Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1642
Springer
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Michael Puschnigg
Asymptotic Cyclic Cohomology
~ Springer
Author Michael Puschnigg Mathematics Institute University of Heidelberg Im Neuenheimer Feld 288 D-69120 Heidelberg, Germany e-mail: puschnig @mathi.uni-heidelberg.de
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Puschnigg, Michael: Asymptotic cyclic cohomology / Michael Puschnigg. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1642) ISBN 3-540-61986-0 NE: GT Mathematics Subject Classification (l 991 ): 19D55, 18G60, 19K35, 19K56 ISSN 0075- 8434 ISBN 3-540-61986-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10520141 46/3142-543210 - Printed on acid-free paper
Introduction
This work is a contribution to the study of topological K-Theory and cyclic cohomology of complete normed algebras. The aim is the construction of a cohomology theory, defined by a natural chain complex, on the category of Banach algebras wlfich a) is the target of a Chern character from topological K-theory (resp. bivariant K-theory). b) has nice flmctorial properties which faithfully reflect the properties of topological K-theory. c) is closely related to cyclic cohomology but avoids the usual pathologies of cyclic cohomology for operator algebras. d) is accessible to computation in sufficiently many cases. The final goal is to establish a Grothendieck-Riemann-Roch theorem for the construeted Chern character which for commutative C*-algebras reduces to the classical Grothendieck-Riemann-Roch formula. In his "Noncommutative Geometry" Alain Connes has developed the framework for a large number of far reaching generalisations of the index theorems of Atiyah and Singer. To motivate the problem addressed in this book and to put it in the right context we recall some basic principles of index theory and noncommutative geometry. The classical index theorem tbr an elliptic differential operator D on a compact manifold M identifies the Fredholm index of this operator with the direct image of the symbol class of the operator under the Gysin map in topological K-Theory: Inda(D) = ~!(a(D)) 7d : K * ( T * M )
--~ K * ( p t . ) ~_ 7Z
In more general situations where one considers not necessarily compact manifolds (for example operators on the mfiversal cover of a compact manifold which are invariant under deck transformations, operators on a compact manifold differentiating only along the leaves of a foliation and being elliptic on the leaves, or elliptic operators of bounded geometry on an open nmnifold of bounded geometry) the considered elliptic operators are not Fredhohn operators anymore. Nevertheless it is still possible to associate an index invariant with them which now has to be interpreted as an element of the operator K-group of some C*-algebra. Moreover, Kasparov and Connes proved a number of very general index theorems of the form: Inda(D)
= 7r[(a(D)) e K o ( C * - algebra)
The C*-algebras occuring in this way can be of quite general type and their Kgroups usually cannot be identified with the K-groups of some topological space as in the classical cases.
vi As far as applications are concerned, the classical index theorem, formulated and proved in the context of topological K-theory, gains its full power only after being translated into a cohomological index formula with the help of a differentiable Grothendieck-Riemann-Roch Theorem. This theorem claims that for any K-oriented map f : X -+ Y of smooth compact manifolds the diagram
I~ >
K*(X)
K* (Y)
~ht
l~h
H3R(X)
I.(-uTd(f))
~ H3R(r)
COlrlnltlt, es. H e r e
ch : K* ~ H~n denotes the Chern character which is given by a universal characteristic class that identifies complexified topological K-theory of a manifold with its de Rham cohomology:
ch : K*(M) |
(~ _E~ H~R(M)"
Under this translation the direct image in K-theory can be identified with an explicit pushforward map in cohomology. Together, the index and Grothendieck-RiemannRoch theorem yield a formula expressing the Fredholm index of an elliptic operator D as integral over the manifold of a universal characteristic class associated to the symbol of D:
I nda ( D ) = /M characteristic class(or(D)) To obtain index formulas from the generalized index theorems above it is necessary to develop a Grothendieck-Riemann-Roch formalism in the context of operator K-theory. This means that one looks for a (co)homology theory on the category of C * - , Banaeh-, resp. abstract algebras, which is defined by a natural chain complex and carries enough additional structure to provide a commutative diagram
K,(A)
ch~
H.(A)
f~ > K . ( B )
~ch
?
> H.(B)
On the subcategory of algebras of smooth (resp. continuous) functions on compact manifolds it should correspond to the classical Grothendieck-Riemann-Roch theorem. So the Grothendieck-Riemann-Roch problem consists of three parts: 1. Define a (co)homology theory for Banach- (C*-) algebras which generalizes the deRham (co)homology of manifolds. 2. Construct a Chern-character from K-theory to this noncommutative deRham(co)homology.
vii 3. Find a cohomological pushforward map and establish a suitable GrothendieckRiemann-Roch theorem. After having formulated this program, Alain Connes also made the first real breakthrough concerning a solution of the problem. In his foundational paper "Noncomnmtative Differential Geometry" [CO] he introduced a generalization of de Rham theory in the noncommutative setting, cyclic (co)homology H C , (resp. H C * ) , which (;an be calculated as the (co)homology of a functorial chain complex vanishing in negative dimensions , and he constructed an algebraically defined Chern character ch : K , --+ H C , .
The dual Chern character pairing ch : K , | HC* -+
generalizes the pairing between idempotent matrices and traces in degree zero and the pairing between invertible matrices and closed one-currents on the given algebra in degree one. Cyclic cohomology proved to be a very powerful tool in many areas of K-theory, as the large nmnber of well known applications shows. The project of constructing characteristic classes for operator K-theory however soon faced serious difficulties. Whereas the 2Z/22~-periodic version H P * := l i m H C *+2k ---~k
of cyclic cohomology of the algebra of smooth functions on a manifold coincides with the deRham homology of the manifold, Hp*(c~
~ H,aR(M),
the periodic cyclic cohomology of its enveloping C*-algebra of continuous functions equals the space of Borel measures on M in even degree and vanishes in odd degree. HP*(C(M))
( C(M)'
0
*= 0
,=1
Thus while the Chern character pairing between reduced K-theory and reduced periodic cyclic cohomology yields a perfect pairing for the Fr~chet algebra C~ (M), it vanishes for its enveloping C*-algebra C ( M ) . (Note that both algebras can be considered as equivalent as far as K-theory is concerned). This example shows how cyclic cohomology and K-theory can behave quite differently in certain situations and that the Chern character from K-theory to cyclic homology can be far from being an isomorplfism.
viii Actually the pathological behaviour of the Chern character pairing for (stable) C*-algebras has nothing to do with the particular structure of cyclic cohomology but is a consequence of the continuity of the Chern character as the following argument shows: Let C, be any cyclic theory, i.e. a funetor from Banach algebras to chain complexes equipped with a Chern character ch : K,A -4 h(C,A) associating a cycle to each idempotent (resp. invertible) matrix over A. Let ~ be an even cocyele for this theory (the argument for odd cocycles is similar). This cocycle yields a map (still denoted by the same letter)
~: {eC A, e Z - e }
-4
which provides the pairing of the cohomology class of ~ with Ko(A). Suppose that the Chern character pairing satisfies the following conditions: (They hold for the Chern character pairings with continuous periodic cyclic cohomology HP* and with entire cyclic cohonlology HC*~.) 1) ~(e) depends only on the homotopy class of e. 2) ~(e) = qo(e') + ~p(e") if [e] = [e'] + [e"] in Ko(A). 3) I~(e)l _< F(II e tl) for some function F on the real half-line. Then if A happens to be a stable C"-algebra, the pairing K,A | h(C*A) -4 equals zero: In fact one observes that. the image of the map ~, viewed as a subset of ~, is closed under addition because A is stable and condition 2) holds. On the other hand this image is bounded by conditions 1) and 3), as any idempotent in a C*-algebra is homotopic to a projector (selfadjoint idempotent) and nonzero projectors in C*algebras have norm 1. So the image of ~ is a bounded subset of 9 closed under addition and thus zero. This fact is quite annoying because the generalized index theorem and the hypothetical Grothendieck-Riemann-Roch are theorems about C*-algebras and do not hold for more general Banach or Fr~chet algebras (bivariant K-theory is well behaved only for C*-algebras). Moreover, it is just the study of the K-theory and the cohomotogy of C*-algebras which is at the heart of the inost important applications: in the index-theoretic approach to the Novikov-conjeeture on higher signatures of manifolds, for example, one has to analyse the K-theory and cyclic cohomology of the group-C*-algebra C~ed(F) of the flmdamental group of the manifold under consideration. Finally another difficulty in establishing a Grothendieck-Riemann-Roch formula is that the pushforward maps of operator K-theory have no counterpart in cyclic homology. Connes and Moscoviei defined in [CM] a modified version of cyclic cohomology, called asymptotic cyclic cohomology, and pointed out that this theory should provide a nontrivial cohomology theory on the category of C*-algebras. Our work can be viewed as attempt to realize this program. This also explains the title of the book. The initial setup of asymptotic cyclic cohomology in [CM] had to be modified in several ways and the theory we are going to develop is however not equivalent to the one originally defined by Connes and Moscoviei.
ix Our aim is to develop a cyclic theory, called asymptotic cyclic cohomology after [CM], which is the target of a Chern character that appropriately reflects the structure and the typical properties of operator K-theory. The theory will generalize ordinary and entire cyclic cohomology providing thus a framework for the explicit construction of (geometric) cocycles and the calculation of their pairing with concrete elements of K-groups. Finally we establish a generalized GrothendieckRiemann-Roch theoreul for the Chern character from operator K-theory to stable asymptotic homology. This will be achieved by the construction of a bivariant Chern character on Kasparovs bivariant K-theory with values in bivariant stable asymptotic cyclic cohomology. The above argument for the vanishing of tile Chern character pairing gives a first hint how one has to modify cyclic cohomology to get a theory with the desired properties. Cochains should consist of densely defined and unbounded rather than of bounded functionals or, as Connes-Moscovici propose in [CM], continuous families of unbounded eochains with larger and larger donlains of definition. To realize our goal we however start from a quite different line of thought. Our point of departure is on one hand the work of Connes, Gromov and Moscovici [CGM] on ahnost flat bundles and of Connes and Higson [CH] on asymptotic morphisms and bivariant K-theory, and on the other hand the work of Cuntz and Quillen [CQ] on cyclic coholnology and universal algebras. In [CH] Connes and Higson made the important observation, that K-theory becomes in a very natural way a functor on a much bigger category than tile ordinary category of Banach (C*-algebras), uamely on the category with the same objects but with the larger class of so called "asymptotic morphisms" as maps. Especially they showed that every pushforward map in K-theory associated to a generalized index theorem is induced from an explicitely constructible asymptotic morphisnl of the C*-algebras involved. A (linear) asymptotic morphism of Banach algebras is a bounded, continuous fanlily (ft, t > 0) of continuous (linear) inaps ft : A --+ B such that linl f t ( a a ' ) -
t --+ o o
ft((t)ft(a') = 0 Va, a' E A
The deviation from multiplicativity w(a, a') := ft(aa') - f t ( a ) f t ( a ' ) is called the curvature of ft at (a, a~). The interest in this notion originates (among other things) from the fact, that tile E-theoretic K-groups, which are a modification of Kasparov's KK-groups, can be described as groups of asymptotic morphisms. A cohomology theory that is the target of a good Chern character on operator K-theory should certainly have tile same fimctorial properties as K-theory itself. Cyclic (co)homology however is by no means a functor on the asymptotic category. Therefore it is no surprise that the Chern character in cyclic homology fails to be an isomorphism in general.
Oil the other hand Connes, Gromov and Moscovici showed in [CGM], that the pullback of a trace T on an algebra B under a linear map f : A --4 B may be interpreted as an even cocycle in the cyclic bicomplex of A: oo
f* T = ~
7)2"
n=0
Moreover its components (7)2".) decay exponentially fast
1~2'~(~ ~ ..... ,~2'~)1 _<
c -n
when evaluated on tensors with entries a ~. . . . . a 2" belonging to a fixed finite subset E of A. The constant C depends on the deviation of f from being multiplicative on E. Cochains with this growth behaviour occur already in the calculations of localized analytic indices of Connes and Moscovici [CM], where the authors point out that a cyclic theory for C*-algebras should be based on such cocycles. Relating this to the approach to cyclic cohomology via traces on universal algebras by Cuntz and Quillen [CQ] suggests that it might be possible to pull back arbitrary cochains in the cyclic bicomplex under linear maps and that in fact every even(odd)dimensional cocycle in the cyclic bicomplex could be obtained as the pullback of a trace (resp. a closed one-current) under a linear map. Thus one might hope to reinterpret cyclic cohomology as being given by a chain complex that behaves functorially under linear maps and to obtain an asymptotic cyclic theory as the envelope under linear asymptotic morphisms of the ordinary cyclic theory. Cochains in this theory should be characterized by natural growth (resp. contimfity) conditions as in the exalnple above. In fact any cyclic theory which is functorial under asymptotic morphisms would possess the pushforward maps necessary to formulate a GRR theorem. So our starting point for the construction of asymptotic cyclic cohomology will be to take ordinary cyclic theory and to extend it to a functor on the linear asymptotic category C. (We restrict ourselves to linear asymptotic morphisms. It would have been possible to dispense with this restriction but only at the cost of making the formulas much more complicated without providing a wider range of applications.) This means the following. First we choose a natural chain complex C* calculating cyclic cohomology, i.e. a functor C*: Algebras --+ Chain Complexes such that H * ( C * ) ~_ H C *
.
xi
a)
Then we consider pairs (C*, (I)) consisting of
a funetor C * : C --+ Chain Complexes, such that the corresponding homology groups define a homotopy flmctor
H C : := H*(C~) : H o m o t C ~
(I]- Vector Spaces
b) a morphism of functors
9 : C* ~
C*IAZ,~b,.=~
on the category of algebras inducing a natural transformation
HC* -4 HC* from ordinary to asymptotic cyclic cohomology. Among all such pairs we look for a nfinimal one, i.e. a pair satisfying the obvious universal property. By an argument due to J.Cuntz any such cohomology theory will be Bott-periodic, so that C* (and C*) should in fact be 7Z/22Z-graded complexes. In [CO] Connes introduced a natural 2Z/22Z-graded complex, the (b,B)-bicomplex CC, of a unital algebra. An equivalent (but, not identical) complex ft PdR, the periodic de Rham complex, has been constructed later on by Cuntz and Quillen [CQ]. These are both complexes of modules of formal differential forms over the given algebra and carry a natural filtration (Hodge filtration), derived from the degree filtration on differential forms. The quotient complexes with respect to the Hodge filtration successively compute the cyclic homology groups H C , and the completed complexes ~,PdR (with respect to the Hodge filtration) calculate the periodic cyclic homology H P , of Connes. The periodic de Rham complex provides in our opinion the best choice for the complex C* above and it is therefore 9 PdR that will be extended to a functor on the linear asymptotic category. The universal problem above can be solved provided that the forgetful fnnctor Banach-algebras
--+ C
has a right adjoint R o An explicit solution would then be given by
5pdR,~ 9
:=
5.dR o R C
--,
If one forgets the topology for the moment and looks at the problem at a purely algebraic level, there is indeed an adjoin< provided by a canonical quotient of the full tensor algebra: R A := T A / ( 1 A - - I r This would lead to
~PdR,o~ (A) = ~~PdR f~. . ( R A9 )
xii 'file algebras R A are of Hochschild cohonlological dimension one, which makes it possible to calculate their periodic cyclic homology via tile quotient complex of tile periodic de Rham complex by tile second step of the Hodge filtration, the so called X-complex of Cuntz-Quillen:
~PdR qis) X . ( R A ) . (RA) where the X-complex is given by X.(A):
-~
A
~
~IA/[f~IA, A]
b
A
--+
In fact,, Cuntz and Quillen [CQ] showed that cyclic hoinology can be developed starting fl'om the X-complex of tensor algebras (resp. quasifree algebras). Moreover one obtains in this way a very natural and advantageous viewpoint of the basic features of the theory. A basic observation is that the tensor algebras R A are canonically filtered by powers of the ideal 0 ~ IA--+ R A .....~t) A ~ 0 So although the algebra R A depends only on the underlying vector space of A, the I-adic filtration on R A makes it possible to recover tile nmltiplicative structure of A. Remarkably, the X-complex of R A with its I-adic filtration turns out to be quasiisomorphic, as filtered complex, to the periodic de-Rham complex of A with its Hodge filtration. So whereas the complex X . ( R A ) is easy to manipulate algebraically it also contains all information encoded in the periodic de Rham complex of A with its Hodge filtration. Especially one recovers the periodic cyclic homology of A as tile homology of the X-complex of the (algebraic) I-adic completion of RA: 0pan(A) (qi, X.(R-'A) A
HP,(A) = H.(X.(RA)) In fact,, the I-adic completion R A of R A is still of cohomological dimension one although quite far from being free. Tile description of periodic cyclic (co)homology using the X-complex of tensor algebras exhibits the functoriality of the (uncompleted) cyclic complexes with respece to linear maps which is crucial for us but somewhat hidden if one uses Connes original cyclic (b, B)-bicomplex. Moreover the Cuntz-Quillen approach enables one to construct product operations and honlotopy operators for cyclic theories on the level of chain complexes by a uniform procedure. One tries to guess the right formulas for the periodic de Rham complex on differential forms of degree zero and one modulo error terms of higher degree. For free algebras, which are of Hochschild cohomological dimension one, the second step of the Hodge filtration is contractible, so that it becomes possible to get rid of the error terlns in this case. This yields by passing to the quasiisomorphic quotient complexes a map of X-complexes of free algebras. For free algebras of the form R A one finally recovers by taking tile associated graded complexes with respect to the I-adic filtration the whole periodic de Rham complex of the initial algebra A,
xiii this time with a globally defined chain map reducing to the initial formula on forms of low degree. As homotopic initial maps on forms of low degree provide homotopie global chain maps in the end, the effect of the constructed chain maps on homology is deternfined by their effect on ordinary cyclic homology of degree zero and one, respectively. There is a "Cartesian square" of flmctors Algebras
fo~g~ A!::;ebras, linear maps
R,I--adicfilt I Filtered Alg.
lR f~
Algebras
on the level of morphism sets. This shows that the I-adic filtrations on the complexes X,(RA) are never preserved by a homomorphism of tensor algebras which is induced by a linear morphism that is not nmltiplicative. Therefore not the degree, but only the parity of an ordinary cyclic cycle is preserved under pushforward by a linear rnorphism. In fact any even (odd) cocycle (in the 2~-graded setting) occurs as the linear pullback of a trace (closed one-current). This explains again why only a 2Z/22Z-graded theory can be defined on the linear asymptotic category. Concerning the original aim of nlaking cyclic coholnology functorial under linear asymptotic morphisms our goal can be described (in terms of the Cuntz-Quillen approach) as follows. Consider the diagram Morphisms:
linear
e-lnult,
mult.
Algebras:
A
=
A
=
A
Algebras:
RA
c
TiA =?
C
RA
Chain complexes:
X,(RA)
C X.(TCA) C X,(RA)
In the right column the Cuntz-Quillen procedure for obtaining the cyclic complex of A is described. The universal way to extend this construction to the category of algebras with linear maps as morphislns is given in the left column: one replaces the given algebra by its tensor algebra and constructs the cyclic complex of the latter algebra. The tensor algebra already being free one can directly pass to its X-conlplex. Tile complex X, (RA) cannot be interesting homologically however. It has to be contractible because every linear map is linearly homotopic to zero. Being interested in a nontrivial homology theory which is functorial under asymptotic morphisms, i.e. a functor on a "category of e-multiplicative linear maps" we have to look for an intermediate theory. One has to find a topological completion of the tensor algebra RA which is not contractible but functorial under e-multiplicative
xiv maps. If it is moreover of cohomological dimension one one can again take its Xcomplex to arrive at a reasonable theory (middle column). Such a completion is constructed as follows. Let .f : A ~ B be an almost muttiplicative linear map of Banach algebras. Then the induced homomorphism R f : RA ~ RB of tensor algebras will not preserve Iadic filtrations but the norms of the occuring "error ternls" will decay exponentially fast with their I-adie valuation. This suggests the following construction: Fix a nmltiplicatively closed subset K of A and consider tensors over A with entries in K. Expand a given element of this subalgebra of RA in a standard basis with respect to the I-adie filtration. A weighted L~-norm for the coefficients of such an expansion is then introduced allowing the coefficients to grow exponentially to the basis N > 1 with respect to the I-adic valuation. Denote the corresponding completion by RA(K,N). It is a Fr6chet algebra and possesses the following crucial property: If f : A ~ 13 is linear with curvature uniformly bounded on K C A by a sufficiently small constant then R f induces a continuous homomorphism R.f : RA(K,N) ~ RB(K,N,) for suitable K ' C B , N ' > 1. Usually f will be a linear asymptotic morphism. As the curvature of an asymptotic morphism is uniformly bounded only over compact sets, the nnfltiplicatively closed subsets K C A used for the construction above will always be assumed to be compact. It turns out that the algebras RA(K,N) are also of cohomologieal dimension one. The Fr6chet algebras RA(K,N) fornl an inductive system with formal inductive limit T/A. This linfit could be called the topological I-adic completion of RA. It shouht be viewed as virtual infinitesimal thickening of A as the kernel of the projection ir : T~A --+ A is formally topologically nilpotent (i.e. the spectrum of its elements equals zero). We define the analytic X-complex X,~ of a Banach algebra to be the reduced X-complex of the topological I-adic completion of the tensor algebra of its unitalization. The eoholnological analytic X-conlplex is closely related to the entire cyclic bicomplex of Connes. It turns out to be convenient to introduce also a bivariant analytic X-conlplex X * ( - , - ) of a pair of algebras as the Hom-complex of the associated analytic X-complexes. The bivariant analytic X-complex is a biflmctor on the category of Banach algebras and its cohomology groups are smooth homotopy biflmctors. There exists an obvious composition product
X2(A, B) | X:(B, C) ~ X*(A, C) . The fimdamental functoriality of the locally convex algebras RA(K,N) under almost nmltiplicative linear maps implies that every linear asymptotic morphism
ft:A-~B,t>O induces a continuous homomorphism of formal inductive limit algebras ~f:
TiA --+ TCB |
Oo~ (Tr
.
Here O ~ ( T ~ ) is the algebra of germs around oo of smooth functions on the asymptotic parameter space 7"r This leads one to define the (cohomological) asymptotic X-complex X*(A) of a Banach algebra A as the cohomological X-complex of the
XV
formal topological I-adic completion T~A with coefficients in the formal inductive limit algebra Oo~ ( T ~ ) . The bivariant asymptotic X-complex X~ (A, B) of the pair (A, B) is introduced as the complex of germs at oc of homomorphisms between the X-complexes of tile formal topological I-adic completions 7~A and T4B (See chapter 6). By construction any linear asymptotic nmrphism defines an even cocycle in the bivariant asymptotic X-complex. Tile composition product carries over to the asymptotic setting and turns X * ( - , - ) into a biflmctor on the linear asymptotic category. Moreovcr bivariant asymptotic cohomology becomes a (continuous) asymptotic homotopy bifimctor. So nmch for the motivation and definition of the asymptotic cyclic theory. We have to be more precise at one point however. Asymptotic morphisms do not consist of a single, but of whole families of linear maps, and one has to keep track of the chain homotopies provided by evaluation at different "parameter values" in such families. We do this hy working throughout in the category of differential graded algebras and differential graded chain complexes. Tile asymptotic X-complex of tile universal enveloping differential graded algebra of the given algebra is large enough to contain the higher homotopy information needed. One obtains then Cartan homotopy formulas for the "change of asymptotic parameters". There are natural maps CC* -+ X ~ ,
CC[ --+ X~
in the derived category yielding natural transformations
HP* ~ H e ; ,
H e : ~ HC~
on eohomology. For the algebra of complex numbers the maps on cohomology above are isomorphisms. More generally, analytic and asymptotic homology coincide: HCi((~,A) ~_ HC*((F,A) . The corresponding cohomology groups are in general quite different however. The well known pairings between cyclic theories and K-theory extend to a pairing K , | HC~, ~ qJ. It is uniquely determined by its naturality with respect to asymptotic morphisms and by demanding that it restricts to the classical pairing between idempotents and traces (resp. invertible elements and closed one-currents) on the ordinary cyclic complex. As for a given value of the asymptotic parameter a cocycle is given by a sequence of densely defined multilinear functionals on the underlying algebra A, the pairing can be defined for this choice of parameter only for special representatives of a finite number of classes in K , A . Taking a family of parameter values which approaches oc in the asymptotic parameter space allows to define the pairing on larger and larger subsets of K , A which finally exhaust the whole K-group and yield the pairing globally. This behaviour explains why the argument at the beginning of the introduction showing the pathological nature of the Chern chracter pairing for the classical cyclic theories on stable C*-algebras does not apply to the
xvi asymptotic theory. Indeed there is a large class of stable C*-algebras for which the pairing of K-theory with asymptotic eohomology is nondegenerate. The most striking new phenonmnon of asymptotic cyclic theory is that inclusions of holomorphically closed subalgebras become cohomology equivalences in many cases. This often allows one to construct asymptotic cocycles on C*-algebras by lifting well known cyclic cocycles from a suitable dense subalgebra. Since these subalgebras are not Banach algebras anymore, we develop the theory for the slightly larger class of admissible. Fr~,chet algebras, i.e. Fr~chet algebras possessing an analogue of the open unit ball of Banach algebras. These algebras seeln to provide the natural framework l~)r our theory. The descent principle to holomorphically closed dense subalgebras can be used to show that asymptotic cyclic cohomology is stably Morita invariant: for any C*algebra A the inclusion A ~-~ AOc-/C('H) induces an asymptotic (co)homology equivalence. In particular {~ HC;(K('H)) =
0
*=0 * :
1
in sharp contrast to the cyclic theories known so far. In order to go further it, is necessary to develop product operations. principles explained above we are able to construct a chain map
By the
x : X , R ( A | B) --+ X , R A @ X , R B
which is associative up to homotopy and yields exterior products X~,~(A)|
--+ X < ~ ( A |
B)
both for analytic and asymptotic cohomology. It behaves naturally with respect to asymptotic morphisms. Moreover, the pairing of K-theory with analytic (resp. asymptotic) cohomology is compatible with exterior products. To be precise, the compatibility of the products in K-theory resp. the cyclic theories holds only up to a factor 2rri if the involved classes are of odd dimension: the cyclic theories are a priori 2g/27Z-graded, whereas the product of odd classes in K-theory has to be defined using Bott periodicity, which causes the "period" factor 27ri. This makes me believe that the exterior product on cohomology coincides up to normalization constants with Connes's product. I have not investigated this point however.
xvii The attelnpt to define an exterior product of bivariant X-complexes was only partially successful up to now. The main difficulty lies in the construction of a homotopy inverse of the exterior product map for the ordinary X-complexes above. (See [P], where meanwhile a natural homotopy inverse has been constructed.) At least it is possible to establish a particular consequence of a bivariant product operation, namely the existence of a slant product
K.(A) | HC*~(A |
B) ~ HC:,~(B)
It is constructed in such a manner that any idempotent (or invertible) matrix over A gives rise to an explicit map X:,~(A | B) ~ X~,~(B) of chain complexes. Its homotopy class depends only on the K-theory class of the given matrix. The slant product behaves naturally with respect to asymptotic morphisms and is compatible with the exterior product. It represents a convenient tool to prove the split injectivity of the exterior product with cohomology classes in the image of the Chern character. As an application we show that the exterior (resp. slant) product with the fimdamental class of the circle yields an isomorphism
HC;(S~,, Sr
~ HC;((U, r
of the bivariant asymptotic eohomology of ~ and its suspension S(IJ = Co(ffi~). Extending this argmnent from I]J to more general admissible Fr~chet algebras A by taking the exterior product with the bivariant cohomology class of the identity on A unfortunately fails: the exterior product is only defined for unital algebras and unitalization does not commute with taking tensor products (the suspension of an algebra is nonunital). In fact it seems to me to be a difficult question, whether an adnfissible Fr~chet algebra is equivalent in asymptotic cohomology to its double suspension (this could be called a cohomological Bott periodicity theorem). In fact such a periodicity theorem would be highly desirable because it necessarily has to hold for any theory with reasonable excision properties. At this point the E-theoretic description of Bott periodicity [CH] fortunately saves us as it realizes the bivariant Bott- resp. Dirac elements inducing the Ktheoretic periodicity maps stably by (nonlinear) asymptotic morphisms. This allows to prove a stable version of cohomological periodicity: there are natural asymptotic cohomology equivalences
O~SA C HC~(S2A, SA),
flSA E HC~(SA, S2A),
inverse to each other under the composition product. Suspending an algebra therefore only produces a shift of its stable asymptotic cohomology groups HC*(S-, S - ) , so that stable asymptotic cohomology becomes in fact a bifimctor on the stable linear asymptotic homotopy category. This opens the way to derive exactness and excision properties of stable asymptotic cohomology which make these groups quite accessible in many situations. By adapting a well known argument fl'om stable homotopy theory, it can be shown that the long cofibre (Puppe) seqnence associated to a homomorphism f : A -+ B of admissible Fr~chet
xviii algebras induces six term exact sequences on (bivariant) stable asymptotic cohomology relating the stable cohomology groups of A and B to those of the mapping cone C I of f. A short exact sequence
O-+.J--+ A & B--+ O of admissible Fr~chet algebras gives rise to six term exact cohomology sequences if and only if stable excision holds. This means that the inclusion of tile kernel J into the cofibre Up of the quotient map p induces a stable asymptotic (co)homology equivalence. Following an argument of Connes and Higson we show that stable excision holds for any epimorphism of separable C*-algebras that admits a bounded linear section. This is the only place where we have to restrict, ourselves to a particular class of adnfissible Fr~,chet algebras, as we need the existence of a bounded, positive, quasicentral approximate unit in the kernel J of p. With all this machinery developed it becomes possible to extend the Chern character to the bivariant setting, i.e. to construct a transformation of bifunctors:
ch : K F : * ( - , - ) --4 H C ; ( S - , S - ) from Kasparov's KK-theory to stable bivariant asymptotic cohomology. In principle it is given by the "composition" (see [CH])
K K * -4 E* " -~ " HC~,~t , where the "arrow" on the right hand side maps an asymptotic morphism to the corresponding bivariant asymptotic cocycle. As the asymptotic morphisms of Etheory are nonlinear however, one has to be careflfl in the actual construction of the bivariant Chern character. In particular, one obtains a Chern character on Khomology defined for arbitrary Fredholm modules and generalizing tile constructions known so far. The Kasparov product on bivariant K-theory corresponds to the composition product on asymptotic cohomology, which is precisely the Grothendieck-Riemann-Roch Theorem: The diagram
KK*(A,B)|
~ > KK*(A,C)
oh| I H C ; ( S A , SB) | H C ; ( S B , SC)
lch ~|
HC~(SA, SC)
commutes. For A = (~ this yields a Grothendieck-Riemann-Roch formula as asked for in the beginning, (The factor 2~ri occurs for the same reason as in the comparison theorem of the ordinary Chern character with products). Consequently the Chern character of a KK-equivalence yields a stable asymptotic (co)homology equivalence. The bivariant Chern character becomes an isomorphism between eomplexified KK-theory and stable bivariant asymptotic eohomology oil a class of separable C*-algebras containing ~ and being closed under extensions with completely positive lifting and
xix KK-equivalences. This shows that with asymptotic cyclic cohomology we have come much closer to the "right" target of a Chern character in operator K-theory. Finally we present two explicit calculations of asymptotic cyclic cohomology groups in concrete examples. In the first the functorial and excision-properties of the theory are used to determine the stable asymptotic homology of separable, commutative C*-algebras. If A is a separable, commutative C*-algebra with associated locally compact space X, then
where on the right hand side sheaf cohomology with compact supports is understood. In the second example we outline and illustrate a procedure to calculate asymptotic homology by standard methods of hmnological algebra. Besides the calculation of asymptotic cohomology groups in concrete examples the most obvious questions not studied in this paper are the determination of more explicit versions of the Grothendieck-Riemann-Roch theorem and their application to generalized index problems. We plan to treat these topics elsewheie. The plan of this book is as follows: In chapter 1 we introduce the linear asymptotic homotopy category and the notion of an admissible F%chet algebra. It is shown that these algebras behave like Banach algebras as far as spectral properties, holomorphic functional calculus and K-theory are concerned. Finally we demonstrate that K-theory becomes a homotopy functor under asymptotic nlorphisms. Chapter 2 begins with a heuristic motivation of the definition of the periodic de Rham complex of an algebra, starting from the desired formal properties of the pairing of K-theory with de Rllam homology. Then the (co)homologieal and bivariant X-complexes are introduced and studied ill the ordinary as well as the differential graded setting. Chapter 3 presents the idea of extending functors from categories of algebras to larger linear categories and develops the approach to cyclic cohomology of Cuntz and Quillen, adapted to the differential graded case. Except for the definition of an admissible Frfichet algebra the first three chapters collect material due to Connes, Connes-Higson, and Cuntz-Quillen. This was done on one hand for the convenience of the reader and on the other hand to document the modifications necessary for our needs. In chapter 4 Cartan homotopy formulas are derived by the method explained in the summary above. They are used to show that derivations on an algebra act trivially on its cohonmlogy, as well as for controlling the change of asymptotic parameters in the differential graded case. Finally some comparison theorems between ordinary, bivariant and differential graded X-complexes are presented. All these results could have been shown by a short abstract argument given at the end of the chapter. We have however decided to go the longer way of giving complete and
XX
explicit constructions on the level of chain complexes whenever possible. This was done to make continuity properties readily accessible and especially to enable one to do explicit calculations in concrete examples. We begin to discuss the analytical aspects of tile theory in chapter 5. Tile topological I-adic completion of the tensor algebra over an admissible Fr6,chet algebra is studied. Then the definition of tile analytic X-complex is given and elementary properties of analytic cyclic (co)homology are derived, The asymptotic X-complex and asyinptotic cyclic cohomology are treated analogonsly in chapter 6. The demonstrations are more involved however. A notable difference is that whereas analytic cohomology is a homotopy flmctor only with respect to smooth homotopies, in the asymptotic case even continuous homotopies may be allowed. Explicit formulas for tile pairing between K-theory and asymptotic cohomology are given at the end of the section. They are a bit more general than the well known Chern character formulas of Connes and Cuntz-Quillen. This will be of use when we analyse tile compatibility of the pairing with products. With the derivation lemma in chat)ter 7, we get the theory off the ground. Tire criteria for tire inclusion of a dense subalgebra to be an asymptotic (co)homology equivalence apply in two cases: for the inchlsion of the subalgebra of smooth elements with respect to the action of a one parameter automorphism group arrd tbr the inclusion of the domain of a positive, unbounded trace on a separable C*-algebra. Some examples are discussed which will reappear frequently in the remaining chapters. The construction of product operations is given in chapter 8. The exterior product B) is defined on the level of chain complexes. The utility of this product depends heavily on the derivation lemma, which enables one to lift the product of two cohomology classes to topological tensor products other than the projective one. This makes it possible for example to prove the stable Morita invariance of the asymptotic cohomology of C*-algebras. After this application the compatibility of the Chern character with exterior products is shown. This finally justifies our choice of constants in the definition of the exterior product. The chapter ends with the construction of the slant product.
