noncommutative geometry and motives (the thermodynamics of endomotives). alain connes, et. al..pdf

8/11/2019 NONCOMMUTATIVE GEOMETRY and MOTIVES (THE THERMODYNAMICS OF ENDOMOTIVES). ALAIN CONNES, et. al..pdf

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NONCOMMUTATIVE GEOMETRY AND MOTIVES:

THE THERMODYNAMICS OF ENDOMOTIVES

ALAIN CONNES, CATERINA CONSANI, AND MATILDE MARCOLLI

Contents

1. Introduction 22. Correspondences for noncommutative spaces 52.1. KK-theory 52.2. The abelian category of -modules and cyclic (co)homology 63. Artin Motives and noncommutative spaces 93.1. Complex coefficients 93.2. Artin motives 103.3. Endomotives 114. Scaling as Frobenius in characteristic zero 194.1. Preliminaries 204.2. The cooling morphism 214.3. The distilled -moduleD(A, ) 244.4. Dual action on cyclic homology 284.5. Type III factors and unramified extensions 335. Geometry of the adele class space 355.1. The spectral realization onH1K,C 355.2. The Trace Formula 375.3. The classical points ofXK\CK 385.4. Weil positivity 396. Higher dimensional virtual correspondences 406.1. Geometric correspondences andK K-theory 416.2. Cycles and K-theory 416.3. Algebraic versus topologicalK-theory 43

7. A Lefschetz formula for archimedean local factors 437.1. Weil form of logarithmic derivatives of local factors 457.2. Lefschetz formula for complex places 487.3. Lefschetz formula for real places 487.4. The question of the spectral realization 507.5. Local factors for curves 51References 52

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2 CONNES, CONSANI, AND MARCOLLI

1. Introduction

A few unexpected encounters between noncommutative geometry and the theory of motiveshave taken place recently. A first instance occurred in [13], where the Weil explicit formulaeacquire the geometric meaning of a Lefschetz trace formula over the noncommutative spaceof adele classes. The reason why the construction of [13] should be regarded as motivic istwofold. On the one hand, as we discuss in this paper, the adele class space is obtainedfrom a noncommutative space, the BostConnes system, which sits naturally in a categoryof noncommutative spaces extending the category of Artin motives. Moreover, as we discussbriefly in this paper and in full detail in our forthcoming work [14], it is possible to give

a cohomological interpretation of the spectral realization of the zeros of the Riemann zetafunction of [13] on the cyclic homology of a noncommutative motive. The reason why thisconstruction takes place in a category of (noncommutative) motives is that the geometricspace we need to use is obtained as a cokernel of a morphism of algebras, which only existsin a suitable abelian category extending the non-additive category of algebras, exactly as inthe context of motives and algebraic varieties.There are other instances of interactions between noncommutative geometry and motives,some of which will in fact involve noncommutative versions of more complicated categories ofmotives, beyond Artin motives, involving pure and sometimes mixed motives. One exampleis the problem of extending the results of [13] toL-functions of algebraic varieties. The latterare, in fact, naturally associated not to a variety but to a motive. Already in [25] it wasshown that noncommutative geometry can be used to model the geometry of the fibers atthe archimedean places of arithmetic varieties. This suggested the existence of a Lefschetz

trace formula for the local L-factors, and at least a semi-local version for the L-function,over a noncommutative space obtained as a construction over the adele class space. In thispaper we give such a Lefschetz interpretation for the archimedean local factors introducedby Serre [47]. In order to obtain a similar Lefschetz formula for the case of several places(archimedean and non-archimedean), it seems necessary to develop a theory that combinesnoncommutative spaces and motives.A further example of an intriguing interaction between noncommutative geometry and mo-tives appears in the context of perturbative renormalization in quantum field theory. Theresults of [18], [17] show the existence of a universal group of symmetries that governsthe structure of the divergences in renormalizable quantum field theories and contains therenormalization group as a 1-parameter subgroup. This universal group is a motivic Galoisgroup for a category of mixed Tate motives. On the other hand, the treatment of divergentFeynman integrals by dimensional regularization, used in this geometric reformulation of

renormalization, is best described in the framework of noncommutative geometry. In fact,using the formalism of spectral triples in noncommutative geometry, one can construct ( cf.[19]) spaces with dimension a complex number z and describe geometrically the procedureof dimensional regularization as a cup product of spectral triples. Once again it is desirableto develop a common framework for noncommutative geometry and motives, so as to obtaina more direct identification between the data of perturbative quantum field theory and theobjects of the category of mixed motives governed by the same Galois group.In a different but not unrelated perspective, Kontsevich recently found another very inter-esting link between noncommutative geometry and motives [37].

The present paper is structured as follows. We begin by discussing the problem of morphismsfor noncommutative spaces. It is well known, in fact, that morphisms of algebras are not

sufficient and one needs a larger class of morphisms that, for instance, can account for thephenomenon of Morita equivalence. We argue that there are, in fact, two well developedconstructions in noncommutative geometry that make it possible to define morphisms as


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NCG AND MOTIVES 3

correspondences, namely Kasparovs bivariant KK-theory on one hand and modules overthe cyclic category (and cyclic (co)homology) on the other.

In KK-theory, one extends the notion of morphism to that of (virtual) correspondencesgiven by formal differences of correspondences defined using bimodules. This yields anadditive category containing the category of C-algebras. The product (composition ofcorrespondences) plays a central role in the understanding ofK-theory ofC-algebras. Inthe approach via cyclic cohomology, one constructs a functorA A from the category ofalgebras and algebra homomorphisms to the abeliancategory of -modules introduced in[10]. Cyclic cohomology and homology appear then as derived functors, namely as the Extand Tor functors, much as in the case of the absolute cohomology in the theory of motives.Working with the resulting abelian category has the advantage of making all the standardtools of homological algebra available in noncommutative geometry. It is well known thatone obtains this way a good de Rham (or crystalline) theory for noncommutative spaces, ageneral bivariant cyclic theory, and a framework for the Chern-Weil theory of characteristicclasses for general Hopf algebra actions [22].

We show, first in the zero-dimensional case of Artin motives and then in a more generalsetting in higher dimension, that the notion of morphisms as (virtual) correspondencesgiven by elements in K K-theory is compatible with the notion of morphisms as used in thetheory of motives, where correspondences are given by algebraic cycles in the product. Infact, we show correspondences in the sense of motives give rise to elements in KK-theory(using Baums geometric correspondences [4] and [23]), compatibly with the operation ofcomposition.

We focus on the zero-dimensional case of Artin motives. We extend the categoryCV0K,E ofArtin motives over a field K with coefficients in a field E to a larger (pseudo)abelian cate-goryEV0K,E ofalgebraic endomotives, where the objects are semigroup actions on projectivesystems of Artin motives. The objects are described by semigroup crossed product algebrasAK = A S over K and the correspondences extend the usual correspondences of Artinmotives given by algebraic cycles, compatibly with the semigroup actions.We prove then that, in the case of a number field K, taking points over K in the projectivelimits of Artin motives gives a natural embedding of the category of algebraic endomotivesinto a (pseudo)abelian category CV0K of analytic endomotives. The objects are noncom-mutative spaces given by crossed product C-algebras and the morphisms are given bygeometric correspondences, which define a suitable class of bimodules. The Galois groupGacts as natural transformations of the functor Frealizing the embedding. The objects in theresulting category of analytic endomotives are typically noncommutative spaces obtained asquotients of semigroup actions on totally disconnected compact spaces.We show that important examples like the BostConnes (BC) system of [7] and its general-ization for imaginary quadratic fields considered in [20] belong to this class. We describe alarge class of examples obtained from iterations of self-maps of algebraic varieties.

We also show that the category of noncommutative Artin motives considered here is espe-cially well designed for the general program of applications of noncommutative geometry toabelian class field theory proposed in [16], [21]. In fact, a first result in the present papershows that what we called fabulous states property in [16], [21] holds for any noncommu-tative Artin motive that is obtained from the action of a semigroup S by endomorphismson a direct limit A of finite dimensional algebras that are products of abelian normal fieldextensions of a number field K. Namely, we show that there is a canonical action of the ab-solute Galois groupGofKby automorphisms of a C-completion

Aof the noncommutative

spaceAK= ASand, for pure states on this C-algebra which are induced from A, onehas the intertwining property

(a) =((a))


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for all a A Sand all G/[G, G].When passing from commutative to noncommutative algebras, new tools of thermodynam-ical nature become available and make it possible to construct an action ofR+ on cyclichomology. We show that, when applied to the noncommutative space (analytic endomotive)F(M) associated to an algebraic endomotiveM, this representation ofR+ combines with arepresentation of the Galois group G. In the particular case of the endomotive associatedto the BC system, the resulting representation ofG R+ gives the spectral realization ofthe zeros of the Riemann zeta function and of the Artin L-functions for abelian charactersofG. One sees in this example that this construction plays a role analogous to the action

of the Weil group on the -adic cohomology and it can be thought of as a functor from thecategory of endomotives to the category of representations of the group G R+. In fact,here we think of the action ofR+ as a Frobenius in characteristic zero, hence ofG R+as the corresponding Weil group.Notice that it is the type III nature of the BC system that is at the root of the thermody-namical behavior that makes the procedure described above nontrivial. In particular, thisprocedure only yields trivial results if applied to commutative and type II cases.The construction of the appropriate motivic cohomology with the Frobenius action ofR+ for endomotives is obtained through a very general procedure, which we describe herein the generality that suffices to the purpose of the present paper. It consists of three basicsteps, starting from the data of a noncommutative algebraAand a state , under specificassumptions that will be discussed in detail in the paper. One considers the time evolutiont

Aut

A, t

R naturally associated to the state (as in [9]).

