The data (N, $, () is determined byMalone.

Lemma. If M is a factor then the restriction of (to the center of N is ergodic (it fixes no nontrivialelement).

Definition. The flow of weights or module for M isthe restriction of ( to the center of N.

This is a refinement of S(M), which is the kernelof this action. Notice that if the action is transitivethen it is determined by its kernel. The intrasitiveactions are (by definition) virtual subgroups of R+


Construction of Factors

Let us begin with:

N, an approximately finite dimensional II" factor

* : R+ / N $ N, an automorphism group whichscales the trace on N.

S $ R+ a closed (virtual) subgroup

For a true subgroup S define

M = NS = { T ( N : *s(T) = T, 1s ( S }.

For a virtual subgroup R+ / A $ A define

M = NS = { T ( M ) A : *s(T) = T, 1s ( R+ }.

It will emerge that these are AFD factors with givenmodule S, and that these are all the infinite AFDfactors (excluding the trivial M = B(H)). Note theanalogy with Galois theory.


Injective FactorsDefinition. M $ B(H) is injective if there is acompletely positive splitting B(H) $ M.

Fairly easy: every AFD von Neumann algebra isinjective.

Stability properties: crossed products by abeliangroups, increasing unions (hence AFD algebras),decreasing intersections, direct integrals.

Observe that if M is injective and P ( M is aprojection then PMP is injective. If M is II" andP is finite then PMP is II1.

Theorem (Connes). There is a unique injective II1

factor.

Corollary. There is a unique injective (and henceAFD) II" factor.

Since injective type III factors are ‘reduced’ to II"algebras and automorphisms by modular theory, aclassification comes into view . . .


Classification of Injective Factors

Theorem. A factor is AFD if and only if it is injective.

Theorem. Type III factors which are injective/AFDare classified by their modules Mod(M).

This is largely due to Connes, but with essentialcontributions from Krieger (type III0) and Haagerup(type III1). Of course, the modular theory of Tomitaand Takesaki is also crucial.

The key methods, beyond those alreadysketched, build on ideas in ergodic theory to classifythe automorphisms of type II algebras.


Type II Di!erential Topology


[email protected], [email protected]

June 26, 2000

AbstractIn this lecture we showcase an early

application of noncommutative geometry —the proof of the Novikov Conjecture forGromov-hyperbolic groups. (Connes-Moscovici,Cyclic cohomology, the Novikov conjecture, andhyperbolic groups, Topology 29(1990), 345–388).


The index theoremTheorem 1. (Atiyah-Singer) Let D be the Diracoperator on a compact even-dimensional spin manifoldM ; then Index(D) =

!M

"A(M).

Heat equation proof. Let K+t and K!

t be the ±-gradedparts of the Heat operator e!tD2

. Then

Index(D) = TrK+t ! TrK!

t

for all values of time t. On the other hand as t " 0 theheat kernels K±

t localize near the diagonal in M # Mand there are asymptotic expansions of the form

tr Kt(x, x) $ am(x)t!m + am!1(x)t!m+1 + · · ·

where m = 12 dimM . The existence of this asymptotic

expansion is what is responsible for the analyticcontinuation of the ! function.

Atiyah-Patodi-Singer identified the di!erence of theterms a+

0 ! a!0 with the "A form. The proof was later

simplified by Getzler using a Cli!ord symbolic calculusfor pseudodi!erential operators on spinors.


Local index theorem

One can regard the heat equation formula asfollows: the index of D defines an element of K0(R),where R is the algebra of smoothing operators;and we are computing the index by pairing with[Tr] % HC0(R). Are there other interesting cycliccocyles for R?

Answer: No. (R is essentially a big matrix algebra.)

However, suppose that we pass to the sub ‘algebra’R! of operators of propagation < ". Every Alexander-Spanier cocycle c(x0, . . . , xq) on M defines a cycliccocyle #c on R! by sending (K0, . . . , Kq) to

# #c(x0, . . . , xq)K0(x0, x1)K1(x1, x2) · · ·Kq(xq, x0).

Theorem 2. (Connes-Moscovici) One has

#c"(IndexD) = const ·#

M

"A(M) $ c.

Proof is an application of the Getzler calculus.


Elliptic families

Let B be a space and let E & B be a fiber bundlewith manifold fibers. A family of elliptic operatorsalong the fibers has an index which belongs to K0(B).