HC*o(A)~HC~,~(B) --+HC*c,(A |
Chapter 9 i8 devoted to exact sequences of bivariant asymptotic cohomology groups. It begins with the stable periodicity theorem and the calculation of the coefficient groups of stable asymptotic cohomology. The proofs are quite tedious as one has to descend several times to different "smooth" subalgebras in order to avoid nonlinearity of the involved asymptotic morphisms. From the stable periodicity theorem the excision theorems for mapping cones of morphisms of admissible Fr~chet algebras and for separable C*-algebras are derived. In chapter 10 we discuss the bivariant Chern character from KK-theory to stable asymptotic cohomology and prove the generalized Grothendieck-Riemann-Roch theorem. In the final chapter 11 the stable asymptotic homology of separable, commutative C*-algebras is computed and a general scheme for the calculation of asymptotic cyclic (co)homology groups is outlined.
xxi For a more detailed overview consult the introductions of the various chapters. A p a r t from the last chapter the text coincides with the authors 1994 d o c t o r a l thesis at the Universit/it Heidelberg.
Acknowledgments: F i r s t of all, I want to thank my advisor Professor Joachim Cuntz most heartily. His constant s u p p o r t and his patience made it possible for me to carry out this project over the last three years. Discussions with him and his advice were of great help b o t h m a t h e m a t i c a l l y and psychologically, especially when I arrived at a point, where it became clear t h a t large parts of the theory had to be redeveloped from a modified t)oint of view, which happened more than once. I a m especially indebted to Prot~ssor Ryszard Nest with wtlom I had very fruitful discussions on the subject during a stay in Copenhagen and in Heidelberg. To him I owe the idea to work with the "simplicial a s y m p t o t i c p a r a m e t e r space" to get a rigid definition of the composition of a s y m p t o t i c morphisms. Finally I want to express my deep g r a t i t u d e to Professor Alain Connes. His courses and seminars in Paris and the enlightning discussions with him were a rich source of motivation and ideas for my s t u d y of n o n c o m n m t a t i v e geometry and cyclic cohomology. I am glad t h a t he was willing to be a referee of this thesis.
Contents
C h a p t e r 1: The asymptotic h o m o t o p y category
1
1-1 Asymptotic parameters 1-2 Asymptotic morphisms
1 4
1-3 Admissible Fr~chet algebras
10
1-4 K-theory of admissible Fr~chet algebras
15
C h a p t e r 2: Algebraic de R h a m complexes 2-1 The periodic de Rham complex
19
2-2 X-complexes
20
2-3 Differential graded X-complexes 2-4 The algebraic Chern character
22 25
C h a p t e r 3: Cyclic cohomology 3-2 The algebra RA
27 27 28
3-3 I-adic filtrations 3-4 Cyclic eohomology
30 34
C h a p t e r 4: H o m o t o p y p r o p e r t i e s of X-complexes
40
4-1 The Caftan homotopy formula
41
4-2 Homotopy formulas for differential graded X-complexes
50
Chapter 5: The analytic X-complex
59
5-1 Behaviour of I-adic filtrations under based linear maps 5-2 Locally convex topologies on subalgebras of RA 5-3 The analytic X-complex
60
5-4 The analytic X-complex and entire cyclic cohomology 5-5 The analytic Chern character
90
3-1 Extending functors
19
66 82
95
xxiii
Chapter 6: T h e a s y m p t o t i c X-complex 6-1 The asymptotic X-coinplex 6-2 Comparison with other cyclic theories 6-3 Functorial properties of tile asymptotic X-complex
97 103
6-4 Hoinotopy properties of the asymptotic X-complex 6-5 The pairing of asymptotic cohomology with K-theory
108 113
C h a p t e r 7: A s y m p t o t i c cohomology of dense subalgebras
118
7-1 The derivation lemma
118
%2 Applications
121
97
107
Chapter 8: P r o d u c t s
127
8-1 Exterior products
129
8-2 Stable Morita invariance of asymptotic cohomology
146
8-3 Compatibility of the Chern character with exterior products 8-4 Slant products
148 154
C h a p t e r 9: Ex act sequences
158
9-1 The stable periodicity theorem
160
9-2 Puppe sequences and the first excision theorem
170
9-3 Stable cohomology of C*-algebras and the second excision theorem
177
C h a p t e r 10: K K - T h e o r y and a s y m p t o t i c cohomology
182
10-1 The bivariant Chern character and tim Ricmann-Roch theorem
183
Chapter 11: Examples
202
11-1 Asymptotic cylic cohomology of commutative C*-algebras
203
11-2 Explicit calculation of asymptotic cohomology groups
217
I n d e x of N o t a t i o n s
232
I n d e x of Symbols
235
Bibliography
237
Chapter 1: The asymptotic homotopy category In this chapter tile asymptotic homotopy category of Connes-Higson [CH] is recalled. It is the natural domain of definition of tile topological K-functor for C*algebras and will play a central role in the book. There are a few differences between the presentation in [CH] and our presentation though. Whereas nonlinear asymptotic morphisms play a crucial role in the work of Connes and Higson we will consider only linear ones. On the other hand the definition of the asymptotic category will be modified such that the composition of two asymptotic morphisms (and not only its homotopy class) can be constructed explicitely. This is done by replacing the positive real halfline /R+ as asymptotic parameter space by a cosimplicial space T ~ given in codimension n by /R~[ U oc (with a nonstandard topology around oc). As objects of the asymptotic homotopy category we introduce the class of adnfissible Fr~chet algebras. A Fr~chet algebra is called admissible if it possesses an analogue of the open unit ball of Banach algebras. An open neighbourhood of zero will be called an "open unit ball" or "small" if the multiplicative closure of any compact subset of this "unit ball" is relatively compact in the given Fr~chet algebra. The most basic examples of admissible Fl'~chet algebras are Banach algebras, for which the open unit ball in the norm sense is an "open unit ball" in our sense. In fact, admissible Fr6chet algebras share a number of formal properties with Banach algebras: the spectra of its elements are compact and nonempty and holomorphic flmctional calculus is valid for admissible Fr6chet algebras. The advantage of this class of algebras compared to Banach algebras is that it is stable under "passage to holomorphically closed dense subalgebras". This is why admissible Fr6chet algebras are taken as the class of algebras considered ill this paper. The chapter ends with the definition and the study of elementary properties of topological K-theory for tile linear asymptotic homotopy category of admissible Fr6,chet algebras.
1-1 Asymptotic parameters In this chapter a category will be considered whose morphisms consist not of single maps but in fact of a whole family of maps ',ff a certain kind. If morphisms f, g are given by families of maps parameterized by a single space X, (to take the simplest case) then tile possible compositions of n morphisms will be parameterized by the space X ~. The natural candidate for the total parameter space of morphisms will then be the union X = [.J,~X n. This is now carried out in the case we will be concerned with.
Let N+ := [0, oo[ denote the closed real halfline. The disjoint union of the algebras C ( ~ _ ) (C~(tg~_)) of continuous ((:ontinuous and smooth on the interior) complex valued functions are denoted by
:= U
) coo
:= Uco ( _s)
n
,o,
The union of the subalgebras of bounded flmctions are denoted similarly by Cb(tg~). A map from an algebra A to C(ff~T) is by definition a map from A to C ( ~ _ ) for some
n,
Two maps f, f ' : A -+ C(~,~~ given by .f : A -4 C ( ~ _ ) , f ' : A -+ C ( ~ ) are identified if there exists an order preserving inclusion i : {1,...,rn} -4 { 1 , . . . , n } such that the triangle
A
.f / --~
ti* C(t~)
/,
eomnmtes. The maps f occurring in that way are called degenerate. Tim algebras C ( ~ )
are augmented by C(/R~_) -+ r
f -~ f(O . . . . . O)
There is a canonical ,nap C ( ~ r | C(/R~) -4 C ( ~ )
given by
~zq-rr~
f | 9
-4
7r;fTc~9
It is compatible with the identifications made above. Remark:
In the terminology of algebraic topology ~ is a cosimplicial space and C(tR~r is a simplicial algebra. A nondegenerate map into C(tR~) in our terminology corresponds to a map into a nondegenerate simplex of C(tR~~ and the identifcations made mean that degenerate simplices are identified with their corresponding nondegenerate simplex. The reason for not using the standard terminology is that the asymptotic parameter space T ~ obtained by introducing a new topology on ~ U {oc} will not be a cosimplicial space anymore (there will be no coface maps). In order that asymptotic morphisms (to be defined later) compose well it is necessary to change tile topology of tile parameter space "R~':
3
D e f i n i t i o n 1.1:
For n E LW the topological space ~ is defined to be the space with underlying set /R,~ U {oc} and the topology defined by the standard topology on ffQ and the fundamental system of neighbourhoods of oo given by Uf
...... f,, :---- ( ( X l , . . . , X n )
e ~;;Xl
> fl(O),x2
> f2(xl),.-.,Xn
)" f,~(z,~_l)} U {oo}
where fi are positive, strictly monotone increasing, convex, unbounded, realvalued flmctions. 7 ~ := [_J,~T/~_ is called the a s y m p t o t i c p a r a m e t e r space. []
The projection map rn + --+ ~ +n x ff~+ .•n+m
extends to a continuous map + -+T/+n xT~+m .]•ng-m and yields therefore a canonical homomorphism
Co(R;) | Co(n ')
Co(nT;+m)
One observes L e m m a 1.2:
a) Under the projection map 7rn :
U f i ..... f,, ~ h~,~_
(Xl,...,Xn)
--4
J~-t-
---}
.Tn
tile inverse images of relatively compact sets are relatively compact. b) If f : A --+ C(ff/~_) is degenerate with associated nondegenerate map f ' : A ~ C(ff~_~) then f E C0(7~_) iff f ' ~ C0(7~) c) The flmdamental open neighbourhoods of oo are convex. []
4 1-2 Asymptotic morphisms [CH] Following Connes and Higson we introduce in this p a r a g r a p h the linear asymptotic h o m o t o p y category.
Definition 1.3: Let A, B be associative ~ a l g e b r a s and 0 : A -~ B a ~ l i n e a r map. The c u r v a t u r e coo of 0 at (x, y) C A • A is defined to be
%(*, y):= ~(:,:.,j)- o(x)o(.v) [] Tile curvature of a linear m a p satisfies the
1.4 Bianchi identity o(a)co(b, c) - w(a, b)o(c) = co(ab, c) - w(a, bc) If A, B, f are unital the curvature descends to a map
w : A/(~.I | A / r
-~ B
Definition 1.5:(Connes-Higson) Let A, B be (augmented) complex Fr~chet algebras. A (smooth) asymptotic
morphism o:A~B is a continuous linear m a p
ot : A -~ C b ( ~ ,
B)
( resp. & : A -+ e ~ ( ~ . ~ , B) A Cb(~.% B))
satisfying i) If A is augmented, then Ot is unital and compatible with a u g m e n t a t i o n maps
ii) wo,(x,y) C C o ( n ~ , B )
V(x,y) E A • A []
L e m m a 1.6:
Let. t) : A --+ B be an asymptotic morphism. Then lim wt(x, y) - 0
t--~ o o
mfifornlly on compact subsets of A x A. Proof." As w t ( x , y ) is bilinear and contimlous on A x A it suffices to remark that any ('ompact subset of a Fr6chet space is contained in the closed convex hull of a nullsequence: L e m m a 1.7:
Let ,,4 C A be a dense subspace of a F%chet space A. Choose a strictly monotone decreasing sequence of positive real numbers OG
(An); A n > ( ) ;
EA.=I n:O
Let K C A be conlpact. Then there exists a countable set B c A with single accunmlation point 0 such that o~
K c {y~A~b~ It), ~ B } i=0
Proof:
As A is dense the balls (in some metric d oil A )
:~:E.A form an open covering of A and therefore of K for all j E / N . Choose ( x ~ , . . . , x{j) E A such that nj
K C Uk=lU(XJk,
2
1 Aj)
and
K N .g(xj, ~1 A2j ) r
Vj, k
One has then 2 z~"" ~ B(z ju- 1 ,Aj_I)
for some k t. Put
Ao
,...
'"~ u { ' Ao
A5-1
;j ~ ~w; d(X,,k, j X 5-1 k, ) ~- A j2_ 1 }
B is a subset of A which has the only accumulation point 0 and is therefore also compact: for any given defining seminorm on A there are only finitely many elements of/3 which in the given seminorm are larger than any given e > O.
6
Let now y C K . Choose for any j E PC an integer k j ( y ) c { 1 , . . . , n j } such t h a t
Then
xO
~
/ xj
_ Xj - 1
~o ~~ + Y-~:~j-1 f ~ ( ~ ) - ~
\
'(~)1
and "
j
1
'
d(3:~i(y),:l:kj_l(y))
._
1
1
2
2
~__ d(xJkj(y),y) @ d(y,.T2kj-ll(y)) ~ --/~2 _~_ 2/~j--1 ~ "~j--1
Therefore
x jk~ (y) - "ckj 9j - 1_ ~(y)
EB
Aj-1 and the l e m m a is proved. [] Every h o m o m o r p h i s m of (augmented) Fr~chet algebras defines an a s y m p t o t i c m o r p h i s m in an evident way. For nontrivial examples see [CH]. (Note however t h a t in their p a p e r a s y m p t o t i c morphisms maybe nonlinear in general).
Composition of asymptotic morphisms Let g : A -+ B, Q~ : B -4 C be (smooth) a s y m p t o t i c morphisms represented by Or: A -4 Cb(~tn~, B ) , gi : B -4 C b ( ~ ' ~ , C )
Define Or o p : A --> C as the composition
(o' oo)~ : A
&
Cb(~LB)
A
-4
C~(~_,B)
c~(o')> -4
C b ( ~ _ , Cb( ~ ' : , C) ) ~-- C b ( ~ ; +m, C)
C~ (at;, C~162 ( ~ ; ', C)) -~ C ~ ( ~ ; +m , C)
Proposition 1.8: (Augmented) Fr~chet algebras and a s y m p t o t i c morphisnls form a category under the composition defined above.
Proof." Let 0 : A -4 B, 0' : B -4 C be as above. It is clear that (0' o O)t : A -4 Cb(ft~_ +m) is b o u n d e d and a u g m e n t a t i o n preserving if 0 and 0 r are. For the curvature of 0 ~ o 0 one finds:
~o, oo(X, y) = o'o(zy) - o'o(z)o'o(y) = = o'(o(xy) - o(x)o(y)) + o'(o(x)o(y)) - 0'O(x)0r0(~) = O'(~&, Y)) + ~;(0(~), O(Y))
7
Curvature estimates: Let I1 I1~, II ll~, be seminorms on B , C such that there exists tl O' lIE Bg with
tl d(b)(~) I1~_<11d Illl b I!~ vb ~ B,r, c ~ + Choose Uf, ..... y,~ C Y/~ such that f
II %(x,.v)II~< 2 II o' t[ o n U f t ..... f , .
For j E fV define
Kj := {Ot(x)(U A .....f. f-I rr~-I ([0, j + 1]), Ot(y)(Uf ...... f. N rrgl([0, j + 1])} C t3 These sets are relatively compact by Lemma 1.2. By the uniformness of curvature estimates (Lemma 1.6) there are open sets U/,{ ..... h{, C 7~2
with
sup,,,.v,eK ' II wr
~
on u a
Construct inductively strictly monotone positive functions
w , . . . ,am c c(~+),,hng.vdt) = +oo satisfying
9t(t) g2(t)
>
sup0_
>
SUPo
gin(t)
> suPo<_i<jh~(t ) for t > q , ~ _ l o " ' o g l ( j )
for
t > j
t>gl(j)
Then
II %'oA~, y)I1~< Oil /Dn+m
The associativity of the composition is obvious. []
For two (augmented) Fr~chet algebras A, B we denote by A | B their projective Fr6chet tensor product, i.e. the (augmented) algebra A | B completed with respect to the (;ross seminorms ~ a~|
i
where II ]]a, I] I1~ ranges over a system of semirlorms defining the topology on A, B.
Lemma
1.9:
There exist natural product and cylinder-maps
| : Hom~(A, B) x Hom~(A', B') -+ Hom,~(A |
A', B @,~ B')
Cyl : Hom~(A, B) --~ Homc,(A[O, 11, B[0, 1]) Proof:
Define the product O @ 0' of 0 : A -~ B, co' : A' --+ B' as the composition A @~ A' o,|
Cb(~TQ, B) |
.~ ,~,~+m , B e~. B') C b ( ~ +m, B , ) --+ ~bt-~+
This is clearly a bounded, linear, (augmentation preserving) map. To obtain curvature estimates, we observe that any element (and indeed any compact subset) of A | A' is contained in the closed convex hull of a nullsequence of elements of the algebraic tensor product A | B (Lemma 1.7), so that by the bilinearity of the curvature it suffices to check the necessary estimates on simple tensors a | a' 6 A | As The curvature formula
Wo| =
'
l
,
~.(~o, ~) | Q (aoa~) +
| ao,al | al) =
Q(aoa~) |
,
,
,
~o, (~o, el) - ~.(~o, ~1) | ~ ' (ao, a~)
shows that if on UI ...... f. and
II ~o'(alo, al) II~< ~ on Ug~,...,~,, for some finite set of seminorms defining the topology on B, B', then II ~| O11
Ufl
(ao | 4, ~1 | al) I1~|
(11Q'(4al) Ib + tl ~(~oal) I1~ +d
,...,f,, ,gl ,...,.qm
The construction of the cylinder map is evident: it is the map given by 0 | I d : A | C[0, 1] --+ B | C[0, 1] on the algebraic tensor product. The above proof of the curvature estimates applies for the completion A[0, 1] of A | C[0, 1] as well as for the projective completion A | C[0, 1]. []
9
D e f i n i t i o n 1.10:
a) Two asymptotic morphisms 00, LO1 : A -+ B are called h o m o t o p i c if there exists an asymptotic morphism X : A --+ B[0, 1] such that O0 = i 0 o x , where
i0,1
ol=ilox
B[0, 1] ~ B are the evaluation maps at 0, 1 E [0, 1] (0, 1 E ~ ) . []
Theorem
1.11"
a) (Augmented) Fr~chet algebras form under linear asymptotic morphisms a category, the l i n e a r a s y m p t o t i c c a t e g o r y 79. b) Similarly, (augmented) Fr~chet algebras form a category under smooth asymptotic morphisms, called the s m o o t h linear a s y m p t o t i c c a t e g o r y 7900. c) The associated homotopy categories are denoted as the l i n e a r a s y m p t o t i c h o m o t o p y c a t e g o r y 79homot, resp. the s m o o t h linear a s y m p t o t i c h o m o t o p y c a t e g o r y 79 h~176 . d) The naturalflmctor 79 h~176 --} ~[~hornot
induces an equivalence of categories. The set of homotopy classes of linear, asymptotic morphisms from A to B will be denoted by [A, B]~. Proofi
The composition of asymptotic morphisms clearly preserves homotopy, smoothness and smooth homotopy. (This follows from the existence of tile cylinder flmctor Lemma 1.9). d) It has to be shown that 79~omot(A,B) --+ 79homot(A,B) is bijective for any augmented FrOchet algebras A, B. Let 0 E 79(A,B); 0t : A --+ C b ( ~ _ , B) Choose a continuous family ~ of positive smooth functions on ~ n+ • ~ with support close to the diagonal A C ffQ x/R~_ and approximating the delta distribution along the diagonal. Then convolution
10
(,-~) C b ( ~ , B) --+ Cb(~_ +1, B) f -+ tz~*f defines an asymptotic morphism canonically homotopic to ~ whose image lies in C ~ ( ~ _ + 1 B). This shows the surjectivity of the map under consideration. The injectivity is clear. [] A
e~ ~
1-3 A d m i s s i b l e Fr~chet a l g e b r a s The appropriate category of Fr~chet algebras for which asymptotic cohomology will be developed is introduced now. It turns out that the category of Banach algebras is not large enough for our purposes because we want to construct asymptotic cocycles by lifting cocycles from a "smooth" subalgebra to the whole algebra and "smooth" (C~ ) subalgebras of Banach algebras are not Banach algebras anymore. On the contrary admissible Fr~chet algebras are stable under passing to smooth subalgebras. D e f i n i t i o n 1.12:
Let A be an algebra and let K C A be a nonempty subset. The multiplicative closure K ~176 C A of K is defined as OG
CO
77,
n=l
1
ailai E K n=l
I::]
D e f i n i t i o n and L e m m a 1.13:
A Fr6chet algebra A is called a d m i s s i b l e conditions is satisfied:
if one of the following equivalent
a) Every nullsequence has an end whose multiplicative closure is relatively compact. b) Every nullsequence has an end whose multiplicative closure is a nullsequence. c) To every compact subset K C A there exists a neighbourhood U of 0 such that K N U has relatively compact multiplicative closure. d) There exists a neighb'ourhood U of 0 such that every compact subset of U has relatively compact multiplicative closure. Open neighbourhoods of 0 satisfying condition d) are called " s m a l l " .
11
Proof:
a)~ b): Let (a,~) be a nullsequence and choose a strictly increasing sequence (An) of positive, real numbers tending to infinity such that (Anan) is still a nullsequence. Choose N big enough so that A,~ > 2 for n _> N and that the multiplicative closure of {)~nan; n > N} is relatively compact. Denote its closure by K. K being compact lim~-~0 Ax = 0 uniformly on K. This shows finally that the multiplicative closure of (a,) = (~-/~nan) n ~_ N is a nullsequence.
b)~ c): Let K C A be compact. If K does not contain 0 one has K • U = ~1 for some neighbourhood U of 0 and the assertion is trivial. So we may suppose 0 E K. According to Lemma 1.7. one can choose a nullsequence (b,~) such that any element y C K can be expressed as oo
y=Eltibi
oo
#i > 0; E # i _ < l
i~0
i----0
for some sequence (Pi) of positive, real numbers. By assumption some end of (bi) has a relatively compact multiplicative closure. By having a closer look at the proof of 1.7. one sees that (in the notations used there) for fixed j = jo every element y E K A B(0, 5Ajo 1 2 ) may be written as
y = ~ A j C j Cj C B J=Jo
=
{bn,n C t'V}
where for J0 large enough the elements cj, j > jo belong to any given end of the o~ Aj <_ ~1 . The assertion becomes then obvious. sequence (bn) and ~J=do c)=~ d): Let (Uk) be a fundamental sequence of neighbourhoods of 0 and suppose that Kk C Uk are compact sets whose multiplicative closure is not relatively compact. As ~k~
Kk (3 {0} is compact, this contradicts c).
d)::v a): Trivial. El
Checking whether a given Fr~chet algebra is admissible is facilitated by the following L e m m a 1.14:
Let A be a Fr~chet algebra and A c A a dense subspace. Then A is admissible if any of the conditions 1.12.a)-d) is satisfied for nullsequences resp. compact sets contained in A.
12
Proof: This is a consequence of Proposition 1.7.
[] The class of admissible Fr4chet algebras is closed under taking subalgebras and quotients. Lemma
1.15:
If A is admissible, then so is .4, the algebra obtained by adjoining a unit. Proof: Let U c A be "small" and let D C G be the open unit ball. We claim that := (1U x 89 C A is "small. Let K C U be compact. Then there exists A > 1 such that 1rl(2AK) C U c A, Tr2(2M~) C D are compact. Let K C A be a compact cone with vertex 0 containing the multiplicative closure of lrl(2)d~). Every ~ 6 K can then be written as ~ = ~ + _1# 2~ with x E K and # E q~, I#1 < 1. Therefore [I~~~i = (2A)-n ~2~Dr Yj where yj either belongs to K C A or D C G. If II - I1' is a seminorm on A corresponding to the seminorm II - II on m then II l-I~xi ]]'<_ (2A)-'~Y~2.te~m~ ]1 Yj ]t<-- A - n o where C : = SUpyEK• ]] y ]]'. Whns ]] (/~)n H,< A-n which proves the claim. []
Examples: a) Every Banach algebra is admissible. b) Let (i : A ~ A be a densely defined, unbounded derivation on an admissible Fr4chet algebra and let A m be the subalgebra of smooth elements with respect to (f with its obvious Fr4chet topology. Then ~4~r is admissible, too. (See 7.4.) [] Although apparently more general than Banach algebras, admissible Fr~chet algebras inherit some basic features from them. Proposition
1.16:
a) The subset of invertible elements in a unital admissible Fr4chet algebra is open and the inversion is continuous. b) The spectrum of an element of an admissible Fr4chet algebra is compact and nonempty. c) Holomorphic functional calculus is valid for admissible Fr4chet algebras.
13
Proof.-
By L e m n m 1.15 we m a y suppose A to be unital in b),c). a) Let U be a "small" neighbourhood of 0. We claim t h a t U' : = 1 - U consists o f i n v e r t i b l e elements. I f x E U then also y := Ax E U for some A > 1. T h e n z := 1 - x E U' is invertible with inverse z -1 = ~
A-ny n
n:O
where the sum converges because the multiplicative closure {yn; rt E ffV} of y is relatively c o m p a c t and thus bounded for any seminorm on A. For the continuity let (xn) E A, limn--+oo x,~ = 1 be a sequence of elements of A converging to 1. Then Xn = 1 -y,~ where (y~) is a nullsequence. Choose a m o n o t o n e increasing, mlbounded sequence of positive real numbers An > 1 such t h a t (Anyn) remains still a nullsequence. After deleting finitely many elements one m a y suppose t h a t {A,,yn; n E PC} is contained in a "small" ball. Therefore there exists a c o m p a c t set K C A containing the multiplicative closure of the sequence (A,~yn). Let II - II be a s e m i n o r m on A. Then supx, e g II x II< C for some C > 0. One finds 0(3
x~ 1 = ( 1 - y ~ ) - I
V TM A-k~A ~k = 2.-, ,~ (nY,~) k=0
and thus
{{~
- i
{{_~ ~
A3~ {{( A . v . ) k
C
{{_~ A , ~ -
1 < c for ~ > > 0
k=l
as (Any,,) ~ E K. This shows the continuity of the inversion. b) The s p e c t r m n of an element is closed by a). Let x E A and choose C > 0 such t h a t {XT, IAI < c } is contained in a "small" ball U. Then Sp(x) C {z E Ig, Iz{ < ~} (by a)) is bounded and therefore compact. Let H - II be a seminorm on A and denote by A' the Banach space obtained by completing A with respect to ]1 - {I. By a) the composition --+ A -+ A '
A
-+
( A - x ) -I
is an analytic flmction on ~ - Sp(x) with values in the Banach space A' which is bounded near infinity. If Sp(x) were empty, one would conclude with Liouville's theorem that the image of (A - x) -I in A' is independent of A for any seminorm on A which is impossible. c) Let A be adnfissible and x E A. Let, f be holomorphic on an open neighbourhood V of Sp(x) C q~. Choose a closed curve F in V not meeting Sp(x) and such t h a t the winding number of F with respect to any point in S p ( z ) equals 1. The integral
f(x)
:= ~ /
S(A) (A - x) -~ d,X
14
yields then a well defined element of A by a). The map A -+ f(A)(A - x) -1 is analytic. If F' is a similar curve not intersecting F, then
JF
IF
for any seminorm on A by the Canehy integral formula and an argument similar to that in b). Therefore f(x) is independent of the choice of the curve F and consequently of the choice of V. If (9(V) denotes the algebra of holomorphic flmctions on V, the map
ix : O(V) -4
A
f
-~ f(:~) is in fact a homomorphism of algebras. The linearity is clear and the multiplicativity can be derived from the identity (A - ~)-i(t~
- ~)-1
_
1
A -.
( ( . - x ) - 1 - (A - x ) - l )
Let f, g E O(V) and choose curves F, F' as above w i t h F' contained in the bounded component of ~ - |7. Then
f(x)g(x) = (~--~ Jrf(A)(A- x) - i d A )
(~/.fr, g(#)(p-x)-ld#)
= (2~z)2 frfr, f(A)g(It)(A-x)-l(#-x)-ldAdl .t 1
-- 21ri ~,g(#) (~-~ .~ f(A) (A- l~)-1d)~) (#- x)-1dl~
+~/
f(A) ~
,g(/~)(/~-A) -~dp (A-x)-ldA
1/;
= 27ri , g(P)f(#) (it - x)-ld# = (fg)(x) as the second integral vanishes because A belongs to the unbounded component of (I;- F'. []
15
L e m m a 1.17:
If A, B are admissible Fr6chet algebras, then so is A | Proof:
Let { 1I - ltk, k E tW } ( { I1 - lift, l e ~W}) be sequences of seminorms defining the topologies of A, B . We may suppose that II - Ilk' -> II - Ilk (11 - I1', -> II - II't) for k' > k (Z' _> l). According to Lemma 1.14 it suffices to check criterion 1.13 a) for nullsequences of elements of the algebraic tensor product A | B C A N,~ B . Let (@) be such a nullsequence. It can be written as nj
nj
b ,~ = y-~a~ | b{: ;4 II "? IIs(j)-> ~ II 0,s IIs(j)ll bL II}(j)} k=l
k=l
where f is a monotone increasing unbounded fimction and lira
j--*~o
II "r IIs(j) = 0; II ~J Ils(j)r 0; II as IIs(j)# 0; II ~ II}<j)# 0 Vj, k E / N
It follows then that I
K ' := {(~ := 2 II 3'j ll~(j)ll a~ II}-~) a~r C A 1
K" := {~g :-- 2 II ~J If}(j)ll bs II}~) ~} c B are mfllsequenees. By hypothesis, K ~ and K ~ possess ends
K'NU' = { c ~ , j > N o } , K " n U " = { ~ , j > _ X , } with relatively compact multiplicative closure. It is then clear by the choices made that {~/J; j>_No+N~} C A| has relatively compact multiplicative closure. [] 1-4 K - t h e o r y of admissible Fr~chet algebras T h e o r e m 1.18:
Let i~[e] and r u -1] be the universal algebras generated by an idempotent and an invertible element, respectively. The abelian groups K o ( - ) := Grotl~lim,.[llJ[e],Mn(-)]
KI(-)::
limn[~[u,u-1],Mn(-)]
define a Bott-periodic homotopy functor oil the category of admissible Fr6chet algebras. (Here Groth(-) denotes the Grothendieck group of an abelian monoid.)
16
Proofi The proof of Bott periodicity for Banach algebras carries over to admissible Fr~chet algebras because holomorphic functional calculus is valid for them as well. See for example [B]. [] It was a basic observation of Connes and Higson that topological K-theory behaves functorially not only under ordinary but in fact under asymptotic morphisms of (Banach) algebras.
Theorem 1.19:(Connes-Higson)[CH] a} For any admissible Fr@het algebra tile natural maps [(IJ[e],A]--+ [r
-+ [(~[u,u-1],A]a
[r
are bijective. b)
Ko(A) := Grothlim~[~[e],Mn(A)]~
KI(A) := lim~ [(F[u,u-1],M,~(A)]~
c) K , ( - ) is a covariant functor oil the (linear) asymptotic homotopy category of admissible Fr~chet algebras.
Proof: Clearly b) ~ c). For a) ~ b) note that A admissible implies M,(A) = M,~(~) | A admissible by 1.17. It remains to show a). For this we proof the
Lemma 1.20: (Cuntz-Quillen)[CQ] Let A be an admissible Fr@het algebra. a) Let g: IIJ[e] -+ A be a unital, linear map. Suppose that
4~(e, e) : 4(~(e) - e(e) ~) is "small", i.e. contained in a "small" neighbourhood of zero in A. Then
1 {~ + i t S }
@(Q(e)) c r and functional calculus with f
: (J~ -- { 89 -~- i.ll~} -+ ([~
f(z)
:~-
0 Re(z) <
1
17
yields an idempotent := F(e) E A It is given by the sum = o(e) +
,
(~(~)-~
k=l
b) Let O: r
1] ~ A
be a unital, linear map. Suppose that CO('U,,'U,-1) ~--- 1 - O(u)O(u -1) a n d co('u.-1,1t) = 1 - O(u-1)O(u)
are "small". Then O(u) is invertible in A and its inverse is given by uO(?Z)-1 :
~O(?.t-1)(,O(U, tt-1)k = Zr.aj('U,--1,u)kLo(u-1 ) k=0
k=0 []
Proof:
We tbllow Cuntz and Quillen: a) Put v := 2 0 ( e ) - I
0(e)=
89
Then @(~(e)) C r
1
Sp(v) c r
{~ + i ~ } ~
{i~}
Moreover 1 - v 2 = 4w(e,e) is "small" so that
t(v+ii 1~.-t)(vy
r,
91/TU-l-t 1-t z~/2-: --) = t ( v 2 + ~ )
= 1 - t4w(e,e)
is invertible for t E]O, 1] which shows the assertion about the spectrum of Q(e). The explicit calcnlation of ~ = F(Q(e)) can be done by using the equality 1
F(z) = ~ ( 1 + a ( 2 z - 1)) with
a:r
{i~}-~r -
c(z)
2 , =z(z)-~
{+1 =
-1
Re(z)>0 Re(z)
18
One obtains =
= 1 (1 + v(1 - 4~(e,e))- 89
= ~_( l + G ( v ) )
F(o(e))
1
1 + (20(e)
2
1)
(-4w(e, e)) k
oo
1 Z(
1 kl-3--'2k-l(_4w(e,e))
k
k=l
= 0(~)+ ( e ( e ) - ) ) ~ k = l ~(2k)! (~(e'e))~ which converges as long as
= e(e) + k=l
(o(e)-)(~(e,e))
k
4co(e, e)is "small" because (~k) _< 4 k.
For part b) one finds that the series OO
Co
Y : : ~-~O('/l,-1)03(?J,,?/,-1) k, ?/):= Z O 3 ( ? Z - I , ~ ) k O ( ' ~ - I ) k=0
k=0
converge provided that a3(u, u -1) and a~(u -1, u) are "small". As O<3
O(u)v = (1 - w(u, u-i)) Z ( w ( u , u-l)) k = 1 k=O (2'O
= ~"~((.d(,U-1 tt))k(1 __ ~ ( , g - 1 U)) = W~O(U) k=0
we are done.