The first step is what we refer to as cooling. One considers the spaceE of extremal KMSstates, for greater than critical. Assuming these states are of type I, one obtains a mor-phism

: AR S(ER+) L1,whereA is a dense subalgebra of a C-algebra A, and whereL1 denotes the ideal of traceclass operators. In fact, one considers this morphism restricted to the kernel of the canonicaltrace on AR.The second step is distillation, by which we mean the following. One constructs the -module D(A, ) given by the cokernel of the cyclic morphism given by the composition of with the trace Trace : L1 C.The third step is thedual action. Namely, one looks at the spectrum of the canonical action

ofR+ on the cyclic homology

HC0(D(A, )).This procedure is quite general and applies to a large class of data (A, ) and producesspectral realizations of zeros ofL-functions. In particular, we can apply it to the analyticendomotive describing the BC-system. In this case, the noncommutative space is that ofcommensurability classes ofQ-lattices up to scaling. The latter can be obtained from theaction of the semigroup N on Z. However, the algebraic endomotive corresponding to thisaction is not isomorphic to the algebraic endomotive of the BC system, which comes fromthe algebra Q[Q/Z]. In fact, they are distinguished by the Galois action. This importantdifference is a reflection of the distinction between Artin and Hecke L-functions.In this BC system, the corresponding quantum statistical system is naturally endowed withan action of the group Gab = G/[G, G], where G is the Galois group ofQ. Any Hecke

charactergives rise to an idempotent pin the ring of endomorphisms of the 0-dimensionalnoncommutative motiveXand when applied to the range ofp the above procedure givesas a spectrum the zeros of the Hecke L-functionL.


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NCG AND MOTIVES 5

We also show that the dualization step, i.e. the transition fromA toA R, is a verygood analog in the case of number fields of what happens for a function field Kin passingto the extension KFq Fq. In fact, in the case of positive characteristic, the unramifiedextensionsKFq Fqn combined with the notion of places yield the points C(Fq) overFq ofthe underlying curve. This has a good parallel in the theory of factors, as we are going todiscuss in this paper. In our forthcoming work [14] this analogy will be developed furtherand will play an important role in developing a setting in noncommutative geometry thatparallels the algebro-geometric framework that Weil used in his proof of RH for functionfields.In the results discussed here and in [14], instead of working with the Hilbert space context of[13] the trace formula involved in the spectral realization of [13] is given in suitable nuclearfunction spaces associated to algebras of rapidly decaying functions, as in the approach of[43].

We also give an overview of some results that will be proved in full detail in [14], wherewe give a cohomological interpretation of the spectral realization of the zeros of the zetafunction as the cyclic homology of a suitable noncommutative space MK. The sought forspaceMK appears naturally as the cokernel of the restriction morphism of functions on theadele class space XK to functions on the cooled down subspace CK of idele classes. Thisoperation would not make sense in the usual category of algebras (commutative or not) andalgebra homomorphisms, since this category is not additive and one cannot take cokernels,whereas this can be done in a suitable category of motives.

Acknowledgement. This research was partially supported by the third authors Sofya

Kovalevskaya Award and by the second authors NSERC grant 024520. Part of this workwas done during a visit of the first and third authors to the Kavli Institute in Santa Barbara,supported in part by the National Science Foundation under Grant No. PHY99-07949, andduring a visit of the first two authors to the Max Planck Institute.

2. Correspondences for noncommutative spaces

It is often regarded as a problem in noncommutative geometry to provide a good notionof morphisms for noncommutative spaces, which accounts, for instance, for phenomenasuch as Morita equivalence. Moreover, it is often desirable to apply to noncommutativegeometry the tools of homological algebra, which require working with abelian categories.The category of algebras with algebra homomorphisms is not satisfactory for both of the

reasons just mentioned. We argue here that the best approach to dealing with a categoryof noncommutative spaces is an analog of what happens in algebraic geometry, when oneconsiders motives instead of algebraic varieties and defines morphisms as correspondences.In the context of noncommutative geometry, there are well developed analogs of corre-spondences and of motivic cohomology. For the purposes of this paper we consider cor-respondences given by Kasparovs KK-theory (cf. [36]), although we will see later that amore refined version is desirable, where for instance one does not mod out by homotopyequivalence, and possibly based on a relative version in the fibration relating algebraic andtopologicalK-theory (cf. [15]).The analog of motivic and absolute cohomology is played by modules over the cyclic categoryand cyclic cohomology (cf.[10], [11]).

2.1. KK-theory. Similarly to what happens in the context of algebraic varieties, C-

algebras do not form an additive category. However, there is a functor from the categoryof all separable C-algebras, with-homomorphisms, to an additive categoryKK, whoseobjects are separableC-algebras and where, for A and B inObj(KK), the morphisms are


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given by Hom(A, B) =K K(A, B). Here K K(A, B) is Kasparovs bivariantK-theory ([36],cf. also8 and9.22 of [5]). Here the notation A for C-algebras comes from the factthat, later on in this paper, we will consider C-algebras that areC-completions of certainnatural dense subalgebrasA, though this assumption is not needed in this subsection.In theKKcategory morphisms are defined through correspondences in the following way.One considers the setM(A, B) of Kasparov modules, that is, of triples (E , , F ), whereEisa countably generated Hilbert module over B,is a -homomorphism ofA to bounded linearoperators onE(i.e.it gives Ethe structure of an (A, B)-bimodule) andFis a bounded linearoperator on Ewith the properties that the operators [F, (a)], (F21)(a), and (FF)(a)are in

K(E) (compact operators) for all a

A. Recall here that a Hilbert moduleE over

B is a right B-module with a B-valued inner product which satisfiesx,yb =x, yb forall x, y E and b B, and with respect to which E is complete. Kasparov modules areMorita type correspondences that generalize-homomorphisms ofC-algebras (the latterare trivially seen as Kasparov modules of the form (B, , 0)).One then considers on M(A, B) the equivalence relation of homotopy. Two elements are ho-motopy equivalent (E0, 0, F0) h(E1, 1, F1) if there is an element (E , , F ) ofM(A, IB),where IB ={f : [0, 1] B | f continuous}, such that (Efi B, fi, fi(F)) is unitarilyequivalent to (Ei, i, Fi), i.e. there is a unitary in bounded operators from Efi B to Eiintertwining the morphisms fi and i and the operators fi(F) and Fi. Herefi is theevaluation at the endpoints fi : IB B. By definitionKK(A, B) is the set of homotopyequivalence classes ofM(A, B). This is naturally an abelian group.A slightly different formulation ofK K-theory, which simplifies the external tensor productKK(

A,

B)

KK(

C,

D)

KK(

A

C,

B

D), is obtained by replacing in the data (E , , F )

the operator F by an unbounded regular self-adjoint operator D . The correspondingF isthen given byD(1 + D2)1/2 (cf.[3]).The categoryKK obtained this way is a universal enveloping additive category for thecategory ofC-algebras (cf. [5]9.22.1). Though it is not an abelian category, it is shownin [44] thatKKand the equivariantKKG admit the structure of a triangulated category.In algebraic geometry the category of pure motives of algebraic varieties is obtained byreplacing the (non-additive) category of smooth projective varieties by a category whoseobjects are triples (X,p,m) withXa smooth projective algebraic variety, p a projector andman integer. The morphisms

Hom((X,p,m), (Y, q, n)) = qCorrnm(X, Y)p

are then given by correspondences (algebraic cycles in the product XY) modulo a suitableequivalence relation. (Notice that the idempotent conditionp

2

= p itself depends on theequivalence relation.) Depending on the choice of the equivalence relation, one gets cate-gories with rather different properties. For instance, only the numerical equivalence relation,which is the coarsest among the various relations considered on cycles, is known to producein general an abelian category (cf.[35]). The homological equivalence relation, for example,is only known to produce a Q-linear pseudo-abelian category, that is, an additive categoryin which the ranges of projectors are included among the objects. One of Grothendiecksstandard conjectures would imply the equivalence of numerical and homological equivalence.In the case of (noncommutative) algebras, when we use KK-theory, the analysis of thedefect of surjectivity of the assembly map is intimately related to a specific idempotent (the-element) in aK K-theory group. Thus the analogy with motives suggests that one shouldsimilarly enlarge the additive categoryKK of C-algebras with ranges of idempotents inKK. This amounts to passing to the pseudo-abelian envelope of the additive

KKcategory.

2.2. The abelian category of-modules and cyclic (co)homology. We now interpretcyclic cohomology in a setting which is somewhat analogous to the one relating algebraic


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NCG AND MOTIVES 7

varieties and motives. The category of pure motivesM(with the numerical equivalence) isa semi-simple abelian Q-linear category with finite dimensional Hom groups. In the theoryof motives, to a smooth projective algebraic variety Xover a field K one wants to associateamotivic cohomologyHimot(X), defined as an object in category MK,i.e.Himot(X) is a puremotive of weighti (cf. [28]). In this way, one views pure motives as a universal cohomologytheory for algebraic varieties. Namely, the motivic cohomology has the property that, for anygiven cohomology theory H, with reasonable properties, there exists a realization functorRHsatisfying

(2.1) Hn(X) = RHHnmot(X).

On a category of mixed motives, one would also have absolute cohomologygroups given by(cf.[28])

(2.2) Hiabs(M) = Exti(1, M).

for Ma motive. Here the Exti are taken in a suitable triangulated categoryD(K) whoseheart is the category of mixed motives. For a variety Xthe absolute cohomology wouldthen be obtained from the motivic cohomology via a spectral sequence

(2.3) Epq2 =Hpabs(H

qmot(X)) Hp+qabs (X).