Example (Lusztig) Let M be a torus, Rn/Zn, neven. The characters % of &1M form a group B (whichis also a torus!) Every such character corresponds to aflat line bundle. Let D be the signature operator of M ;then for each character % % B we may form the twistedsignature operator D", and in this way we obtain afamily D of elliptic operators parameterized by %. Thuswe get a ‘higher signature’ Index(D) % K0(B).

Theorem 3. The higher signature Index(D) is aninvariant of the oriented homotopy type of M .

Compare the homotopy invariance of the ordinarysignature. . .


Noncommutative families

For a di!erent view of Lusztig’s construction,consider the lift D of D to the universal coveringspace M of M . Let ! = &1(M) ( = Zm in thisexample.) Let $R(M) be the algebra of smoothingoperators on M which are of finite propagation andalso !-invariant (this is the groupoid algebra of thefundamental groupoid of M). Then D is invertiblemodulo $R(M). Consequently there is defined an indexIndex(D) % K0( $R(M)).

Lemma 1. The C"-algebra completion of $R(M), inthe norm topology of B(L2(M)), is Morita equivalentto C"

r (!).

In the case of the torus, C"r (!) = C(B) and the

index of D defined by operator algebras equals thehigher index in K0(B) defined on the previous slide.

Theorem 4. The higher signature Index(D) %K0(C"

r (!)) is an invariant of the oriented homotopytype of M for every fundamental group !.


The Novikov Conjecture

• Belongs to the realm of surgery theory, in which thefundamental question is: What is a manifold?

• Conjecture (and its relatives) reduce theclassification of manifolds with given fundamentalgroup ! to a question of relative homology.

The Novikov Conjecture is this: Let M be acompact manifold with fundamental group ! and letc % Hk(!; R) be a group cohomology class. Then theNovikov higher signature

'L(M) $ f"(c), [M ](, f :M & B!

is an invariant of oriented homotopy type.

Program of proof. Obtain the Novikov highersignatures from Index( $D) % K0(C"

r (!)).


Application of the local index theorem

Let c % Hk(!; R). Then f"(c) may be representedas an Alexander-Spanier cohomology class on M andtherefore defines a cyclic ‘cocycle’ 'c on R!(M) forsu"ciently small ".

Lemma 2. For su!ciently small " there is a1 : 1 correspondence between R!(M) and $R!(M).Moreover, under this 1 : 1 correspondence, 'c

corresponds to a genuine cyclic cocyle #c on the wholealgebra $R(M).

Hence

Theorem 5. For each group cocycle c for M there isa cyclic cocycle #c on $R(M) such that #c"(Index(D))is the Novikov higher signature 'L(M) $ f"(c), [M ](.


Almost a proof of Novikov

We have nearly proved the Novikov conjecture.

‘All’ we need to do is to fill in the dotted arrow inthe diagram below

K0( $R(M)) !!

#c"""!!!!!!!!!!!!!!!!!

K0(C"r (!))

##

R

Problem: The cocycle #c need not, a priori, be ak-trace.


Rapid decay

We ask: When is a sequence {cg}g#! the sequenceof Fourier coe"cients of some element of C"

r (!)?

For groups ! of polynomial growth it is clear thata su"cient condition is |cg| = O(|g|!N) for N largeenough. But such groups are not very interesting.

Haagerup (An example of a non nuclear C"-algebra which has the metric approximation property,Inventiones 50(1979) 279–293) proved: Let ! be a freegroup. Then there is an N > 0 such that if the functiong )& |cg||g|N belongs to (2(!), then

%cgg belongs to

C"r (!); in fact he estimated the C"

r (!) norm of thesum in terms of the (2 norm of g )& |cg|(1 + |g|)N .

A group enjoying this property is said to be of rapiddecay (RD).


Proof of RD

The key lemma in Haagerup’s proof of RD for thefree group is the following:

Lemma 3. Let ) and ( be given. There exists aconstant C > 0 such that, for any g % !, there are atmost C ways of decomposing g = g1g2 with |g1| = (and |g1| + |g2| * |g| + ).

Exercise: Prove this.

Gromov defined a hyperbolic group to be one inwhich all geodesic triangles are thin. Hyperbolic groupshave the large scale qualitative features of free groups.Jolissaint extended Haagerup’s calculation to prove

Theorem 6. Hyperbolic groups enjoy property RD.