[] With the help of tile lemma it is now easy to demonstrate Theorem 1.19. Let [Q] e [G[e], A]a be represented by
Q: r
-~ c ~ ( n ~ ~, A)
Choose a connected, punctured neighbourhood U of cc in T/~ such that 4w(e, e) is "small" on U. Then Q': r -+ C~(U,A)
--~
F(o(e))
defines a homotopy class [0'] E [r A] which is independent of all choices made and provides an inverse to the natural map [r
A] -+ [r
A]~
The odd case is similar. []
19
Chapter 2: Algebraic de Rham complexes 2-1 The periodic de Rham complex [CQ] A de Rham complex for arbitrary Frfchct- (Banach-, C*- ) algebras A should have the folh)wing properties: a) There should be a pairing
h,(~dR(A)) |
--~ r
generalizing the pairings {Traces on A} | Ko(A) -+ (~ {Closed traces onf~IA} | KI(A) --+ q2 b) It should be periodic because any stable homotopy functor on the category of C*-algebras has to be Bott-periodic (Cuntz). The first condition asks that traces should be zero-cocycles and closed 1-traces should be l-cocyclcs in such a complex. If one starts in analogy with ordinary de Rham cohomology with the universal enveloping differential bigraded algebra ~tA of a graded algebra A:
f~A aOdal...dan d(a~
la~
~-~
~,~ A | A / r ~ "
++ a o | 1 7 4 "~) := da~
n
:= 1.,~ + lall + " " + I~nt
and asks that the periodic (cohomological) de Rham complex should consist of linear functionals on ~A:
~*pdR(A) C H o m ( ~ A , r then the trace condition on 0,1-cocycles is satisfied if one takes as differential the transpose of the operator b~ : ~]~A
-+
~tn-lA
([, ] denoting the supercommutator) satisfying b~ = o
2O
A trace on ~ I A would in addition be closed if it were annihilated by the transpose of d: ~Y~A -+ ~ ' + I A So a first, guess would be to try to form a (now 2~/2-graded ) complex by putting
?*(A) := H o m ( ? , ( A ) , r
?,(A) := (ftA, b + d )
This however clearly fails as
(b + d) 2 = bd + db ~ 0 So one has to alter the definition and try ??,(A) := { flA/(bd + db)~2A, b + d} This is indeed a complex but one sees soon that it cannot give the right cohomology: h(??,(A)) contains bB(f~A). (In the notations of [CO]). A modified attempt however leads to a reasonable theory: one replaces the exterior differential by the operator
N d : ~'*A
-+
~**+IA
-~
(n + 1) da~
and defines
Definition 2.1: (Cuntz,Quillen) Let A be a graded unital algebra. The p e r i o d i c de R h a m c o m p l e x of A is given by ~PdR(A) := {~A/(b(Nd) + (Nd)b)fM, b + Nd} It is a filtered 28/2-graded chain complex. The Hodge filtration by the subcomplexes Fnf~P, dR generated by differential forms of degree at least n yields a sequence of quotient complexes [-~PdR , (A ) / F n~PdR , ( A ) that approximate successively the periodic de Rham complex itself. We will be especially interested in the smallest of these quotient complexes, the X-complex:
2-2 X-Complexes [CQ] Definition 2.2: a) The X - c o m p l e x of a graded, unital (~algebra is the periodic chain complex X.A:=
f~.Pdn (A ) / F f2~ , PdR (A )
Explicitely, it is given by
X , A : ~ A -~ ~ A d(a) := lda
:=~XA/[d,P.lm]~ b b(xdy) := [x,y]s
d -~
21
The r e d u c e d X - c o m p l e x is defined as
X,A := X,A/q2.1 b) The ( r e d u c e d ) c o h o m o l o g i c a l X - c o m p l e x is defined by duality:
X*A := Homr
(X*A := Homr
c) The ( r e d u c e d ) b i v a r i a n t X - c o m p l e x of the pair (A, D) of nnital algebras is the Horn-complex
X*(A,B) := Honh*~n(X,A,X,B) (X*(A,B) := Hom~n(X,A,X,B)) If -~ is obtained from A by adjoining a unit then X , A ~- X,A if A was already unital. []
The ordinary (eohomological, bivariant) (reduced) X-complex defines a covariant (contravariant, bivariant) flmctor from the category of unital (augmented) complex algebras to the category of ~/2-graded chain complexes. One easily verifies in the notation of Connes [CO] that
ho(X*A) = HC~
hi(X'A) = HCI(A)
for trivially graded algebras (look at the Connes-Gysin sequence
ItHI(A) --~ HC~
~ HC2(A) for the calculation of ho(X,A)).
Let A now be a Fr4chet algebra. The enveloping differential graded algebra f~A can be topologized by declaring ~A
~-
~-o
A | ~ok
aOda1...da k +._ ao|174174 to be a topological isomorphism, where each summand on the right hand side is given the projective tensor product topology and the whole sum is given the product topology. We denote by ~}iA b the quotient of ~ l A by the closure of the commutator subspace [A, f~iA] c ~IA. It is a Fr4chet space in a natural way. The differentials of the X-complex d : A --+ ~IA~ b : ~ i A b --+ A are continuous so that the X-complex becomes a complex of Fr4chet spaces. D e f i n i t i o n 2.3:
The composition of linear homomorphisms defines a bilinear map of chain complexes
X*(A,B) | X*(B,C) ~ X*(A,C) Tile induced map oil cohomology is called the c o m p o s i t i o n p r o d u c t . []
22
Let 6 be a (graded) derivation on A. It extends canonically to an action on the complex X , A by putting
6(a~ 1) := 6a~ l + (-1)lSIla~176 and defines a homomorphism of the super-Lie algebra of graded derivations on A to the super-Lie algebra of graded endomorphisnls of X,A. 2-3 Differential g r a d e d X - C o m p l e x e s Our final aim will be a version of the X-complex that behaves flmctorially under asymptotic morphisms, i.e. under morphisms that are given by a whole family of maps. In order to get a reasonable theory the homotopy information of such a family has to be encoded into higher homotopy information in the associated X-complexes. This can be taken care of by replacing an algebra by its resolution as a differential graded algebra and by working in a differential graded setting throughout. Let (A, 0, N) be a differential graded algebra (0 denotes the differential and N the number operator multiplying homogeneous elements by their degree). 0 and N are both derivations on A of degree +1 and 0 respectively. The remark following 2.3 about the action of derivations on X-complexes implies then L e m m a 2.4: If (A, 0, N) is a differential graded algebra, then X , A becomes a 2Z/2 graded complex of differential graded (DG)-modules. The differentials in the X-complex are morphisms of DG-modules:
b, d E HOmDG(XiA, X~+IA) The same holds for the topological X-complex and the reduced X-complex provided that O1 = O. []
D e f i n i t i o n 2.5: Let A be a (trivially graded) unital Fr~chet algebra and let V. be a locally convex topological DG-module. a) The d i f f e r e n t i a l g r a d e d X - c o m p l e x of A with coefficients in 17. is the complex X;G,v.(A ) := H o , t ~ t ( x , ( a A ) , V . ) of degree preserving continuous DG-homomorphisms from the X-complex of the differential graded Fr~chet algebra flA to V..
23 b) The b i v a r i a n t d i f f e r e n t i a l g r a d e d X - c o m p l e x of the pair (A, B) with coefficients in V. is the complex
XbG,v. (A, B) := Hom*DG.... t(X,(~2A), X,(f~B)@,,V.) where ~ denotes graded tensor products. [] The composition product extends to a map of complexes
XbG,v.(A,B) | X~G,w.(B,C) ~ XDG,vb~w.(A,C) Moreover, there i8 a natural pairing
X~G,v.(A , B) | X~)G,w.(B) --4 XDG,V.~.w.(A ) The basic example of a locally convex DG-module will be the following E x a m p l e 2.6: Let U be a smooth manifold and let | be the algebra of smooth exterior differential forms on U. If V C U is a relatively compact open submanifold, if x t , . . . , xn are coordinates on V, Y1,..-, Yk E F(TU) are smooth vector fields and is a multiindex, then F(AkT*U)
~
~+
s,,p vI2@(iyl..iy )(x)l is a seminorm. The differential graded algebra F(A*T*U) equipped with the topology defined by all these seminorms will be denoted by E(U). [] One has $(U)@,~g(U') ~- g(U x U') as locally convex DG-modules. In the sequel we will be interested in the simplicial DG-module V. := g ( t g ~ ) . We denote the associated X-complexes by
X~)a(A ) := X~)a,e(z~r)(A) X~)G(A, B) := X~)G,e(~T)(A, B) Under the identification g(h~)@~g(h~,~) _~ g ( ~ sition product yields maps
x h~)
-~ g ( N ~ ) the compo-
X•G(A, B) | X~)G(B, C) -+ X]Da(A, C) XSG(A,B) | X~)a(B ) -+ X]DG(A) It is possible now to replace in our considerations single homomorphisms by whole families of algebra homomorphisms.
24
L e m m a 2.7:
Consider unital Fr6chet algebras and form a new category AIq ~ by defining the set. of morphisms f,o be c'o77,t morAt,~. (A, t3) := Hornet 9 (An C o<9 (ff~+, B))
Then there is a canonical map
morA,g~ (A, B) -+ X~
B)
defining a natural transformation of biflmctors
rrtO?'Alg~ ~ X~ Especially, the (bivariant) differential graded X-complex becomes a contravariant (bivariant) flmctor on the category Alg ~. Proof:
Recall that C ~ 1 7 6
A) _~ A |
C~176
Let
fl C morAlg~(A, B) = Homalg(d, B | We define 0, E X~
X.(~A)
C~176176
B) to be the composition
x,(~o)>
X.(Ft(B|
x,(~(~)6~(c~(t~)))
-~
X.(Ft(B)~fl(Coo(j~)))
-~ X.(~(B)6~E) & x,(~(t~))6~E
The map r
X . ( ~ ( B ) 6 ~ E ( t ~ T ) ) -~ X . ( a ( B ) ) |
E(~T)
is given by
aQw
a| w
(a o | wO)d(al | ~1)
-+ (-1)l~~176
on X0
l |176 1 on X1
One verifies easily that O, C X ~ B) (i.e. it preserves degrees and intertwines the differentials 0 on FtA and ddR on s Moreover, the identity ~1. o Qo. = (01 o 00). becomes obvious once the two expressions are written down explicitely. []
25
The component of degree zero of the map above is just the family of maps of Xcomplexes induced by a given family of algebra homomorphisms. The components of higher degree however contain higher homotopy information which will be necessary to obtain homotopy formulas comparing the cohomology classes of members of a family of cocycles for different parameter values. 2-4 T h e a l g e b r a i c C b e r n c h a r a c t e r [CO] The interest in cyclic homology comes from the existence of the well known Chern character map D e f i n i t i o n 2.8:
There is a natural transformation
ch: K, ~ h ( X , ( M ~ ( - ) ) ) on the category of adnfissible Fr~chet algebras from topological K-groups to the homology of the X-cornplex of stable matrices. It is defined on idempotent (resp.invertible) matrices by [e];e 2 = e E A' := MooA [~t]; 1tIt - 1
:
It--lit :
1
E A'
-+
e E A'/[A', W] = ho(X.A')
--+ u-ldu E ftl(A')/([A',f~l(A')] + dA') = hl(X.A') []
So by duality there are pairings with cyclic cohomology: L e m m a 2.9:
There exists a natural pairing
K , A | h(X*(M~A)) --~ of flmctors oil the category of algebras. It extends to a pairing
K , A @ h(X~)c(M~A) ) ~ on tile extended category of Fr~chet algebras with smooth families of algebra homomorphisms as morphisrns (2.7). Proof:
The latter pairing is clearly given by
K . A | h(X~a(MooA)) ch| i|
h(X.MooA) | h(X~a(MooA)) -~
h(X, (f~(M~A))) | h(Hom(X. (~(MocA)), f ~ d R ( ~ ) ) ) --~ f ~ d n ( ~ )
It, remains to be shown that the image of tile pairing which may be a priori any differential form on ~ is in fact a constant function which can be identified with a complex number. This follows however from the following
26
L e m m a 2.10: Let A be a graded algebra, A0 the subalgebra of elements of degree 0. Then any graded derivation on A annihilates the image of
ch : K . A o ~ h ( X . ( M ~ A ) ) []
Applying the lemma to f~A, 0 shows for [x] E K , ( A ) , [~] E h(X~)c(A))
ddR((Ch(x), ~}) = (ch(x), daR o ~) = (ch(x), ~ o O) = (O(ch(x)), ~) = 0 []
Proof of Lemma
2.10:
Let 0 be a graded derivation on A, 0 | 1 =: 0' its extension to M ~ A =: A '. Even case: Let e = e 2 E A'. Then
a'(ch(e)) = 0% = O'(e 2) = (a'e)e + e(O'e) = [(O'e)e, e]8 + [e, e(0'e)]8 E [A', A']~ because
O'e = (O'e)~ + e(O'e) implies e(O'e)e = e(O'e)e + e(O'e)e = 0 Odd case: Let uu -1 = u - l u = 1 E A r. Then
O'ch(u) = O'(u-ldu) = O'(u-1)du + u-ld(O'u) = - u - l O ' u u - l d u
+ u-ld(O'u) =
= - [ u - l O ' u , u-ldu]s - (u--lduu-t)O' u + u-ld(O'u) = = - [ u - l O ' u , u-ldu]s + d(u-1)O'u + u-ld(O'u) = - [ u - l O ' u , u-ldu]~ + d(u-lO'u) E [A', f~I(A')]8 + dA' [] The lemma above illustrates the role played by the higher homotopy data contained in the differential graded X-complex.
27
C h a p t e r 3: Cyclic c o h o m o l o g y To proceed in the construction of an asymptotic cyclic cohomology theory, i.e. a 2Z/22~-graded chain complex for admissil)le Fr~chet algebras that behaves functorially under asymptotic morphisms, it is necessary to make the complexes introduced in the last chapter functorial under (families) of linear maps. This can be done in a universal way by replacing a unital algebra A by its tensor algebra RA. Working with tensor algebras also reduces the algebraic complexity of the involved chain complexes a lot: tensor algebras are of Hochschild cohomological dimension one and therefore the periodic de Rham complex of a tensor algebra becomes quasiisomorphic to the much smaller X-complex under the natural quotient map.
The cohomology of RA is uninteresting though because its X-complex depends only on the underlying based vector space of the given algebra. However, the tensor algebra of a given algebra comes equipped with a canonical adic filtration, defined by the kernel of the universal homomorphism RA ~ A, which evidently allows to recover A from its filtered tensor algebra. The filtration on RA induces a filtration on the associated X-complexes and a fimdamental result of Cuntz and Quillen states that the X-complex of RA and the periodic de Rham complex of A with its Hodge filtration are quasiisomorphic as filtered complexes. So the complex X.(RA) is easy to manipulate algebraically on one hand but contains all information encoded in the whole periodic de Rham complex of A if its filtration is taken into account. Especially, the algebraic I-adic completion of this complex defines the well known periodic cyclic homology groups of Connes. While the X-complex of RA is evidently functorial with respect to linear maps of the underlying algebras, the completed X-complexes (and therefore the periodic cyclic groups) will only be functorial under homomorphisms of algebras, because a based, linear map induces a filtration preserving homomorphism of its tensor algebras iff it is multiplicative, i.e. an algebra homomorphism. This chapter is taken from Cuntz-Quillen [CQ] (who studied the complexes X . (RA) from a somewhat different point of view.) The reason for repeating their proofs is that we work in a graded setting and we want to document the calculations in this case, too.
3-1 Extending functors The problem of extending functors from a given to a larger category (in our case from the category of unital algebras to the category of unital algebras with based linear maps as morphisms) will now be formulated in full generality.
28
Let j : C + C' be a (covariant) functor between categories C, C' and let F : C -4 7? be any (contravariant) fimctor. By an e x t e n s i o n of F from C to C' we mean a pair ( F ' , r consisting of a functor F ' : C' -4 D and a n a t u r a l t r a n s f o r m a t i o n r : F -4 F r o j
C
2+
F$ Z)
C'
;F' ~+ r
"l?
An extension ( F ' , r is called u n i v e r s a l if, given any extension (G, r of F , there exists a unique transformation r : F ~ -4 G making the following diagrmn commutative
C
F$ D
-J~
~ r
C'
SF' 29
II
1)
][ --4 r
The correspondence r ~ r
"NG ~ r
D
yields then a bijection
Transfc(F, G o j) ~_ Transfc, (F', G) A sufficient criterion for the existence of a universal extension provides
Proposition 3.1: Suppose t h a t j a d m i t s a left adjoint (j*, 4p): j* : C' -4 C
: Homc,(X, j Y ) ~- Homc(j* X, Y) Then any functor F : C -4 l? a d m i t s a universal extension to C'. Moreover these extensions are n a t u r a l with respect to F . I n s t e a d of proving the proposition we will describe the construction of universal extensions in the following relevant case. 3-2 T h e a l g e b r a R A [CQ] Let C,C~o be the categories with augmented tT~-algebras (Fr~chet algebras) as objects and morphisms
morc (A, B) : = b a s e d ( = a u g m e n t a t i o n preserving) linear m a p s A -4 B morc~ (A, B) := based bounded linear maps A -4 B |
C~(~)
29 Lemma
3.2:
The forgetflfl flmctors
j : Alg -~ C (Alg~ -+ C~) adnfit an adjoint
R :C --+ Alg (C~ ~ Alga)
Home (A, j B) ~- HornAO (RA, B)
H o m c ~ (A, j B ) ~- HomAtg~ (RA, B)
Proof" In the first case the quotient
R A := TA/(1A - 1r A| does the job. If f : A ~ B is of the total tensor algebra T A := | linear and based, then the algebra homomorphism T f : T A ~ B descends to an (augmentation preserving) homomorphism R A -+ B. In the second case, the goal is achieved by a completion of R A in the topology described in 3.6. []
Let.
o : A -~ RA,
7r : R A -+ A
be defined by
0 ++ IdRA
IdA ++ 7r
H o m c ( A , RA) ~- HomAO(RA, RA)
Homc(A, A) ~- H o m A o ( R A , A)
One has
j o t ) o ~ = IdA and any based linear map f C morc(A, B) admits a unique factorization
AG, RA~B into the universal based linear map ~ : A -~ R A and an algebra homomorphism R A - + B.
3o There is a canonical h o m o m o r p h i s m of algebras
i: R A ~ R ( R A ) corresponding to the composition of universal based linear maps
A v ~ R A on% R ( R A ) under the bijection
Homc(A, R(RA)) = Homma(RA, R(RA)) P r o p o s i t i o n 3.1 yields in this case
L e m m a 3.3: The assignment V -~ ( F o R, F ( ~ ) ) yields a n a t u r a l extension of ally contravariant functor F : AIg -+ I9 to C. If G : C ~ i9 is any contravariant functor the maps
T r a n s f ( F , Gj)
~
T r a n s f ( F o R , G)
c(o) x o F(~r)
r
o
X
are bijeetions inverse to each other. Therefore tile above extensions are universal. [] The algebra R A depends only on the underlying vector space of A (and the a u g m e n t a t i o n m a p A --+ ~). R A however admits a canonical filtration which enables one to recover the algebra structure of A.
3-3 I-adic filtrations [CQ] D e f i n i t i o n 3.4: The I - a d i c f i l t r a t i o n of R A is the filtration associated to the twosided ideal I A c R A defined by the exact sequence
O-* I A ~ R A ~ A - * O
31 L e m m a 3.5:
Tile conmmtative diagram of flmctors Alg
(aa-~dic nlt)~ Filtered Alg
Jj.
1 f~
C
R
>
Alg
induces on Horn-sets a pull-back(cartesian) square:
Hotnalg(A,B) (a(-),I-adic flit.)) Honlfilt.pres.(RA,RB)
1
l
Homc(A,B)
~
Hom~Ig(RA,RB)
Proof:
Let (.f, ~) be a pair consisting of f C Homc(A, B) such that R f = ~ preserves the I-adic tilt.rations of R A resp.RB. Then there is g E Hom~lg(A, B) given by 0
IA
--+ R A
~
A
~
0
0
IB
--+ R B
~,
B
--+ 0
From the commutativity of A
A~
RA
-24
A
&
RB
4
g
;I B
we obtain from the identity j(rr) o 0 = Id the claimed equality f = .q E Homalg(A, B). El
If the curvature (see 1.3) of the universal based linear map o:A-+ RA is denoted by w co(a, b) := o(ab) - o(a) | o(b) the I-adic filtration on R A can be described explicitely as follows.
32
Proposition 3.6:[CQ] There is an isomorphism of vector spaces RA
+%
o ( a O ) w ( a l , a 2 ) . . . a ; ( a 2 ' * - l a 2~)
+--
~A a O d a l . . . d a 2~
under which the I-adic filtration on R A corresponds to the degree (Hodge)-filtration on ~ A (IA)-~ ~Z_ ~ ) t~2kA k--m
The product on R A corresponds to the Fedosov product
on f ~ A . If A is a Fr6chet algebra then R A becomes a locally convex topological vector space under this isomorphism by giving ~ A the topology of Chapter 2.
Proofi Consider the subatgebra G[Q,w] C R A generated by the elements Q(o,),
~ ( a ,' a
"
), a . a .' a " E A
This subalgebra in fact equals R A because R A is generated by the elements Q(a). By the Bianchi identity its elements can be written in the form Z
Pw k
finite
So the map under consideration is surjective. The injectivity follows from the fact that any sum a~ I . . . da 2k + forms of lower degree is mapped (modulo tensors of degree < 2k-1 in R A ) to the element a-6 | . . . | a 2k + (a ~ - aO)a I |
| a 2k
in R A (see the explicit formula for w(a, b)). The formula for the powers of the ideal I A is clearly true for re=l; it follows in general by taking m-th powers and using the Bianchi identity to bring elements of R A into normal form. The Fedosov product on differential forms is associative, so, by induction, it suffices to check that it corresponds to the product on R A for pairs of elements of da2'*). For those it is clear from the form (a, a ~ a ~
a~
da 2n __+ p(a~
p(~)
1, a 2 ) . . . w ( a 2 n - 1 , a 2n)
33
a * a~
1 . . . da 2n = aa~ o(aa~
1 . . . da 2n - dada~
'~ - co(a, a~
1 . . . da 2n --+
n = o(a)v(a ~
n
We still need the structure of the spaces t ~ R A , ~2IRA~
Proposition 3.7:[CQ] There is a canonical isomorphism of vector spaces flRA+--
~
(A|174176174174174174174174
n,io,...,i,~ ocoi~
Q . . . O O n O c o i'~ t--- O,0
@ a~-@... @ a n + 2 i o + ' ' + 2 i , ~
[] The algebra [ ~ R A is canonically filtered by the powers of the ideal 0 -+ I ( ~ R A )
-+ ~ R A
a.) ~ A -+ 0
It is still called the I-adic filtration of ~ R A . Under the identification above the degree of an elementary tensor is io + il + " " + in. The commutator quotient adnfits the following description:
Proposition
3.8:[CQ]
There is a canonical isomorptfism of filtered vector spaces ~ I RA~ (o(a~
+-
f~odd A
+-
aOdal...da2k+ 1
Under this isomorphism the I-adic filtration of the R A bimodule [~IRA~ on the left corresponds to the degree (Hodge)-filtration on the right side. []
34
3-4 Cyclic c o h o m o l o g y [CO],[CQ] Let us reformulate what we obtained so far in this chapter
Proposition 3.9: The universal extensions of the functors X,, X~)a from the categories of algebras (Fr~chet algebras with smooth families of morphisms) to the based linear categories C(Coc) are given by
A -+ X,(RA)
A ~ X~a(RA )
Both of these complexes consist of filtered vector spaces (under the I-adic filtration). The morphisnls of complexes ~ . : X . ( R A ) -~ X.(RB)
~ * : X b a ( R A ) e- X~)a(RB )
induced by based linear maps ~ E C(A, B); ~ E C~(A, B) preserve I-adic filtrations
iff ~, ~/, are homomorphisms of algebras: ~ ~ Horn(A, B), r E Hom~(A, B). Tile behaviour of the differentials in the X-complex with respect to I-adic filtrations was determined by Cuntz and Quillen. T h e o r e m 3.10:[CQ] Under the isomorphisms of vector spaces
RA ~_ fleVA
~Y RA b "-" f~~
A
the complex X,(RA) is transformed into the complex
X.(RA): ~ IYVA -~ gt~ with
d = - (.~r~ 2i b+ \i=O
nJ d
/
oil ~]2"A j3=b-(1
+~)d
Both differentials lower the I-adic valuation degree by 1.
[] (For tile definition of tile Karoubi operator t~ see tile proof of theorem 3.11.) The latter complex X,(RA) is closely related to the periodic de Rham complex as well as to Connes normalized (b, B) bicomplex of A.
35
T h e o r e m 3.11:[CQ]
a) The linear map ~A
-+ f~A/(bNd + Ndb)~A
o~2n
__+
(_ 1)n?t! (~2n
(y2n+l
___}
(_l)nn[ oe2n+l
(I):
induces a filtration preserving map of complexes
X , ( R A ) ~_ X , ( R A ) -+ ~P, dn(A) b) Under this map the periodic de Rham complex of A becomes a deformation retract of X , (RA) in a canonical way, i.e. there exists a homomorphism of complexes
02: ~ P d ' ( A ) --+ X , ( R A ) such that
~ 2 o ~ = Id~yR(m ) ~ o ~
= P C Q " Idx,(RA)
where PCQ is the Cuntz-Quillen projection [CQ]. c) The periodic de Rham complex of A is isomorphic to the normalized (b, B)-bicomplex of A ([CO]). []
P r o o f o f t h e o r e m 3.10:
We repeat the calculation of [CQ] for the convenience of the reader and to verify them also in the graded case. Odd degrees:
13(a~
..da 2n+1) = bs(O(a~
[Qw'~,g]s = o(a~
2n+l) =
2n+1) - (--1)]~']tQ[~(a2n+l)Q(aO) wn =
= own-lw(a 2n-1, a2na2n+ 1) _ Qwn-l~o(a2n-la 2n, a2n+ 1)
+ o(a~
1, a2a3)w n-1 _ o(a~
+ (Q(a~
2, aa)w n-1
n - Q(aOal)w ~) + Q(a~
(--(-l)]~162176176
"
(-1)]e~"l[e[Q(a2n+la~
- ( - 1)IQ~~ I1~1o(a2n+la~
n
n)
3 6
by the Bianchi identity. The two s u m s in brackets are equal to
-- co(a ~ al)co n and ( - 1 ) le~~ II~
2n+ l, a~
n respectively
T h e whole s u m corresponds therefore to 2~'z
E(-1)i
a~
. . . d ( a i a i + l ) . . . d a 2n+1
i=0
_ ( _ l ) l a 2"+* I(la~I+..+1~, 2'' I ) a 2 n + l a O d a l . . . da 2n
_ daOda 1 . . . da 2n+1 + ( - 1 ) l ~2"+11(la~
+la 2'' I)da2n+ldaOda 1 . .. da 2n
da 2n+l) - (1 - t % ) ( d a ~
= bs(a~
da 2n+l)
E v e n degrees: n,--1
5(a~
I . . . d a 2'~) = d(oco '') = dow n + E
owidww ~-i-1
i=0
For a single s m n m a n d of the l a t t e r s u m one finds
flw~ d w w n - i - 1
= owi do( a 2i+ l a2i+ 2 ) w n - i - 1
_ owidooco n - i - 1
_ Owiodow n - i - 1
=
= ( _ 1) lao~d0(J~+~a2~+2)ll~o'. . . . . i w n _ i _ l o w i d o ( a 2 i + l a 2 i + 2
_ ( _ l ) l O J d o t l o ~o'. . . . . I o w n - i - l o w i d o _
(_l)lOw~odollw '. . . . ~ l w ~ - i - l o w i o d 0
la~l)(la~+~ I + . . . + la2"l) as c~,~+t)
T h i s corresponds (by d e n o t i n g (la~ + . . . + (-1)c'-'~+2.2~+a ( ( d a 2 i + 3 . . . da 2n) , ( a ~ -
_
)
da2i)) d(aZi+la 2i+2)
(-1)~2~+1.2~+~ ((a2i+2da2~+3.. "da2n) , ( a O d a l . . . da2i)) d(a2i+l) (_l)~2~+2.2,+3(da2i+3...da 2~ , a ~ , d a l . . . d a = (-1)c-'~+2.2i+* ( d a 2 i + 3 . . . a ~
2i , a2i+l)d(a 2i+2)
da2id(a2~+la 2i+2) _
_ da2i+ 3 . . . a ~ . .. da2ia2i+tda 2~+2 + da 2~:+3 . .. d a ~ -
da2ida2i+lda 2i+2)
(-1)c2~+~.2~+ 2 (a2i+2da2,:+a.. " a O . . . da2ida 2i+1 _ _ da2i+2.., da~
da2ida 2i+1)
= - bs ( ( - 1 ) c~+ .... + ~ d a ~ i + a . . a O . . . a2i+l-da2i+2 ) + (-1)c~+>a~+~da2i+a... d a ~
da 2i+2
+ ( - 1 ) c~+ .... + ~ d a 2 i + 2 . . . d a O . . . d a 2i+1 = = ( -bs(~(",-~-~))
+ ~ 2(;'z-i-i) d q- t~"2(n-i-1)+1 d ) ( a ~
1 . . d a 2n)
to
37
S u m m i n g up yields E-b 6 = n,s2 " d +
(~2(n-i-1)]
st's
2(n--i--I).
] + ~,.~
. 2(n--i--1)+l d =
a + ~s
i=0
"2j+ld
(j~=Oi'~j bs -~- E i'~ d
j=0
i=0
on f~2nA []
Proof
of theorem
3.11:
We recall from [CQ] t h a t the Karoubi operator ~8:=
~s(a~
1 -
db~ -
b~d : 9 A ~
f~A
... da n) = ( _ l ) n - : ( _ l ) ( l a ~
aOdal ... dan-1
satisfies the identity (my-
1)(~;; +: - 1) = 0 0 1 : a n A
The K a r o u b i o p e r a t o r commutes with the differentials/~, 6 of X, (RA) so t h a t X . ( R A ) splits under the generalized eigenspace decomposition
X . ( R A ) ~_ a A ~_ ker(1 - ~,,)2 0 ker(1 + gs) (9 1 ~ ker(gs - r r
into tile direct sum of three complexes. The first is isomorphic to the periodic de R h a m complex
~*PdR(A) := (f~A/bs(Nd) + (Nd)b~, b + N d ) hi fact
b , ( g d ) + (Nd)b~ = (n + 1 ) ( b , d + d b j + (1 - dbs) - 1 = (n + 1 ) ( 1 - gs) + g~+: - 1 on f~nA and tile greatest common divisor of the polynomial x n+l - ( n + 1 ) x + n and the nfinimal p o l y n o m i a l
(xn+l
1)(z '~
_
of ~ equals (x
-
1) 2
-
:)
38
So tile canonical projection of 12A onto the periodic de R h a m complex ~*PdR may be identified with the projection onto ker(1 - ns) 2 in the spectral decomposition of X.(RA).
The differentials fl, 5 on this suinmand simplify to (~[kr
= -nb~ + (2n+l)d
=-nbs
+ Nd on~2nA
1 = b~ - 2d = b~ - - - N d n+ 1
fl[kr
9
on~2n+lA
'
so that the map in a) defines in fact, a map of complexes. The inverse of the rescaling map in a) yields an inclusion of coinplexes g, := ~P. dRA --% ker(1 - n,) 2 C X . ( R A ) In order to show b) it suffices to prove that the complementary subcomplexes ker(1 + K,,) and
G
ker(ns - ~)
r in X . ( R A )
are contractible.
The differentials on the subcomplex ker(1 + ~ ) are
given by 51~:e,-(l+~) = - n b s
+ d
o n l ~ 2n
fl[ker(l+a~) = bs o n ~ ~
A contracting nullhomotopy is provided by
h =
Gd
on f W '
--~fGd
o n ~ 2n+l
(The notations are those of [CQ]). The differentials on k e r ( ~ - ~); ~ r +1 equal 5lk,~(~_ O = 0 o n ~ ~v /31k~,,(~,_r
= b~ - (1 + ~ ) d o n ~ ~
A contracting nullhomotopy is provided in this case by h = G ( d - (1 + r
The claim c) is clear because N d = B on ker(1 - ns) 2 by [CQ]. []
39 C o r o l l a r y 3.12: a) The differentials on X, (RA) are continuous with respect to the I-adic topology. Therefore one can define the following two complexes b) The I-adic completion X,(RA) of the X-complex of RA. X,(RA) is canonically quasiisomorphic to the periodic cyclic Connes-bicomplex CCP~(A) of A. Its homology equals the periodic cyclic homology of A:
h(X.(RA)) ~_ PHC.(A) c) The complex X~in(RA ) of linear functionals on X.(RA) vanishing on terms of high I-adic valuation, X~i,~(RA) is canonically quasiisomorphic to the cohomological Connes-bicomplex CC*(A) of cochains of finite support. Its cohoinology groups equal the periodic cyclic cohomology groups of A:
h(X;in(RA)) ~_ PHC*(A) [] Both complexes )~.(RA), X~in(RA ) behave fimctorially under homomorphisms of algebras but not under arbitrary based, linear maps anymore.
Definition 3.13:[CO2],[CQ] a) Let 1r
-+
Q(e):0(lr
be the universal based linear map. According to 1.20 there exists an idempotent A
r(e(e))
~
9 Re
in the I-adic completion of R e obtained from Q(e) by functional calculus. Its image in Xo(Rr will be denoted by A
~
oh(e) := Q(e)+ ~
(2:)(Q(e)-
~)w(e,e)~E Xo(R,~)
k=l b) Let Q : r a) we put
u -z] -+
Re[u, u -z] be the universal based linear map. Analogous to
ch(u) : : ~0(?A-1)~d(U,U-1)kd~0(U) 9 XlRr k=O
,u -1] []
40
Chapter 4: H o m o t o p y properties of X-complexes In this chapter various Cartan hoinotopy operators expressing the triviality of the action of deri;eations on the cohomology of ordinary and differential graded X-complexes are constructed. The homotopy formula is found by guessing it for the periodic de Rhaln complex of A in analogy with the classical homotopy formula for the Lie derivative along a vector field acting on the exterior differential forms on a manifold. This yields an operator h that works for algebraic differential forms of degree zero and one modulo error terms of higher degree:
s
-
(hO + Oh) = ~/, : f~v,~R _~ F2ftP,~R
In the case of a tensor algebra however, the latter complex F2ft,v~n is contractible, so that ~/~ is nullhomotopic: ~/; = Oh/ + h~O, which provides a true homotopy operator H := h + h/ on ftv, an(RA), respectively on the quasiisomorphic quotient complex X , (RA). Considering finally the I-adic filtration on X , (RA) allows via the identification
GrI_~gir
qi'~>GrHodge(f~p.dR(d))
to obtain a Cartan homotopy formula on the whole periodic de Rham complex of A which coincides in degrees zero and one with the formula guessed in the beginning. We comment on this procedure in such detail because it is typical for the way one works with X-complexes and "lifts" constructions on algebraic differential forms of low degree to the full cyclic complexes. Another example for this technique will be the construction of exterior products on the chain level in chapter 8. Beside this another homotopy fornmla for the action of a vector field on the asymptotic parameter space for the differential graded X-complex X~)G(RA ) is obtained which uses the higher homotopy information encoded in the differential graded X-complexes. Also we compare the differential graded to the ordinary Xcomplex which will be needed to construct natural transformations between the different cyclic theories encountered in later chapters. Although all formulas and calculations are quite explicit it is not easy to develop a thorough understanding of the behaviour and the properties of the differential graded X-complex. This is provided in a final remark by calculating the cohomology of differential graded X-complexes using a universal coefficient spectral sequence in the abelian category of differential graded modules (DG-modules). The reason for not using the machinery of homological algebra from the beginning is that explicit formulas are needed as soon as topologies and growth properties are taken into account.