In the context of noncommutative geometry, we may argue that the cyclic category andcyclic (co)homology can be thought of as an analog of motivic and absolute cohomology.In fact, the approach to cyclic cohomology via the cyclic category (cf. [10]) provides a wayto embed the nonadditive category of algebras and algebra homomorphisms in an abeliancategory of modules. The latter is the category ofK()-modules, where is the cycliccategory and K() is the group ring of over a given field K. Cyclic cohomology is thenobtained as an Ext functor (cf. [10]).The cyclic category has the same objects as the small category of totally ordered finitesets and increasing maps, which plays a key role in simplicial topology. Namely, has oneobject [n] for each integer n, and is generated by faces i : [n 1][n] (the injection thatmissesi), and degeneracies j : [n+ 1][n] (the surjection which identifies j with j + 1),with the relations

(2.4) j i = ij1 fori < j, j i= ij+1, i j

(2.5) ji = ij1 i < j1n ifi = j ori = j + 1

i1j i > j+ 1.

To obtain the cyclic category one adds for each n a new morphism n : [n] [n], suchthat

(2.6)

ni= i1n1 1 i n, n0 = nni= i1n+1 1 i n, n0 = n2n+1n+1n = 1n.

Alternatively, can be defined by means of its cyclic covering, through the category .The latter has one object (Z, n) for each n0 and the morphisms f : (Z, n)(Z, m) aregiven by non decreasing maps f :ZZ , such thatf(x+n) =f(x) +m , xZ. Onehas =/Z, with respect to the obvious action ofZ by translation.

To any algebra A over a field K, one associates a module A over the category by assigningto eachn the (n + 1) tensor powerA A A. The cyclic morphisms correspond to thecyclic permutations of the tensors while the face and degeneracy maps correspond to the


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algebra product of consecutive tensors and the insertion of the unit (cf. [10]). A trace: A K gives rise to a morphism ofA to the -module K such that(2.7) (a0 an) = (a0 an).Using the natural isomorphism of with the opposite small category o one obtains bydualization that any algebra A gives rise canonically to a moduleA over the small category, by assigning to each integern 0 the vector space Cn ofn + 1-linear forms(x0, . . . , xn)onA, with the basic operationsi : Cn1 Cn andi : Cn+1 Cn given by

(2.8)

(i)(x0, . . . , xn) = (x0, . . . , xixi+1, . . . , xn), i= 0, 1, . . . , n 1

(n)(x0, . . . , xn) = (xnx0, x1, . . . , xn1)

(j )(x0, . . . , xn) = (x0, . . . , xj , 1, xj+1, . . . , xn), j = 0, 1, . . . , n

(n)(x0, . . . , xn) = (xn, x0, . . . , xn1).

These operations satisfy the relations (2.4) (2.5) and (2.6). This shows that A is a -module.It is then possible (cf.[10], [40]), to interpret the cyclic cohomology H Cn as Extn functors,namely

(2.9) HCn(A) = Extn(A,K) = Extn(K, A),for A an algebra over K. Both terms express the derived functor of the functor which assignsto a -module its space of traces.

The formula (2.9) may be regarded as an analog of (2.2), except for the fact that here theExtn are taken in the abelian category ofK()-modules.Similarly to (2.9), one also has an interpretation of cyclic homology as Tor n functors,

(2.10) HCn(A) = Torn(K, A) .A canonical pro jective biresolution ofZ is obtained by considering a bicomplex of -modules(Cn,m, d1, d2) (for n,m > 0) where Cn,m = Cm for all n and the component Ckj ofC

k is

the free abelian group on the set Hom(k, j ) so that Hom(Ck, E) =Ek. See [10] for thedifferentials d1 : Cn

+1,m Cn,m and d2 : Cn,m+1 Cn,m. This projective resolution ofZ determines a bicomplex computing the cyclic cohomology Extn(Z, E) of a -moduleEanalogous to the (b, B)-bicomplexCnm(A).All of the general properties of cyclic cohomology such as the long exact sequence relatingit to Hochschild cohomology are shared by Ext of general -modules and can be attributed

to the equality of the classifying spaceB of the small category with the classifying spaceBS1 of the compact one-dimensional Lie group S1. One has

(2.11) B =BS1 = P(C)

This follows from the fact that Hom(n, m) is nonempty and that Hom(0, 0) = 1, whichimply that B is connected and simply connected, and from the calculation (cf. [10]) ofExt(Z

,Z) = Z[] as a polynomial ring in the generator of degree two. The cohomologyringH(B,Z) is given by Z[] where corresponds to a map f :B P(C) which is ahomotopy equivalence.

Finally we note that there is a natural way to associate a cyclic morphism to a correspondenceviewed as anAB bimoduleE, provided that the following finiteness holds :Lemma 2.1. Let

E be an

AB

bimodule, which is finite projective as a rightB

-module.Then there is a canonically associated cyclic morphism

E Hom(A, B).


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NCG AND MOTIVES 9

Indeed the left action ofA gives a homomorphism fromA to EndB(E) andE is thecomposition of with the trace map from EndB(E) toB which is also a cyclic morphismas follows from the more general Proposition 4.7 below.

All of the discussion above, regarding the cyclic module associated to an algebra, extends tothe context of locally convex topological algebras. In that context one needs to work withprojective tensor products in the construction ofA and of continuous multilinear forms inthe construction ofA.Moreover, an important issue arises, since the ranges of continuous linear maps are notnecessarily closed subspaces. In order to preserve the duality between cyclic homology and

cyclic cohomology we shall define the cokernel of a cyclic map T : A B as the quotientofB by the closure of the range ofT. In a dual manner, the kernel of the transposed mapTt : B A is automatically closed and is the dual of the above.

3. Artin Motives and noncommutative spaces

In this section we show how certain categories of motives can be embedded faithfully intocategories of noncommutative spaces. We first recall some general facts about motives.Historically, the roots of the idea of motives introduced by Grothendieck can be found in theL-functionsL(X, s) of algebraic varieties, which can be written as an alternating product

(3.1) L(X, s) =n

i=0

L(Hi(X), s)(1)i+1

.

Typically, such combined L-function as (3.1) tends to have infinitely many poles, corre-sponding to the zeros of the factors in the denominator, whereas in isolating a single factorL(Hi(X), s) one avoids this problem and also gains the possibility of having a nice functionalequation. It is convenient to think of such a factor L(Hi(X), s) as the L-function, not of avariety, but of a pure motive of weight i, viewed as a direct summand of the variety X.In algebraic geometry, the various constructions of categories of pure motives of algebraicvarieties are obtained by replacing the (non additive) category VKof algebraic varieties over afield K by a (pseudo)abelian category MKconstructed in three steps (cf.[42]): (1) Enlargingthe set of morphisms of algebraic varieties to correspondences by algebraic cycles (moduloa suitable equivalence relation); (2) Adding the ranges of projectors to the collection ofobjects; (3) Adding the non-effective Tate motives.

3.1. Complex coefficients. Since in the following we deal with reductions of motives byHecke characters which are complex valued, we need to say a few words about coefficients.In fact, we are interested in considering categories of motivesMK,E over a field K withcoefficients in an extension E ofQ. This may sometime be a finite extension, but we arealso interested in treating the case where E= C, i.e. the categoryMK,C of motives over Kwith complex coefficients.By this we mean the category obtained as follows. We letMK denote, as above the(pseudo)abelian category of pure motives overK. We then define MK,Cas the (pseudo)abelianenvelope of the additive category with Obj(MK) Obj(MK,C) and morphisms(3.2) HomMK,C(M, N) = HomMK(M, N)

C.

Notice that, having modified the morphisms to (3.2) one in general will have to add newobjects to still have a (pseudo)abelian category.


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Notice that the notion of motives with coefficients that we use here is compatible withthe apparently different notion often adopted in the literature, where motives with co-efficients are defined for E a finite extension of K, cf. [50] and [27]2.1. In this set-ting one says that a motive M over K has coefficients in E if there is a homomorphismE End(M). This description is compatible with defining, as we did above,MK,E asthe smallest (pseudo)abelian category containing the pure motives over K with morphismsHomMK,E(M, N) = HomMK(M, N) E. This is shown in [27]2.1: first notice that to amotive Mover K one can associate an element ofMK,E by takingME, with the obviousE-module structure. IfM is endowed with a map E End(M), then the correspondingobject in

MK,E is the range of the projector onto the largest direct summand on which

the two structures ofE-module agree. Thus, while for a finite extensionE, both notionsof motives with coefficients make sense and give rise to the same categoryMK,E, for aninfinite extension like C only the definition we gave above makes sense.

3.2. Artin motives. Since our main motivation comes from the adele class space of [13], wefocus on zero-dimensional examples, like the BostConnes system of [7] (whose dual systemgives the adele class space). Thus, from the point of view of motives, we consider mostlythe simpler case of Artin motives. We return to discuss some higher dimensional cases inSections 6 and 7.Artin motives are motives of zero-dimensional smooth algebraic varieties over a field K andthey provide a geometric counterpart for finite dimensional linear representations of theabsolute Galois group ofK.We recall some facts about the category of Artin motives (cf. [1], [50]). We consider in

particular the case where K is a number field, with an algebraic closure K and with theabsolute Galois groupG = Gal(K/K). We fix an embedding : K Cand writeX(C) forX(C).LetV0K denotes the category of zero-dimensional smooth projective varieties over K. ForX Obj(V0K) we denote by X(K) the set of algebraic points ofX. This is a finite set onwhich G acts continuously. Given X and Y inV0K one defines M(X, Y) to be the finitedimensionalQ-vector space

(3.3) M(X, Y) := HomG(QX(K),QY(K)) = (QX(K)Y(K))G.

Thus, a (virtual) correspondenceZ M(X, Y) is a formal linear combination of connectedcomponents ofX Ywith coefficients in Q. These are identified with Q-valuedG-invariantfunctions on X(K)Y(K) by Z =

aiZi

aiZi(K), where is the characteristic

function, and the Ziare connected components ofX

Y.