Remark: Not all groups have RD — e.g. noexponential growth solvable group has RD.


Novikov for hyperbolic groups

Lemma 4. (Gromov) Every cohomology class for ahyperbolic group may be represented by a boundedcocycle.

Choose such a representative for c % H"(!; R).Then we have

Lemma 5. The homomorphism #c":K0( $R) & Rextends to a homomorphism K0(C"

r (!)) & R.

The proof uses Haagerup estimates to extendthe domain of #c to a dense subalgebra of C"

r (!)which is smooth (closed under holomorphic calculus).Consequently

Theorem 7. The Novikov conjecture holds forhyperbolic groups.


Discussion

This proof is a classic application of cyclic theoryto obtain maps which did not (a priori) come fromK-homology.

There now exist K-theoretic proofs of NCfor hyperbolic groups, using hypereuclideanness,contractibility of the Gromov compactification,amenability of the boundary action, . . . But these seemto use rather di!erent features of hyperbolic geometry— it is an open question to relate the key features ofthese proofs.

V La!orgue has shown that uniform lattices inSL(3, R) enjoy RD, and this is a key step in his proofof the Baum-Connes conjecture for such lattices. Onthe other hand SL(3, Z) does not enjoy RD since itcontains an exponential growth solvable subgroup.

Conjecture (Valette) All uniform lattices inSL(n, R) enjoy RD.


Type III Di!erential Topology



June 28, 2000

AbstractThis lecture is an exposition of the work

of Connes et al on the transverse fundamentalclass. Basic reference: Cyclic cohomology andthe transverse fundamental class of a foliation,in ‘Geometric methods in operator algebras’,Pitman Research Notes volume 123.


Topology of foliations

A foliation on a manifold M can be completelydescribed by the subbundle F ! TM of vectors thatare tangent to the leaves. However, not all subbundlescan occur, but only those that are integrable in on ofthe following equivalent senses:

• The sections of F form a Lie algebra of vector fields;

• The 1-forms annihilating F form a di!erential ideal.

Bott (around 1970) asked: Are there topologicalobstructions to integrability?

He proved the following theorem: If Q = TM/F isthe normal bundle to a foliation of codimension q, thenthe Pontrjagin ring of Q vanishes in degrees k > 2q.

This is carried out by means of Chern-Weil theoryapplied to a special connection — a basic connection— on q.


Secondary characteristic classes

Bott vanishing allows the construction of secondarycharacteristic classes of a foliation. Namely, one cancompare the representatives for primary characteristicclasses of Q built from a Riemannian connection, onthe one hand, and a basic connection, on the other.Using Bott vanishing one obtains canonical closedforms which implement homotopies between theserepresentatives — secondary characteristic classes.

Example Let F be a foliation of codimension one.Then it is defined by a single 1-form !. Frobenius’theorem gives d! = ! " " for some 1-form " (whichmay be interpreted as the connection form of a basicconnection). Then " " d" is a closed form (its exteriorderivative (d")2 is a Chern-Weil representative of p1

relative to a basic connection, hence vanishes.) Itscohomology class is independent of the choices and iscalled the Godbillon-Vey invariant GV # H3(M ; R).

(Thurston) GV measures ‘helical wobble’. It canvary continuously.


The transverse fundamental classproblem

Let M be a compact manifold. One can define an‘integration map’ in K-theory by

x $%!

Mchx, K!(M) % R.

This integration map is defined cohomologically. Onecould also find a family of integration maps reflectingthe various characteristic classes of TM .

Alternatively one could use index theory (K-homology) to produce an integration map.

Investigate a foliated manifold (M, F ), oralternatively an action of a group ! on V . In eithercase we would like to produce similar ‘integration maps’from K!(C!

r (M,F )) to C.


About the solution

This problem is solved using cyclic cohomology inthe transverse fundamental class paper. Moreover onecan reflect the secondary characteristic classes (likeGV ) of the transverse space in the integration map.The continuous variation of GV shows that these can’tarise from K-homology.

The problem is entirely one of analysis.

The natural approach would involve a ‘transverseDirac operator’, but this makes no sense without aninvariant (transverse) Riemannian structure. As wehave seen, the secondary classes (like GV) measure theobstruction to the existence of such a structure.

We are in a type III situation.


What use is the transverse fundamentalclass?

One needs (a) things to pair with it, and (b)methods to evaluate the result.