41 4-1 T h e C a r t a n h o m o t o p y
formula
Let 0 : A -+ A be a graded derivation on A. It induces a graded derivation on This action is denoted by
R A which acts on X , ( R A ) .
X , ( R A ) -+ X , ( R A )
Co:
The claim of this paragraph is to show that L:o acts trivially on the cohomology of X , ( R A ) . A naive attempt would be to generalize the homotopy formula
s
= ixd + dix
for a vector field X acting on the de Rham complex of a smooth manifold. Here
i x : z"X o antisymmetrization with
i~ ( f ~
df n) = f ~
df '~
As antisymmetrization will yield reasonable results only for de R h a m complexes of commutative algebras, it is better to start by generalizing the operator i~. Definition 4.1:[CQ]
For 0 a graded derivation on A put
io :
~nA
--+
a~
n
-+
~n-lA (_l),~-l(-1)la~
n
When we consider in how far this operator can be used as homotopy operator in the noncommutative de Rham complex respectively the homotopy equivalent (b, B) bicomplex the first observation is Lemma
4.2:
[io, b~] = 0 Proof:
[io~ bs](a~ = b8 ( ( - 1 ) n - l ( - 1 ) l a ~ 1 7 6 1 7 6
n)
2 . . . d a n) - io ( ( - 1 ) n - l [ a ~
= + ( - 1 ) ' ~ - 2 ( - 1 ) io(a~ _
(_l)n-l(_l)la~176176176176
an]~)
an da n-1
_ (_1) ,~-1 io(aOdalL, a n)
+ (-1)'~-l(-1)(]'~~176176176
da n-1
42
:
(io(a~
(--1) n-1
n - io(a~
It has to be shown that the expression in brackets vanishes. One clearly may suppose n=2. In this case
io(a~ = (-1)la~176176
2
2 - io(a~
(-1)la~176176
2) =
2) + (-1)(ta~176
2 ----- 0 []
With respect to the operator B however io behaves as a homotopy operator only in degrees less than two where the differential geometric formula is recovered. This would be not so bad, as in the X-complex of A only forms of degree less than two are considered, but unfortunately the operator io does not preserve the Hodge filtration on the noncommutative de Rham complex and does therefore not descend to an operator on the X-complex. For the algebras R A however this drawback can be overcome as the X-complex is not only a quotient of tile de Rham complex but there is also a map of complexes
X,(RA)
x.i) X , ( R R A ) -+ fiP, dR(RA)
defining a section of the quotient map and enabling one to construct a candidate for a homotopy operator on X , ( R A ) . T h e o r e m 4.3: Let 6 be a graded derivation on A. Define
h5 : X , ( R A ) --+ X , + I ( R A ) to be the composition
X,(RA)
x.(i)~ X , ( R R A ) ~- f~RA ~+ f~RA ~_ X , + I ( R R A )
Then for the action s is valid:
x.(~)~ X , ( R A )
of 6 on the complex X , (RA) the following homotopy formula
s
= hc~Ox. + Ox.h~
The homotopy operator is given explicitely in terms of standard elements of
X , ( R A ) (3.6, 3.8) as follows: h~ : X , ( R A ) --+ X , + I ( R A ) 5(o)d(
+
Eo
- Eo
o~o"d o --* oa;'~~ ( O)
o
5(o)do " - "
43
Proof: Tile crucial step consists in showing the identity X,(~) o (i6a + oi6) : s
o X,(~)
of operators on ~ R A . It is trivial in degree >3 because the operator (i6c9 + 0i(~) shifts degrees by at most two and X,(Tr) vanishes in degrees 71. We find In degree 0:
X,(Tr) o (i6O + Oi6)(a) = X,(Tr)(i6da) = = X , ( ~ ) ( S a ) = 5X,(~r)(a) = s o X.()T)(a) In degree h ') = (-i6(1 + ~ ) d + di~)(a~
(/60 + a/~)(a~
-i6(da~ = 5a~
1 - (_l)ta~
t _ (_l)l~~
o) + (-1)la~176 o + (-1)l~%61da~
= E6(a~
~) =
1) - (_l)la~
1)
~ + (-1)la~176
1
1, da~
And so
X.(~r)(i60 + Oi6)(a~
t) = X,(~r)(~C6(a~
= f~6(X.(Tc)(a~
In degree 2:
X.(~) o
(i6a +
ai6)
= X,(~)
((i6(-bs
+ (1 + as + a~)d) + (b~ -
(1 + g,)d)i6)
= X,(Tc)(-i6bs + b.~i6) by degree considerations = 0 by the lemma. In degree 3 the reasoning is the same. Finally one gets therefore h6o + 0h6 = X,(Tr) o i6 o X , ( i ) o 0 + 0 o X , ( ~ ) o i~ o X , ( i )
= X,(7c) o ( i 6 o 0
+ Ooi6) o X , ( i )
= s
oX,(i)
= s
= s
The explicit form of h6 will be calculated in 4.4. []
44
C o r o l l a r y 4.4: The homotopy operator constructed iu the theorem above extends to an operator on the complexes X,(RA) X*f " (RA) of periodic chains (cochains of finite support). So the homotopy formula
is valid on the complexes )~, (RA), X~i,(RA ) calculating the periodic cyclic (co)homology of A. Proof."
It has to be verified that the homotopy operator shifts I-adic valuations by a finite amount only, i.e.
hs : FNX,(RA) --+ F N - C x , ( R A ) for some C c Z W Denote the curvature of ~ A : A -'-+ R A
by w, that of :-~ ~ R A : R A
-+ RRA
by It. As X,(Tr) o i~ annihilates forms of degree >2 in 12RA ~_ X,(RRA) one obtains
h~(Q~ n) = X , ( ~ ) o i~ (~(e)(~(~) + , ( e , 0))~)
= x,(~)oi~
(
1
~(o)~(~) ~ + ~-~.~(o)~(~)i~(o, o)~(~) n - t - i
)
0
= x . ( ~ ) o i~
- ~
~(o~J).(~, ~n-'-~)
- x . ( ~ ) o i~ (.(4, ~ ' ) ) +
0
n-2
= Z oc~
n-1
4- 5(o)d(w n) - E gwis(o)down-l-i
0
which is of valuation >_ n - 1. In odd dimensions the calculation is even simpler:
0
45
This shows that the above claim holds for C = 1. [] As an application we show that the Caftan homotopy formula can be used to calculate the cyclic (co)homology of a direct sum of algebras.
P r o p o s i t i o n 4.5: Let A, B be unital. The canonical homonmrphism
p : R ( A | B) --+ R A O R B adjoint to the based, linear map
A|
B
oAeo.> R A | R B
splits after I-adic completion, i.e. there exists a natural, continuous map
such that
~o s = Id~Ar B and such that s o ~ is canonically homotopic to the identity. C o r o l l a r y 4.6:
X , R ( A | B) ~
X, RA | X, RB
is a quasiisomorphism.
[]
C o r o l l a r y 4.7: Let, A be unital and let A be obtained fl'om A by adjoining a unit. Then the canonical projection X , R A -~ X , R A is a quasiisomorphisnl.
[]
46
P r o o f o f P r o p o s i t i o n 4.5: The proof proceeds in several steps. First of all we define the splitting s. 1) Construction of a pair of orthogonal idempotents in/~(A @ B): P u t o(a):= Q((a, 0)) 9 R(A | B), 0 ( b ) : = 0((0, b)) 9 R(A @ B). Let then
ch(1A) := F(a(1A)) 9 R(A @ B) ch(1s) := F(Lo(1B)) E R(A | B) F(x) := x + ~=~ ( ? ) ( x - 1 ) ( x - x2) k ch(1A), ch(1B) are idempotents in R(A @ B) (1.20). In fact they are orthogonal. Claim:
ch(1A) ch(1B) = 0 ch(1A) + ch(1B) = 1 To verify the claim note first that p(1A) and Q(1B) commute: [Q(1A), 0(1.)] = [p(1A), 1 -- 0(1A)] = 0 E R(A @ B) Consequently ch(1A) and ch(1B) commute, too, so that e' := ch(1A)ch(1B) is an idempotent in R(A@B) satisfying rr(e') = 1AIB = 0 E A@B. Thus e' 9 "I(A@ B) and if e' 9 ~ ( A @ B) for some k > 0 the equality e' = e '2 9 ~ k ( A 9 B) shows that
ch(1A)ch(1B) = e' 9 5 ~ ( A @B) = 0 k=l
This identity being established, it follows that e" := ch(1A) + ch(1B) is an idempotent, as well as 1 - e". The identity ~r(e") = 1A + 1B = 1, ~r(1 -- e") = 0 leads as above to the conclusion 1 - e" = 0, i.e. ch(1A) + ch(1B) = 1 2) Construction of s: We want to construct a homomorphism s : RA | RB ~ R(A @ B) satisfying S(1A) = ch(1A), sOB ) = ch(1B). To do so, put first of all ~o1:
A
-+
a
--~ Q(a)+ Ek~l (Ik)(9(1A) -- 89
Sl : A a
/~(A|
--4
R(A @ B)
--+
~l(a)ch(1A)
1A)k-lW(1A, a)
and define ~ : B -+ R(A @ B), s2 : B ~ R(A | B) by the analogous formulas. L e m m a 4.8:
si(a)----si(IA)Si(a)=
si(a)Si(IA)
s2(b) = s2(IB)S2(b) = s2(b)s2(IB)
47 Proof: First of all note that s~(1A) = V l ( 1 A ) ~ h ( 1 A ) = c h ( 1 A ) c h ( 1 A ) = c h ( l ~ )
This shows already sl(a)sl(1A) = ~l(a)ch(1A)ch(1A) = ~l(a)ch(1A) = sl(a)
Furthermore the Bianchi identity (1.3) implies that ch(1A)~,(~) =
= ch(1A)
0(a)+ E
( p ( 1 A ) - - ~ ) W ( 1 A , 1 A ) k - l w ( 1 A , a)
k=l oo
= p(a) + E(AkQ(1A) + #k)W(1A, 1A)k-lW(1A, a) k=t
with universal coefficients )~k,Pk E ~ . Taking a := 1A, the identity ch(1A)~I(1A) = ch(1A)ch(1A) = ch(1A)
shows
so that the whole stun equals in fact ~l(a). Consequently st (1n)sl(a) = ch( l a)~ol (a)ch(1A) = ~pl (a)ch( l a) = sl(a) []
Continuation of the proof of 4.5: The lemma shows that Sl(1A)~ (resp. S2(1B)) acts as unit on the subalgebra of R(A @ B) generated by Sl(A), (resp. s2(B)). Consequently there are homomorphisms of full tensor algebras sl : T A ~ R ( A @ B) a -+ sl(a) 1r --~ Sl(1A)
s~ : T B ~ R ( A @ B) b -+ s2(b) 1r ---+ 82(18)
which annihilate (1A -- 1r (resp. (1B -- 1r and descend therefore to homomorphisms s t : R A -~ R ( A O B) s2: R B ~ fi~(A O B) Furthermore, the images of these inorphisms annihilate each other due to the identity sl(a)s2(b) = st(a)sl(1A)S2(1B)s2(b) = sl(a)ch(1A)ch(1B)s2(b) = 0
48
because ch(1A) and ch(1B) are orthogonal. Thus
s:
RA|
--+
(~,y)
-~
R(A|
s~(.) + s~(~)
is an algebra homomorphism. It is in fact unital because s(1) = s((1A, 10)) = Sl(1A) + s2(1B) = ch(1A) + ch(1B) = 1 Finally s preserves I-adic filtrations as
s(w(a ~ al)) = 81(a~ 1) = qol(a~
- s I (a~
(o, 1) =
-- g)l(a~
projects to zero under rr : R(A 9 B) -4 A @ B. Thus s extends to I-adic completions
s: ~(A) ~ ~(B) ~ ~(A 9 B)
3) !~o s = Id~(A)$~(B ) As i~o s is a continuous homomorphism of algebras it suffices to check the identity on a set of generators of a dense subalgebra, i.e. on o(A) C R A (and similar for B). One finds s(~o(a) ) = sl(a) = qol (a)ch(1A) =
=
Q(a)+Exkw(1A,a)
ylW(1A,1A)
O(1A) +
k=l
/=1
for some xk, Yl E R ( A | Now under the canonical m a p p : R ( A O B ) --+ R A | C0(1A) --+ 1RA, 0(1B) --+ 1RB S0 that p(ca(1A, a)) = WRA(1, a) = O, p(W(1A, 1A)) = WRA(1, 1) = 0 and therefore
~(s(o(a) ) ) = p(o(a) )p(O(1A) ) = o(a) IRA = o(a) which proves the claim.
4) S o ~ ~, Id~(A~B) As it suffices to describe a homotopy oi1 the generators p(A | B) of R(A | B) we consider s o p(0(a, b)) = sl(a) + s2(b). Put
F ( - , t ) := F l ( - , t ) + F 2 ( - , t ) F~(a,t) = g(a) + t(sl(a) - co(a)) F2(b,t) -- co(b) +t(s2(b) - o(b))
Fl(b,t) = 0 F2(a,t) = 0
Then F ( - , t) defines a smooth family of endomorphisms of R(A (9 B) as is seen by the equality F(1, t) = F1 (1A, t) + F2 (1B, t)
49
= O(1A) + t(ch(1A) -- ~O(1A)) + LO(1B) + t ( c h ( l u ) - O(1B))
= l + t(ch,(1A) + c h ( l u ) - 1)
1
and the estimate F(~o(a, a'), t) = o(a,z') + t(<(o,a')
- o(a,,') ) - (o(a) + t... ) (0(o,') + t . . . ) =
= o(aa') - O(a)O(a') = co(a,a') m o d [ ( A | B) []
P r o o f o f c o r o l l a r y 4.6: If A and B are unital observe first that the map
X,(A@B)
(x.(~o),X.(~,))~ X , A |
is ill fact an isomorphism of complexes. This is obvious for X0 and follows for X1 vanish in the c o m m u t a t o r from the fact that the mixed terms adb, b'da' of ~ I ( A | quotient: adb b = ad(blu)~ = abd(lu)~ + luad(b)~ = 0 From this we see that
X , ( f ) : X , R , ( A | B) --+ X , R A | X , R B X , ( s ) : X , R A | X , R B -~ X , f ~ ( A O B) satisfy X , ( ~ ) o X , ( s ) = Id whereas X , ( s ) o X,(~) = X , ( s o~) is chain homotopic to the identity by the Cartan homoto W tbrmula, applied to the smooth homotopy connecting s o ~ and ld~(AOB) and therefore a quasiisomorphisnl. It follows that X,(/~) is a quasiisomorphism itself. []
P r o o f o f c o r o l l a r y 4,7: For mfital A one has A _~ A | Ig as algebras. Thus qis
by corollary 4.6 and
X, RA|
X.Rr
1~ex.~) X , R A | X , r = X , R A | qis
r
by Lmmna 5.17 are natural quasiisomorphisms mapping the subcomplex
5o
X.I~ ~_ ([J1RA C X . R A to (P,(1RA, le) C X.fi~A | OJ. Thus the composition of the maps above descends to a quasiisomorphism
X.RA ~
X . ~ d ~ r162
1r = X , ~ A []
4-2 H o m o t o p y formulas for differential graded X-complexes Preliminary remarks: For any unital algebra A one can fi)rm the differential graded algebras ~)RA and
R(f~A). They are related by canonical maps as tbllows: Lemma 4.9: a) There is a canonical homomorphism of DG-algebras
j : i~RA -~ R(f~A) defined by
HomDcA(fIRA, R(~A)) j
Z+
HomAlg(RA, R(f~A)o)
++ Id: RA--+ RA = R(f~A)o
b) There is a canonical homomorphism of DG-algebras k:
R(~M) -+ ~)RA
defined by
HOmDGA(R(i~A),ftRA)
~
HomDG-lin~.ar(f~A, ftRA)
k
~
k'
kt(a~
n ) := o(a~
e) k o j = Id~RA d) If A is a Fr6chet algebra, then so are ftRA, R(~A) and the morphisms j, k are continuous. e) Tile morptfisms j, k preserve I-adic filtrations. [] Now the first result of this section can be fornmlated:
51
T h e o r e m 4.10:
Let fi : A ~ A be a graded derivation on the Fr4chet algebra A. Denote by 5' the corresponding derivation on ~]A commuting with the exterior differential on this algebra. Let be the operator constructed in Theorem 4.3. Define
h~ : X~)G(RA ) ~ X~)c(RA ) to be induced by the composition
X,(f~RA) x.j> X,(Rf~A) ~+ X,(R~IA) X.k X , ( D R A ) Then the following Cartan bomotopy formula is valid in the differential graded X-complex X~G(RA):
s
= Oxb(; o h~ + h5 o Oxb a []
Proof:
It is clear that h~, commutes with tile action of the exterior derivative 0 and the number operator N on X.(f~RA) because j, k are homomorphisms of differential graded algebras and the differentials (V, 0 on ftA commute. Therefore ho yields in fact a chain map on X ~ G ( R A ). If we denote the differential of the periodic complex X . (ftRA) by dx., we find
dx. hd + hddx. = dx. o X . ( k ) o h x o X . ( j )
+ X.(k) o h x o X . ( j ) o d x .
=
X . ( k ) o ( d x . o h x + h~, o d x . ) o X . ( j ) = X . ( k ) o E x o X . ( j ) = X . ( k o j ) o E x = 12~, []
Now we prove a second homotopy formula for the differential graded X-complex. It concerns the homotopy properties in the "parameter space" ~ and corresponds to the condition, formulated by Connes-Moscovici [CM], that the "time derivative" of an asymptotic cocycle should be an asymptotic coboundary.
52
Theorem
4.11:
Let V. be a DG-module, i v : V . - + V._ 1
a derivation of degree -1 and
s
:=
Oiy q- i y O
the associated Lie-derivative. It is a derivation of degree O. Then s acts on
HomDa(X,(f~RA), V.) and H o m S c ( X , ( ~ R A ), X,(f~RB) | V.) in an obvious way. Put
hv : HomDc(X,(~}RA), V.) ~ HomDo(X,_z(f~RA), V.) ( b y : Hom*Da(X,([]RA), X,(ftRB) | V.) --+ Hom*D-al(X,(f~RA), X,(ftRB) | V.)
hv(r
: = (coiv) o ~, o ( ~1 h,N) + iv o ~' o ( N1
hN) OCO
where N : X,(f~A) --+ X,(~A) is the number operator and hN is the C a f t a n h o m o t o p y o p e r a t o r associated to the action of N on X, (f~RA) via T h e o r e m 4.3. T h e n hy satisfies the C a f t a n homotopy formula
~.y = hy o cOHomBc, + OHomba
o hz
[]
Proof."
We t r e a t the second case, the first being similar. Let r E Hom*DG(X,(~RA), X , ( ~ R B ) | V.) A m o n g the operators used to define b y , -~, 1 hN,q) preserve internal degrees, cO increases t h e m by 1 and iv decreases them by 1 so t h a t hy(q)) is still degree preserving. Furthermore [0, h r ( r
1 = (Oir) o ~ o ( ~ hN)
so t h a t hy (qS) is a D G - m a p .
o O -
(Oiv)
o r o
( ~1 hN) o O = 0 iv
53
Let us check the homotopy fornmla:
hv OOHom*Dc, + OHOm*Doo hy = ( 1 = OHOm*DO (Oiv) Oq2o(~h,N) + i y O r +by
- ( - ~ ) deg~) . o, . u
(~oOx.A
1 )OOX. = (Oiy)o~o(~hN
) =
A + i y O ~ ) o ( ~ 1 hN) OOOOX. A
B o (Oiv) o , ~ o ( ~1h N )
- (-1)d~~r
1
o @ o(Po ( ~1 hN) o 0
- ( 1)~g|
1
1
+(Oiy) og2oOX.A o ( ~ h N ) + iyoq) oOX.A O ( ~ h N ) o O -- ( - 1 ) g ~ g r
o Ox..
o q? o
1
(-~ hN)
-- ( - - 1 ) d ~ g ~ ' i v o O X . B o g? o
1
(-~ hN)
o O
---- (Oiy) o ~ o l (h N o OX.A + OX.A o hN) 1
+ivo(~o~(hNOOX.
A + Ox. A O h N ) oO
as l a x . , o] = [ @ . , ~1]
= (Oiy)o(~ + iyo(~oO
=
[OX.B,iy] ~
= 0
= (Oiy + iyO) o(~ = s
as
[~, 0] = o []
C o r o l l a r y 4.12: Let U C Tr be an open submanifold and let Y r F(TU) be a smooth vector field o11 U. Denote by
iy : F(AkT*U) -+ F(Ak-IT*U) tile contraction by Y and by s
s
-~ s
tile Lie derivative along Y. (See 2.7.) Then Ey acts trivially on the cohomology groups
h(Hom*Dc(X.(nRA), X.(nRB) |
f(U))) and
h(HomDc(X.(gtRA), f(U))) []
54 Corollary 4.13: Let
X~G,/i,,(RA) C X~G(RA) be tile subcomplex of flmctionals vanishing on elements of high I-adic filtration (i.e.
on Ffl(f~RA) for some N > > 0). Let as before Y he a smooth vector field <m ~ o . Then s
preserves the subcomplex of cochains with finite support
s
: X~G,Im(RA ) -~ X~)G,Iin(RA)
and acts trivially on its cohomology. [] We will now study the natural projection of the differential graded onto the ordinary X-complex and will exhibit a natural contracting homotopy on the kernel of this projection. T h e o r e m 4.14: Let A, B be algebras and let V. be a DG-module. a) There exist natural maps of complexes
HOm*Dc(X, (aRA), X, (f~RB)) -+ Horn*(X, (RA), X, (RB)) Hom*DG(X,(gRA), X,(f~RB) | V.) --+ Hom*Dc(X,(f~RA), X,(RB) | V.) obtained by restricting functionals to the degree zero subspace X, (RA) of X, (~RA), and by projecting X,(f~RB) onto X,(RB), respectively b) The underlying maps of vector spaces split naturally. e) Denote the kernels of the maps in a) by
Hom*Do,+(X,(f~RA), X,(I-~RB)) c Hom*Dc(X,(f~RA), X,(f~RB)) and
Hom*DG,+(X,(12RA), X,(f~RB) | V.) C Hom*DG(X,(~RA), X,(12RB) | V.) respectively. Then there exist contracting homotopies
H: Hom*DG,+(X,(DRA),X,(f~RB)) --+ Hom*D~.+(X,(f~RA), X,(~RB))
H' : Horn*DC,+(X,(~RA ), X,(f~RB)|
--+ Hom*D~+(X,(~RA),X,(~RB)|
)
55
Id Id
= =
OHorn=o H + H OHom" o H' + H'
o OHom. o OHom*
which are natural in A, B (resp. A, B, V.). []
Proof:
We treat the second case, the first being similar. The complex Hom*Dc,+(X.([~RA),X.(~RB ) | V.) is naturally filtered by the subcomplexes
Xr;G+ , (RJA*,
RB)v. := {~ r XDC+ , (RA*,
where
RB)v. I(P(X.(f~RA)) C FJ(X.(f~RB)|
oo
Fi(W.) := {~W,~ = { x 6 W., d e g x > j } n=j
for any DG-module W.
XDG,+(RA, RB)v. = XDG,+(RA, RB)v. Define a natural map
hj
:
*d XDG(RA, RB)v. ff~
-+ XD-GI'J(RA, RB)v. --+ hj ff2
by the commutative diagram
X. (f~RA)j+2 o'~ X.(f~RA)j+I
0 --+
(X. (f~RB) | V)j+2 to ,Oo-~(hNo+)oho> (X.(f~RB) | V)j+I
hjd2 : X.(~RA)j ot X.(flRA)j_I
~r(htv~ 0 --+
(X.(flRB) | V)j to (X.(f~RB) | Y)j_~
Here
ha: X,(f~RA). --+ X.(~RA).-1 is the natural contracting homotopy oil the DG-module ((~, Xi(f~RA), (9) and
h~: X.(aRB) -+ X._I(aRB) is the homotopy operator associated to the number operator acting on f~RB via Theorem 4.3.
56
(*)
Put
now
*'J (RA, RB)v. -+ "'DG,+~-"*, Y * ' / + I / / ? A RB)v. fj := I d - (OHom* o hi + hj o OHom*) : XDG,+ .fj is a natural map of chain complexes homotopic to the identity. These maps can be used to obtain successively the desired nullhomotopy. Define r, -H : XDa,+(RA, R / ~ )v ~ X~-al,+(R A,RB)v
~-~hjofj_lofj_2o...ofl
H:=
j=l Each term of the sum is well defined by (*) and no problems of convergence arise because tile maps fk preserve degrees and hj vanishes except in degrees j, j + 1. Finally OHo,,~" o H + H o OHo~*
=
oo
= ~__OHom* 0 hj o fj-1 o fj-2 o . . . o fl + hj o fj-1 o fj-2 o . . . o fl o Omom. j=l 0(3
= ~-~(OHom" ohj + h j O O H o m . ) O f j _ l o f j _ 2 o "
of 1
j=l oc
= Z(Id-
f~)ofj_lofj_2o...ofx
= Id . . . .
ofjo..ofl
= Id
j=l
because . . . o f j o . . . o f l maps X ~o, + (RA, R B )v. into oo
N XDO,+(RA, ,,j RB)v. = 0 j=l
[]
T h e o r e m 4.15:
For any Fr6chet algebra A the natural maps of complexes
X*(RA) -~ X~,c(RA ) X*(RA, RB) +- Hom*Da(X.(~RA), X . ( ~ R B ) ) --9 X b a ( R A , R B ) are quasiisomorphisms, i.e. induce isomorphisms on cohomology groups.
57 Proof: This is a consequence of the contractibility of the " p a r a m e t e r space" ~ . If one chooses a contraction, for example F~(z) : =
for some interior point x0 E ~ , any (P E X~) G (RA)
(1 - t ) x + txo
the second C a r t a n h o m o t o p y formula yields for
F~+ - + = F~+ - F~cp = =0
(/o 1h ~ F ~ d t
g~ (f;+)dt
) + /o1h~F~(O~)dt
= OH~ + HO~
with H ~ :=
/0
h~F(~dt
As F~ retracts X~)c(RA ) onto the subeomplex X * ( R A ) , this shows t h a t X * ( R A ) is a deformation retract of the differential graded X-complex and the assertion follows. In the bivariant case Theorenl 4.14. has also to be used. []
C o r o l l a r y 4.16: The inclusion
X~i,~(RA ) -+ XSG,fin(RA ) is a quasiisomorphism and therefore
h,(X~)c,f,n(RA)) ~ PHC*(A) []
Remark: The comparison theorems between the ordinary and differential graded X-complexes were formulated so explicitely ill order to be able to t r a n s l a t e t h e m into a topologized setting in the next chapters. If we stay in a purely algebraic context however, the results above can be explained in a more conceptual way.
58 It is shown easily that the category of DG-modules is an abelian category with enough injectives and projectives: a DG-module (V, 0) is injective (projective) iff (V,O) is contractible as chain complex and tile subobjects Vj,j E fV are injective (projective) in the underlying category of ~ v e c t o r spaces. Especially X,(ftRA) is a 2Z/2-graded chain complex of projective DG-inodules. It is therefore possible to calculate the cohomology of the bivariant differential graded X-complexes via the universal coefficient spectral sequence [GO]
E rp'q) : E p'q := Extq(H.+p(C.). H.(D.)) ~ arPHP+q(Hom*(C., D.)) In our case this yields the E2-terms
ExtqDa(g.+, (X. (~RA)). H. (X. (9.RB))) GrPHp+q(Hom*Da(X, (f~RA ), X, (VtRB) ) ) Ex@a (H.+p(X. (~RA)), H. (X, (~RB) | V.)) GrPHP+q(Horn*DG(X.(~RA),X.(~RB) | V.)) As DG-modute H. (X.(~RA)) is concentrated in degree zero because the number operator N acts trivially on it by the Cartan homotopy formula 4.3. To calculate the desired Ext-groups, we note that any DG-module /14o concern trated in degree zero has a canonical projective resolution by acyclic DG-modules of length two (i.e. concentrated in two consecutive degrees). Using this resolution provides isomorphisms
EXt*Da(Mo, (V, 0)) --~ H*(Homr
V), 0) -~
Homr
H*(V, 0))
Applying this to our examples yields
ExtqDa(H.+p(X.(~RA) ), H.(X.(~RB) ) ) ~{ Hom~(H.+v(X.(RA)),H.(X.(RB))) 0
q=0 q>0
ExtqG(H.+p(X.(~RA) ), H.(X.(~RB) | V.)) ~_ Hom~( H.+p(X.( RA )), H.( X.( RB) ) | Hq( (v. 0))) So we see that the projections
Hom*Da(X. (~RA), X. (~RB)) --+ Horn*(X. (RA), X. (RB)) Hom*DG(X.(~RA), X.(~RB) | V.) --+ Hom*DG(X.(YtRA), X,(RB) @ V.) induce isomorphisms on the E2-terms of the associated universal coefficient spectral sequences. Therefore their kernels have to be acyclic. []
59
Chapter 5: The analytic X-complex The complexes X, (RA) studied up to now have a rich algebraic structure but are uninteresting fi'om a cohomological point of view: they behave functorially under linear maps of algebras and the Cartan homotopy formulas imply then the vanishing of their cohomology groups because any linear map is linearly homotopic to zero. Already in the algebraic setting it was necessary not to consider the X-complex of the tensor algebra RA, but that of its algebraic completion RA with respect to the I-adic topology. In this chapter we suppose that A itself comes equipped with a (Frdchet)-topology and will construct a (formal) topological I-adic completion 7~A of the tensor algebra RA in the case that A is admissible. The choice of topology on RA is dictated hy the demand that 1) The completed tensor algebra ~ A should still be of cohomological dimension one and the cohomology of tile completed X-complexes X, (TiA), X ~ a ( R A ) should be nontrivial. 2) A linear map f : A -~ B which is almost multiplicative should still induce a continuous homomorphism f : 7~A --+ T/B (at least on a subalgebra that depends on tile deviation of f from being multiplicative, i.e. its curvature). The difficulty with 2) is that the induced homomorphism R f : RA ~ RB of a linear map does not preserve I-adic filtrations unless f is multiplicative. In fact it may move the I-adic valuation of tensors by an arbitrary large amount. On the other hand tim norm of these "correction terms" of different degrees decays exponentially fast with the I-adic valuation if the curvature of f becomes smalh If a E I'~A/I'~+IA, then one obtains
Rf(a) = ~ bk bk E IkB/Ik+lB k=0
with
Il bk ll <- Cn-k ]l a tl
where C depends only on the maximmn of the curvature of f on the entries of a and becomes arbitrarily small if the curvature does so. Thus an ahnost multiplicative map will "ahnost" preserve I-adic filtrations if norms are taken into account. This suggests the following construction: Fix a multiplicatively closed subset K of A and consider tensors over A with entries in K. Expand a given element of this subalgebra of RA in a standard basis (Chapter 3) with respect to the Iadic filtration. A weighted Ll-norm for the coefficients of such an expansion is then introduced allowing the coefficients to grow exponentially to the basis N with respect to the I-adic valuation. Denote the corresponding completion by RA(K,N). It is a Fr~chet algebra and possesses the following crucial property: If f : A ~ B is linear with curvature uniformly bounded on K C A by a sufficiently small constant, then R f induces a continuous homomorphism R f : RA(K,N ) -+ R B ( K , , N , ) for a suitable multiplicatively closed subset K ' C B and N ' _> 1. In practice f will be an asymptotic morphism as studied in chapter one. As the curvature of an asymptotic
60
morphism is unifornfly bounded only over compact sets, the multiplicativety closed subsets K c A used for the construction above will throughout taken to be relatively compact. To guarantee the existence of sufficiently many multiplicatively closed compact sets the underlying Fr~chet algebra will be supposed to be admissible. In this case, the completed tensor algebras RA(K,N) will be admissible Fr6chet algebras, too. Moreover, as the algebraic I-adic completion RA, the algebras RA(K,N) are of cohomological dimension one, i.e.quasifree. The study of these completed tensor algebras will make up most of this chapter. The topological I-adic completion will finally be the formal inductive limit 7~A := "
lira
-+(K,N)
"RA(K,N)
of the algebras constructed above. Tile kernel of the projection 7r : 7ZA -+ A is formally topologically nilpotent, so that 7~A defines in fact a formally topologically nilpotent extension of A. The chapter ends with the introduction of analytic cyclic (co)homology of A as the (co)homology of the complexes X.(TCA), X*(TiA). This is justified by the cohomological dimension of 7r being equal to one. The resulting complex turns out to be closely related to the entire cyclic bicomplex of Connes [CO2]. 5-1 B e h a v i o u r o f I-adic filtrations under based linear m a p s The isomorphism RA ~- f~evA of (3.6) allows one to expand tensors over A in a sum of standard elements corresponding to homogeneous differential forms. The algebra structure of RA and the behaviour of the I-adic filtration under homomorphisms induced by linear maps will now be analyzed with respect, to this standard presentation. The first and nlOSt basic result is L e m m a 5.1: Let f : A + B be a based, linear map. Denote the curvature of f by
t~(a, a') = f(aa') - f(a)f(a') Then
RI
', a2)...
=
n
M
= o(f(a~
= Z 1
n~
Ol:c~ 11 tc,~2i-1,ca2i)
c~=l
where 1) the entries %i are products of terms
~(a 2j-1, a 2j) and f(a k) and each entry contains at most 2 factors of the form f(a k) .
1
61
2) In each s u m m a n d O(c~ l-X1~ co(c(, ' , 2 i - - 1 ,c~) 2 i , of the right hand side, if we put ~w := n~ and denote by tin the total number of factors n(a2J-l,a 2j) in the entries co, one has 3) The total number of summands M is bounded by M = ~{~ _< 8" Proof."