We denote byCV0K the (pseudo)abelian envelope of the additive category with objectsObj(V0K) and morphisms as in (3.3). Thus,CV0K is the smallest (pseudo)abelian categorywith Obj(V0K) Obj(CV0K) and morphisms HomCV0K(X, Y) = M(X, Y), forX, Y Obj(V0K).In this case it is in fact an abelian category.Moreover, there is a fiber functor

(3.4) : X H0(X(C),Q) = QX(K),which gives an identification of the categoryCV0K with the category of finite dimensionalQ-linear representations ofG = Gal(K/K).

Notice that, in the case of Artin motives, the issue of different properties of the categoryof motives depending on the different possible choices of the equivalence relation on cy-cles (numerical, homological, rational) does not arise. In fact, in dimension zero there

is no equivalence relation needed on the cycles in the product, cf. e.g. [1]. Nonetheless,one sees that passing to the categoryCV0K requires adding new objects in order to have a(pseudo)abelian category. One can see this in a very simple example, for K=Q. Consider


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NCG AND MOTIVES 11

the field L = Q(

2). Then consider the one dimensional non-trivial representation of theGalois groupG that factors through the character of order two of Gal(L/Q). This represen-tation does not correspond to an object in Obj (V0Q) but can be obtained as the range of theprojector p = (1 )/2, where is the generator of Gal(L/Q). Namely it is a new objectinObj(CV0Q).3.3. Endomotives. We now show that there is a very simple way to give a faithful em-bedding of the categoryCV0K of Artin motives in an additive category of noncommutativespaces. The construction extends to arbitrary coefficients CV0K,E (in the sense of3.1) and inparticular to

CV0K,C. The objects of the new category are obtained from pro jective systems

of Artin motives with actions of semigroups of endomorphisms. For this reason we will callthem endomotives. We construct a categoryEV0K,E of algebraic endomotives, which makessense over any field K. For K a number field, this embeds in a category CV0K of analyticendomotives, which is defined in the context ofC-algebras.

One can describe the category V0Kas the category of reduced finite dimensional commutativealgebras over K. The correspondences for Artin motives given by (3.3) can also be describedin terms of bimodules (cf. (3.31) and (3.32) below). We consider a specific class of non-commutative algebras to which all these notions extend naturally in the zero-dimensionalcase.

The noncommutative algebras we deal with are of the form

(3.5) AK = A S,

where A is an inductive limit of reduced finite dimensional commutative algebras over thefield K and S is a unital abelian semigroup of algebra endomorphisms : A A. Thealgebra A is unital and, for S, the image e = (1) A is an idempotent. We assumethat is an isomorphism ofA with the compressed algebra eAe.

The crossed product algebraAK= A S is obtained by adjoining to A new generators UandU , for S, with algebraic rules given by

(3.6)

U U= 1, UU =(1), S

U12 = U1U2 , U21 = U

1 U

2 , j S

U x= (x) U, x U = U(x), S,x A.

SinceSis abelian these rules suffice to show that ASis the linear span of the monomials

U1 a U2 fora A and j S. In fact one gets(3.7) U1a U2U

3b U4 = U

133(a) 23(1) 2(b) U24, j S , a , b A .

By hypothesis,Ais an inductive limit of reduced finite dimensional commutative algebras Aover K. Thus, the construction of the algebraic semigroup crossed productAK given aboveis determined by assigning the following data: a projective system{X}I of varieties inV0K, with morphisms, :X X, and a countable indexing set I. The graphs (,) ofthese morphisms areG-invariant subsets ofX (K)X(K). We denote byXthe projectivelimit, that is, the pro-variety

(3.8) X= lim

X,

with maps : X

X.

The endomorphisms give isomorphisms

: XXe


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ofXwith the subvariety Xe associated to the idempotent e = (1), i.e. corresponding tothe compressed algebraeAe.

The noncommutative space described by the algebra AK of (3.5) is the quotient ofX(K) bythe action ofS, i.e. of the .

By construction, the Galois group G acts on X(K) by composition. Thus, if we view theelements ofX(K) as characters,i.e.as K-algebra homomorphisms : A K, we can writethe Galois action ofG as

(3.9) () = : A K, G= AutK(K), X(K) .This action commutes with the maps . In fact, given a character

X(K) ofA, one has

((1)) {0, 1} (with((1)) = 1 Xe) and ( ) = ( ).Thus, the whole construction is G-equivariant. Notice that this does not mean that G actsby automorphisms onAK, but it does act on X(K), and on the noncommutative quotientX(K)/S.

The purely algebraic construction of the crossed product algebraAK = A Sand of theaction ofGand Son X(K) given above makes sense also when K has positive characteristic.One can extend the basic notions of Artin motives in the above generality. For the resultingcrossed product algebrasAK = ASone can define the set of correspondences M(AK, BK)usingAKBK-bimodules (which are finite and pro jective as right modules). One can thenuse Lemma 2.1 to obtain a realization in the abelian category ofK()-modules. This maybe useful in order to consider systems like the generalization of the BC system of [7] to rankone Drinfeld modules recently studied by Benoit Jacob [34].

In the following, we concentrate mostly on the case of characteristic zero, where K is anumber field. We fix as above an embedding : K C and we take K to be the algebraicclosure of(K) C in C. We then let(3.10) AC= A K C = lim

A K C, AC= AK K C =AC S.

We define an embedding of algebras AC C(X) by(3.11) a A a, a() = (a) X.Notice thatAis by construction a subalgebraA C(X) but it is not in general an involutivesubalgebra for the canonical involution ofC(X). It is true, however, that the C-linear spanAC C(X) is an involutive subalgebra ofC(X) since it is the algebra of locally constantfunctions onX. TheC-completionR = C(X) ofAC is an abelian AFC-algebra. We then

let A = C(X) S be the C

-crossed product of the C

-algebra C(X) by the semi-groupaction ofS. It is theC-completion of the algebraic crossed productACSand it is definedby the algebraic relations (3.6) with the involution which is apparent in the formulae ( cf.[38] [39] and the references there for the general theory of crossed products by semi-groups).

It is important, in order to work with cyclic (co)homology as we do in Section 4 below, tobe able to restrict from C-algebrasC(X) Sto a canonical dense subalgebra

(3.12) A =C(X)algS A =C(X) SwhereA = C(X) C(X) is the algebra of locally constant functions, and the crossedproduct C(X) Sis the algebraic one. It is to this category of (smooth) algebras, ratherthan to that ofC-algebras, that cyclic homology applies.

Proposition 3.1. 1) The action (3.9) of G on X(K) defines a canonical action of G by

automorphisms of theC

-algebra A = C(X) S, preserving globally C(X) and such that,for any pure state ofC(X),

(3.13) (a) = (1(a)) , a A , G.


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NCG AND MOTIVES 13

2) When the Artin motives A are abelian and normal, the subalgebras A C(X) andAK A =C(X)Sare globally invariant under the action ofG and the states ofR Sinduced by pure states ofR fulfill

(3.14) (a) = (()(a)) , a AK , () =1 , Gab = G/[G, G]

Proof. 1) The action of the Galois group G on C(X) is given, using the notations of (3.9),by

(3.15) (h)() = h(1()) , X(K) , h C(X) .

For a A C(X) and given by the evaluation at X(K) one gets

(a) = ( )(a) = ()(a) = a(()) = (1(a))using (3.15). Since the action ofG on Xcommutes with the action ofS, it extends to thecrossed product C(X) S.2) Let us show that A C(X) is globally invariant under the action of G. ByconstructionA = A where the finite dimensional K-algebrasA are of the form

A =

Li

where each Li is a normal abelian extension ofK. In particular the automorphism ofAdefined by

= 1 is independent of the choice of

X. One has by construction fora

A with a given by

(3.11), and b = (a), the equality

b() = (b) = (a) = (a) = 1(a)()

so that b = 1(a) and one gets the required global invariance for A. The invariance ofAK follows. Finally note that () = 1 is a group homomorphism for the abelianizedgroups.

The intertwining property (3.14) is (part of) the fabulous stateproperty of [16], [20], [21].The subalgebraAK then qualifies as a rational subalgebra ofA= C(X) Sin the normalabelian case. In the normal non-abelian case the global invariance ofA no longer holds.One can define an action ofG on A but only in a non-canonical manner which depends onthe choices of base points. Moreover in the non-abelian case one should not confuse thecontravariant intertwining stated in (3.13) with the covariant intertwining of [16].

Example 3.2. The prototype example of the data described above is the BostConnessystem. In this case we work over K = Q, and we consider the projective system of theXn = Spec(An), where An = Q[Z/nZ] is the group ring of the abelian group Z/nZ. Theinductive limit is the group ring A = Q[Q/Z] ofQ/Z. The endomorphism n associated toan elementnSof the (multiplicative) semigroup S= N = Z>0 is given on the canonicalbasiser Q[Q/Z],rQ/Z, by(3.16) n(er) =

1

n

ns=r

es

The direct limit C-algebra is R = C(Q/Z) and the crossed product A = R S is theC-algebra of the BC-system. One is in the normal abelian situation, so that proposition3.1 applies. The action of the Galois groupG ofQover Qfactorizes through the cyclotomic

action and coincides with the symmetry group of the BC-system. In fact, the action onXn= Spec(An) is obtained by composing a character : An Qwith an element g of theGalois groupG. Since is determined by the n-th root of unity(e1/n), we just obtain the


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cyclotomic action. The subalgebraAQ A coincides with the rational subalgebra of [7].Finally the fabulous state property of extreme KMS states follows from Proposition 3.1and the fact (cf.[7]) that such states are induced from pure states ofR.