One pairs the transverse fundamental class withindices of elliptic operators (this can be systematizedby using the Baum-Connes assembly map). And oneevaluates the result by an appropriate index theorem.

Example Let M be a manifold provided with aspin foliation. Then there is a transverse fundamentalclass functional #:K!(C!(M, F )) % R such that#(IndexDF ) = "A(M), where DF is the leafwise Diracoperator. (Of course, the right hand side need not bean integer.)

Corollary If M has nonzero "A genus, then itadmits no spin foliation whose leaves are of positivescalar curvature.

Challenge: Give an elementary proof.


The cocycle

It is easy to produce a cocyle on the smoothgroupoid algebra which represents the transversefundamental class. In fact, it is just given by anequivariant version of the usual de Rham formula forthe fundamental class:

(f0, . . . , fq) $%!

Vf0df1 · · · dfq.

Here we must take the f ’s to be elements of thegroupoid algebra C"(V ) ! !; the operations takeplace in the algebra "!(M) ! ! whose elements areformal sums

#!gUg, the ! being di!erential forms:

and the algebraic rules are

• (#

!gUg)(#

$hUh) =#

!g " g!($h)Ugh;

• d(#

!gUg) =#

(d!g)Ug;

•$(#

!gUg) =$!e.


The extension problem

This cocycle gives a homomorphism K!(C"c (G)) %

C, we need to extend it to a homomorphismK!(C!

r (G)) % R. (Compare lecture 7.)

First attempt is the theory of q-traces (lecture 4).These are cocycles for which one has an estimate like

|%(a1dx1 · a2dx2 · · · aqdxq)| ! Cx1,...,xq&a1& · · · &aq&.

(the left side can be interpreted in terms of themultilinear functional %).

Theorem 1. A q-trace on a dense subalgebra of aBanach algebra extends to define a homomorphismfrom the K-theory of the whole algebra to R.

This theory solves the problem for q = 1.


An awkward example

Let ! = Z acting on V = T2 via the hyperbolic

matrix $ =%

1 11 2

&. (Notice that the crossed

product is then the C!-algebra of an exponentialsolvable group Z2 ! Z.)

Then the transverse cocycle is not a 2-trace. Thisresults from the exponential growth of the powers of$ which are introduced when one tries to commute x1

and a2 in the expression %(a1dx1a2dx2).

Thus one cannot expect that the theory of q-traceswill be adequate by itself.


Triangular structures

It is easy to deal with the case when ! acts byisometries (invariant Riemannian structure).

Next simplest case — ! acts by triangular

matrices

%A B0 D

&, where A and D are orthogonal

transformations. We refer to a triangular structure.(May also assume the A direction is integrable. . . )

Theorem 2. The transverse fundamental class canbe defined in the case that ! preserves a triangularstructure.

Even in this case the proof is highly non-trivial. Ituses the theory of q-traces, applied not to the C!-algebra but to a Banach subalgebra B defined as thedomain of a certain unbounded operator (generalizedmodular operator); and B is shown to be smoothbecause of the existence of the triangular structure.


Some K-theoretic technology

Recall the Thom isomorphism in K-theory: if Wis, say, a complex vector bundle over B then there isan isomorphism K!(B) '= K!(W ).

Proof: The isomorphism is given by taking theproduct with the ‘Bott element’ & # K0(Cn), and theinverse isomorphism by taking the product with the‘Dirac operator’ D # K0(Cn).

In his work on the Novikov Conjecture, Kasparovgave an abstract theory of such ‘Dirac’ and ‘dual Dirac’elements. One need not restrict attention to vectorbundles; one can build them for any bundle whosefibers are contractible nonpositively curved spaceson which the structure group acts by orientation-preserving isometries.


Transfer to the bundle of metrics

Let ! act on V as usual, and let W be the bundleover V whose fiber at x is the space of positivedefinite quadratic forms on TxV . The fiber of W is anonpositively curved symmetric space GL+(q)/SO(q),and ! acts on it. Kasparov theory then provides aThom map

K!(C(V ) ! !) % K!(C0(W ) ! !).

Moreover

Lemma 1. The action of ! on W preserves atriangular structure.

Now we may define the fundamental class in generalby first using the Thom map to reduce to the casewhere a triangular structure is preserved, and thenusing Theorem 1.

Remarks: (1) The construction is inspired by themodular theory of type III von Neumann algebras; (2)There is also an analogy to the ‘transfer to the discbundle’ used by Farrell-Jones.