One has
e(b~ I-I(o(a) +
b2 -1,
=
1
(*)
E
IZl n2 lt2k-1 n2k ~(bO)(EO(~C~))(H~&)(bC~2'bC~'~+l))'H'( LO(~{~k-- l )) (l~ ~J (b(3~k ' b~/24-kI ) )
2" terms
1
1
1
1
If such a product is reduced via the Bianehi identity
o(a)Q(b) = -w(a,b) + o(ab) to standard form (r.h.s. of 5.1.) the nmnber of n-factors remains constant while the number of co-factors may increase, but certainly cannot decrease. As in the initial term we have the first part of 2) is proved. The second follows from induction over n. By having a closer look at the Bianchi identity, it becomes also clear, that the entries ca of the r.E.s.of 5.1. are of the form claimed in 1). It remains to estimate the number of terms of (*) in standard form. We proceed by induction over k. First we apply the Bianchi identity to
?~2k t 1
and obtain a sum < 2"2~-t terms
j
Next, the Bianchi identity applied to
([[
(2k--2
1
yields a SUlII <2n2k
2+lterms
,b
O{2k 2
62
Tile initial expression of a s u m m a n d of the r.h.s, of (,) is thus reduced to a sum of 2"2k-l(2n2k_2 + 1) terms of the form ~1
712
1
1
TL2k
3
t
which establishes the induction step. We end up with at most 2.2n~+l(2n2 + 1)2 ~a+l . . . (2n2k-2 + 1)2 n2k-~ terms. Using 2k+1
< 2 k+l < 4 k, k > 1
we o b t a i n 2.2nt+t(2n2 + 1 ) 2 n 3 + t . . . (2nzk-2 + 1)2 '~2~-' _< 4 ~ + n 2 + + n 2 k - ~
_< 4 n
As the initial expression (,) consists of 2 '~ s u m m a n d s of the form above we find 3). [] It will t u r n out to be necessary to do functional calculus on the ideal IA. First of all, the s t a n d a r d presentation of high powers of a given tensor is needed. Lemma
5.2:
Let A be an augmented algebra and let
{aj = ~ AflOZco~' Jj e J} Z be a set of elements of IA with entries in K U {1} C A and A~ C q; such t h a t
IAr~l <_ CN" i~ = n
for some C > 0, N > 1. Then r i k aj can be represented as a sum k
H aj : ~ 1
A7~70)it
,7
with entries in the multiplicative closure of K t2 {1} and
63 satisfies
C(k,n) <
nk (~C)k(~N)n~.
[]
Proof:
Modulo (IA) n+l one finds
• Iaj
=
Z
1
(fll,''',/~k) ifh + . . . + i ~ <_ n if~ >_ 1
which, t)y the proof of Lemma 5.1. equals a sum
Z if~1 4-..-4-iBk <_n; ioj
()~1"" "~k )( Z ~)~O']i-f) -->1 e(ioI ,...,iok ) terms
where c(io . . . . . . i ~ ) _< (2iz, +2)...(2i~k +2)
i v >_ i~1 + . . - + i ~ k
and the entries of ~co i~ belong to the multiplicative closure of K t2 {1}. It remains to estimate tile number of terms and the coefficients A~. One finds
Z IA l <_
i-)=n
<-
(/~1,''', f~k) i~1 + "'" + i;3k <_n i~j > 1
Z
IAI
CkNi'~+"+i~(2i~ + 2)...(2i~k +2)
i~1 +-..+i~k
1
<- CkUn2k -< (2C)kN~
Z (i~ + 1)...(i~k + 1) if31+...+i~ _1 2i~+'"+/~k ~
Z
(2C)k(2N)nF(k,n)
ioa +...+iok <_n;i~j ~ 1
whP,re F(k,n) = # { ( i l , . . . , i k ) E fVklij > 1, il + . . . +ik <_n} We claim F(k, n) <_
nk
64 the case k = 1 being obvious. For k > 1
F(k,n) <_ F ( k - l,1) +...+ F( k -
1k-1 +...+ _< - ( k - 1)!
1,n-l)
(rt -- 1) k-~ (k - 1)!
by the induction hypothesis
~on xk-i)l dx
<-.
(k-
.
nk -
k!
[]
Then power series can be treated Lemma
5.3:
Let
oo
f(z)
=
~'z=l
be a complex power series with radius of convergence R > O. Let a = E A~0~wi' E I A (i~ _> 1) 2 be all element of the I-adic completion of IA with entries in the multiplicatively closed set K U {1} C A and AZ C ff~. Assmne that
Z JAzl <_ CN i~=u
for some C > O , N > 1. Then c~
f(a)
E IA
:= ECna" n=l
is well defined and can be represented as
f(a) : E A'7Q'7wi~ "Y
with entries in K U {1} and where for any R' < R ,
n
i,~ : n
for some C ' > 0 depending only on f and RL
2C
9
65
Sinfilarly, if a E IA, b E I'B are of the same forni as above with entries in K C A (resp. K ~ C B), then f(a | b) 9 "[(d | B) is well defined and can be written as
S(a | b) = ~ A3`(03`,coi'' | 03`=wi'=) 3`
where the entries of 03`,wi~, (resp.0-~a/,~) belong to K U {1} C A(resp. K ' U {1} c B) and 2' / 11. Z IX3`l -< C (2N) e,mp(~Tn ) iv :i'vl +i~2 = n
[]
Proofi
The sum f(a) = V'~176 Z - ~ n = l c n a '~ converges ill I'A because a E I'A. If one brings f(a) in standard form using the Bianchi identity as ill Lemma 5.1 and Lemma 5.2 one finds oo k=l
3`k
with entries in the multiplicative closure of K W {1} and such that k
n l~k
la3`~l _< (2c) (2N) g It, follows that k,Tk
3'
with oo i-r~-n
nk
k=l
The radius of convergence of f being equal to R, one has lim[cki ~_ z k
1 R
--
which yields
I~kl _< C'(~,) k for ally R ' < R and some C'(f, R') > 0 so that oo
I),-~1 i.~=n
< C'(2N)'~ Z
-
k=l
(~y)k k!
66
,
n
2Cn
< C (2N)exp(~7-
)
wifich yields the claim. The proof ill the case of a tensor product is similar. []
5-2 L o c a l l y c o n v e x t o p o l o g i e s on s u b a l g e b r a s o f R A Formal inductive limits D e f i n i t i o n 5.4:
Let C be a category. The category IndC of I n d - o b j e c t s or f o r m a l i n d u c t i v e l i m i t s over C is tile category with functors from ordered sets to C as objects, i.e.
ob(IndC) = {X = " lira/"Xi} X = {Xi, i C I, .fi,i' : Xi --+ Xi,, i <_ i'} where I is an ordered set, Xi E ob(C), fi,i' E mot(C) and fv,i,' o f
is another object ill IndC, the morphisms in IndC from X to 3; are defined as
Homtnd C(X, y ) :-- lira lira Homc (Xi, Yj) + - I ---+J
wimre the projective (resp.inductive) limit above is taken in the category of sets. []
It is easily verified that with this definition IndC is in fact a category, i.e. that morphisnls compose well. T h e t o p o l o g i c a l I-adic c o m p l e t i o n o f R A D e f i n i t i o n 5.5:
Let A be an admissible Fr~chet algebra and fix a "snlall" open, convex neighbourhood U of zero ill A (see 1.13.). Denote by )~ = ~(A, U) the set of pairs (K, N ) , where K C A is a compact subset of U and N >_ 1 is a real number. K: becomes a partially ordered set by putting (K,N) _ (K',N')
*=> K c K ' a n d N < N '
To ally pair (K, N ) , (K', N ' ) E/(7 there exists (K", N " ) C K: such that (K, N) _< (K", N " ) and (K', N ' ) _< (K", N " )
67
Definition and Proposition 5.6: (Topological I-adic completion) Let A be an adnfissible Fr6chet algebra and let (K, N) E K(A, U). 1) Put R A K := { a ~ ~ A I " = Z
A~~ ~k~ }
where An e G and the entries of OZwk:~ belong to K ~ U {1} where K ~ denotes the multiplicative closure of K. (It is relatively compact by definition.) RAg is a unital subalgebra of RA. 2) For any m E fV the Nnctional [I-IINK,m= RAK a
~ --+
~+ infa=~2~os
~-~
y']~fllAfll(l+kfl)mN -k~
defines a senfinorm on RAg where the infimum is taken over all presentations of a with entries in K ~176 U {1} as in 1). Denote the completion of RAg by these seininorlllS
aS
RA(K,N) := Compl(RAK, II- I[~,m, m
PC)
It is a Frfichet space. 3) The multiplication on RAK satisfies K
I] xy IlN,m <
2m + 1
K K II ~= tlN.m+~ll y IIN,m
so that RA(K,N) becomes a Fr~chet algebra in fact. 4) The Fr~chet algebra RA(K,N) is admissible. A "small" open set is given by U := {x C RA(K,N)I
1
II ~ IIN,~< ~}
5) For (K, N) <_ (K', N') the natural inclusion
RAK -+ RAK, of subalgebras of RA extends to a contimlous homomorphism of Fr~chet algebras
RA(K,N) ~ RA(K,N,) 6) The Ind-Fr~chet algebra 7-/A := "
lira
--+K(A,U)
" RA(K,N )
is called the topological I-adic completion of RA. It is independent of the choice of the small open set U C A. []
68 Proof:
1): follows from the Bianchi-identity (5.1). 2): Evident. 3): We show that for x, y ~ R A K the inequality
II x y II.... 1 _< 2 ''~ II x I1~ II Y Ilm< is valid where we write II - ll.~ instead of tl Choose presentations x = ~kf~#~w <~
-HN,rn'K
y = y~'k~,8~,w~ '
such that
Y~l~l(l+~x~) mN-~ ~--~l;~'l(l+k~') m-~x
~ll~llm +~
~ ' -
Then and we find
where ~wk~ has entries in K ~ U {1}, the number of summands equals 2k~ + 2 and the valuation k.y satisfies k~ + k~, _< k~ _< k~ + k~, + 1. Therefore II xY II,,~-x -< ~ IA;~A~'I(2A:;~+ 2 ) 0 + k,y)"~-xN-~'~ ~,~'
_< ~
IA~II~,I 2(1 + k~)((~ + k~) + (a + k Z , ) ) m - l N -kn-kn'
~,~' -< Z IA~I[A~'] 2(1 + k~)2m-l(1 + k~)m-l(1 + k ~ , ) m - l g - k z g fl,fl'
-< 2m(y~ I:~1 (1 + k ~ ) ~ N - k ' ) ( Z
I:~,l (1
-k~'
+ kj3,)rn-1N -k~' )
_< 2m(ll x lira +~)(1I V lira-1 +~) where we have used a + b <_ 2ab for a, b >_ 1. 4): By L e m m a 1.14. it suffices to check that the multiplicative closure of any compact subset V C R A K N U is relatively compact. Moreover it is easily seen that one may suppose that every element of V is homogeneous, i.e. corresponds to a differential form of a single degree under the isomorphism R A ~- i~eVA. Let m E 2Z+ and put C ( m ) := s u p p e r I] x IIN.... Clearly C(1) < c < i for sonic g. Let x l , . . . , x ~ ~ V. Then xi = ~ A i j & j w ~ ~ with entries in K ~176Fix i0 such that kio = m a x i ki.
69 Choose presentations x~ = ~ A~ OZ~w~' s.t. IA~I(1 + k~)N -k' <_ ~ i r io
II xi IIN,I_< ~
II :cio IIN.~+~< E IA~;0I(1 + kr176 /3io
<- C ( m + 1) + 1
then
II X l ' . . X n
IIN,m~ ~
I~Xf~,I...[A~.I II o ~ , ~ , . . .
~
~
~ql,..-,fl,, <_
~
fl~,...,/~,,
IA~,l...t~r~l(2kl+2)...(2k.n+2)IIo~k~ II~,m
where 0,w k~ has entries in K ~ and kl + "." + k,~ <_ k.y < kl + "" + kn + ( n - 1). Therefore
II ~ . . . x ~ IIN,,o._< _< 2"
Z
I ~ , I . . . I~,~ I(1 + k l ) . . . (1 -~- kn)(n -[- ~1 4-*.. ~- ~n)mN -(k14-'''4-k')
Bl,-..,fl,~
2~n m
~ IA~,l...]A~,~l(l + k l ) . . . ( l § k**)(l + kio) m N - ( k * + + k ' ) fh .....I~,,
<_ 2 n n m e n - l ( C ( m + 1) + 1) = e - l ( C ( m +
1) +
1)(2e)nn m
t this shows
Ase<
lim ( sup
n--+oc x i E V
II x x . . . x,, I!N,m) = 0
for all m C 2~+. 5): Evident. 6): First of all note that every algebra of the form RA(K,N) is isomorphic to an algebra RA(K,,N,) for which K ' is a multiplicatively closed, compact subset of any given neighbourhood of 0. It suffices to choose M > 1 large enough that [0, ~ ] K ~ =: K t C W. K ~ is then a multiplieatively closed, compact set. Moreover, tim subalgebras R A K C RA, RAK, C R A coincide and the identity R A K ~-~ R A g , extends to a topological isomorphism I~A(K,N ) -~ RA(K,,NM 2) as is readily seen by applying the obvious homothety to the entries of tensors in RAK. Choosing W :-- U the argument above shows that, as long as the order structure on K;(A, U) is ignored, one may suppose that for an algebra of type RA(K,N) that in fact K C U is multiplicatively closed. P u t t i n g W := U N U ~ where U' is another "small" convex neighbourhood of 0 the argument proves that "
tim "RA(K,N) ~ - "
IC(A,U)
lira
K.( A , U n U ' )
" I~A(K,,,N,, ) ~+ "
lira
~:(A,U')
" RA(K,,N,)
as claimed. []
70
The next two lenmms concern technic,al results that will be needed for the study of the cohomology of a direct sum and for the construction of exterior products. L e m m a 5.7: The notations are those of 5.6. Let e = e 2 E A be an idempotent acting as a unit on the compact set K C A (e C ~ K ) and let x c RA(K,N) ,
K
satisfy the following condition: For every fl all except at most No entries of O~w~e are equal to e. T h e n for any y E RA(K,N) K
<
K
K
1[ xy HN. . . . . (2No + 2) II * t1>,,,11 y 1IN,., Proof: The same as for 5.6.3). The better estimates are due to the fact that ahnost all w-terms in x equal ~v(e, e) and if the product xy is brought into s t a n d a r d form via the Bianchi identity the majority of the arising terms cancels due to the identity
w(e, e)fl(a) = w(e, ca) - w(e 2, a) + 0(e)w(e, eL) = ~ ( e , 0) -
~(~, ~) +
o ( e ) ~ ( ~ , a) = o ( ~ ) ~ ( e , a) []
Lemma
5.8"
Let A be an admissible Fr6chet algebra and .4 c A a dense subalgebra. Let U C A be a "snlall" neighbourhood of zero. Define K'(A) := { ( K ' , N ' ) } where K p is a nullsequence in A ~ U and N t > 1. Then " h~n"RA(K,,N,) ~-~ " l iiCm " R A ( K N)
71 Proof:
We are going to construct an inverse of the obvious morphism " lira "RA~, -~ " lira "RA~
-+K:'
-+~
So let ( K , N ) E ]C(A). The notations of Lemma 1.7. are used throughout. For a subset S C A we denote by (S) its linear span and by Con(S) its convex closure. Construct a nullsequence K " ( = B) for K ~ as in the proof of Lemma 1.7. Choose
for all N > > 0 such that
YN C B(0, 2AN) (x~,.
9
~
x'ng N ) N B(0, AN) C Conv(YN)
where B(O, r) denotes the r-ball around 0 in a suitable translation-invariant metric
on A. Let,
xNxN_ N Z N :~__ { * 3)~N Xk Id(x,N xjN ,x N k) <
AIr3 }
and put finally K':=K"U
U
(YNUZN)
N>>0 K ' is a nullsequence. Replacing K ' by ~1 K K ' C A N U. The desired map
t
for some C > 1 we may assume
l : RA(K,N) -~ RA(K,N,) (for a suitable N') will be defined oil generators ~(y), y c K ~ by
l(o(y)) := ~ AnQ(bn) E R A ( K ' N ' ) n:O where
y = ~-~n~-_oAnbn
bn -
X-§ fi~n "k"+l(Y)-" A. k,,(,)
is a presentation of y as in 1.7. We note first that l(y(y)) does not depend on the choice of such a presentation: Let
i(y) = Ao ~"ko(y) + E Aj-1 j=l
0
it(Y)--~ " Xkt~
f'0 ,~--~--
oo
+ E
j=l
)~J--1
"'kj(y)
4---[ ~_ (y)
( Xj "k~(y) Z X ~ l , ( y ) )~j--I
]
72
be two presentations of y constructed as in the proof of L e m m a 1.7. Let N ' be such that O(3
A N ' - I < e,
E
An <
7,t:N I
Then II
.i(y)
g -
i'(u)
N,m
f x N ,"
=
t ,
~
(3o
.N
j=N'+I
j=N'+I N' .N' -< I[ X,k~,(.~) -.~%,(~) II +2c~
Now Nt
N'-I
'
N~-I
dA (XkN, (y), Xk,N,_~(y)) < dA (xN;,(y), V) + dA (y, X '-,(Y)) A2N,_I < @' + - 2
N'
' -1
XkN,(y ) x~,;, ,(y) AN,_I -
E K'
and
II i(y)- ~:'(y)H ,N'
1] ,~N'--I
Let ff~(K ~
__ x N ' - I
N' _ xN'- I ~'kN,(~) 'k'~, ~(v) xk'~,(v) k'~, ~(~) II + It ( - - - ~ N ' - I ) AN,-I AN,-I < 2CAN,_1 + 2Cc <_ 4C~
-~-2C~
be the free vector space generated by the set K ~ and let l
ff~(K ~
--4
RA(K,,N,) be the linear extension of I. We claim t h a t it factors l : (~(K ~ -+ AK~ --+ RA(K,,N,) where A K ~ C A is the linear span of K ~ inside A. So let a l , . . . , a m E K ~162 be such t h a t ~ # , a i = 0 i n A . #il(Q(ai)) = 0 in RA(K,,N, ). Now
We have to show t h a t
o(3 i
j=O
[-tiXkN(ai) q-
i
i
j=N+I
and thus
II ~-~ml(o(ad)II(K,,N,)_
"
j=N+I
73
for all N. Moreover, for any seminorm
o,1
A
N II }-~'~,~X kN(o,)IIA=II Y~ i
-
x N
i
<_ ( ~ Ip~l)ma.~:~il X~N(a,)
--
i
ai IIA
which is small. Thus
~--~#i(Xkg(a~) = ~ i
vkNykN
k
with yN e YN and limy--,oo Y~-kIvgl = 0 so that, by the choice of g '
H~--~#il(o(ai)"(K' ) ,N')<--(~'vN') §
( s Ai) \j=N+I
9
<~
for N > > 0. A similar estimate, using ZN C K' shows
~(kk')
= l(k)l(k')
W, k' 9 K ~o
Moreover, the map 1 is bounded: Let a 9 RAK be represented as
a = ~ IZflflflcakz with entries in K ~176 U {1} and such that K
~-~lPf~](l + kf~)m g - k '
<-If a ][N,m Q-e
Then
K It /3,io,...,i2k~
-< ~ I.~1 c ~ + '
(1 + k~) m (C~N) - ~
fl
_< c(~
I~,~1(1 + k~)"
N -ks
<_
C(ll
a
K IIN,,-,, + e)
This implies finally, that l : AK~ -+ RA(K,,N,) extends to a continuous homomorphism of algebras 1 :
RA(K,N) --+ RA(K,,N,)
provided that N' >>N is choosen large enough. It is easily checked that this construction provides a morphism of Ind-algebras " lim"RA~c -+ " lira "RAm --4K:
--+/C'
74
inverse to the ol)vious inclusion.
[]
Lemma
5.9:
a) The canonical quotient map rr : R A--+ A extends to a continuous map rc : RA(K,N) -+ A for all (K, N) E/(7. b) The curvature of the map O: l l ( K ~ U {1}) -+ RA(K,N)
is bounded by tl aJo(x,y) [t(N,,~) K --< 2 " N
1 Vz, y E K ~
[] It was mentioned in the introduction that the topological I-adic completion is a (formally) topologically nilpotent extension of A. We want to make this precise. For an ideal in a Banach algebra topological nilpotence means that the spectra of all its elements reduce to zero. Consequently topological nilpotence is equivalent to the condition that holomorphic functional calculus can be applied with any function holomorphic near zero. This means that if f is a complex power series of strictly positive radius of convergence and z belongs to the topologically nilpotent ideal 27, the series f ( z ) will converge in A. It is this condition for topological nilpotence that can be verified for the kernel of the projection TeA -+ A. This result will be used when we investigate the compatibility of the Chern character with the multiplicative structures on K-theory and cyclic cohomology.
L e m m a 5.10: Let
OO
f(z)
=
be a complex power series of radius of convergence R > 0. Let A, B be admissible and B(0, C)y,0 := {x e RA(K,N), [1 x < c}
I1~,0
B'(O, C)N,O := {Y E RB(K,,N),
K II Y g,o
< C}
75
Then f defines continuous maps
f : IA(K,N) A B(0, C)N,O -+ -~
X
f : IA(K,N)
~ J~(O,
IA(K,M) Zn~=l anx n
C)N,O X IB(K,,N ) CI B'(O, C ) N , O (x,y)
-+ I(A @~ B ) ( K | --+
)
En%, a . ( x | y)"
for any
M > 2N exp(~)2C where IA(K,N) denotes the closure of IAK C RAK in RA(K,N). []
Proof:
Choose e > 0 and let a C IA(K,N) be presented as a sum
a = EA~O~wk~ with entries in K ~ U {1} and such that ZIA,] N-k' Z
-<1t a ]IN,O + e
and therefore ]AZ[ -< (]] a []N,O +e)N '~
Z k~=n
L e m m a 5.3 shows then that
f(a) = Z A.yO~wk~ ,y
with entries ill K ~ U {1} and such that for R' < R
[AT]-< C'(2N)nexp(2(H a t]N,0R t -~- s
E k 7 ~7~
for some C' depending only on f, R' (and not on a). Let M t)e any real number satisfying
M > 2near( 211 a Ilu,o) R
If e > 0, R' < R are choosen such that
M' = 2gexp( 2(1[ a IIN,O R' + ~)) < M
76
one finds II f(a)
K II.,m_< IA I(I+
k.y)mM-k~
7
C'Mm(l+n)mM -n = C ' E~( I + n )
<_ which proves that
f(a) E RA(K,M).
To show continuity choose M > 2 N c x p ( ~ ) to a. Then f(b) E RA(K,M) is well defined and oo
and let
k=l
be close
b)bk-l-j
j=0
so that
II m , <_
II f ( a ) - f(b)
E]Ckl E 2m+l Haj
b E RA(K,N)
k-1
f(a) - f(b) = E c k E a J ( a -
k=l
raM, ~ (~-) < oo
I]M,rn+l 2 m + l
11
(a-
bk-l-j I]M,m
b)[JM,m+li]
/=0
by L e m m a 5.3 oc k - 1
-< 22(m+1)II
(a-b)]IM,,~+i
.
C'~(~7,)
k II aJ [tM,m+ltl bk-l-j
[IM,m
k=l j=0
(for the choice of C', R ' see 5.3.)
-< 4('~+1)II
(a-b)IIM,,,~+I ~
"
(
)
)J' II a" JiM,m+1
(~/)k' II bk' [IM,m
\ j ' =0
The estimates of Lemma 5.2 allow now to conclude for
a = EA~O~wkZ and a similar presentation
that ~ j'=0
< --
1 j
j'=0
oo E (-~/)jC(j'n)(l+n)m+l(2N)-nexp( j=0,n=l
9
2(It a tiN'0 "~g) n)C"" R
where C" = 2 N e x p ( 2([1 a
IIN,OR,+e))M-1 < 1
!
77
1 9 II a IIN,0 4-~) j u7.T( g 2(11 a IIN,o (~)~(2 1 4 - ??.)m+lexp( R! 4 - e ) n ) C , , n
< j=0,n=l
by Lemma 5.2 oo
<- ~e~p(2(lt
a
IIN,O 4-~) n ) e x p ( - 2(11 a IIN,O R' R' +c)
n)(1 4- n)m+lcttn
rt:l Go
< Z(t
4-n)m+ac ''n = C'"(f,R',e~,II a tlN,o,M, rr~,) < oo
If b is very (:lose to a a similar calculation holds for the second sum O~3
k=O
and the same choice of
M > 2Nexp( 2 II a ]IN,O) R so that finally
II f(a) - f(b) IIM,m<<_ K C ' " ' ( f , R ' , e , M , m ) C ' II a - b
IlN,m+e
which is what we wanted. The proof in the tensor product case is similar. []
It will now be verified that the algebras RA(K,N) are quasifree, resp. of cohomological dimension one in the category of Fr~chet algebras. This means that they possess tile lifting property with respect to nilpotent extensions of Fr~chet algebras with continuous linear splitting. The result guarantees that the cyclic (co)homology of the algebras RA(K,N) can be calculated by their X-complex. This gives the theoretical justification for the definition of the analytic resp. asymptot.ic cyclic cohomology of admissible Frdchet algebras presented later in this and the next chapter. The proposition is also needed for establishing the topological versions of tile Caftan homotopy formulas and the exterior products. Proposition
5.11:
The natural homomorphism
i : RA ~ R(RA) extends to a continuous homomorphism
i : RA(K,N ) --~ ~(RA(K,N)) i.e. for each k >_ 0 the natural map
RA ~ R(RA) ~_ l]cV(Rd) P~) ~ k ( R d )
78
extends to a continuous linear map
RA(sr,N) --+ fI2k(RA(~r,N)) ~ RA(K,N) |
RA(K,N) |
Proof:
The h o m o m o r p h i s m i : R A --+ R ( R A ) is given on s t a n d a r d elements by
i : e~" ~ ~l~I (~(~) + re(o, ~)) 1
where 0 : A -+ RA, p : R A ~ R ( R A ) are tile n a t u r a l inclusions, we are interested only in the p a r t of I(RA)-adic valuation k if the right hand side is written in s t a n d a r d form. Modulo I ( R A ) k+l we find
i(ooo'~) = ~ go(oo)'~
~DJo... ~(~o)"~(o, o)J'
where J0 + 999jl _< k and i0 + J0 + 999il + Jl _< n so t h a t the number of terms is bounded by ~-~ ( ; ) _ <
( k + l ) nk
j =0
It is not difficult to verify t h a t a power fm of a simple tensor can be written modulo I ( R A ) k+l as k
-~m ~
j=0
~ ~k+l--j~j
with c(j) <_ (,,-jk-1). Thus k
-~(~)m -.~~ ~ ~(cd)k-f-l-j-~(O21~dl')j j=O<_c(j)terms = y~?(J)m(J',U")J,
j <
k
where the number of terms grows for fixed k only polynomially in m. Using the calculations and estimates of 5.1. one sees further t h a t the original expression for i(pw '~) can be written as
i(o<-') = ~ ~(~jo )~(~j, Q~,~)... ~(~j~j-,, Q~o~) where j _< k and the nunlber of terms grows for fixed k p o l y n o m i a l l y in n. This however implies
II i(~)il(N,m)|
< C ( m , m ' ) I t ~ tlN,,~'
for c~ E RAtr and m ' > > 0 which proves the claim. []
79
The crucial property of the completed tensor algebras RA(K,N), which motivated their construction is derived in
P r o p o s i t i o n 5.12: Let
f:A--+B be a bounded, based, linear map of adnfissible Fr~chet algebras. Let K C A, K ' c B be compact sets contained in "small" convex balls U C A, U' C B and 0 < A <
1, i t > 1 such that
.f(aa') - f(a)f(a') c AK' Va, a' E K ~ f(a) E IrK' ga E K ~ Suppose that A satisfies A<
1 -8N
Then for M > 8 N # 2 the linear map f induces a continuous morphism of ~ e h e t algebras
R f : RA(K,N)
-+
RB(K' M)
Proof: Suppose first that tt = 1. Let a E RAK be presented as a sum
a = E A~of~wk~ with entries in K c~ t2 {1} such that
E ] A ~ [ ( I + k z ) ' ' N - k "
.7=0 with entries in K r~ U {1} and CZ _< 8 k' and
[Av[ < Aj~ with J r + k v = lJ~-y+flwv> k~, k-, < k~ We therefore find the estimate K t[ R f ( a ) [u,m <- E I A Z l E [ A ~ [
( 1 + k.r)mM -k"
Z <- E
]AZ[(1 + k~) m E A Y ~ M -k~
8O
-<
Z IA~I(1 + k~) ~(sN)-J~(sN) - ~ TM
-< ~
lair( 1 + k ~ ) m ( # ~ ) ( 8 N ) -*~ K
For the general case let K(K~) be the cone over ~1 K ( K )r with vertex 0.
Then
c U, ~ r c U t are compact sets satisfying
,,s(~ ~ • ~ )
c ~ K , ~f(K
) c
Thus R f induces for M > 8Np 2 by the first case considered above a continuous morphism
R f : RA(K,N ) ~-- RA(~,N~2 ) --+ RB(-~,,M ) ~-- RB(K,,M ) which proves the claim. []
Remark:
Tlle proposition shows how the seminorms H -- ][N,m K come up by starting from the L 1 norm II - ][1K,0and looking for a natural, locally convex topology on R A such that for any almost nmltiplicative, linear m a p f : A -~ B the induced morphism R f : R A -+ R B becomes continuous. []
Our constructions suggest two possible natural topologies on the c o m m u t a t o r quotient (~IRA)~, one induced from the topology of RA(K,N), the other obtained by expanding elements of (~IRA)b in a series of standard elements (3.7) and taking weighted/1-norms of the coefficients. We show that both topologies coincide. Lemma
5.13:
Consider the linear space
(~IRAK)~ := (owndQ) with entries in K . Define seminorms II - IIK,,~ on ([~IRAK)~ by II a rIN,~ K := i n f E
I~,l(1 + k~),,,N-k,
81
where the infimum is taken over all presentations
with entries in K ~ U {1}. Denote by ~I(RAK)o,lv the completion of with respect to the topology defined by these seminorms. Then
(f~IRAK)~
[~I(RAK)~,N ~_ Ftt(RA(K,N))~ as Frfichet spaces. Proof:
It is clear that the natural map
~! (RAK)t~,N --+ ftl(RA(K,N))tl is continuous. We construct an inverse map on the dense subspace
Ftl(RAK)~ c ~I(RA(K,IV))~ and show that it is continuous providing thus a global inverse. So let,
oa = ~
aidb~ C ~I(RAK)b
finite
with ai = Y~e~ A;3,0~,w k'~
bi = ~ ,
ItT~07~wk''
Assume that this presentation of w satisfies K
E
[I ai ]lN,m+21l
bi HKm+2 <_Hw
K
IIN,m4.2,~]I(RAK,N)O
-~-s
i
I.~l(1 + k~,)m+2N -k~ _< II bi IIN,~+~ +~' Then
aidbi = E AI~l~-r~0~ wk~ d(o'r~wk'~ ) fli ,%
reduces modulo commutators to a sum
E Ae,#'~i E O'~wk~do~ ~,~
with entries in K ~ U {1} where for fl, "/given the number of interior summands can be estimated by
#~ _< c(1 + k~)(1 + k~) 2
82
tor s()me universal constant C > O. The latter sum defines an element of in canonical form and for the norms we find
K
]l w IIN,m,~,(.AK)~.N
= [I ~ aidbi K
(~21RAK)~
(RA,~h.~
i
_< C ~
<- C N ~
IA~llp.y~l(1 + k/~)(1 + k.y) 2 (1
+ kz + k.~)'" N -(k~+k~-l)
(~'AZ~l(l+k;~,)"+tN-~:'~') ( ~
as 1 + kl~ + k.y _< (1 + kz)(1 + k.~) _< CN}-~(II ai IlN,m+l + d ) ( l l bi IIN,.~+2 + d ) i
<-- (CN~llail'N'm+2Hbi ' . HN'm+2) 4if e' is choosen correctly
<_ C N
K II ~ [[N,m+2,~(RA~.N)u + (1 + CN)e
[]
The (lifferentials of the X-complex extend to continuous operators on Their norm satisfies the following estimates
X.RA(K,N).
L e m m a 5,14:
For a C RA(K,N)
and
w E -~I(RAK)~,N 2_N II a IlN,m+3
II do, IIN,m <_ '2 II a IIN,m+l + 3 II b~
]]N,m<-- 2 [l w [IN,m+x + 2 m+l N - 1 II ~ IIN,.~ [] 5-3 T h e a n a l y t i c X - c o m p l e x
Let A, B be admissible Fr6chet algebras, and let A, B be obtained from A, B by adjoining a unit. Denote by g A := " lim~:"RA(K,N)
7~B := " lim~:,
"RB(K,N,)
the topological I-adic completions of RA, RB. They are formal inductive limits of t~milies of adnfissible Fr6chet algebras of Hochschild-eohomological dimension one. Let V, W be locally convex, topological vector spaces.
83
D e f i n i t i o n 5.15: a) The h o m o l o g i c a l a n a l y t i c X - c o m p l e x of A with coefficients in (V, W) is defined as
X~.'v'w (A)
:=
Horn(V, X.(7CA) |
W) = limHom(V, X,(Rflt(K,N)) | --+;C
W)
We put X.~(A) :=
X:'*'e(A)
= n,nX.(RA(~,N)) --+~2
b) The c o h o m o l o g i c a l a n a l y t i c X - c o m p l e x of A with coefficients ill (V, W) is defined to be
Xo*v.w(A )
:=
Hom(-Z.(n_~.) |
V, W) = limHom(X.(R_~(~: N)) | e--]C
V, W)
We put
X:(A)
:= X$,e,,(A) =
Uom(-X,(r~),),r
= limHom(X,(RA(KN)),r +--IC
c) The b i v a r i a n t a n a l y t i c X - c o m p l e x of the pair (A, B) with coelficients in (V, W) is defined as
X~*y,w(A, 13) := Hom.(-X.(TiA) | = lira lira
+--/~ --+K:'
Hom.(-X.(R~i(K,N)) |
V,-R.(nB) |
W)
V,-X.(RB(K,,N,)) |
W)
The homological, (cohonmlogical, bivariant) X-complexes define covariaitt,
(contravariant, bivariant) flmctors from the category of admissible Fr~chet algebras to the category of 2Z/2-graded chain complexes of vector spaces. [] The c o m p o s i t i o n p r o d u c t defines a bilinear map of chain complexes
X~*y,w(A, 13) | X*,w,u(13, C) -4 X*,v,F(A, C) As a special case, there exist p a i r i n g s of complexes
X~'v'W(A) | X*.w,F(A ) -4 Horn(V, F) X , ~(
A ) 0 X*(A) -4 (~
D e f i n i t i o n 5.16: The homology groups
Hc~'V'W(A)
:=
h(x~'V'W(A)) HC..vw( * A)
HC~,v,w(A, B)
:=
:=
h(X~*v,w(A, B))
h(X*v,w(A) )
84 are called the a n a l y t i c cyclic h o m o l o g y , ( c o h o m o l o g y , b i v a r i a n t c o h o m o l o g y ) g r o u p s of A, resp. (A, B).