Remark 3.3. Fourier transform gives an algebra isomorphism

C(Q/Z) C(Z)which is compatible with the projective systemZ = limn Z/nZ, while the endomorphismsnare given by division byn. It is thus natural to consider the example given byYn= Spec(Bn)where Bn is simply the algebra of functions C(Z/nZ,Q), with the corresponding action of

S = N. It is crucial to note that the corresponding system is not isomorphic to the onedescribed above in Example 3.2. Indeed what happens is that the corresponding action ofthe Galois groupG is now trivial. More generally, one could construct examples of that kindby taking Sto be the ring of integersO of a number field and consider its action on thequotientsO/J, by ideals J. The action is transitive at each finite level, though the actionofOon the profinite completion Ois not. The resultingC-algebra isC(O)O. The basicdefect of these examples is that they do not yield the correct Galois action.

Example 3.4. As a corollary of the results of [20] one can show that given an imaginaryquadratic extension K ofQ there is a canonically associated noncommutative space of theform (3.5), with the correct non-trivial Galois action of the absolute Galois group ofK. Thenormal abelian hypothesis of Proposition 3.1 is fulfilled. The rational subalgebra obtainedfrom Proposition 3.1 is the same as the rational subalgebra of [20] and the structure of

noncommutative Artin motive is uniquely dictated by the structure of inductive limit offinite dimensional K-algebras of the rational subalgebra of [20]. The construction of thesystem (cf.[20]) is much more elaborate than in Remark 3.3.

We now describe a large class of examples of noncommutative spaces (3.5), obtained as semi-group actions on projective systems of Artin motives, that arise from self-maps of algebraicvarieties.We consider a pointed algebraic variety (Y, y0) over K and a countable unital abelian semi-groupSof finite endomorphisms of (Y, y0), unramified over y0 Y. Finite endomorphismshave a well defined degree. ForsSwe let deg s denote the degree ofs. We construct aprojective system of Artin motives Xs over K from these data as follows. Fors S, we set(3.17) Xs= {y Y : s(y) = y0}.For a pair s, s S, with s =sr, the map s,s: Xsr Xs is given by(3.18) Xsr y r(y) Xs.This defines a projective system indexed by the semigroup Sitself with partial order givenby divisibility. We letX= lims Xs.Sinces(y0) = y0, the base pointy0 defines a componentZs ofXs for all s S. Let1s,s(Zs)be the inverse image of Zs in Xs . It is a union of components of Xs . This defines aprojection es onto an open and closed subsetXes of the projective limit X.

Proposition 3.5. The semigroup Sacts on the projective limitXby partial isomorphismss: X Xes defined by the property that(3.19) su(s(x)) = u(x),u S, x X.

Proof. The map s is well defined as a map XXby (3.19), since the set{su: uS}is cofinal and u(x) Xsu. In fact, we havesuu(x) = s(y0) = y0. The image ofs is in


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NCG AND MOTIVES 15

Xes , since by (3.19) we have s(s(x)) = 1(x) = y0. We show that s is an isomorphism ofXwithXes . The inverse map is given by

(3.20) u(1s (x)) =su(x),x Xes , u S.

In fact, for x Xes , we havesu(x) Xu.In this class of examples one has an equidistribution property, by which the uniformnormalized counting measures s on Xs are compatible with the projective system anddefine a probability measure on the limitX. Namely, one has

(3.21) s,ss= s ,

s, s

S.

This follows from the fact that the number of preimages of a point under sSis equal todeg s.

Example 3.6. LetYbe the affine group scheme Gm (the multiplicative group). LetSbethe semigroup of non-zero endomorphisms ofGm. These correspond to maps of the formu un for some non-zeron Z. In fact one can restrict to N Z {0}.

Proposition 3.7. The construction of Example 3.6, applied to the pointed algebraic variety(Gm(Q), 1), gives the BC system.

Proof. We restrict to the semi-group S = N and determine Xn for n N. One has byconstruction Xn = Spec(An) where An is the quotient of the algebra Q[u(n), u(n)1] of

Laurent polynomials by the relation u(n)n = 1. For n|m the map m,n from Xm to Xn isgiven by the algebra homomorphism that sends the generators u(n)1 Antou(m)a Amwith a = m/n. Thus, one obtains an isomorphism of the inductive limit algebraA with thegroup ring Q[Q/Z] of the form

(3.22) : limAn Q[Q/Z] , (u(n)) = e 1n .Let us check that the partial isomorphisms n (3.16) of the group ring correspond to thosegiven by (3.19) under the isomorphism . We identify the projective limit X with thespace of characters ofQ[Q/Z] with values in Q. The projection m(x) is simply given bythe restriction of x X to the subalgebra Am. Let us compute the projections of thecomposition ofx Xwith the endomorphism n (3.16). One has

x(n(er)) = 1

n ns=r x(es) .This is non-zero iff the restrictionx|An is the trivial character,i.e.iffn(x) = 1. Moreover,in that case one has

x(n(er)) = x(es) , s , ns= r ,and in particular

(3.23) x(n(e 1k

)) = x(e 1nk

).

For k |mthe inclusion Xk Xm is given at the algebra level by the homomorphismjk,m : Am Ak, jk,m(u(m)) = u(k).

Thus, one can rewrite (3.23) as

(3.24) x n jk,nk = x|Ank ,

that is, one hasnk(x) = k(x n),

which, using (3.20), gives the desired equality of the s of (3.16) and (3.19).


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The construction of Example 3.6 continues to make sense for Gm(K) for other fields, e.g.fora field of positive characteristic. In this case one obtains new systems, different from BC.

Example 3.8. LetYbe an elliptic curve defined over K. LetSbe the semigroup of non-zeroendomorphisms ofY. This gives rise to an example in the general class described above.When the elliptic curve has complex multiplication, this gives rise to a system which, inthe case of a maximal order, agrees with the one constructed in [20]. In the case withoutcomplex multiplication, this provides an example of a system where the Galois action doesnot factor through an abelian quotient.

In general, we say that a system (X, S) with X a projective system of Artin motivesand Sa semigroup of endomorphisms of the limit X is uniform if the normalized countingmeasures on X satisfy

(3.25) ,=

as above.

For data (X, S) and (X , S

) as above, we consider the Q-vector space

Corr((X, S), (X , S

))

of linear combinations of correspondences given by projective systems Z, of subvarietiesin the product X X , invariant under the left action ofSand right action ofS. Wealso require that the projectionZ Xis proper.For M= (X, S), M

= (X

, S

), andM

= (X

, S

), the compositionCorr(M, M) Corr(M, M) Corr(M, M)

of correspondences Z, Corr(M, M) and W, Corr(M, M) is obtained usingthe composition Z, W, of correspondences for Artin motives and passing to theprojective limit. This inherits all the required properties.

Definition 3.9. The categoryEV0K,E ofalgebraic endomotives with coefficients inE is the(pseudo)abelian category generated by the following objects and morphisms. The objects areuniform systems (X, S) as above, withX Artin motives overK. ForM = (X, S) andM = (X , S

) we set

(3.26) HomEV0K,E

(M, M) = Corr((X, S), (X , S

)) E.

When taking points over K, the algebraic endomotives described above yield 0-dimensionalsingular quotient spaces X(K)/S, which can be described through crossed product C-algebrasC(X(K)) S. This gives rise to a corresponding category of analytic endomotivesgiven by C-algebras and geometric correspondences. This category provides the data forthe construction in Section 4 below.

LetXbe a totally disconnected compact space, San abelian semigroup of homeomorphismss : X Xs, with Xs a closed and open subset ofX, and a probability measure on Xwith the property that the RadonNykodim derivatives

(3.27) ds

d

are locally constant functions on X.

We consider totally disconnected locally compact spaces Zwith a left action of S and aright action ofS, with equivariant maps fX :Z X and gX :ZX. We assume thatfX is proper. We consider the pullback of the measure Z :=gX

onZ(with the counting


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NCG AND MOTIVES 17

measure on the fiber). We require that, under the left action ofSon Z, the RadonNykodimderivatives satisfy

d(sZ)

dZ=

d(s)

d .

We consider the Q-vector space

Corr((X,S,), (X, S, ))

of linear combinations of such correspondences Zmodulo the equivalence relationZ Z =Z+ Z for disjoint unions.

For M = (X, S, ), M = (X, S, ), and M = (X, S, ), the composition of corre-spondences

Corr(M, M) Corr(M, M) Corr(M, M)is a special case of the composition of geometric correspondences considered in [23]. Infact, a correspondence, in the sense above, gives rise to a bimodule MZover the algebrasC(X)Sand C(X)S and the composition of correspondences translates into the tensorproduct of bimodules.

Definition 3.10. The categoryCV0E ofanalytic endomotives is the (pseudo)abelian cate-gory generated by objects of the form(X,S,)with the properties listed above and morphismsgiven as follows. ForM= (X,S,) andM = (X, S, ) we set

(3.28) HomCV0E

(M, M) = Corr((X,S,), (X, S, ))

E.

Notice that, for an abelian AF algebra R = C(X) with Xa totally disconnected space,the points of X are uniquely specified by the corresponding linear forms on K-theoryK0(C(X)) Z, and the K0-group (ordered and pointed) completely determines the al-gebra. Elements in KK-theory can be specified by homomorphisms of K0. Projectivesystems of correspondences

(3.29) u, ZXY

define elements u Hom(K0(R), K0(R)), hence an element in KK(R, R), since R andR are AF algebras. When taking into account the actions of semigroups S and S itwould also be possible to define a category where morphisms are given by KK-classesKK(R S, R S). The definition we gave above for the categoryCV0E is more refined.In particular, we do not divide by homotopy equivalence. In fact, correspondences Z inCV0E give rise to bimodulesMZ. The associated K K-classk(Z) is simply given by takingthe equivalence class of the triple (E , , F ) with (E, ) given by the bimoduleMZ andF= 0. There is a functor k : CV0E KK to the category KK described in Section 2 above,which maps correspondencesZk(Z). The version of Definition 3.10 is preferable to theKK-formulation, since we want morphisms that will act on the cohomology theory that weintroduce in Section 4 below.