Hypoelliptic theory

There is a way to define the transverse fundamentalclass as a K-homology cycle, using the theory ofhypoelliptic operators (example: the heat operator).This approach also proceeds first by using the transferto the bundle of metrics to show that we may assumean invariant triangular structure.

Basic idea: as well as the familiar ‘first ordersignature operator’ d + d! on a manifold M , thereis also a ‘second order signature operator’ dd! ( d!dwhich has the same index.

If M is a triangularized manifold we may nowdefine a ‘mixed signature operator’ which is equal tothe second order operator in the A directions plusthe first order operator in the D directions. Thenthe deviations from ! invariance become lower orderterms, and we can define a K-homology class.

Ref: Hilsum and Skandalis, Morphismes K-orientesd’espaces de feuilles et fonctorialite en theorie deKasparov, Ann Sc Ec Norm Sup 20(1987), 325–390.


The Godbillon-Vey Class

Our construction of the fundamental class startedwith a !-invariant closed current on V .

Such a current is a particular example of a cocylein the groupoid de Rham cohomology of G = V ! !.All such cocyles give cyclic classes for the groupoidalgebra, and a particular example is the ‘Jacobiancocycle’ (we assume codimension one)

'(() = log'Transverse distortion

(.

This cocycle gives rise to a ‘Godbillon-Vey transversefundamental class’.

Theorem 3. The GV transverse fundamental classcan be reconstructed from the action of the modulargroup on the standard transverse fundamental class.

Corollary 1. Suppose GV )= 0. Then the vonNeumann algebra of the foliation is of type III (infact one can prove it is of type III0).


Metric geometry and local indexformulae



June 29, 2000

AbstractIn this last talk we’ll summarize some

of Connes’ ideas about the metric aspect ofnoncommutative geometry and associated localindex formulae. Main Reference: Connesand Moscovici, The local index formula innoncommutative geometry.


Metrics and noncommutative geometry

What does it mean to give a metric on anoncommutative space described by an algebra A?

According to Connes, this data is given by a spectraltriple (a.k.a. unbounded Fredholm module), thatis, a representation ρ:A → B(H) together with anunbounded selfadjoint operator D on H, with compactresolvent, such that [D, ρ(a)] is bounded for a denseset of a ∈ A.

Why? Consider the example of a Riemannianmanifold M . Then one can recover the distancebetween two points x, y of M as

sup{|f(x)− f(y)| : ‖[D, ρ(f)]‖ ≤ 1}.

Passing from D to F = D|D|−1 should be thoughtof as passing from metric to conformal structure.


Summability

An unbounded Fredholm module is p-summable ifthe resolvents (D ± i)−1 belong to the Schatten idealLp. (This implies the corresponding condition for F .)Exercise: The unbounded Fredholm module definedby the length function on a group is finitely summableif and only if the group has polynomial growth.

We will focus attention on p-summable modules.But one should not think that this condition isuniversal. In fact one has:

Theorem 1. If Γ is a nonamenable group then thereare no finitely summable unbounded Fredholm moduleson C∗

r (Γ).

There do exist bounded p-summable Fredholmmodules on C∗

r (Γ) even for some property T groups Γ.Question: Are there examples in rank > 1?


Constructing the character

For a finitely summable bounded Fredholm modulewe have seen how to construct a cyclic character:

τq(a0, a1, . . . , aq) = Tr′(a0[F, a1] · · · [F, aq]).

Recall that this construction produces many charactersfor the same Fredholm module, which are related bythe periodicity operator S:HCq(A) → HCq+2(A).Ask: What is the lowest dimension in which the cycliccharacter exists?

By Connes’ exact sequence to ask whether τq canbe ‘desuspended’ is the same as to ask whether theimage I(τq) in Hochschild cohomology HHq(A) iszero. How to calculate this image?

The answer involves the Dixmier trace.


The Dixmier trace

Let H be a Hilbert space. The Macaev ideal M isthe collection of all compact operators T on H whosecharacteristic values µi (the eigenvalues of |T |) satisfy

supN

1log N

N∑i=1

µi < ∞.

Define the Dixmier trace

Trω(T ) = limω

1log N

N∑i=1

λi

where ω is a (suitable?) ultrafilter on N. This formuladefines a linear (Surprise!) functional on M, which isclearly a trace. Moreover it annihilates all operators inthe ordinary trace class L1.