[] The coefficient groups of these theories are nontriviah
Lemma 5.17:[CO2] a) Tile universal Chern character cycles of an idempotent (resp. all invertible) element of definition 3.13. satisfy oh(e) C
Rr
) =-]~,(IJ({1,et,4+6 ) eh(u) E R r
..... -~ },1+6)
for all 6 > 0 b) The homomorphisms of Ind-Fr6chet algebras e -4 ch(e) satisfy sort =:p
rros=Ide
where p is an idempotent endomorphism of R ~ canonically homotopic to the identity whose image is the one-dimensional subalgebra generated by ch(e) ~ Tr c) The complex x.(r162
=
r ~-
,
0
*=1
is a deformation retract of X, (R~?), i.e. X , Tr o X , s = Id, X , s o X, rc = X , p
and X , p is homotopic to the identity: X , i d - X . p = hO + Oh
for some operator
h: X,(n~) -4 X . _ l ( ~ )
0
85
Proof:
b) The homotopy between tile identity and p is defined by -+ ( 1 - s ) o(e)+sch(c) Cn~, s 9
Rr
which extends to a continuous homomorphism F : Ts162 -~ C~176 1], 7 ~ ) . c) The tlomotopy operator h is provided by the composition
X.(nr
x.i> X.R(~r h'
--+ X,_IR(nr
X.RF>X.R(COo([O' 1],n~)) X,
7r>
X,_l(nr
where X,i, X, RF and X,~ are continuous by 5.11., 5.17.b), and 5.9. The homotopy operator h' is constructed in Theorem 6.12. where also its continuity is shown. [] L e m m a 5.18: There exist canonical homotopy equivalences of chain complexes
X~.,v,w(A) ~+ X~,v,w(r
)
X*v,w(A ) ~+ X*v,w(A,r ) which show that the analytic homological, resp. cohomological X-complexes are deformation retracts of the corresponding bivariant complexes. [] Proof." By Lemma 5.17 the maps
X: (A) = Horn* (X.r X.nA) -oX. 5 Horn* (X.nr X.nA) = X~ (r A) X:(A, r = Hom*(X.nA, X . n r
x . . . . >Ho,n*(X.nA, X.r = X*(A)
are quasiisomorphisms and X~(A), X*(A) become deformation retracts of the bivariant complexes X~ (@, A), X* (A, ~). [] C o r o l l a r y 5.19:
HC~(r *
r
_,= HC*(r
_~ HC.~(r _~ J" (~
/,
0
*
=
0
*=1
[]
86 Theorem 5.20: (Homotopy invariance) Bivariant analytic cyclic cohomology is a smooth homotopy biflmctor on the category of admissible Fr~chet algebras, i.e. if
f,g:A-+B are smoothly homotopie morphisms of admissible Fr6chet algebras, then
f. = g. C HC~'.v,v(A,B) and consequently
f* = g* : HC:.v.w(B,C) ~ HC2,v,w(A,C) f. = 9. : HC*,v,w(D, A) ~ HC:,v,w(D, B) []
This is an immediate consequence (see also 6.15.) of
Theorem 5.21:(Caftan homotopy formula) Let A be an adnfissible Fr~chet algebra and
5:A~A a bounded derivation. a) Let K C U C A be a multiplicatively closed, compact subset of A contained in a"small" ball U around 0. Choose M > 1 large enough that KU-~5(K) C K' C U for some compact set K'. Then the Lie derivative
s
X , ( R A ) -+ X , ( R A )
and the associated Cartan homotopy operator
hs : X , ( R A ) --~ X . _ I ( R A ) of Chapter 4 provide continuous operators
s
X.(RA(K,N)) --4- X.(RA(K,,N))
hs : X.(RA(K,N)) --+ X._I(RA(K,,N))
87
b) Therefore they define elements
s
E X~
A)
and
h~ E X~(A,A)
satisfying
OHom" h~ = s and consequently, the Cartan homotopy formula
s
= h~ o Ox. + Ox. o h.~
is valid. []
Proof."
b): Follows from a) and the definition of tile bivariant analytic X-complex. a): From the definition of the Lie derivative 2n
=
J'(a~
a
0 2n+l
s176
a2n+l)) = E
oJ~do(a~ .... ' 5 a i ' ' ' " a2"+l)
0
the estimate
II s
K K IN,m <-- 2M II a IIN,m+l
follows readily for a E X . ( R A K ) . Especially 5 acts as bounded derivation oll RA(K,N). The Cartan homotopy operator is given by the composition (see 4.3)
X . ( R A ) z.i) X . ( R R A ) ~ ~ R A -% ~ R A ~- X , _ I ( R R A ) If we take topologies into account, we have
X , i : X.(RA(K,N)) --+ ~(RA(K,SN)) by Proposition 5.11.
i5 : ~(RA(K,SN)) --~ ~(t~A(K',SN)) by what we just showed about 5 and
X , ~ : ~(RA(K, SN)) -+ X,(RA(K, SN)) by Lemma 5.9. Together this yields h~ : X,(RA) --+ X._I(T~A) which is what we wanted.
x.,) X,_IRA
88
[]
P r o p o s i t i o n 5.22: Let A, B be mfital, admissible Fr~chet algebras. Let
p : Tt(A @ 13) (nv,,nv2)~ TtA | TtB be the canonical homomorphism of topological I-adic completions induced by the projection of A | B onto its factors. There exists a hmnonlorphisnl s : T~AOTtB -+ Tt(A| which splits p: pos = Id and such that s o p is smoothly homotopic to the identity. []
Corollary 5.23: The natural maps
X,J'C(A | B) --+ X, TCA @ X,J-CB X , R ( A ) -~ X . n A are quasiisomorphisms. C o r o l l a r y 5.24: Let A, B, C, D be (not necessarily unital) admissible Fr~chet algebras. Then
HC* (C, A ~ B) ~, HC: (C, A) | HC* (C, B) HC*(A, D) 9 HC*(B, D) ~ HC*(A 9 B, D) []
89
P r o o f o f 5.22-5.24: A splitting s with the desired properties has been constructed in a purely algebraic setting in Proposition 4.5. We show that this splitting extends to topological I-adic completions as well as the homotopy connecting s o p and the identity. The two corollaries follow from this as in 4.6, 4.7. From now on the notations are those of Proposition 4.5 and its demonstration. We consider
s: R A -+ R ( A | t3)
S(Lo(a)) = s l ( a ) = 991(a)ch(1A) ( 8 t ( 1 A ) =
ch(1A))
( 0 ( 1 A ) - - ) W ( 1 A , 1 A ) k - l W ( 1 A , a)
~l(a) := Lo(a)+ E k=l and
s(w(a, a')) = sl(aa') - 81(a)sl(a')
=
sl(aa')
~l(a)sl(1A)Sl(a') =
-
= si(aa') - qot(a)sl(a') = (~l(aa') - qol(a)qol(a'))ch(1A) Let K C U C A be compact and contained in a "small" ball U C A and denote by K ' its image in A q~ B. Suppose 1A C C K for some C >_ 1. The formula for qol(a) above and the estimate (2k) _< (1 + 1) 2k = 4 k show that
ch(1A), ~l(a), qol(aa') - ~ l ( a ) ~ l ( a ' )
K' 9 R ( A q~ B)N,m
for a, a' 9 K ~162 and N > 4C. Moreover all except at most two entries of the elements above equal 1A which is an idempotent in A | B that acts as a unit on A C A @ B. Therefore Lemma 5.7. applies for ai 9 K ~ and provides the estimates
II s(own( aO. . . . ,a2n)) i KN,m =
n
=]l
~al(aO)ch(1A)1--[(qal (a2~-la2') 9
.
Kr
-- qol(a2i-i)qol(a2i))ch(lA) ]lIV,m
1 K K \n+l a2i) K 62n+1 11 ~1 (aO) lN,m (l[ ch(1A) N,m) f i II 0")~O1 ( a 2 i - l ' [N,m 1 with
II ~l(a ~ gy,~,llch(1A) liN,m, g' l l ~ , ( a ~, 1 a2i) -
K N,
< C(K,N,m )
for some constant C. Then the estimate K I[ X K NM,m ~ M - k [] X [N,m Vx ~ I k ( A | B)(K,,N) shows that
H8(~OOdn) NM,m ~ 6 6 2
- -
9O
which proves that given (K, N') E ]C(A) s : RA(K,N, ) -+ R(A @ B)(K,M, ) is continuous provided that M ' is large enough. The map s being defined naturally, it extends over the formal inductive limit
s : R,A | TiB ~ 7r
| B)
A similar reasoning shows that the algebraic homotopy F connecting s o p to the identity extends to a continuous honiomorphisnl F : T~(A @ B) -+ C([0, 1], 7~(A 9 B)) If we denote by t the coordinate flmction on [0, 1], then in fact the inlage of x E R(A | B)(K,N) can be expanded in a formal power series in t: oc
F(x, t) = x + ~ x,,t" n=l
where x . E In(A | B)(K,N) by definition of F and the assignements x -+ xn form a bounded family of continuous selfmaps of R(A 9 B)(K,N). One finds therefore for the time derivatives of F: 0k
K < II b~F(~, t)[IN,m-
E?~k
K tl t"-%,~ IIN,~
rz=l
_<
(1 + n ) -2 l1 (1 +n)k+2x n K n=l
<
K ~=I
oO K < y~(1 + n)-~c(k) l{ z IlK,re+k+2 n=l
which shows that the homotopy F is in fact smooth. []
5-4 T h e a n a l y t i c X - c o m p l e x a n d e n t i r e cyclic c o h o m o l o g y Before discussing the relation between analytic and entire cyclic cohomology, the normalization procedure of Cuntz-Quillen has to be extended to the analytic situation L e m m a 5.25: The Cuntz-Quillen projection defines a natural continuous map of complexes
PCQ : X.(RA(K,N)) --4, X.(RA(K,N)) The subcomplex ( 1 - PcQ)X.(RA(K,N))
C
X,(I~A(K,N))
is naturally contractible. []
91 Proofi
We use throughout the notations of Theorem 3.11.
Especially, the complex
X . ( R A ) is identified with the differential graded envelope ftA of A as in (3.6), (3.7) and all operators (d, b, t~) are supposed to act on ~A. The norm of a differential form is meant to be the norm of the corresponding element in X.RA(K,N). The Cuntz-Quillen projection PCQ equals the spectral projection onto the generalized 1-eigenspace of the Karoubi-operator t~: Each an C s n decomposes as an
=
E aA . (A'-I)(X~+1-1)=0
with ~(a~) = Aa~ A r 1
( ~ - 1 ) 2 ( a 1) = 0 A = 1
As
~ : RAK --+ RAK satisfies for a C RAK
II ,d(a) ]lN,m ~: ~ (J + 1) II a IlN,~ K we find
II a n
ItK
: I[ (n'~ -- 1)(8 n+1 -- 1) (a..)
..N,,~
< ( 0(n2)
'~ - ~
,,
1I an
o(~ 3) It a,~ t1~,,,,
-~ =
so that
HPcQ(a) HN,~ K ~ C [I a,~ tiN,m+3 K which proves the continuity of the Cuntz-Quillen projection. The contracting homotopies for the complex (1 - PcQ)X.(RA) on the eigenspaces of the Karoubi-operator for the eigenvalues A =~ 1 are
hx :=
Gd -n-~Gd G ( d - (1 + A)-lb)
for A = - 1 on X0 for A = - 1 on X1 forA :fi - 1
where Greens operator equals 1 Glker(~-~) = ~ A ~ 1
so that [I h - l ( a n 1)
K < IlN,m-
K 2roW [I an- 1 ]lN,m
and
[I h:~(a~) IlN,m K <- O(n2) 2raN II an:~ IIN,m K
1
92 for a l l A #
-1.
Altogether II
IIN,m ~ ~< O(n2)2mN~-~.lla,,llN,m'x K
h(an) II~,m K =ll~h,~(a~)
K --< 0(~r 2ran II a~ IIN,., K <- O(~2)O(n2)2mN ~ II a,, IIN.m so t h a t we m a y conclude
II h(a) IIg.mK< c(N,m) II a ]IN,~+5~" for a E RAK []
D e f i n i t i o n 5.26: (The notations are those of [CO2].) Let A be a Banach algebra. For ~ E C'~(A, A*) and K C A put
II ~ ILK:-- sup t ~ ( a ~ aicK
A cochain C C~V(A) (r
(r
E c~
is called locally e n t i r e if
Z
I] ~2n I[K ~1
[I ~/'2n+l IlK ~I
is an entire function of z for every compact subset K C A. The locally entire cochains form a 2Z/2-graded chain complex under Connes's differential 0 which is denoted by CC*lo~(A). []
T h e o r e m 5.27: For any Banach algebra A the normalized, locally entire Connes-bicomplex is a natural deformation retract of the cohomological analytic X-complex.
X:(A)
~) CC*,toc(A)
quis
A natural retraction is provided by the maps (I), q~ of Theorem 3.11. []
93
Proofi
Because the algebras RA(K,N) are completions of the subalgebras R A K c R A of
RA and R A = ( U R A K ), () = l i n e a r s p a n K
an element
X E X 2 ( A ) = l+--IC i m H o m . t X . ( R A ( K . N ) ) , q;) is uniquely determined by the fimctionals
X2n(a~
2n)
X(Own(a~
:=
X(O~ndo(a~
x2,~+~(~0,..., a :'~+x) :=
a~'~+~))
On the other hand, a eoehain
71 E CC:,tor
)
is by definition given by a family
~1 = (Tin) 'In E C " ( A , A ' ) The maps ~, 9 between these two complexes can be described as follows (see Theorem 3.11. and [CO2]) r
k9 :
CC~*loc(A)
--+
~]2.
~
712,~+i
-~
X:(A) (-1) ~ ~ (-1) ....
n! ~J2,~ !
712n+1
X*(d)
--+
CC*,lo~(A)
x2n
--+
( - l ) " (~'")~ PcQ(X~.)
X2n+l
-~
(-1) n ~
n!
PCQ(X2n+I)
so that
l~o~ = Id
B o O = PCQ
Let now
71 = (7],~) E CC*loc(A) Choose as basic "small" open set the open unit ball B(0, 1) in A and let K be a compact subset of B(0, 1). Then there exists for any N > 1 some C ( N ) > 0 such that
I] ?]2n ItK ~ ~ C(N, TI) N - 2 n n !
[t ?~2n+l HK ~ ~_ c(g,~l) y-(2'~+l) n!
94
Let a E X o ( R A K ) be presented as a snii1 a = EA/~c~ with entries in
{1} such that
K~ U
I'~1N-2k~ ~< II a I1~,o + Then
I(~rD(a)l = I~-~ ~ n=O
n=O
~Xz(@rt)2~(~oz~k~)l
kf~=n
?2
ko=n
n=O
E
ko=n
\ 7~ /
nX2,0 +s
< C ( N , TI)([] a
SO that (I)(7/) E X * ( A ) . The odd case is similar. To show that tI, maps the cohomological analytic X-complex to the locally entire Connes-bicomplex it, suffices in light of the preceeding proposition to prove that for It
=
Pcqx
C X~(A)
the functionals It2n(a, 0 , " . , a 2'~) := ( - 1 ) ' ~ ,
It2n+l(a
0
,a2n+l) ''
'
:=
(-1)'~
It(own(a 0 ' . . . , a2n))
(2n + 1)! It(ewndp(aO, " . a2.+,) ) n!
" '
form the components of a locally entire cochain. First of all t 0 sup IIt,,(a ,...,an)]
< no
aiEK
for every compact set K C A and every n E zW which already implies that the maps It" extend to bounded, linear functionals on A | For fixed K there exists C > 0 such that, 1 --K C
C K' C B(0,1)
for some multiplicatively closed, compact set K ' . Then 1
t
0
1 c2n+
~lit~da ,..., a"~)l _< 7.,
1
(2n)!
a0
a 2n
,~! I#(~"(~-,...,
K'
w n a~
II ~ II~,mll Q
a2n (~,...,-U)
c ))1
K
N,m
95
for some m C s
because # is continuous on RA(K,N) for all N > 1 K
<- C2"+14'~ I 1 ' N,m ( l + n ) mN-'~
K' -
so that
N C(l+n)m(~5)
-
n
1
lira,,
1
II [t2n ' I1~
-~
-
-
N
for every N _> 1. The conclusion follows.
[] C o r o l l a r y 5.28: There exists a natural map of complexes
CO: (A) --+ X: (A) from Connes's entire cyclic-bicomplex to the cohomological analytic X-complex given by the composition
c c : -~ CC5o~. 2+ x: []
5-5 T h e a n a l y t i c C h e r n c h a r a c t e r
Definition and Proposition 5.29:(Chern Character) [(302] Let A be a unital admissible Fr6chet algebra. The C h e r n C h a r a c t e r
ch: K , ( - ) -+ H C * ( r
: HC~,(-)
is defined by ch 0 :
Ko(A) = [•[e],Moo(A)]
[f] ehl : Ka(A) = [(~[u,u-I],M~o(A)]
[f]
-+
HC~(Moo(A))
-~
[f,(a~(~))] -+
HC~(M~(A))
~
[f,(ch(u))]
•
HC~(A)
•
HC~(A)
The Chern character is a natural transformation of smooth homotopy functors. The exterior product with the trace is defined in Chapter 8 and coincides with that
of [co].
96
Proof; In the definition of the Chern character care has to be taken of the fact that 9 [e], ~[u,u -1] are not equipped with topologies. Tim analytic X-complexes are still defined for arbitrary algebras however by declaring
E(A) := {(K, N)]K C A, K a finite set} Therefore [eh(e)],
[ch(u)] may be identified with classes
[ch(e)] 9 HC~162
[ch(u)] 9 HCl~(r162
The only thing to be shown is that any continuous homotopies e0 ~ el : (l][e] --+ A (resp. u0 ~" ul : q][u, u -I] -+ A) may be replaced by piecewise smooth ones. The rest follows then from the fact that bivariant analytie cohomology is a smooth homotopy functor. Tile assertion is clear for morphisms (l][u, u -I] --+ A because the invertible elmneilts of A form an open set and the case g[e] -+ A follows by applying functional calculus to a smooth path close to a contimu)us path of idempotents connecting e0 and
e i.
[]
C o r o l l a r y 5.30: There exists a natural p a i r i n g for unital adnfissible Fr6chet algebras
K, | HC~ ch|
HC~, | HC~ --+ (~ []
For an explicit description of the pairing see 6-2.
97
Chapter 6: The asymptotic X-complex Our main goal, the construction of covariant (contravariant, bivariant) chain complexes on the category of admissible Fr~chet algebras that behave functorially under linear asymptotic morphisms will be achieved in this chapter. The eohomological asymptotic X-complex of an admissible Fr~chet algebra A is the differential graded X-complex of the topological I-adic completion IRA of the algebra RA of tensors over A with coefficients in the DG-module of germs around infinity of smooth differential forms on the asymptotic parameter space T ~ . Asymptotic cocycles in the sense of Connes-Moscoviei [CM] do not yield cocycles in our sense and vice versa, but many concrete examples turn out to be asymptotic cocyeles with respect to both definitions. Asymptotic cocycles in the sense of Connes-Moscovici are globally defined (on X,(RA)) functionals with uniform growth conditions imposed only on tensors with entries belonging to a given finite set. Our cochains "live" only on virtual neighbourhoods of infinity in the parameter space and in fact their components in the corresponding (b, B)-bicomplex will not be families of globally defined, but only of densely defined multilinear functionals in general. Uniform growth conditions are imposed however on tensors with entries in a given multiplicatively closed compact set. The bivariant asymptotic X-complex is then introduced in analogy with the differential graded bivariant X-complex. The composition product of bivariant Xcomplexes carries over to the asymptotic setting. This is no formal matter anymore, because it corresponds to the Kasparov product in bivariant K-theory as we will see later. The fundamental observation is that every linear asymptotic morphism defines an even cocycle in the bivariant asymptotic X-complex whose cohomology class depends only on the continuous (!) homotopy class of the morphism. This correspondence is compatible with compositions, i.e. functorial. Consequently the (bivariant) asymptotic X-complex defines a (bi)functor on the linear asymptotic category and asymptotic cohomology is an asymptotic homotopy (bi)functor. It turns out to be a nontrivial theory as the calculation of the asymptotic cohomology of the algebra of complex numbers shows. Finally the Chern character and the natural pairing between K-theory and asymptotic eohomology is treated and explicit formulas for the values of the pairing are derived.
6-1 The asymptotic X-complex Before coming to tim definition of the asymptotic X-complexes some notations and definitions have to be fixed. First of all, the topological I-adic completion of chapter 5 has to be extended to the differential graded setting.
98
Definition 6.1: Let A be
an
admissible Fr6chet algebra and let (K, N) E/C(A, U).
a) Define f~A
:= " lim" ~: ~RA(K, N)
where the differential graded algebras ~}RA(K,N) are topologized as in 2.4, i.e. by declaring ~ B ~ (~ B | ~|
to be a topological isomorphism.
b) P u t
R(eA)K := {~-~ A,0~c@ e, kf~ C (I;} where tile entries of Of~,cof~ are of the form a~
with ai C K ~ tO {1}.
Denote the subspace of elements of degree k by R(DA)(~ ). Analogous to 5.6 there are weighted/1-seminorms II - lINK.... on R(~A)(~ ). The algebra R(~A)(K,N) is then defined by declaring
R(~A)(K,N) k to be a topological isomorphism where tile direct sum on the right hand side is given the product topology. Finally we put
T~(~A) := " lim"R([~A)(K,N) ~c
[] We still have to treat a technical point, (necessarily coming up when the bivariant asymlltotie X-complex is defined), which concerns the notion of a function with values in a formal inductive limit. If X is a topological space (a smooth manifold) and if A denotes a topological vector space, then the assignment
U --+ C(U,A)
(U -+ C~176
defines a sheaf of topological vector spaces on X and C(X, A) (resp. C~176 A)) coincides with its module of global sections. On tile contrary, for a formal inductive limit of topological vector spaces " lim, Ai" the presheaf
(*) U -+ "liml"C(U, Ai)
("limi"goo(U, Ai))
will not be a sheaf in general. Therefore we define a continuous (smooth) flmction on X with values in " lira, A~-" to be a glol)al section of the sheaf associated to the presheaf above.
99
Definition 6.2: Let X be a topological space (a smooth manifold) and let " limj Ai" be a formal inductive limit of topological vector spaces. a) Define
C(X, " limi "Ai)
:=
r(sh(U -+ " limi "C(U, Ai)))
C ~ ( X , " limi "Ai)
:=
F(sh(U -+ " limt "C~(U, Ai)))
C~176 |
" limz"Ai
:=
|
F(sh(U -+ " limt"d~
Ai)))
where F ( s h ( , ) ) denotes the space of global sections of the sheaf associated to the presheaf ,. b) If X is a locally compact space (manifold) then the module of global sections of the sheaf above can be described explicitely as
C(X," liju" Ai) = ~-vlimlim~iC(V, Ai) where V runs over all relatively compact open subsets of X ordered by inclusion. c) Similarly, if g ( X ) denotes the algebra of smooth differential forms on the manifold X equipped with the topology of 2.6., we define " lim"Ai | I
s
:= r(sh(U -+ " lira"A, | I
s
[] Sometimes we will write " limi "Ai | E(X) instead of " limi "Ai | renfind the reader that a sheafification has been carried out.
g ( X ) to
Finally let us recall tile asymptotic parameter space ~oo + = U,R+ n (see (1.1).) The punctured neighbourhoods of cc c ~ _ form a directed set by inclusion, denoted by/4.
Definition 6.3:
Coo(/g) := " lin~"Coo(U)
C(L/) := " lirnu"g(U)
Now we can construct the central objects of our study, the asymptotic X-complexes of an admissible Fr~chet algebra.
100
Definition
6.4:
Let A, B be admissible Fr6chet algebras. a) Tim cohomological a s y m p t o t i c X-complex of A is defined as X~,(~*A ) : = HornDa(X.(f~TiA),E(ld)) = m im lilHomDR aX t-(fA.K ()N ,E,)U (+ )_.p_c.+u b) The cohomology of the asymptotic X-complex of A
UC~(m) := h(X*(A)) is called the a s y m p t o t i c cyclic eohomology of A. c) The bivariant a s y m p t o t i c X-complex of the pair (A, B) is given by X ; (A, B)
:=
HOmDG ( X , (~T4A), X , (f~TgB) |
:= limlimHomDa(X,(f~RA)(K,N), X,(~7-CB) | *-~ir --+lA
E (ld))
g(U))
d) The cohomology of the bivariant asymptotic X-complex of (A, B)
HC;(A,B) := h(X*(A,B)) is called the bivariant a s y m p t o t i c cohomology of (A, B). 1"3
Tile cohomological (bivariant) asymptotic X-complex defines a contravariant flmctor (bifunctor) from the category of admissible Fr@het algebras to tile category of 2Z/2-graded chain complexes of vector spaces. Every morphism f : A -+ B of admissible Fr6chet algebras provides a cocycle f . := X.(f~Rf) C lira lira HOm~ 9
Theorem
e-K:
--+ K;'
6.5:
The c o m p o s i t i o n p r o d u c t for the bivariant, differential graded X-complex (2.6) yields a natural and associative composition product
X*(A,B) | X*(B,C) ~ X*(d,C) and consequently a natural and associative product
HC~(A,B) | HC*(B,C) -+ HC*(A,C) on cohomology. []
101
This pairing makes H C * ( A , A) into an associative algebra with unit, I d A c HC~ Under this p r o d u c t f : A --> B, g : B ~ C and ~ ~_ H C ~ ( B , D ) , ~ E H C * ( E , B ) satisfy
f . | g , = .q. o f . = (go f ) ,
.f. | (p = ~o o .f. =: f *qo ~b|
= g. otb =: g.~b
T h e o r e m 6.6: There is a similar n a t u r a l composition product X~(A,B) |
X ~ ( B ) -+ X(:(A)
inducing a n a t u r a l product
HC;(A,B)
| H G ( B ) -+ H C ~ ( A )
on cohomology.
[] This p r o d u c t makes X ~ ( A ) into a module under X~,(A, A). Under this product again we p u t f,|
= q o o f , =: f * ~
for f : A -~ B and ~o E HC~, (B, D).
Definition 6.7: A cohomology class x E H C * ~ ( A , B) is called a s y m p t o t i c a l l y
(analytically)
HC-invertible or an asymptotic (analytic) HC-equivalence if there exists y E HC,~,~(B, A) such t h a t yoz
= I d A E H C ~ 1 6 2 A)
xoy
= IdB. C HC~
t3) []
102
Proof of Theorem
F i x (I) E
6.5;
X~(A, B),q2 E X~(B, C). Let (I) be represented by functionals (O(K,N) E HomDcJor
X,(~RB) |
E(U))
for some p u n c t u r e d neighbourhoods U(K, N) of oc E "R~ Let U f ...... I,, : = U' C m U / C U be a flmdamental punctured neighbourhood of oc E 7"/+ as described in (1.1). By Lelnma 1.2 the sets
--
u,~ := v ' n ~,7)([0,,~}) are relatively compact in U, so t h a t the restrictions of (I)(K,N) to U,~ are of the form
q)}'K,N) := (Sd | ~b,, ,v,) o q)(/~,N)
c Uomna(X.(ftRA(r4,N)), X.(VtRB(K,,N,)) | for some
1
I
(K,., N~) E
)~1
g(Un))
as follows from 6.2.b).
Choose now representatives
~(K;,N: ) e Hom*Dc(X.(f~R[3(K:,,N: )),X.(ftTgC) |
g(Vn))
for suitable fundamental neighbourhoods t
o f 0(3.
C o n s t r u c t an open set W(K.N ) C ,~m+m' ''-tas follows: Define inductively pieeewise continuous, strictly monotone increasing flmctions k l , . . . , kin, on ~ + by !
kll[n-l,n[ =
s u p g:~ n t ~V/,
k2l[k,(~-l),k,(~)[ = sup g2 nl~n
i
kmt][kmtl .....
k l (7/,-- l ) , k , n /
1.....
kl(n)[
sup
g m'~ t
rd<(n
and choose h i , . . . , hm' continuous, strictly monotone increasing, convex and satisf y i n g h j >_kj ; l <_j <_m'. P u t .T2m+m r
W ( K , N ) :~- U f ...... .f.,,hl ..... h~, C ,~+
We claim t h a t
tp o '~(K,N) C Hom*Da(X.(~RA(~:,N)),X.(ftZCC) |
E(W(K,N)))
103
Choose x = (Xl, . .., x,~, x~,. . . , x m) ', i4" of x in W satisfying
r W and a relatively compact neighbourhood
7rm(W') C [0, [x.~ + 1]] =: [0, M] Then 7r(W') C V CI 7rml([0, M]) = U M
.'(w') c v~
are relatively compact. So W t C UM X VM
and on this set .* (Id | *w',u^l XVM)(~ o (~(K,N))
= kO(K~,N~4 ) o
e HOm*DG(X,(f~RfiI(K,N)),-X,(anC) |
M Cf~(K,N )
$(UM • VM))
as
~
N) ~ HomDv (X, (~/r
X, (~RB(K~,,N;,)) |
E (UM))
and
q(K'~,N'~,) E HOm*Do (X, (~2RB(K,M,N;,)), X , ( a n C ) |
s
Restriction to possibly smaller open sets shows that the construction is independent of all choices. Passing to the limit over (K, N) tile conclusion follows. The demonstration of Theorem 6.6 is similar. []
6-2 Comparison with other cyclic theories Proposition 6.8: a) There exist natural transformations of (bi)fimctors
H e : ( - ) ~ HC*(-)
H C : ( - , - ) --+ HC~(-,-)
compatible with composition products. b) Moreover, there is a natural equivalence of functors
HC~(-,(~) ~ HC~(-) Proof:
a) The complex X* (A) = lim~_~: Hom(X, (RA(K,N)), (~) can naturally be identified with the subcomplex of X*(A) = lim~_~: lim__,u HOrrtDG(X,(~RA(K,N)),E(U)) of functionals which are constant on U (and vanish in positive degrees).
104
For the bivariant complexes recall that there is a natural projection
Hom*Dc(X.(f~RA),X.(~RB)) -+ Hom*(X.(RA),X.(RB)) which gives a diagram of maps of complexes
X*(A, B) +- lim lim HomDv(X.(~RA(K,N)),-R.(f~RB(K, N,))) -+ X~,(A, B) ,,--]C _+~t Recall that the projections
Hom(X.RA(K,N), X ,
Rh(K,,N,))
+-
HomDc(X. (f~RA(K,N)), X. (f~Rh(K,,N,)))
are quasiisolnorphisms because they split naturally as a map of vector spaces and there exists a natural contracting homotopy oll their kernel. By naturality these sections and holnotopies fit together so that the projection
X*(A, B) +--lira lim Homoc,(X.(nRA(K,N)), X.(~RB(K,,N,))) +--IC --+K.' also splits linearly and has a contractible kernel and is therefore a quasiisomorphism, too. The map lim lira
.,--IV. --~ lC '
HomDa (X. (~RA(K.N)), -X. (12RB(K,,N,))) -+ X* (A, B)
identifies liln~_~: l i m ~ : , HomDc (X. ( ~ R A ( K , N ) ) , X. (~RB(K,,N,))) with the functionals in X*(A, B) that are constant on U. The induced maps on cohomology provide tile desired transformation.
b) Recall (4.10) that the underlying map of vector spaces of the projection
Hom*Dc(X.(~RA), X.(12RB) | V.) ~ gom*Dc(X.(~Rd ), X.(RB) | V.) splits naturally and that the kernel of this map of complexes is naturally contractible so that the map
X* (A, r --+lira lira HOmDG(X. (f~RA(K,N)), -X. (Tt~) | ,,- ]C -+ l d
E(U))
becomes a quasiisomorphism. The projection (5.17)
x,~:-Z,(nff;) contracting X.($r the complexes
-~ X,(ff:)
to X . ~ _~ r provides then a homotopy equivalence between
lira lim HomDa (X. ~-K:-~L/
(~nS, r~:N~),X* (n~) | " '
"
E(U))
and lira lira HomDa(X. (~RArK NO, X . ~ |
F(U))
105 = lira lim HO~DG( X , (~RA(K N)), ~'(U)) +--/c-+/4
[] The next result shows that asymptotic cohomology is a nontrivial theory.
T h e o r e m 6.9: Analytic and asymptotic homology coincide, i.e. the transformation of Proposition 6.8 defines a natural equivalence
HC*(q~,-) -~ HC~(q2,-) Corollary 6.10: 9
HCT~(q;'q3) ~-
{~
*=0
0 ,=1
[]
P r o o f of Theorem 6.9: We divide the demonstration into several steps. 1) Recall that ~ is a deformation retract of the Ind-Fr~chet algebra ~ via the holnomorphisms ~ : n ~ --+ ~, s : ~ ~ 7 ~ (see 5.17). The argument in (5.17.c)) extends to the differential graded setting and shows that
X,(~n~) x.% -R,(~) is a deformation retraction with section
Consequently the number operator N acts trivially on the cohomology of X,(12~) because the Cartan homotopy formula for the action of N on X , ( ~ t T ~ ) (see 4.3.) carries over to a homotopy formula for the action of N on X , ( ~ ) . 2) According to 1) and Theorem 4.10. the maps X~(q~, A) =
gom*Da(-X,(gWt~), X,(~7"i,4) |
-+ gom*Da(X, (fl7r -oX, n=
X. (hA) |
E(l~)) --+
8(lg)) -ox. ~=
Hom*DG(X,(fl~),-R,(T~A) |
s
=
= lira gom*Da(X, (~r ~ , (7~A) | 1 6 2~'(U)) -+/4 are quasiisomorphisms. Here we may suppose that the open subsets U of 7 ~ involved in the inductive limit are convex (Lemma 1,2.).