Theorem 3.11. The categories introduced above are related as follows.

(1) Taking points overK defines a faithful tensor functorFfrom the categoryEV0K,E toCV0E.

(2) The Galois group G = Gal((K)/(K))acts by natural transformations ofF.(3) The categoryCV

0K,E of Artin motives embeds as a full subcategory ofEV

0K,E.

(4) The composite functork F maps the full subcategoryCV0K,E ofEV0K,E faithfully tothe categoryKKG ofG-equivariantKK-theory.


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Proof. (1) The functorF maps an object (X, S) to the object (X, S, ) with X =lim X(K) and the measure is the projective limit of the normalized counting measures.Given a correspondence Z, , its image underF is the geometric correspondence given byZ = lim(,) Z,(K). This is compatible with composition of correspondences becausethe geometric correspondencesZobtained in this way sit in the product X X, hence thefiber product agrees with the intersection product. The fuctorF is faithful. In fact, forgiven (X, S) and (X

, S

), a morphism Z, is uniquely determined by the correspondingelements in

(3.30) u,

(EX(K)X

(K))G.

Thus, it suffices to show that the elements u, of (3.30) can be recovered from Z =F(Z,). The correspondenceZ induces in particular a correspondence with coefficients inE between the abelian AF algebrasC(X) andC(X). Thus, it gives a morphism

K0(C(X)) E K0(C(X)) E,which is the inductive limit of the u, . These can be recovered from the direct limit sinceK0(C(X)) is in fact the union of the K0(C(X(K))).(2) We know from (3.9) that the group G acts by automorphisms onF(X, S). This actionis compatible with the morphisms by (3.30). This shows thatG acts on the functorF bynatural transformations.(3) The category of Artin motives is embedded in the category EV0K,E by the functorJ thatmaps an Artin motive Mto the system with X = M for all and S trivial. Morphismsare then given by the same geometric objects.(4) Let X = Spec A, X = Spec B be 0-dimensional varieties. Given a componentZ =SpecCofX X the two projections turn C into anA B-bimodulek(Z). Moreover, thecomposition of correspondences translates into the tensor product of bimodules i.e.one has

(3.31) k(Z) Bk(L) k(Z L) .We can then compose k with the natural functor A AC = A KC which associates toanA B-bimoduleE theAC BC-bimoduleEC =E K C. Thus, we view the result as anelement inK K.Recall that we have, for any coefficients E, the functorJwhich to a variety X Obj(V0K,E)associates the system (X, S) where the systemXconsists of a single Xand the semigroupS is trivial. Let U HomCV0

K,E(X, X) be given by a correspondence U = aiZi , withcoefficients in E. We consider the element in K K(with coefficients in E) given by the sum

of bimodules

(3.32) k(U) =

aik(Zi).

Using the isomorphism (3.11) we thus obtain a G-equivariant bimodule for the correspondingC-algebras, and hence owing to finite dimensionality a corresponding class in K KG. Thusand for any coefficients Ewe get a faithful functor

(3.33) k F

HomCV0K,E

(X, X)

KKG(kF(X), kF(X)) E

compatibly with the composition of correspondences.Since a correspondence U =

aiZi , with coefficients in E, is uniquely determined by the

corresponding map

K0(AC) E K0(BC) Ewe get the required faithfulness of the restriction ofk F to Artin motives.


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NCG AND MOTIVES 19

The fact of considering only a zero dimensional setting in Theorem 3.11 made it especiallysimple to compare the composition of correspondences between Artin motives and noncom-mutative spaces. This is a special case of a more general result that holds in higher dimensionas well and which we discuss in Section 6 below. The comparison between correspondencesgiven by algebraic cycles and correspondences in KK-theory is based in the more generalcase on the description given in [23] of correspondences in K K-theory.

The setting described in this section is useful in order to develop some basic tools thatcan be applied to a class of zero dimensional noncommutative spaces generalizing Artinmotives, which inherit natural analogs of arithmetic notions for motives. It is the minimal

one that makes it possible to understand the intrinsic role of the absolute Galois group indynamical systems such as the BC-system. This set-up has no pretention at defining thecomplete category of noncommutative Artin motives. This would require in particular tospell out exactly the finiteness conditions (such as the finite dimensionality of the center)that should be required on the noncommutative algebraA.We now turn to a general procedure which should play in characteristic zero a role similarto the action of the Frobenius on the -adic cohomology. Its main virtue is its generality. Itmakes essential use ofpositivity i.e.of the key feature ofC-algebras. It is precisely for thisreason that it was important to construct the above functorFthat effects a bridge betweenthe world of Artin motives and that of noncommutative geometry.

4. Scaling as Frobenius in characteristic zeroIn this section we describe a general cohomological procedure which starting from a non-commutative space given by a pair (A, ) of a unital involutive algebra A over Cand a state onAyields a representation of the multiplicative group R+. We follow all the steps in aparticular example: the BC-system described above in Example 3.2. We show that, in thiscase, the spectrum of the representation is the set of non-trivial zeros of Hecke L-functionswith Grossencharakter.What is striking is the generality of the procedure and the analogy with the role of theFrobenius in positive characteristic. The action of the multiplicative group R+ on the cyclichomologyH C0 of the distilled dual system plays the role of the action of the Frobenius onthe-adic cohomologyH1(C(Fq),Q). We explain in details that passing to the dual systemis the analog in characteristic zero of the transition from Fq to its algebraic closure Fq. Wealso discuss the analog of the intermediate step F

qF

qn . What we call the distillation

procedure gives a -module D(A, ) which plays a role similar to a motivicH1mot (cf.(2.3)above). Finally, the dual action ofR+on cyclic homologyH C0(D(A, )) plays a role similarto the action of the Frobenius on the -adic cohomology.While here we only illustrate the procedure in the case of the BC-system, in the next sectionwe discuss it in the more general context of global fields and get in that way a cohomologicalinterpretation of the spectral realization of zeros of Hecke L-functions with Grossencharakterof [13].There is a deep relation outlined in Section 4.5 between this procedure and the classificationof type III factors [9]. In particular, one expects interesting phenomena when the noncom-mutative space (A, ) is of type III and examples abound where the general cohomologicalprocedure should be applied, with the variations required in more involved cases. Othertype III1 examples include the following.

The spaces of leaves of Anosov foliations and their codings. The noncommutative space (A, ) given by the restriction of the vacuum state onthe local algebra in the Unruh model of black holes.


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One should expect that the analysis of the thermodynamics will get more involved in general,where more that one critical temperature and phase transitions occur, but that a similardistillation procedure will apply at the various critical temperatures.The existence of a canonically given time evolution for noncommutative spaces (see [9]and subsection 4.5 below) plays a central role in noncommutative geometry and the proce-dure outlined below provides a very general substitute for the Frobenius when working incharacteristic zero.

4.1. Preliminaries. LetAbe a unital involutive algebra over C. We assume thatAhas acountable basis (as a vector space over C) and that the expression(4.1) ||x||= sup||(x)||defines a finitenorm onA, where in (4.1) ranges through all unitary Hilbert space repre-sentations of the unital involutive algebraA.We let A denote the completion ofAin this norm. It is a C-algebra.This applies in particular to the unital involutive algebras A = C(X)alg Sof (3.12) andthe correspondingC-algebra is A = C(X) S.A state onAis a linear form : A C such that

(xx) 0, x A and (1) = 1 .One letsH be the Hilbert space of the GelfandNaimarkSegal (GNS) construction, i.e.the completion ofAassociated to the sesquilinear form

x, y =(yx) , x, y A .Definition 4.1. A state on aC-algebra A isregular iff the vector1 H is cyclic forthe commutant of the action ofA by left multiplication onH.LetMbe the von-Neumann algebra weak closure of the action ofAinH by left multipli-cation.The main result of Tomitas theory [48] is that, if we let be the modular operator

(4.2) = S S ,

which is the moduleof the involution S, x x, then we haveitM

it = M , t R .

The corresponding one parameter group t Aut(M) is called the modular automorphism

group of and it satisfies the KMS1 condition relative to (cf. [48]). In general, given aunital involutive algebraA, a one parameter group of automorphisms t Aut(A) and astate onA, the KMS-condition is defined as follows.Definition 4.2. A triple (A, t, ) satisfies the Kubo-Martin-Schwinger (KMS) conditionat inverse temperature0


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NCG AND MOTIVES 21

4.2. The cooling morphism. Let (A, ) be a pair of an algebra and a state as above.We assume that the modular automorphism group t associated to the state leavesthe algebraA globally invariant and we denote by t Aut(A) its restriction toA. Byconstruction the state is a KMS1 state for the quantum statistical system ( A, ). Here Ais the C-completion in (4.1) as above.The process of cooling down the system consists of investigating KMS-states for >1 (i.e.at lower temperatures).The general theory of quantum statistical mechanics shows that the space of KMS-states is a Choquet simplex. We letE denote the set of its extremal points. Each E isa factor state on

A.

We require that, for sufficiently large , all the E are type I factor states.In general, the original state will be of type III (it is of type III 1 for the BC-system), sothat the type I property we required means that, under cooling, the system simplifies andbecomes commutative. In fact, under this assumption, we obtain a cooled down dualalgebra C(, L1), using the set of regular extremal KMS states (cf. (4.20) below),that is Morita equivalent to a commutative algebra.

Lemma 4.3. Let be a regular extremal KMS state on theC-algebra A with the time

evolutiont. Assume that the corresponding factorM obtained through the GNS represen-tationH is of type I. Then there exists an irreducible representation ofAin a HilbertspaceH()and an unbounded self-adjoint operatorHacting onH(), such thateH is oftrace class. Moreover, one has

(4.4) (x) = Trace((x) eH

)/Trace(eH

) , x A,

and

(4.5) eitH (x) eitH = (t(x)).