(Connes denotes M by L1,∞.)


Wodzicki residue

One can show that a pseudodifferential operatorof order −n (for instance (1 + D2)−n/2) on an n-dimensional manifold belongs to the ideal M. (Lookat the circle for an example.)

Theorem 2. Let P be such a pseudodifferentialoperator with principal symbol σ(x, ξ) (a homogeneousfunction of degree −n on T ∗M .) Then (for any choiceof the ultrafilter ω),

Trω(P ) = cn

∫S∗M

σdv.

(In other words, the trace depends only on the principalsymbol. The right hand side is the Wodzicki residue;Wodzicki extended it to a trace on the algebra of allpseudodifferential operators. It is the residue at s = 0of the meromorphic function s 7→ Tr(P∆−s/2). )

Proof Vanishing on L1 and symmetry.

Key Point: The symbol is a local invariant.


Classical Limit v. Dixmier Trace

Let F be the Fredholm module defined by the Diracoperator D over a spin manifold Mn. It is p-summablefor all p > n (as we saw above), so it has a characterin HCn(C∞(M)).

Recall from lecture 4 that

HCn(C∞(M)) = Zn(M)⊕Hn−2(M)⊕Hn−4(M)⊕· · · .

Theorem 3. The character of the Dirac modulehas top-dimensional component equal to the current(f0, . . . , fn) =

∫M

f0df1 · · · dfn, and the homologycomponents are those of the Poincare dual of theA-genus.

The proof uses the classical limit. Connes’ idea:Replace this by the Dixmier trace.


The residue theorem for the cycliccharacter

Let (ρ,H, D) define an unbounded Fredholmmodule over A, and suppose that this module isLn,infty summable (in the obvious sense).

Then for a0, . . . , an ∈ A the operatora0[D, a1] · · · [D, an]|D|−n belongs to the Macaev idealM, and

φω(a0, . . . , an) = Trω

(a0[D, a1] · · · [D, an]|D|−n

)defines a Hochschild cocycle on A.

Theorem 4. This Hochschild cocycle ‘is’ the imageof the n-dimensional cyclic character τn under themap I:HCn(A) → HHn(A). Consequently, if thecohomology class of φω is not zero, then τn cannot bedesuspended — it does not belong to the image of S.

Proof Exercise?! An easier exercise — compute inthe manifold example.


Dimension spectrum

To get a more refined formula one needs toincorporate lower order terms in asymptotic expansion.

Definition 1. The dimension spectrum of a spectraltriple (if it exists) is a discrete subset Σ ⊆ C with theproperty that all the zeta functions

ζb(s) = Tr(b|D|−s),

where b is a useful operator, continue analytically toC \ Σ.

(The ‘useful’ operators are just all the things like[|D|, a] which are going to appear in the formulae.)We’ll assume that the continuations have simple poleson Σ; there is a more elaborate version of the theorywhich allows for multiple poles.

Example: The Hilsum-Skandalis hypoellipticoperator corresponding to a triangular structure.


Extended Dixmier trace

One can define a notion of abstractpseudodifferential operator, given by expansions like

bq|D|q + bq−1|D|q−1 + · · ·

where the b’s are useful operators. For any suchpseudodifferential operator P , the function ζP (s) =Tr(P |D|−s) is meromorphic on C, and the residueχ0(P ) of ζP (s) at zero is an extension of the Dixmiertrace on M to all pseudodifferential operators —exactly as in Wodzicki’s work.

Remark: The above is really true only in the caseof simple poles. For multiple poles there is a moreelaborate theory involving all the negative coefficientsχ−i−1(P ) in the Laurent series; they are related bya bunch of combinatorial identities of which only thehighest one is the trace property.


Local index formula

Theorem 5. In the situation above the followingformulae define a cocycle in the (b, B)-bicomplex ofA. Moreover the cyclic class of this cocycle coincideswith the cyclic character of D.

In degree n we have up to a universal constant∑q≥0,kj≥0

ck,qχq

(a0∇k1(da1) · · ·∇kn(dan)|D|−(n+2

Pkj)

),

where da = [D, a] and ∇a = [D2, a].

According to Connes-Moscovici it is an easycalculation using the Getzler calculus to show that thisformula defines the A genus in the ordinary manifoldcase. Here the terms with k > 0 do not contribute.


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