106
3) Fix such a punctured, convex neighbourhood U of oo and choose a point x0 c U. Let Y be the vector field on U generating the linear contraction [0,1] • u
~
(t,x)
-+
U
(1-t)x+tx0
Then iy is a derivation of degree -1 on ~'(U). Theorem 4.11. can now be applied to conclude that the action of s on tile complex Hom~v (X, (f~),-X, (TiA)@~hg ( V ) ) is nullhomotopic. No difficulties due to the sheafification arise because the total orbit of a compact neighbourhood of a point in U under the contracting homotopy remains a compact subset of U. By integrating this action we learn that the constant selfmap of U with image x0 induces a chain map homotopic to the identity on the complex above. Consequently
HOm.DG(X,(~t~),X,(Tt~) | (id|
)o-
E(U))
Hornbc(X.(~r ), --X.(nA) |
= HomDc(X.(~r
= Hom*(X.(r
(~d| C(x0)) =
= X~.(A)
is a quasiisomorphism. 4) As the inductive limit is an exact flmctor on the category of directed systems of vector spaces the quasiisomorphisms
X:(A) (qis Hom*(X,((F),X,(I~A)) qi~ Hom~c(-R,(~),X,(Tt~)|
E(u))
of 2) yield a quasiisomorphism
X:(~,A)
q.is) X~.(A) = limX.~(A ) --+g(
q~ limSom*DG(-X,(~),X,(n~ ) | q% UOm*DG(X,(nnr
~,(nn2) |
E(U)) q%
E(u)) = x*(r A)
under taking the inductive limit over the ordered set of convex, punctured neighbourhoods of co C T~+~ which coincides with the transformation of Theorem 6.8.
[]
Remark: The contracting homotopies off'(U) do not fit together for different U (one cannot find a common basepoint for all U) and do not provide thus a global homotopy formula for the whole asymptotic X-complex. Therefore the exactness of Iim_~ is essential in the demonstration above. As the projective limit functor lim~ is very far from being exact, the asymptotic cohomology groups can in fact be quite different from the analytic ones.
[]
107
6-3 Functorial properties of the asymptotic X-complex Now we will see that the asymptotic X-complex achieves the goal set out in the beginning: it extends the known cyclic theories and is flmctorial under asymptotic morphisms. This may be viewed as the central result of the paper.
T h e o r e m 6.11: Let A, B be admissible Fr6chet algebras. Then any asymptotic morphism
f:A--+ B defines an even cocycle in the bivariant, asymptotic X-complex of (A, B):
f. 9 X~ Algebraically f . is given by the DG-map of complexes (see Chapters 2 and 4)
--+ X . ( ~ ( R / } ) 6 ~ f ~ C ~ ( 7 ~ ) ) -+ ~ . ( ~ ( R / } ) ~ E ( n ~ ) )
r
This correspondence makes the cohomological (bivariant) X-complex into a contravariant (bivariant) functor on the linear, asymptotic category of admissible Fr6chet algebras.
Proof: Let f : A ~ C~(Tir~, B)fq Cb(Kl'~, B) be an asymptotic morphism from A to B. Fix a "small" open neighbourhood U C B of zero. Let (K, N) 9 K(A, U) and choose punctured open neighbourhoods V(K,N ) Of OO 9 ~,~ for any (K, N) 9 /(:(A) such that
{8Nwy(a,a')(x)ja, a ' 9 1 4 9
V(K,N)} C V
We claim that
X.(f~Rf) 9 Hom~ Let W
C
|
C(V(K,N)))
Y(g,g ) be a relatively compact, open subset. Then f provides a map
where C ~ ( W ) is the admissible Fr4chet algebra of smooth functions with bounded derivatives on W. Moreover
8NwI(K ~ • g ~) C {h 9 C~(W) |
9 U, Vx 9 W}
108
which happens to be a "small" open set in C~(W) | then that f induces a continuous morphism
B. Proposition 5.12 implies
R~(K,N ) Rf) j-.~(C~z(W) ~)~r B) By Proposition 8.10. the canonical morphisin of topological I-adic completions
is bounded, too. Consequently the linear maps
are continuous and natural so that they provide an element
X.(f~Rf) 9 lira Hom~
|
+--W
= Hom~
(~Rft(K,N)), X . (127~h) |
E(W))
g'(I/~K,N)))
which yields the claim. The naturality and compatibility of the construction with the composition product is obvious. []
6-4 H o m o t o p y p r o p e r t i e s of t h e a s y m p t o t i c X - c o m p l e x T h e o r e m 6.12:
Let
6:A-+ A be a bounded derivation on the admissible Fr~ehet algebra A. Then the Cartan homotopy operators of Theorem 4.10 carry over to the asymptotic setting and provide operators
1 Ca E XO~(A,A) h,~ E X~,(A,A)
that satisfy 0h~ = s and consequently the Caftan homotopy formula
s
= Ox. o ha + ha o Ox.
holds. []
C o r o l l a r y 6.13:
Bounded derivations act trivially on (bivariant) asymptotic cohomology. []
109 Proofi
In light of Theorems 4.10, 5.21 it will be sufficient to show the continuity of ~ X.(Ti(~A))
X.(j) : X,(~A) X,(k): X,(~(~M)
~ X.(~TiA)
So the theorem follows from the Lemma
6.14:
The morphisms j :f~RA(K,N) ~ R(f~A)(K,N))
k R(f~A)(K,N)) -~ f~RA(K,N)
are continuous. They give therefore rise to morphisms j : aTCA --+ Ti(aA)
k : 7~(aA) --9 fl'/ZA
of differential graded topologcal I-adic completions. Proof:
a) Let Ow"~ . . . . , gw TM C R A K . Then
;(o~,oo(o~,~)...o(Q~n~)) =
~
~,~,~o~,~,~...o,~,~
_<(2n~ +l)...(2nk + 1) terms
with entries in K ~ U O ( K ~ ) . Therefore IIu,m
<-
~
II ~ ' ~ " ~
'~' ..- o'~ ' ~ It~,~ ~
_<(2nl+l)...(2nk +1) terms
-<
~
(2m+b ~ l l ~ / J n ~
K
IIN,m+a
'"
, ' ~ [IN,m+i K
II 0 ~
_<(2hi +1)... (2nk -I-1) terms
_< (2hi + 1 ) . . . (2nk + 1)(2m+l)k(1 + n 0 ) m + l N - ~ ~ ... (1 + n k ) ' ~ + l N -"~ _< (2m+2)k(1 -t- ~'~o)m+2N - n ~ . . . (1 + n k ) m + 2 N - n k
so that the estimate II J(ew"~
99 .O(oJ~k))
K --< Ilu,m
(2m+2)k II a~
follows for a ~ . . . , a k E R A K which proves the continuity of (k) j : ~kRA(K,N) --+ R(~A)(K,N)
" ""
oak
K IlU,m+~
110
j,b ~
b) Let a ~
C K ~ . Then k(w(a~
= o(a~
. . . Oa j, b~
. . . ObJ') )
OQ(aJ-1)Ow(a j, b~
OQ(t~') +
j-1
+ E(-1)i+Jg(a~
Ow(a ~, a~+~)... Oo(aJ)Og(b~
Op(b j' ) +
1 + ( - - 1 ) J w ( a O, a l )Oao(a2) . . . o p ( a j ) O p ( b O ) . . . oO(b j ' )
as an easy calculation shows. Therefore
[tk(w( a~
. ..OaJ, . . bOOb . 1
.ObJ'))[iN,m < ( j + l ) 2 m N - 1
To proceed further note that the inequality
K II ~b I1~ . . . . < 2 ''+~ II a I/N,m+lll b IlN,m+l o n R A ( K , N ) implies
]1 (c ~
,Oc~)(c'~Oc '~ " Oc'J) ]IN,~
C ( i , llt) l[ cOOcI''" OCi HKN,m+I]I ctOOctl "'" OCtj ]]N,rn+lK
on ~ R A ( K , N ) . Let now
~C~n = Ow'~'~z'w"2w'...w'w nt C R ( ~ A K ) (k) where we suppose that w '~ E R A K and at least one entry of the terms w' has strictly positive degree. Clearly I < k + 1. Then
II k ( ~ " )
K
_ l
K
K
0 ~'~ IlN,m+2k < C'(k, ,,~)II o~ '~ [IN,~o.+2k I t II k(~') I[~,m+2kll ~ 2 l
<_ C'(k, rn)(1 + nt)m+2kN -'~ 1-[(k + 1)2m+2k(1 + rti)m+2kN -hi 2 l
<_ C"(k, rn) H ( 1 + ni)m+2kN - ~ 1 < C t ' ( ~ , ~'~)(1 + ~ ) ( k + l ) ( m + 2 k ) x - ( n - k )
<_ C'"(k, N, m)(1 + n)(k+l)(m+2k)N -n so that
[] k(a) ]]N,m+2k K K < C'"(k, N, rn) [J a IIN,(k+l)(m+2k)
for a E R ( F t A ) ~ ) []
lli As a consequence of the Cartan homotopy formula, one obtains the homotopy invariance of asymptotic cohomology groups. T h e o r e m 6.15:
Asymptotic (bivariant) cyclic cohomology is a bifunctor on the linear, asymptotic homotopy category. In other words, if
f,g:A
-4 B
are continuously homotopic, smooth, asymptotic inorphisms between admissible Fr~chet algebras A, B, then [f.] = [g.]
9 HC~ []
This result is in sharp contrast to what is known for other cyclic theories because in the asymptotic case even continuous and not only smooth homotopies are allowed. Proof:
If f and g are homotopic there is a factorization A ~
B[0,1] : ~ B il
where F is a smooth asymptotic morphism such that f = i0 o F, g --= il o F and i0,1 is given by the evaluation at the endpoints of the unit interval. Here B[0, 1] is adnfissible if B is. So it has to be shown that [/0.]--[il.]
9 HC~
1],B)
The inclusion C~176 1], B) ~ B[0, 1] being an asymptotic HC-equivalenee by the derivation lemma (see 7.7.) it suffices to prove [i0.1 = [i~.] e HC~176
1],B),B)
From now on the arguments are valid for the analytic X-complex as well and provide a demonstration of Theorem 5.21. Tile time derivative o acts as bounded derivation on C ~ ([0, 1], B) and yields an action/2~, on X.(T/C~~ 1], B)). Tile evaluation maps define a h o m o m o r p h i s m
eval : R(C~
1],B)) -4 C~([O, 1],RB)
which extends to a continuous homomorphism
eval : 7~(C~([0,1],B)) -4 n B |
1])
as the latter m a p can be written as composition n(C~176
B)) = R(C~176174
B) - ~
112 ~+ n B |
TCC~
1]) id|
7~B |
C~176 1])
which is continuous by Proposition 8.10. We obtain thus a commutative diagram
cs
T/(C~ ([0, 1], B))
> n(C~([O, 1],B))
~ eval
eval ~ 7EB | Recall that the action of s formula
C~176 1])
Z:id|
> ~B |
oil X,(7~(C~
C~
1])
1], B))) satisfies a Cartan homotopy
s 0"7 = hOx. + Ox.h
for some operator h e X~(C~([0, 1], B), C~162 1], B)) Define now X E X)(C~
1], B), B) as the composition
x: x.(n(c~([o,1],B))) & x. ~(~(c~([0,1],B))) x.evo~ X. eval) X._I([email protected]([0,1}) ) ~ X._I(T~B)| id| 2 -dr
X,_I(ZCB)
(For the definition of r see 2.7.) Then OX = X o O x . + O x . oX =
/o 1o f =
f0
r o eval o s 37o dt =
(~boevalo(hoO+Ooh))dt
I'
O 0 s174
0 eval dt
(g, o eval) dt = (r o eval)l - (r o eval)o = il. - io.
= .1o
The asymptotic case is similar. []
Corollary 6.16: Let A, B, C, D be admissible Fr6chet algebras. Then HC*(A, C) @ HC*(B, C) -% H C ~ ( A @ B, C) HC~(D, A @ B) -% HC~(D, A) @ HC*(D, B)
113
Proof." This follows from Proposition 5.22, the Cartan homotopy formula 6.12 and the fact that if U C T ~ , V C 7~_~ are neighbourhoods of oc then so is U • V c Tpn+m ,~+ (1.1). []
6-5 Pairing of asymptotic cohomology with K-theory Definition and Proposition 6.17:(Chern Character) [CO2],[CQ] Let A be a unital admissible Fr~chet algebra. The Chern Character
ch: K , ( - ) ~ HC*((~,-) = HC~,(-) is defined by
cho:
Ko(A)=[qJ[e],M~(A)]~ If]
ch,l:
KI(A)=[r [f]
~
0 HC~(IIJ, M~(A))
-~
[f,(ch(e))]
~
HC~(~,Moo(A))
-~
[f,(ch(u))]
xTr
) HC~
xT'~
HCI(q~,A )
The Chern character is a natural transformation of asymptotic homotopy functors. This means that if f : A ~ B is an asymptotic morphism, the diagram
K,A
/* ~
K,B
ch i HC*(r
1 ch I.
) HCs
A)
coInnnltes and depends only on the homotopy class of f.
Proof: The proof is similar to that of Proposition 5.29.
[]
114
C o r o l l a r y 6.18: There exists a n a t u r a l pairing
( , - ) : K. | HC(*~ ,:h|
HC~. | I:tC(*~ --4 r
of functors on the linear a s y m p t o t i c homotopy category. Consequently, for x E K,(A), ~ E HC~(B) a n d any a s y m p t o t i c mort)hism f : A -+ B, the equality ( c h ( f , ( x ) ) , : } = (d~(x),f*(w)) holds. Moreover the two quantities in the equality depend only on the holnotopy class of f .
[] The value of the pairing can be calculated by an explicit formula. In fact not a single but a whole class of formulas will be derived. This additional flexibility will turn out to be useful in later chapters, for example when we check the c o m p a t i b i l i t y of the Chern character with products. Theorem
6.19:[CO2],[CQ] (Explicit f o r m u l a for the Chern character)
Let A be a unital admissit)le Fr~chet algebra. a) Let e = e 2 be an i d e m p o t e n t in A. Suppose that
a E lim RA(K,N) -+K lifts e:
~(~)
=
e
T h e n for any ~a E Z~
(ch([e]), [~]) = ~a(a) +
~ ( ( a - 5) (a - a2) k) k=i
whe,'e the sum converges to a constant flmction oil any sufficiently small neighbourhood of oc E 7~_~. b) Let u : uu -1 = u - l u = 1 be invertible in A. Suppose t h a t
v,w E limRA(K,N) lift u, u-l: T h e n for any r E Z~(A)
k=O
115
where the sum converges to a constant fimction on any sufficiently small neighbourhood of oc E 7 ~ . c) The choices a = 0(e), v =
O(u) w = O(u -1) yield the well known formulas
<~h@]), [~]> = ~0(~) +
~k(~-
~,~ .... ,~.)
k=l oc
: Zl/Qk+l(tt-l'?z''''"t/'
-1,1/,)
k=O
where the sum converges to a constant flmction on any neighbourhood U of oc E 7r ~ such that
(fl E HOm~ r E HomaDG(X,(~RA({,.,u ,},t+~)),g(U)) Remark: For a given "parameter" t E ~ , the pairing with a fixed asymptotic cocycle will in general not be defined on the whole K-group of A but only for finitely many classes. It will even not be defined for all idempotent (invertible) matrices representing a given class in K-theory. (Consider for example the asymptotic cocycle (~t) on ]C(7/) constructed in the proof ofT.10: For any given t E ~ + - I W one can find easily rank one projections in ]C(7/) for which the sum defining the coupling with the cocycle diverges.) However the domains of definition of (~ot) grow as the asymptotic parameter t approaches infinity so that the desired coupling is eventually obtained "at infinity". This explains why the argument in the introduction showing the triviality of the pairing between K-theory and tile classical cyclic theories for stable C*-algebras does not apply to the pairing with asymptotic cohomology. Indeed we will exhibit later a class of C*-algebras for which this pairing is nondegenerate. []
Proof." c): The formulas for the Chern character of an idempotent and an invertible element are immediate from the definitions. The assertions about the domain of convergence follow from tile fact that
ch(e) ~ X0(n~({1,~t,4+~/) and ch(u) E Xl(Rr (see Lemma 5.17). The functions
-1
]({~,~-~},l+~/)
are constant on the claimed
116
We are now going to show a), the demonstration of b) being similar. Let a E
RA(K,N) lift, e E A (suppose that Ae E K for some A > 0). Then s --+ a(s) := (1 - s) 8(e) + sa E RA(K,N), s E [0, 1] is a fanlily of elements lifting e E A and ~(a(s)
-
o(s)
~) =
e -
e~ =
0
so that a(s) - a(s) 2 E "fAN RA(K,N ) V s E [0, 1]. Recall that if a n clement a of a n admissible Fr6chet algebra is close to an idempotent, i.e. if 4(a - a 2) is "small", then a true idempotent can be obtained from a by applying functional calculus with F(.)
c0 ( 2 ; ) ~ := 9 + X : (x-)(.-
,~)~ = .
+ (.-
1 G( x x 2) ~) -
k=l
1 Lemma (see L e m m a 1.20). The power series G(y) has radius of convergence R = a. 5.10 allows to apply functional calculus also to the family a(s): For any M satisfying M > sup 2 N exp(S
II
~z(s) - a(s) 2 IIN,0) ~
8
M > s u p 2 N e x p ( 8 l] O ( a ( s )
-
a(s) 2)
I 1~~ , o )
8
one has
G ( a ( s ) - a(s) 2) E RA(K,M)
~ G ( a ( s ) - a(s) 2) E RA(K,M)
and consequently F(a(s)) E RA(K,M) is a continuously differentiable one parameter family of idempotents in RA(K,M). Let now U C ~ ) be a punctured neighbonrhood of oc such that the asymptotic cocycle ~ provides a map
E Hom~
$(U))
Then, for given s, (F(a(s)), ~} is a constant flmction on U: for any vector field Y Oil U
Y(F(a(s)),~) = (F(a(s)),s
= (F(a(s)), (Ohy + hyO)Vl
by the second C a f t a n homotopy formula (4.11, 5.21) = < F ( a ( s ) ) , 0 ( h y v ) ) = = (0, hyqo> = 0 as 0 F ( a ( s ) ) is a sum of commutators (see the following argument or 2.9, 2.10). The value of the pairing is also independent of s E [0, 1] as the equality F ( a ( s ) ) = [F(a(s))F(a(s)), V(a(s))] + [F(a(s)), F(a(s))iF(a(s))]
117
and the fact that ~a is a trace on the equation
I~A(K,M) show. Evaluation at s = O, 1 gives finally
([ch(e)],[~]} = ~(a) +
~,(2~:) , ~((a-
1 (a_a2)k) ~)
k=l
Tile independence of the value of the pairing fl'om the choice of the idempotent (invertible) matrix over A has already been shown in 5.29, respectively follows from the Cartan homotopy formula applied to the class of e, u in [~[e], M~A]~, [ff~[u,u - l ] , M~A]~. []
118
Chapter 7: A s y m p t o t i c cyclic cohomology of dense subalgebras The phenomenon we are going to consider in this chapter lies at tile heart of asymptotic cyclic cohomology and accounts for most of the properties that distinguish asymptotic theory from the cyclic theories known so far. It concerns the comparison between the cohomology of a fixed algebra and of dense subalgebras of "smooth" elements: if there is a regularization procedure to approximate any element in the full algebra by "smooth" elements of the dense subalgebra, then the two algebras are asymptotically cohomology equivalent.
7-1 The Derivation Lemma T h e o r e m 7.1: (Derivation Lemma) Let i : .4 ~-+ A be an inclusion of admissible Fr~chet algebras with dense image. Suppose that the following two conditions are satisfied: 1) There exists a neighbourhood U of 0 in A such that i - l U is "small" in A. 2) There exists a smooth family
s
It : ~ + ~ of bounded linear maps such that lim l o f t
t -* oo
= IdA
lim ft, o i = I d A
t --~ c'x~
pointwise on A resp..4. Then the inclusion i induces an asymptotic HC-equivalence: [i,1 9 H C ~
[] Let us illustrate the first condition above by noting the following implication
Lemma 7.2: Suppose that condition 7.1.1) holds for the inclusion .4 C A of admissible Fr~chet algebras with dense image. Then the subalgebra .4 is closed under holomorphie functional calculus in A. The same is true for the inclusions M~(.4) C M n ( A ) n > O.
119
Proof.It is clear by Lemma 1.15 that condition 7.1.1) holds for ,4 c A iff it holds for the inclusion A c ft. obtained by adjoining units. So one may suppose that `4 c A is a unital inclusion. It suffices to prove that tile spectra of x E M in the algebras `4 and A coincide. For this one has to verify that x is invertible in `4 iff it is invertible in A. So let x -1 E A be an inverse of x. As `4 is dense in A one can find y, y~ E `4 close to x -~ in A such that xy E I + i - ~ ( U ) , y'x E I + i - ~ ( U ) . The demonstration of 1.16.1) however shows that 1 + i - I ( U ) c `4 consists of elements invertible in A. The conclusion follows. For the last assertion note that condition 7.1.1) hohts for the inclusion M,,(`4) C Mn(A) n > 0, tot), provided it holds for `4 C A. []
P r o o f o f T h e o r e m 7.1: By Lemma 1.15 the inclusion A C .4 obtained by adjoining units also satisfies the conditions of the theorem. The family of regularizations
f : ~ -~ c~(~§ ~) does not define an asymptotic morphism in general but the induced element
f. E HornDG(-X.(f~RA),X.(~2RA) |
C~
belongs nevertheless to the asymptotic X-complex:
f. E X~ Tile class [f.] E HC~ is an asymptotic HC-inverse to [i.] because f t o i and i o f t are asymptotic morphisms (the families ft o i and i o f t of continuous linear maps are bounded by the theorem of Banach-Steinhaus) smoothly homotopic to the identity and thus [f.] o [i.] = [ ( f o i ) . ] = [idA,] E HC~ [i,] o [f,] = [(io f ) , ] = [idA ] E HC~ In fact, tile assertion follows from the Lemma
7.3 :
L e t / / / b e the ordered set of punctured neighbourhoods of c~ in ~ + U {c~}. Then under the conditions of Theorem 7.1.1
Rf E limlimgom(RA(g,g),R~| +--~ -+hi
h C~176
120
Proof:
Let, (K, N) C K;(A) and choose U - ] t o , oo[C LT~+such that
{8Nw(ioS,)(a,a')la, a' E K ~ ' , t E U} is contained in a ball W in A satisfying the hypothesis 1) of Theorenl 7.1. This is possible because the family i o .ft is bounded by the theorem of BanachSteinhaus so that its curvature decays uniformly on compact sets. (Lemma 1.6) Consequently
8Nwf~(a,a') E i - I ( w ) Va, a' E K ~, t
E
U
which happens to be a "small" ball in A. As in the proof of Theorem 6.11. one obtains then that RI :
~A(K,N) -~ "R-.A| h c~176
is continuous. The claimed result follows now from the naturality of the construction. []
The name "Derivation Lemma" stems from the following observation Lemma
7.4:
Let A be a Fr6chet algebra and let {6i,i E I} be an at most countable set of unbounded derivations on A. Suppose that there is a common dense domain A of all compositions l-Ij 6ij 9Then every at most countable set of graph seminonns
I1~ IIk,s,m:=
~} JC{1 ..... k}
II (H(~is(,))a I1~ jEJ
defines the structure of a Fr~chet algebra on A, where [I - lira ranges over a set of seminorms defining the topology of A, J runs over the ordered subsets of {1, . . . , k} and f is a map from the finite set { 1 , . . . , k} to the index set I. If A happens to be admissible, then the inclusion A r 7.1.1). Especially A is admissible, too.
A satisfies condition
121
Proof: We t r e a t for simplicity the case k = l , the reasoning in the general case being similar. Therefore tile topology on ,4 is defined by the seminorms
tl a
f
IIm:--II Oa II,, + II a lira "~ 9 ~V
Let U C A be "small". We claim that U / := i-l(U) will be "small" in A. Let K C U' be compact and choose A > 1 such t h a t AK C U ~ which is possible by the compactness of K . One finds for aj c K
[I H a j I1-=11 ~ a 1 ...O(a~)...a,~ lira + II 1
i=1
fi
n
A ] - ~ H (Aal)...O(ai)...(Aa,~)lira
<-
aj lira 1
+~-'~
II H ( A a j ) I l m j=l
i=1
By hypothesis i(AK) C U has relatively compact multiplicative closure in A. Moreover O(K) C A is compact. An estimation of the sum above yields therefore
II fl aj II: -< (~,~Co+ Cx)A-~ j=l
If one treats the case k > 1 one sees t h a t the number of sumnmnds after differentiating the product k times equals n k which is of subexponential growth in n so t h a t the assertion holds then as well. []
7-2 A p p l i c a t i o n s Theorem
7.5:
Let a : ~ --+ Aut(A) be a continuous (in the topology of pointwise convergence) o n e - p a r a m e t e r group of automorphisms of the admissible Fr~chet-algebra A. Let A m :=
~z~ Ak H
k=0
be the s u b a l g e b r a of s m o o t h elements under a where A k denotes the algebra of k-fold continuously differentiable elements with respect to a. T h e n the inclusions A m ~-+ A k ~-~ A induce a s y m p t o t i c HC-equivalences for any k C JW.
122 T h e o r e m 7.6: Let M be a smooth, compact manifold without boundary and A an admissible Fr6chet algebra. Then the inclusions of admissible Fr6chet algebras
C~
r
Ck(M,A) ~-~ C~
induce asymptotic HC-equivalences. Proof: Apply the derivation lemma where you choose the following regularization maps: For Theorem 7.5 take the convolution with a smooth family of smooth, positive functions ut on ~ with support near 0 which approximate the &distribution at O:
ft : a -+
L't(s) as(a) ds 9
O0
For Theorem 7.6 take convolution with a smooth family of smooth, positive functions on M x M which approximate the &distribution along the diagonal
AcMxM []
Proposition
7.7:
Let (M, OM) be a compact manifold with boundary and let, M :-- M UOM OM x [0, oc[ be the noncompact manifold obtained by adding a collar. Let p : M --+ M be the map which equals the identity on M and projects the collar to the boundary. Define
C~(M, OM) := {f c C~(M) If is locally constant outside M} C~(M) := C~(M)/(g, glM = O) Then for any admissible Fr~chet algebra A the natural inclusions
C~(M, OM, A) ~ C~176 are asymptotic HC-equivalences.
~-+ C(M,A)
123
Proof:
Analogous to 7.6. it is easily shown t h a t the inclusions
C~176 OM, A) ~ C(M,A) +-~ C~(M,A) satisfy condition 7.1.1) and t h a t all three algebras are admissible if A is. To o b t a i n the needed regularization maps identify a neighbourhood of OM c M with M • oc, 0] so t h a t a neighbourhood of the collar M • [0, oc[ can be identified with M • ~ . Let vt E C ~. ( ] - 7', 1 7' 1 D, t E ff~+ be a smooth family of s m o o t h kernels with s u p p o r t concentrated near 0 and approaching the 5-distribution at O. Let furthermore (I)t, t _> 1 be a smooth fanfily of diffeomorphisms of t g with c o m p a c t s u p p o r t t h a t equal the translation L{ : x -+ x + 7'1 on a large c o m p a c t interval around 0 and such t h a t limt--,ec Ot = Id pointwise as operators on C ~ (tg). i _ r where r = r is smooth, For example one m a y take (I)t(x) = x + 7, vanishes oll [ - 1 , 1], equals 1 outside a large interval [ - C , C] and satisfies I ~ 1 6 2 < i on ~ . Then we define
Xt:
C(M,A) f
~
C(M,A)
~
C~(M, OM, A)
p*.f
-->
vt * (p o (I)~ )* f
--+
Oil the other h a n d
X~ : C(M,A) g
-~
C~(M,A)
-~
.~ 9 (q~_~)*g
preserves the ideal of functions vanishing Oil M C M and descends thus to a family of m a p s
X't : C(M,A) -+ C~176 It is easily shown t h a t Xt, )t~ are regularization maps for the inclusions
C~
OM, A) ~-+ C(M,A), C~176
C(M,A)
So the derivation l e m m a m a y be applied to t h e m and yields the claim. []
Proposition
7.8:
Let ,SC denote the algebra of s m o o t h functions on the closed unit interval which vanish at the endpoints. For any adnfissible Frdchet algebra A the canonical m a p
SA := S(~| induces an a s y m p t o t i c HC-equivalence.
A -+ SA
124
Proof:
This follows from the previous proposition because the tensor product algebra
S A can easily be identified with the algebra C~([0, 1], A). One only has to nmdify the regularization maps of 7.7 so as to preserve the ideals of functions vanishing on the endpoints of the unit interval. [] There is still another situation where the derivation lemma can be applied. T h e o r e m 7.9: Let A be a separable C*-algebra and let r be an (unbounded), densely defined, positive trace on A. Let ll(A, T) be the domain of r . It is a twosided ideal in A which becomes a Banach algebra under the graph norm I1Y112:=
sup
zeA,llzN<_l
Ir(YZ)l + [l y ll
Then the inclusion ll(A,T) ~
A
induces an asymptotic HC-equivalence. Proofi
It is well known that the domain of a positive trace on a C*-algebra is a twosided, dense ideal ([D]) which is complete in the norm II - II1 described above. The inequality
Ilyzlll <-IlylllllZ[IA
VyEll(A,T);zE
A
is obvious from the definitions and shows that hypothesis 1) of the derivation lemma is satisfied for the inclusion i : ll(A, r) ~ A . To find a regularization map ft : A --+ l l ( A , T ) one observes that, as A is separable, there exists a smooth one-parameter, positive, bounded approximate unit (ut)te~ c A which consists of elements of the ideal ll(A, T). The family
ft:=
A
~
/l(A,r)
Z
--~
Ut Z
yields then a regularization map with the desired properties: Clearly lim l o f t ( z )
t --4 c<:~
= z Vz E A
because (ut) is an approximate unit for A.
125 On the other hand, for y C ll(A,~ -) one finds [[ ut y -
[11 ~
y
[T(~ttyz -- YZ)t + ~ +
[[ ?ttY
-- Y
[[
for sonic
z=
~
acbj E A;IIaj II<_ t, Ifbj Il<_ 1
finite
(every element of a C*-algebra is a finite linear combination of positive elements, i.e. of squares). So
t (,,tyz - yz)l _<
\
(L(bj,,t-
I<
which tends to 0 as t tends to infinity. Therefore lim f t o i ( y ) = y r y e l~(A,r) so that i satisfies the second hypothesis of the derivation lemma, too. []
T h e o r e m 7.10:
For any C*-algebra A the inclusion A|
~
A|
K(7/)
induces an asymptotic HC-equivalence. Here the tensor products are supposed to be the projective Banach tensor product on the left and tile C*-tensor product on the right hand side respectively. Proofi
Take 7/ = 12(~W) and let M,.(r
= End(@1,..., en}) C s
Suppose that F is a densely defined, positive, unbounded operator on 7/ with F -1 EK:(7/) ] 1 F - I [I<-1 such that {ek;kC~W} forms a complete system of eigenveetors of F. If A is faithfully represented on 7/' then A | on 7/@7/
~(7/) is faithfully represented
126
L e m m a 7.11: 1) a
~ II (Id |
defines a norm on Moo(A) C A |
F)a(Id | F) I ] ~ , ~ IC(7-l) Denote the completion by .A.
2)
I,A < II r176IIA C~ ~ .J IIA| .~:(.)) tl a '~ ItA
/I a~
3) If F -1 is of trace class, one has a commutative diagram of inclusions A
~ A|
11(7/)
1 A
~A| []
Continuation of the proof of Theorem 7.10: Let (ut) C /1(7/) be a bounded approximate unit for K(7/). Suppose that (in the notations of the lemma above) F and ut commute, that Fur = utF extends to a bounded operator on 7/ and that t --+ Fut : ~ + ---+E(7/) is smooth. For example one may take
uN := PN = the orthogonal projection onto { e l , . . . , eN) and suitable convex combinations for intermediate values of t. Define Xt :
A|
K(7/) a
--+
.4
-+
(1 | ut)a(1 | ut)
1 | ut E AJ(A) |
1C
The maps Xt are obviously bounded. Because of Lemma 7.11,2) the derivation lemma may be applied to the inclusions A ~-+ A |
A ~-+ A Q c . K(7/)
This shows that two of the three inclusions in diagranl 7.11,3) induce asymptotic HC-equivalences and therefore also the third inclusion A|
~-~ A G e . 1C(7/) []
127
Chapter 8: P r o d u c t s The next basic step after having defined the various cyclic theories consists in developing operations and especially product operations for them. The first one, the exterior product corresponds to the cup product on cyclic cohomology. To define it it is necessary to construct a natural chain map
• : X , R ( A | B) -~ X , R A ~ . ~ , R B on the algebraic, resp.