Proof. Let 1 H be the cyclic and separating vector associated to the regular state in the GNS representation. The factor M is of type I hence one has a factorization

(4.6) H = H() H, M= {T 1|T L(H())}.The restriction of the vector state 1to Mis faithful and normal, hence it can be uniquelywritten in the form

(4.7) (T 1)1, 1 = Trace(T ),for a density matrix , that is, a positive trace class operator with Trace() = 1. This canbe written in the form = eH. It then follows that the vector state of (4.7) satisfies theKMS condition on M, relative to the 1-parameter group implemented by eitH.The factorization (4.6) shows that the left action of A in the GNS representationH is ofthe forma (a) 1. The representation onH() is irreducible.Let H be the generator of the 1-parameter group of unitary operators t acting onH.Since M is the weak closure of A inL(H), it follows that the extension of the state to M fulfills the KMS condition with respect to the 1-parameter group implemented by

eitH. This extension of is given by the vector state (4.7). This shows that the 1-parameter

groups implemented byeitH andeitH agree onM. This proves (4.5). Equation (4.4) follows

from (4.7), which implies in particular that the operatorH has discrete spectrum boundedbelow.

In general, we let E be the set of regular (in the sense of Definition 4.1) extremalKMS states of type I.


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The equation (4.5) does not determineHuniquely, but only up to an additive constant. For

any choice ofH, the corresponding representation of the crossed product algebra A =ARis given by

(4.8) ,H(

x(t) Ut dt) =

(x(t)) e

itH dt.

Let us define carefully the crossed product algebra A = A R by specifying preciselywhich mapst x(t) Aqualify as elements ofA. One defines the Schwartz spaceS(R, A)as the direct limit of the Schwartz spaceS(R, V) overfinite dimensionalsubspacesV A:

(4.9) S(R, A) = VS(R, V).Notice then that, since the vector spaceA has countable basis and the action of t isunitary, one can find a linear basis ofA whose elements are eigenvectors for t and suchthat the corresponding characters are unitary. It follows that the Schwartz spaceS(R, A) isan algebra under the product

(4.10) (x y)(s) =

x(t) t(y(s t)) dt .

We denote the algebra obtained in this way by

(4.11) A =A t R =S(R, A).The dual action Aut(A) ofR+ is given by

(4.12) ( x(t) Ut dt) = it x(t) Ut dt.By construction, the set of pairs (, H) forms the total space of a principal R-bundle over , where the action ofR is simply given by translation ofH i.e. in multiplicative terms,one has a principal R+-bundle with

(4.13) (, H) = (, H+ log ) , R+,which gives the fibration

(4.14) R+ .Notice that (4.14) admits a natural section given by the condition

(4.15) Trace(e H) = 1 ,

and a natural splitting

(4.16) R+ .The states are in fact uniquely determined by the corresponding representation,thanks to the formula (4.4), which does not depend on the additional choice ofH fulfilling(4.5). The property (4.4) also ensures that, for any functionf S(R) with sufficiently fastdecay at, the operator f(H) is of trace class.We let A be the linear subspace ofA spanned by elements of the form

x(t) Ut dt where

x S(I , A). Namely, x S(R, A) admits an analytic continuation to the strip I ={z :z [0, ]}, which restricts to Schwartz functions on the boundary ofI .One checks that A is a subalgebra ofA. Indeed the product (4.10) belongs toS(I , A)provided thatx

S(R,

A) and y

A . One then gets the following.

Proposition 4.4. (1) For(, H) , the representations,Hare pairwise inequiva-lent irreducible representations ofA =A R.


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NCG AND MOTIVES 23

(2) Let Aut(A) be the dual action (4.12). This satisfies(4.17) ,H = (,H), (, H) .

(3) Forx A, one has

(4.18) ,H(

x(t) Ut dt) L1(H()).

Proof. (1) By construction the representations ofA are already irreducible. Let usshow that the unitary equivalence class of the representation

,H determines (, H). First

the state is uniquely determined from (4.4). Next the irreducibility of the restriction to Ashows that only scalar operators can intertwine ,Hwith,H+c, which implies thatc = 0.

(2) One has ,H(Ut) = eitH, hence,H((Ut)) =

it eitH = eit(H+log ), as required.(3) Let us first handle the case of scalar valued x(t) C for simplicity. One then has

,H(

x(t) Ut dt) = g(H) =

g(u)e(u),

where thee(u) are the spectral projections ofHand g is the Fourier transform in the form

g(s) =

x(t) eits dt .

Sincex S(R) admits analytic continuation to the strip I ={z : z[0, ]}and sincewe are assuming that the restriction to the boundary of the strip I is also in Schwarz space,one gets that the functiong(s) es belongs to S(R) and the trace class property follows fromTrace(eH)< . In the general case with x S(I , A), with the notation above, one gets

,H(

x(t) Ut dt) =

(x(t))

eitse(s) dt=

(g(s)) e(s)

where, as above, e(s) is the spectral projection ofHassociated to the eigenvalue s R.Again||g(s) es|| < C independently of s and es Trace(e(s)) 1.Indeed, what happens is that the set of irreducible representations of A thus constructeddoes not depend on > 1 and the corresponding map at the level of the extremal KMS states is simply obtained by changing the value ofin formula (4.4). In general the size of is a non-decreasing function of :

Corollary 4.5. For any > the formula (4.4) defines a canonical injection

(4.19) c, : which extends to anR+-equivariant map of the bundles .

Proof. First Trace(eH)


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Corollary 4.6. Under the above conditions, the representations ,H, for (, H) ,assemble to give anR+-equivariant morphism

(4.20) : A C( , L1)defined by

(4.21) (x)(, H) = ,H(x) , (, H) .

The continuity can be checked directly in the case of the BC-system where = E iscompact for the weak topology and where all the representations ,Htake place in the sameHilbert space. In the general case one deals with a field of elementaryC-algebras whichis best understood from the corresponding maximal closed ideals of the C-crossed productB= A R. Indeed each,Hdefines an irreducible representation ofB whose range is theelementary C-algebra of compact operators inH(). Thus, by Corollaries 4.10 and 4.11 of[31], this representation is characterized by its kernel which is a maximal closed ideal. Thekey property then is the automatic semi-continuity of the function (, H)Trace ,H(x)for xB , x >0, which follows from Proposition 3.5.9 of [31]. This implies that the set ofx B, x > 0 for which this function is continuous forms a face, and hence defines a twosided ideal J ofB by Lemma 4.5.1 of [31]. Using the continuity of (x) forx A, it isthen possible to obtain the density ofJ in B and to proceed as in the theory of continuoustraceC-algebras ([31], Section 4.5).

4.3. The distilled -module D(A, ). The above cooling morphism of (4.20) composeswith the trace

Trace : L1 Cto yield a cyclic morphism from the -module associated to A to one associated to a com-mutative algebra of functions on . Our aim in this section is to define appropriately thecokernel of this cyclic morphism, which gives what we call the distilled -module D(A, ).A partial trace gives rise to a cyclic morphism as follows.

Proposition 4.7. LetB be a unital algebra. The equalityTrace((x0 t0) (x1 t1) . . . (xn tn)) = x0 x1 . . . xnTrace(t0t1 tn)

defines a map of-modules from(B L1) toB.

Proof. We perform the construction with the -module of the inclusion

B L1

B L

1

and obtain by restriction to (B L1) the required cyclic morphism. The basic cyclicoperations are given by

i(x0 . . . xn) = x0 . . . xixi+1 . . . xn , 0 i n 1,

n(x0 . . . xn) = xnx0 x1 . . . xn1,

j (x0 . . . xn) = x0 . . . xj 1 xj+1 . . . xn, 0 j n,

n(x0 . . . xn) = xn x0 . . . xn1 .

One has, for 0 i n 1,Tracei((x0 t0) . . . (xn tn)) =

x0 . . . xixi+1 . . . xnTrace(t0 t1 tn) =i Trace((x0 t0) (x1 t1) . . . (xn tn)).

Similarly,Tracen((x0 t0) . . . (xn tn)) =


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NCG AND MOTIVES 25

xnx0 . . . xn1Trace(tnt0 t1 tn1) =n(x0 . . . xn) Trace(t0 t1 tn) =nTrace((x0 t0) . . . (xn tn)).

One has, for 0 j n,Tracej ((x0 t0) . . . (xn tn)) =

x0 . . . xj 1 xj+1 . . . xn Trace(t0t1 tn) =j (x0 . . . xn) Trace(t0t1 tn) =j Trace((x0

t0)

. . .

(xn

tn)).

Finally, one hasTracen((x0 t0) . . . (xn tn)) =

xn x0 . . . xn1Trace(t0t1 tn) =nTrace((x0 t0) . . . (xn tn)).

When dealing with non-unital algebrasA, the maps x x 1 in the associated -moduleA are not defined. Thus one uses the algebra A obtained by adjoining a unit, and definesA as the submodule ofA which is the kernel of the augmentation morphism,

: A Cassociated to the algebra homomorphism : A C.We take this as a definition ofA

for non unital algebras. One obtains a -module whichcoincides withAin degree 0 and whose elements in degree n are tensorsa0 an , aj A ,

(a0) (an) = 0

While the algebra A corresponds to the one point compactification of the noncommutativespace A, it is convenient to also consider more general compactifications. Thus if A Acompis an inclusion ofA as an essential ideal in a unital algebra Acomp one gets a -module(A, Acomp) which in degreen is given by tensors

a0 an , aj Acomp ,where for each simple tensor a0 an in the sum, at least one of the aj belongs toA.One checks that there is a canonical cyclic morphism

A

(A

,A

comp)

obtained from the algebra homomorphism A Acomp.Thus, we can apply the construction of Proposition 4.7 to the case withB= C( ).The next step is to analyze the decay of the obtained functions on . Let us look at the

behavior on a fiber of the fibration (4.14). We first look at elements

x(t) Ut dt A withscalar valued x(t) C for simplicity. One then has, with g (s) = x(t) eits dt,

Trace(,H+c(

x(t) Ut dt)) = Trace(g(H+ c))

and we need to understand the behavior of this function when c . Recall from theabove discussion that both g and the function g(s) es belong toS(R).

Lemma 4.8. (i) For allN >0, one has

|Trace(g(H+ c))

|= O(e c cN), forc

+

.

(ii) For any elementx A , one has|Trace(,H+c(x))|= O(e c cN) , c + .


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Proof. (i) Let g(s) = e s h(s) where h S(R). One has|h(s+ c)| C cN fors Spec H, hence|g(s + c)| C e c cN e s, fors Spec H, so that|Trace(g(H+ c))| C e c cN Trace(e H).(ii) In the general case with x S(I , A) and the notation as in the proof of Proposition4.4, one gets

Trace(,H+c(x)) =

Trace((g(s + c)) e(s)),

whose size is controlled by||g(s + c)|| Trace(e(s)). As above one has||g(s+ c)||

C e c cN e s, for s Spec H, so that using Trace(e(s)) e s 0}, so that Trace(g(H+ c)) =

nN g(log(n) + c). Let f(x) = g(log(x)) be extended

to R as a continuous even function. Then one has f(0) = 0 and, for = ec,

Trace(g(H+ c)) =1

2

nZ

f( n) .

The Poisson summation formula therefore gives, with the appropriate normalization of theFourier transform f off,

nZ

f( n) =nZ

f(1 n).

This shows that, for c

, i.e. for

0, the leading behavior is governed bynZ

f( n) 1

f(x) dx.

This was clear from the Riemann sum approximation to the integral. The remainder iscontrolled by the decay offat,i.e.by the smoothness off. Notice that the latter is notguaranteed by that ofg because of a possible singularity at 0. One gets in fact the followingresult.

Lemma 4.9. In the BC case, for any >1 and any elementx A , one has(4.22) Trace(,H+c(x)) = e

c (x) + O(|c|N) , c , N >0,where is the canonical dual trace on A.We give the proof after Lemma 4.16. Combining Lemmata 4.8 and 4.9 one gets, in the BCcase and for x A , (x) = 0, that the function(4.23) f(c) = Trace(,H+c(x))

has the correct decay for c in order to have a Fourier transform (in the variable c)that is holomorphic in the strip I . Indeed one has, for all N >0,

(4.24)|f(c)| = O(e c |c|N), for c +|f(c)| = O(|c|N) for c .

This implies that the Fourier transform

(4.25) F(f)(u) =

f(c) e

icu dc

is holomorphic in the strip I , bounded and smooth on the boundary. In fact, one hasF(f) S(I ), as follows from Lemma 4.10 below, based on the covariance of the mapTrace .


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NCG AND MOTIVES 27

We let Hol(I ) be the algebra of multipliers ofS(I ),i.e. of holomorphic functions h in thestrip I such that h S(I ) S(I ). For such a functionh its restriction to the boundarycomponent R is a function of tempered growth and there is a unique distribution h suchthat

h() it d= h(t).

One then has in full generality the following result.

Lemma 4.10. 1) For h Hol(I ) the operator (h) =

h() d acts on A . This

turnsA into a module over the algebraHol(I ).2) The map F Trace is anHol(I )-module map.Proof. 1) One has

(h)(

x(t) Ut dt) =

h() it x(t) Ut dt d

=

h(t) x(t) Ut dt

andh x S(I , A).2) The equivariance of , i.e.the equality (cf.(4.17))

,H = (,H+log ) , (, H) ,gives

(Trace )(x)(,H) = (Trace )(x)(,H+log ).Thus, with f as in (4.23), one gets

(Trace (h))(x)(,H) = h() f(log ) d.ReplacingHbyH+ cone gets that the function associated to (h)(x) by (4.23) is given by

g(c) =

h() f(log + c) d

.

One then gets

F(g)(u) =

g(c) e

icu dc=

h() f(log + c) e

icu dc d

=

h() iu d

f(a) e

iau da= h(u) F(f)(u) .

This makes it possible to improve the estimate (4.24) for the BC-system immediately. In-deed, the function h(s) = s belongs to Hol(I ) and this shows that (4.24) holds for allderivatives off. In other words it shows the following.

Corollary 4.11. In the BC case, for any >1 and any elementx A Ker , one hasF Trace (x) C( , S(I )) .

Thus, by Lemma 4.10 we obtain an Hol(I )-submodule ofC( , S(I )) and as we will seeshortly that this contains all the information on the zeros ofL-functions with Grossencharaktersover the global field Q.When expressed in the variable = ec, the regularity of the function Trace (x) can bewritten equivalently in the form Trace (x) C( , S (R+)) where, with () =, R+, we let

(4.26) S (R+) = [0,] S(R+)be the intersection of the shifted Schwartz spaces ofR+ R.


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We letS( ) be the cyclic submodule ofC( ) whose elements are functions with re-striction to the main diagonal that belongs to C( , S (R+)). We then obtain the followingresult.

Proposition 4.12. Assuming (4.22), the map Trace defines a cyclic morphism : A,0 S( ) .

The proof follows from Proposition 4.7.

Definition 4.13. We define the distilled -module D(A

, ) as the cokernel of the cyclicmorphism.

For a general pair (A, ) the decay condition (4.22) will not be fulfilled. One then needsto proceed with care and analyze the behavior of these functions case by case. One cannevertheless give a rough general definition, just by replacingS C( ) above by thesubalgebra generated by Trace (Ker ).

4.4. Dual action on cyclic homology. By construction, the cyclic morphism is equi-variant for the dual action ofR

+ (cf. (4.17)). This makes it possible to consider the

corresponding representation ofR+ in the cyclic homology group, i.e.

(4.27) () Aut(HC0(D(A, ))) , R

+.The spectrum of this representation defines a very subtle invariant of the original noncom-mutative space (A, ).Notice that, since the -moduleS( ) is commutative, so is the cokernel of (it is aquotient of the above). Thus, the cyclic homology group H C0(D(A, )) is simply given bythe cokernel ofin degree 0.It might seem unnecessary to describe this asH C0 instead of simply looking at the cokernelof in degree 0, but the main reason is the Morita invariance of cyclic homology whichwipes out the distinction between algebras such as B andB L1.We now compute the invariant described above in the case of the BC-system described inExample 3.2 of Section 3. Recall that one has a canonical action of the Galois groupG =Gal(Q/Q) on the system, which factorizes through the action on the cyclotomic extension

Qab, yielding an action ofGab = Z as symmetries of the BC-system. Thus, one obtains adecomposition as a direct sum using idempotents associated to Grossencharakters.

Proposition 4.14. Let be a character of the compact group Z. Then the expression

p=

Z

g (g) dg

determines an idempotentp inEndD(A, ).Notice that the idempotent p already makes sense in the category EV0K,C introduced abovein Section 3.The idempotents p add up to the identity, namely

(4.28) p= Id,and we get a corresponding direct sum decomposition of the representation (4.27).


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NCG AND MOTIVES 29

Theorem 4.15. The representation ofR+ in

(4.29) M = HC0(p D(A, ))gives the spectral realization of the zeros of theL-functionL. More precisely, letz I beviewed as a character ofHol(I )andCz be the corresponding one dimensional module. Onehas

(4.30) M Hol(I) Cz= {0} L(i z) = 0.

One can give a geometric description of the BC-system in terms ofQ-lattices in R (cf.

[16]). The Q-lattices in R are labeled by pairs(, ) Z R

+ and the commensurabilityequivalence relation is given by the orbits of the action of the semigroup N = Z>0 given by

n (, ) = (n , n ). The Q-lattice associated to the pair (, ) is (1Z, 1) where isviewed in Hom(Q/Z,Q/Z). The BC-system itself corresponds to the noncommutative spaceof commensurability classes ofQ-lattices up to scale, which eliminates the (cf. [16]). The

Q-lattices up to scale are labelled by Z and the commensurability equivalence relationis given by the orbits of the multiplicative action of the semigroup N. The algebraAof theBC-system is the crossed productC(Z)N whereC(Z) is the algebra of locally constant

functions onZ. By construction every f C(Z) has a level i.e. can be written as(4.31) f= h pN, pN : Z Z/NZ.General elements ofAare finite linear combinations of monomials, as in (3.7). Equivalentlyone can describe

Aas a subalgebra of the convolution algebra of the etale groupoid Gbc

associated to the partial action ofQ+ onZ. One has Gbc ={(k, ) Q+ Z | k Z} and(k1, 1) (k2, 2) = (k1k2, 2) if1 = k2 2. The product in the groupoid algebra is givenby the associative convolution product

(4.32) f1 f2(k, x) =

sQ+

f1(k s1, s x)f2(s, x),

and the adjoint is given by f(k, x) = f(k1, k x). The elements of the algebraA arefunctionsf(k, ) with finite support ink Q+ and finite level (cf.(4.31)) in the variable .The time evolution is given by

(4.33) t(g)(k, ) = kit g(k, ) , g A .

The dual system of the BC-system corresponds to the noncommutative space of commensu-

rability classes ofQ-lattices in R (cf.[16]). Thus, the dual algebra A for the BC-system isa subalgebra of the convolution algebra of the etale groupoid Gbc associated to the partialaction ofQ+ onZR+. The product in the groupoid algebra is given by the same formulaas (4.32)

noncommutative geometry and motives (the thermodynamics of endomotives). alain connes, et. al..pdf

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