• : X,n(A |
B) -4 X , T Z A ~ X , R B
on the topological and
x : Y,,~R(A |
B) -+ X , ~ R A ~ , . ~ , I 2 n B
on the differential graded level. The construction is again based on the interplay between the cyclic bicomplex (resp. periodic de Rham complex of A) and the Xcomplex of the tensor algebra R A presented in Chapter 3. It is not difficult to derive, starting from the natural homomorphism
~(A | B) -+ ~ A ~ I ) B of enveloping differential graded algebras a chain map
f~Pdn , (A|
-+ X , A ~ X , B
which induces products on cyclic cohomology of degree less or equal to one. Passing to tensor algebras one can derive a chain map
X , R ( A @ B -+ X , R A ~ X , R B which preserves I-adic filtrations and yields by taking the associated graded complex a product oll (periodic) cyclic cohomology which coincides ill degrees less or equal to one with the product we started from. Moreover the product carries over to the analytic and the asymptotic X-complex. The product thus obtained is not associative on the chain level but associative up to homotopy by an explicit chain homotopy involving only a fixed finite number of nmltilinear algebraic operations in the entries of the tensors under discussion. This allows to carry over the chain homotopy to the topological and differential graded setting. To get some information about the nature of the exterior product on cyclic cohomology obtained in this way we compare it with the product on K-theory via the Chern character. The formula finally obtained is
(ch(a • b),~ • g,) -
1 (27ri)ij(ch(a),~>(ch(b),r
128 fOr
a E K,:(A), b E Kj(B), ~g e HC~,~(A), r E HC~,~(B); i,j E {0, 1}. which shows that both products correspond to each other up to the factor 2~ri appearing when all classes involved are of odd dimension. The period factor 27ri will necessarily come up in comparing any kind of multiplicative structure on K-theory and periodic (analytic, asymptotic) cyclic cohomology for the following reason. The cyclic theories involved are a priori defined by 2Z/2~ graded chain complexes whereas any product of two classes in K1 will a priori lie in K2 and can only a posteriori be identified via Bott periodicity with a class in K0. It is the fact that Bott periodicity (a deep transcendental result) is involved which is responsible for the "period factor" 2~i. It is also not possible to get rid of the constant 2~i by changing the Chern character by introducing normalization constants: the naturality of the Chern character under asymptotic morphisms implies that the only freedom of choice one has is to multiply the global formulas for cho resp. chl by a constant. The introduction of any constant in front of cho would destroy the multiplicativity of the Chern character in even degrees whereas the only reasonable modification in odd dimensions would be to replace chl by ch~ " 2-A~chl which destroys the purely algebraic character of the definition of chl but makes the character of the flmdamental class in KI(C~(S1)) integral. This would however change the character formula for a product only to
(ch(a • b),~ • r
= (2~i)iJ(ch(a),~} {ch(b),r
which is similar to the one obtained originally. The last remaining possibility is to change the chain map • in order to make the Chern character strictly multiplicative. Experience shows however that it seems to be difficult to construct an explicit chain map
X * R A ~ X * R B -4 X*R(A | B) which has the same effect on cohomology as • if at least one factor is evendimensional but differs from • if both factors are of odd dimension. So we leave the product as it stands. From the Eilenberg-Zilber theorem in periodic cyclic cohomology it is known that
• : ,~.R(A | B) --+ X . R A ~ , ~ . R B is a quasiisomorphism so that there has to exist a chain map
X . R A O X . R B ~ X . R ( A | B) providing an inverse up to homotopy of • Such an inverse would yield an exterior product for the bivariant analytic and asymptotic cohomology theories. Here we treat however only a simple consequence, namely the existence of a "slant" product
\: K.A | HC;,.(A o. B)
129
which can easily be established directly. It is usefld for checking the injectivity of the exterior product with cyclic cohomology classes that lie in the image of the Chern character. As an application of the exterior product operation we show the stable Morita invariance of asymptotic cyclic cohomology: For any C*-algebra A the inclusion A --+ A |
K(7-/)
is an asymptotic HC-equivalence. This is in sharp contrast to the behaviour of periodic or entire cyclic cohomology as it provides nontrivial cocycles on stable C*-algebras. It should be noted that in the meantime a natural homotopy inverse to the chain map x has been constructed. It can be used to show that there is an Eilenberg-Zilber quasiisomorphism
X,R(A |
B)
qis) X,T~A @~ X , R B
and to prove the existence of an associative exterior product
HC2,,~(A , B) | HC*,~(C, D) --+ HC*,~(A |
C, 13 |
D)
on bivariant analytic, resp. asymptotic cyclic cohomology. (See [P]). 8-1 Exterior p r o d u c t s
We work at first on a purely algebraic level and begin by considering the effect of the desired chain maps on cohomology. Algebras are supposed to be unital throughout. Recall the following remark: L e m m a 8.1:
Let C,, D, be chain complexes (bounded from below) of vector spaces. Then a map of complexes q~ : C, + D, is determined, up to chain homotopy, by its effect on homology: ~ , : h(C,) ~ h(D,) . Conversely, any homomorphism from the homology of C, to the homology of D, arises in this way. Proof."
A morphism of complexes (of degree d) q~ : C, ~ D, is the same thing as a cocycle (of degree d) in the Hom-contplex Hom,(C.,D.) Moreover, two cocycles in the Horn-complex are homologuous if and only if the associated maps of chain complexes are chain homotopic. Tile assertion follows then from the fact that tile universal coeIticient spectral sequence collapses for complexes of vector spaces yielding
h,(Hom.(C., D.)) ~_ Hom(h(C.), h(D.)) []
130
Tile lemma shows immediately that there is ill general no reasonable map
X*(A) | X*(B) -+ X*(A | B) because the X-complex takes care only of the cyclic cohomology up to degree 1 (see 2.2) and the product of two classes of degree 1 ought have degree 2. However there have to exist reasonable product maps if one considers better approximations of the periodic de Rham complex than tile very crude X-complex: The homology of the complex
Horn( ~'~PdR , (A|
3 ~, ~PdR (A
| B ) , qJ)
equals
h(Hom (f~Pdn(A | B)/FaflPdR(A | B), (I;))
I HC2(A| I HCI(A | B)/kerS
whereas the homoh)gy of the tensor product of complexes X* (A)|
h(X*A|
{ HC~ = HC~
*=0
, =1 (B) equals
| HC~ | HC~(A) | HCI(B) | HCI(B) 9 HCt(A) | HV~
*=0 *----1
as Cuntz and Quillen show [CQ]. The preceding lemma yields therefore P r o p o s i t i o n 8.2:
a) There exists a unique homotopy class of morphisms of complexes
: ~P,d~(A | B)/Faf~P,~R(A | B) ~
X,A | X,B
inducing a product on the cohomology of the dual complexes that coincides up to stabilization by S with the Yoneda product [CO] of the corresponding Hochschild cohomology groups. This means that it is given by the following table where {l denotes Connes's product [CO]
HC~
| HC~
HC 1 | HC 1 HC~ | HC 1 HC 1 | HC~
So~)
H C2 HC 2 HCl/kerS HC1/kerS
b) There is a map of complexes representing the honlotopy class described in a) defined by making commutative the following diagram of 2Z/22Z-graded vector
131 spaces
~Pdn(A | B)
* X.(R(A|174 --+
Sx $ ~P. dR(A | B)/F3~P, dR(A|
~A~B $7r X,A | X,B
~
The map 9 and the isomorphism of the upper line are those of Theorem 3.11. The map X : l](A | B) ~ ~ A ~ B is the identity in degree 0 and the composition
gt(A | B) N-') ft(A | B) -~ ~ A ~ f t B where N is the number operator and v is the morphism of differential graded algebras which corresponds to the inclusion A | B -+ f t A ~ t B via the adjunction
HomAlg( A, Bo) = HornDa(~A, B) []
Proof:
The proof is lengthy but straightforward. Let us translate the preceding proposition into terms of X-complexes: C o r o l l a r y 8.3: a) By composing the canonical map X , R ( A | B) --~ ~Pdn(A | B) with the canonical projection and the map above one obtains a map of complexes
#:
X , R ( A @ B) -+ X , A ~ X , B
b) An explicit representative of the homotopy class of this map is given by # := #o @ #i
#o : XoR(A | B)
-~
XoA~XoB ~ X1A~X1B
o(a ~ | b ~
-~
a 0 | b~
o(a ~ | b~
I | b1, a 2 | b2) ~"
~tl :
X1R(A | B)
o(a ~ | b~ Owmd~
1 • b1)
--+
_ 89
~ @ b~
2 _ aOalda 2 | b~
-~
o (n > 1)
-+
XoA~XIB 9 X1A~XoB
-+ -~
t _o_1 ~(~ ~ +ala~174176 + a ~ 1 7 4 0 (m > O)
2)
t(bOb1 + bib o)
132
The same is true in the case of graded algebras and the graded periodic de Rham complex. The graded tensor product @ indicates that switching the factors in the product yields the commutative diagram
X , R ( A | B)
) X, A6X, B
x.(s.,,) t >X,B~X,A
X , R ( B | A) where the graded switch map is given by
= (-1)a~gx)a~gy)(y@x) []
Proof:
Elementary. Whereas our construction so far does not yield anything interesting for general algebras because cyclic cohomology above degree one is ignored, it already provides a product on tile chain level for the cyclic complexes of tensor algebras which are of cohomological dimension one respectively a product oil the chain level for the quasiisomorphic X-complexes of tensor algebras. Recall that the exterior product of asymptotic morphisms resp. maps is well defined and yields via the adjunction
Home(A | A'; RA | RA') OA @ cOA'
=
based linear
HomAtg(R(A | A'), RA | RA')
~
71"l
a homomorphism of algebras
m: R ( A |
I) --4 R A Q R A '
L e m m a 8.4:
There exists a natural map of complexes
p; : X . R ( A | B) ~ X , R A ~ X . R B which is defined as tile composition
X,R(A|
x.(~)~ X , R ( R ( A |
x.(nm))X,R(RA|
X, R A Q X , R B
If A, B happen to be differential graded algebras and A | B is the graded tensor product, viewed as differential graded algebra in the obvious way, then the above map of complexes becomes a DG-map. []
133
The product #~ is not associative on the level of chain complexes, but associative up to a canonical chain homotopy.
Proposition 8.5: The following two maps of chain complexes
~o : X , R ( A | 1 7 4
--+ X , R ( R ( A | 1 7 4
--+ X . R ( A |
9~ : X , R ( A | 1 7 4
--+ X , R ( A | 1 7 4
~ X.A~X.R(B@C) ~ X,A~X,[email protected]
~o = (# | Id) o # o X,R(OA|
| Idc)
~ X,A~X,[email protected]
6~21 ---- (Id @ p) o # o X , R ( I d A @ OB|
are chain homotopic. An explicit homotopy is provided by the degree one map O : X , R ( A | B | C) --+ X , A ~ X , B ~ X . C
Owk --+ 0 k r l owJdo -+ 0 j r O
Q(a~ | b~ | c~
1 | b1 | c 1)
!4 (aOdal@[bo, bl]~cOdc 1) o(a ~ | b~ | c~
88 (a2a~176176
1 | b1 | c 1,a 2 | b2 | c 2)
1 + a2a~176
~
+al a2 da~@b~ db2~c2c~dc 1 + a~al da2 @b2b~dbl@c~cl dc 2 -aOalda2~b2bOdbl~clc2dc o - ala2daO~b2bOdbl~cOcldc 2) []
134
Proof:
Lengthy but elementary. []
L e m m a 8.6:
The diagrams
X.R(A | B | C)
"-~ X.R(A | B)~X, RC
(u':l)) X.RA~X, RBOX, RC
[[ X,R(RA | RB | RC)
X.R(A| B | C)
X, RA~X, RB~X,RC
~o -~
X.RA~X.R(B | C)
(1.u')~ X.RA~X.RB~X.RC
+
II
X.R(RA | RB | RC)
---+
X.RA~X, RB~X.RC
commute. Proofi
This follows by combining the commutative diagrams
X,R(A | B | C)
X.R(Oa|174
x.n(e4|
X,R(R(A | B) | RC)
~ X. R( R(eA |
~
X,R(RA | RB | RC) X.n(~RA|
)|
)
X,R(R(RA | RB) | RC)
and
X,R(R(A | B) | RC) $ X,R(R(RA | RB) | RC)
2+
X,R(A | B)~X, RC $ 2+ X,R(RA | RB)@X, RC J..#| Id X, RA~X, RB~X, RC
obtaining thus the first diagram of the lemma. The commutativity of the second diagram is shown similarly. []
135 It is important that only multilinear algebraic operations are involved in the chain homotopy. This will enable one to carry the homotopy-associativity over to the topologized setting. In the algebraic case we obtain by taking I-adic filtrations into account the T h e o r e m 8.7: Let X~i,~R be the complex of linear fimctionals on X , R that vanish on elements of high I-adic valuation (3.12.) Then tile chain map #r provides a map of complexes
It I : X I*i n R A |^ X I*i n R B
--4 X / *m R ( A |
which is homotopy associative and induces thus an associative " e x t e r i o r p r o d u c t "
•
PHC*(A)SPHC*(B)
--4 P H C * ( A |
on its cohomology groups. Proof:
Among the maps used in defining the chain map #' (8.4) m : R ( A Q B ) --4 R A | preserves I-adic filtrations if the right hand side is given the product filtration as the commutative diagram
R ( A | B) ~l A|
m -~ R A Q R B 1 ~~ ~
A|
shows. The map # (8.3) vanishes on elements of I-adic valuation bigger than one. Taking this into account, we conclude with Lemma 5.1 (where the effect of i : R A -4 R R A is investigated) that #~ : X , R ( A | B) --4 X , R A | X , R B shifts I-adic valuations by at most 1. The conclusion follows. The homotopy-associativity is shown by a similar analysis of the chain maps of 8.5, 8.6. [] Tile exterior product carries over to the differential graded setting:
136
T h e o r e m 8.8: a) The composition of maps of complexes (see chapter 4 for the definitions)
X.(flR(A|
x:
x.j>
X.R(f~A~flB)
~~%
x.(a,,)~
X,R(t2(A| X,R(~A)@X.R(flB)
X.R(flA~B)
x.k@x.~: X.t2RA~X, flRB
induces a natural map of differential graded X-complexes
•
* A | XSG(RA)~XSG(RB) ~ XDa(R(
b) The maps
XDcRA ~ X~cR B ~ X~cR C (•
X.DGR(A|
~ X.DGRC __%X~GR(A|174
)
and
X ~ c R A ~ X ~ G R B ~ X ~ c R C (~d.• 9X,DcRA |174 A ,
C ) ~ XDGR(A|174 *
are naturally chain homotopic. []
Proof: a): is obvious from the definitions and by the multilinearity of p which turns #' automatically into a DG-map if the involved algebras are differential graded, b): we divide the proof into several steps: L e m m a 8.9:
The diagrams
X , flR(A | B | C) X,j
--+ •
.l.
X, f2R(A | B ) ~ X , t2RC .l.
X,R(f~(A| B | C))
--~
X,R(ti(A | B ) ) ~ X , RflC
X,R(UA~flB~C)
~
X,R(t2A~I2B)~X, Rf~C
X,j~X,j
137
(x,~)~
X,QR(A| B)~X, fiRC X,j~X,j
X, flRA~X,~RB~X,f~RC
$
$
X,R(fi(A | B))~X.R~C
~
X,R(~A~flB)~X.R~C
X,j
X.R~AbX.RfiB~X,R~C ~ X.R~A~X.RflB~XoR~C
(~'.1)
commute up to homotopy. There is a similar diagram showing that
X,~R( A | B | C')
--4 X,~RA~X,~R( B | C)
$
~
X.~RA~X,flRB~X,flRC
$
X,R(~A~flB~C)
~
$
X,R~A~X,R(~B~C)
~
X , R ~ A ~ X , RflB~X, RflC
commutes up to homotopy.
Proof; While the lower squares commute strictly, one finds for the upper ones:
( Z , j ~ X . j ) o x = (X,j~X,j)o(X.k~X.k)opoX,j = (X.jk~X,jk)o#oX.j ... #oX,j because X,(jk) is chain homotopic to the identity. The chain bomotopy even preserves I-adic filtrations because j and k do so (Lemma 4.9) (See also 8.16).
P r o o f of T h e o r e m 8.8,b): Combining the preceding lemmas yields the following diagrams which commute up to homotopy:
X,(nR(A | B | C)
(x,l)ox)
X,(~RA)~X,(~RB)~X, (flRC)
X,(Q | o j) 3.
~. X , j ~3
X,R(R~A@R~B@R~C)
X, RflA~X, RflB~X, RflC $ X.k ~ X. (~RA)~X. (~2RB)~X, (~2RC)
138
X. (f~R(A | B | C)
(l,x)ox>
X. (f~RA)SX. (f~RB)@X. (ftRC) $ X.j |
X.(o | o j) $ X.R(Rf~A@Rf~B@Rf~C)
X.Rf~A~X.Rf~B~X.R~C
)
$ X, k | X. (f~RA)@X. (f2RB)@X. ([~RC) Following tile diagrams one way, one obtains (x, 1)o •
resp.( (1, •
• ) because
koj = Id. Therefore the explicit chain honlotopy (9 between q50 and q~l, constructed in Proposition 8.5 yields an explicit chain homotopy between (x, 1)o x and (1, x)o x . Because the homotopy operator (9 is nmltilinear, it is compatible with gradings and derivations and provides therefore a homotopy operator
(9'-PC : X~aR(A | B | C) -+ X b c R A @X~)cRB @X~aRC[-1] [] The algebraic construction of the exterior product being achieved, topologies can be taken into account. Proposition 8.10: Let A, B be admissible Fr~chet algebras. The natural homomorphism adjoint to the product of the universal based linear maps
m: R(A|
-4 R A |
induces continuous morphisms m 6 lira
lira
Hom(R(A |
B)(K,N),RA(K,,N,) |
RB(K,,,N,,))
of Fr~chet algebras, i.e. a homomorphisms m : 7~(A |
B) -~ 7~A |
7~B
of topological I-adic completions. [] In order to prove the proposition we show first the
139
L e m m a 8.11:
Let A, B be admissible Frdchet algebras and suppose that. K is a multiplicatively closed compact subset of a "small" open ball U in A | B. Then there exist nnfltiplicatively closed compact sets K ' C U' C A, K " C U" C B such that, with
K'|
:=
{a|
6 K ' , b E K"}
the following hohts: For any N _> 1 there exists M > 0 such that the identity on R(A | B) induces a continuous nmp (see 5.6)
R(A |
B)(K,N) -+ R(A |
B)(K,|
)
Consequently . tim .
.|
. B)(K,N) ~-- t:'|lira "R(A |
B)(K,|
where on the left hand side the limit is taken over all compact subsets of U C A. Proof."
We may assume that K is a nullsequence in A | B contained in the algebraic tensor product, A | B by L e m m a 5.6. Choose increasing sequences of seminorms 1[ [[J, [[ ][~ defining the topologies of the admissible Fr~chet algebras A, B such that the open unit balls Ur II~' VII 1t5 are "small" for all j , j ' E PC. Denote by I] HJ| the projective cross norm associated to [I I1~,
II I1~ on
A|
~.
P u t /~ := Un ! K T h e n / f is a nullsequence in the algebraic tensor product A O B, too, and because we work with the projective tensor product, K may be written (after exclusion of finitely many elements) as n,j
/~:=
{TJ -- E
nj
aj |
k=0
'~J where r
nj
. I E I l a ~ . IICA(J)llb~ I I.C B ( J ) .< 2 1 l E a ~ | k=0
rA|
"t
k=0
"
"
" r
= 0
tends to cx~ with n.
Then the sets "fl'J
.
.
CU) .~ 89
A| rtj
.
K [ := {/~J := (2 II E a ~ |
.
rA|
~ 89
a
~
.ltCA(j)l j e t Y , O < k < n j }
C n
b~ . H~(j)lj e fV, O < k < nj}
C
B
are nullsequences contained in the open unit balls Ull t~, (resp. ~l I1~)" These being "snlall", it follows that K~ := mult. closure of K~ C A
140
/ f " := mult. closure of K~ C B are compact, too. For some C _> 1, the cones K ~(K") over o1K--t ( o1K~tt ) with vertex 0 will be multiplicatively closed and contained in "small" balls U c A, U ~ C B. As any element of K is in the linear span of K t @ K " there is a natural inclusion of algebras
R(A | B ) K C R(A |
B)K,|
C
R(A | B)
Let
x = E A'YOTwk~ E R(A | B)K = R(A | B)R 2/
be such that
R +~ IA, I (1 + k~) m X -~, < II ~ IIN,m 7
Now
0w,(70,...,72,~) = 0 w " ( . . . , E a ~
| b~,...)
k
= ~
II a~ II~a<J)ll~ II; (j) k 2 II ~ u = o k, | A|
k
so that K'|
7
ko,...,k2k~
1-12k~ II a~ II~(J)tl b~ F~(j) ) ( 89 2 II W"J ~k'=o atk' | b~, *U)A|
,O:k|
C2N,m
R _< Y~ IA~Ic ~ + ~ ( 1 + k~) m (c~N) -~" <_ C(ll ~ I1~,~ + ~) 3'
As there exists C' > 1 such that
R(A | B)(K,N) --+ R(A | B)(yr is continuous, the conclusion follows. [] The demonstration of Proposition 8.10. can now be achieved by L e m m a 8.12: Let K ' C A, K " C B be multiplicatively closed "small" compact subsets of admissible Fr6chet algebras A, B. Suppose that K ' | K " is "small" in A @~ B. Then the canonical map
m : R ( A @ B ) -+ R A | induces a continuous morphism
R(A @~ B)(K,|
) "-+ RA(K,,M ) |
RB(K",M)
141
for any N , if M is choosen large enough. Consequently one obtains m : " lira " R ( A | K'|
B)(K,|
-+ T~A |
T~B
Proof: Under the canonical morphism m : R ( A | B) --+ R A | R B
~(a | b)
-,
~@,) | ~(b)
w(a | b, a' | b')
-+
w(a, a') | o(bb') + o(aa') @ w(b, b') - w(a, a') @ w(b, b')
Thus
m : R(A |
B)K,|
-4 R A K , | RBK,,
algebraically and
Own(a ~ | b~. . . . . a 2'~ | b2'~) a ~ E K ' , bj 6 K " nlaps to
Z
o'o~',o',...~,,(...,~,
...) | d o J , d ~ . . . J - , ( . .
~,...)
~*l ~er?rt8
L
0~k( .... a' .... ) | 0 J ( . . . , b ' , . . .
_~(3 8 2 )" temr~s
by L e m m a 5.1, where k < n, k ~ _< n, k + k ~ > n. and the entries of a ~ (b9 belong to K ' , ( K " ) Therefore
II-4~) IINII ,~' ~92N,,,,,eli II ~.,,
~.2N,~,,
K' | K"
< c II 9 IIN,,~'+r~-
-
and the lelnma is proved. []
P r o p o s i t i o n 8.13: Let A, B be unital, admissit)le Fr~chet algebras. The chain m a p
p' : X . R ( A | B) -+ X . R A ~ X . R B extends to a continuous m a p of X-colnplexes of topological I-adic completions
~': X.n(A
|
B) -+ X . T ~ A 6 ~ - X . n B
142
Proof: Recall (8.4.) that, #' was defined as the composition
X.R(A|
x.(i)> X.R(R(A|
x.(R.~)> X.R(RA|
~+ X.(RA)@X,(RB)
The morptfism X.(i) induces a map of Ind-objects X.(i) : X.7~(A| by Proposition 5.11. objects
B) -~ X.R(T~(AQ~ B))
The universal homomorphism m yields morphisms of Indm : T~(A |
B) -~ "/-r |
T~B
by Proposition 8.10. and thus a map of complexes
X.Rrr~ : X.R(T~(A |
B)) -+ X . R ( n A |
riB)
The map
#: X . R ( A Q B ) ~ X . A @ X . B involves only multiplication and summation in A and B and vanishes on elements of I-adic valuation > 1 so that it also yields a morphism of formal inductive limits
#: X.R,,(nA |
riB) ~ X . n A Q,~ X.Tr
Composing all these maps provides finally the morphism of formal inductive limit complexes
~' : X.TC(A |
B) --+ X.TCA ~ X.TCB []
The aim of this paragraph, the construction of an exterior product for analytic and asymptotic cohomology can be achieved now. The involved algebras are not supposed to be unital anymore.
T h e o r e m 8.14: a) The map
#': X.R(A@B) -+ X . R A | induces natural chain maps of analytic X-complexes •
X*,v(A)@X*,w(B ) -+ Xr174174 • : X ~ ( A ) ~ X ~ ( B , C ) ~ X~(A|
B) B,C)
143
b) The maps
X~(A)|
(x,1)) X:(A|
~+ X * ( A | 1 7 4
and
X:tA) @X:(B) @X:(C) (x,•
X:(A) 6 X * ( B |
C) ~+ X'~tA |
B|
C)
are naturally chain homotopie. c) A similar statement holds if one of the complexes involved on the left is a bivariant one.
(t) The chain maps x define associative " e x t e r i o r p r o d u c t s "
HC*(A)@HCg(B ) ~ HC*(A|
x:
B)
x : HC*(A)@HC:(B,C) ~ ItC*(A|
B,C)
e) Naturality means that for any algebra homomorphisms f : A --+ A t, g : B --+ B ~ the square
Xr (A)|
(It)
> X2(A' @B')
,'|162~ Xr* (A)| ^
l (,| * (B)
)
X2(A|
B)
colniniltes.
Proof:
a): Follows from Proposition 8.13. b) The morpifisms in tile (tiagrams of L e m m a 8.6 extend to morphisms of the correspon(ling Ind-objects, where one has to take X,R(TIA | TIB | TiC) in the lower left corner. The maps 4)0, (Ih and the chain homotopies of Proposition 8.5 vanish on elements of high I-adic filtration and involve only a fixed finite number of additions and multiplications and extend therefore also to the corresponding formal inductive limits. []
Theorem
8.15:
a) Tile m a p
x : XDvRA|
-+
cR
of Theorem 8.8 induces c h a i n m a p s o f a s y m p t o t i c
• •
X*(A)@X*(B) ~ Xs174 X*(A)@X,~(B,C) -+ X~(A|
X-complexes
B) B,C)
144
b) The maps
X~(A)^|
* ' ^|
(x,1)>
*
X*(A|
B)~X~(C) -~- X*(A@~ B|
C)
and
X;(A) ~ X;(B) 5 x~(C) (1,x)>
X*~(A) |^ ,Y* ~(B
|
C) ~ X:,(A |
B|
V)
are chain homotopic. c) A similar statement holds if one of the complexes involved on the left is a bivariant one.
d) The chain maps x define associative " e x t e r i o r p r o d u c t s "
HC~(A)A|
x:
,
~ HC~(A * |
HC~(A) QHC*(B,C) -+ HC~(A|
x:
B) I3, C)
e) The exterior product is a natural transformation of linear asymptotic h o m o t o p y functors, i.e. if [f] E [A, A']~ [g] e [B, B']~ are asymptotic morphisms, the diagram
HC~(A )| *
I
^
HC~,(A' |
*
f'|
B')
(/|
>
HC~,(A)@HC~,(B) *
~
HC*(A |
$
B)
commutes.
Proofi a): All maps in the definition of x (8.8) are continuous: X,j by (6.14), X, Ru by definition of the topology on ~A, #' by (8.13) and X,k by (6.14). Therefore x induces maps X ; ( A ) |^
* ) --+ lira lira Hom~a(X,(f~R(A @~B)(K,N)),E(U)~E(V)) Xc,(B <..-/(: - ~ U x V
= lira lim
e-K: ---+tO x 1,~
Hom*Dc(X,(aR(A|
B)(K,N)),F(U X V))
X*(A) | X*(B, C) --+ lira lira Hom*Dc(X,(~R(A~~ B)(K N)), X , ( ~ t / Z C ) | ,'-.- ~ --,'./g x "P
if U and V are so in 7~_, 7~2 (1.1), we As U x V is a neighbourhood of o c c -pn+m ,.+ are done. b): The demonstration is similar to that of the preceding Theorem 8.14 b) with the single difference that diagram 8.9 has to be used in addition. Its maps are continuous and yield morphisms of the associated formal inductive limits as well, but it commutes only up to homotopy. The proof is thus completed by the following
145
L e m m a 8.16:
Let A be a unital admissible Fr6chet algebra and consider the natural morphisms j : f~gCA --+ TO(f/A), k : T/(f~A) --+ fW/A of differential graded algebras (see 6.14).Then jok is smoothly homotopie to the identity id~(~A). Consequently (j o k), is chain lmmotopie to the identity on X,~(f~A). Proof:
The homomorphism j o k is given by
R(f~A)
j o k:
o(a~
R(aA)
-~
n)
O(,~
. . . o(Oa ~)
Note that
j o k IRA = IdRA so that the curvature of j o k vanishes on A x A C f~A x ftA. This implies that the homotopy
R(f~A) 4 RR(f~A
RE) R(f~A)[0, 1]
induced by the linear homotopy F:=
(1-t)jok
+ tId
R(aA)-+ R(aA)[O, 1]
factorizes, when restricted to R(~2A)(k) as
R(f~A) -~ RR(flA) ~ RR([~A)/IkR([~A) RE R(f~A)[0, 1] Taking topologies into account, one knows [rom Lemma 6.14 that F extends to a continuous linear map
F : TC(f~A) --+ ~r
1]
Thus we obtain with the help of Lemma 5.11 that the composition n(ftA)(k) 2+ RT4(~A) ~ RT4(~A)/~TC(f~A) = k
= O n ( f ~ A ) | n(f~A)|
nF) U(aA)[0, 1]
0 provides a continuous homotopy F ~ connecting j o k with the identity. Moreover it is clear that the restriction of F ' on T~(f~A)(k) is a polynomial function in t of degree at most k. It follows that 7r (a) decomposes into a topological finite direct sum k
= On, 0
146
of weight spaces (k), F ' ( x , t) = tJx, t E if2}
n j := {x C 7r
As the homotopy F ' is evidently smooth on each weight space T~j we are done. Parts c) and d) of Theorem 8.15 are clear and a) foilows readily from the definition of the exterior product. Finally e) can be verified by a lengthy diagram chase. []
8-2 S t a b l e M o r i t a invariance o f a s y m p t o t i c c o h o m o l o g y P r o p o s i t i o n 8.17: For any adnlissible Fr~chet algebra the natural inclusion i : A ~ M,(A) induces both all analytic and asymptotic HC-equivalence. Proof: See tile next theorem, where the analogous "infinite dimensional" statement is treated.
The following stability result for asymptotic cohomology is in sharp contrast to the behaviour of ordinary (resp. entire) cyclic cohomology. T h e o r e m 8.18: For any C*-algebra A the natural inclusion i : A ~
A|
of C*-algebras defines an asymptotic HC-equivalence.
147
Proof: The inclusion iA factors via i'
i"
2A~ A |
A 2~ A|
K~(7-l)
where i~ is an asymptotic HC-equivalence by the derivation lemma (Theorenl 7.1, 9 is provided by 7.10). We claim that an asymptotic HC-inverse to z.A j A := (IDA. x Tr) o [i~.]-1 e H C ~
. IC(?-I),A)
In fact j A o z,A 9 = (Id A • Tr) o (iA") -1 o (i A'') o ( ~-At. ) -- (IdJ
• T,.)o (,,)
=
(IDA,
•
T r ) o (Id A |
zr "
= Id A x (Tr o i l . ) by the naturality of the exterior product =
[
I d,A x To = I d,A
d,A x ( z,/*e T r ) =
with TO C H C ~ 1 6 2 the normalized trace on ~. To show that
.A --A| ~, o jA : lCl,
~C(n)
we use Atiyah's "rotation trick": The two inclusions iAec"PC(n) : A | =A|
,
K;(7/)
K ( N ) ~- A |
: A |163
-~ A |
Is
r |
|
r ~ A |
K;(?/) --4 A |
K(N) |
K(7-l)
/s
~(1-/)
|
are canonically horn| By the naturality of the exterior product (8.14) one has jA|
o =**A |
jAec,.~c(n) o =A| ~.
~C(n) ~ jA|
"l.d , 0
jA
K~(Tt)
so that .A 0 ~,,
jA
~
~:(n)
.AGe*JC(7-t)
0 "l,
by tlomotopy invariance = idA| hy what we just proved. []
148
C o r o l l a r y 8.19:
]" r
HC~(K(~))
0
9= 0
,=1 []
The,'e are many ways to construct nontrivial cocycles on/C(7/). One possibility was described in the proof of Theorem 7.10. 8-3 C o m p a t i b i l i t y of t h e C h e r n character w i t h exterior p r o d u c t s
In this paragraph the behaviour of the pairing K.| with respect to exterior products is investigated. The final result (Theorein 8.22) is not completelyobvious because the X-complex is a priori 7Z/2-graded whereas K-theory be('onms two periodic only by the Bott periodicity theorem. So although hi(Iden in the statements, the periodicity theorem shows up in tile demonstrations. All algebras involved are assumed to be unita]. Consider frst of all a special case: the coinpatibility of the canonical isomorphism
a2: KIA ~
KoSA = Ko(SA)
with the paMng with asymptotic cohomology. (For the definition of SA see 7.8.) Recall that the isomorphism dp is constructed as follows:
Let
u E GL.,(A) and v(0) =
choose a snlooth path v : [0, 1] --+
(o 0) u- 1
As an example, one may take V := W
0
with
o
v ( 1 ) = ( lnO 1.0)
1),//,_1(10
0 '~t--1
)
(cos~t -sin~t~ w(t):=
Then the class r
and
GL2,~(A)connecting
E
KoSA can
,~t
~os~t ]
be represented by the formal difference of the
idempotents
[,,(lo,,
Oo)]
149
L e m m a 8.20:
Let ~/J C XZA C X 1 R A be a cyclic 1-cocycle oil A and let vl C X1(802)
Tl((fdg)~) = f~ fdg XoR(A | t3) -+ X1A |
be the fundamental class of the circle,. Recall the map # X1/3 (Proposition 8.3). Then for any u E GL(A) ( c h ( ~ ( u ) ) , p * ( r | 7~)) = (ch(u),r
u-~du,r
=
where (I)(u) is supposed to be given by the element constructed explicitely above. Proof:
Lengthy but straightforward. []
P r o p o s i t i o n 8.21:
Let A be a unital admissible Fr6chet algebra and let r E HCI,~(A). Let Tr • r • T1 C HC~ be the exterior product of Tr • r with the fundamental class of the circle. Then, under the isomorphism 9 : K1A ~ Ko($A)
{ ca((I)(u)),
T F x if) x T 1 > ---- { c h ( ~ t ) ,
T~" x ~) >
for all u C K1A. Proof:
Let [u] E K1A be represented by u c GLn(A) and let v:_= w ( 0
01)w_l(10
u -10 )
e :---- v (I0n
00) v-1
be as above. Then
----
= with F(x) = x +
( x - 5) k=l
150 (See Lemma 1.20) Remember that the exterior product on cyclic cohomology is induced by the map of complexes
X.R(A|
X.(nmo~) X . R ( R A | 1 7 4
(Lemma 8.4) so that
(F(o(eo)), Tr • r • 7-1} = ( (Rm o i) F(o(eo)),/L*(Tr x r | 7c* ~-1) } = ( (R(Id @ ;r) o R m o i) F(0(eo)), p*(Tr x r @ 7-1) ) by the naturality of p,
= with a := v 1 ( 1 ; ~ ~ ) v~-I Vl:= w ( L)(u)0 ~ ) w - ~ ( :
O(u-~) )0
where a exists and the sum defining F(a) converges in SRA(K,N ) provided that (K, N) E K is large enough. On the other hand, Q(u) becomes invertible in RA(K,N) for (K, N) large and the equality
:
becomes valid by Lemma 8.20.
By definition
<eh( ~(O(u) ) ), #*(Tr x g, | 7-1) } =
(10 0)v ,
0) ('0 0)
0
1 w-1
L)(u)-1
b C SRA(K,N) The projections
(Tr | Id)(a) = e = (Tr | Id)(b) being equal to the idempotent e = r representing O(u), we find by using the linear homotopy connecting a and b and the Cartan homotopy formula as in the proof of Theorem 6.19 that
=
which proves the proposition. D
151
The main result of this paragraph is Theorem
8.22:
Let A, B be unital admissible Fr6chet algebras. Let a
e K i ( A ) , b 9 K j ( B ) , ~ 9 HC~,~(A),r
Then
9 ucJ,~(B);i,j 9
1}.
1 {ch(a x b),~ x ~/,) - (2~ri)ij{ch(a),9~}(ch(b),r
Remark:
The appearance of the factor 27ri has a conceptual reason: While the product of two one-dimensional cohomology classes is autonmtically a zero-dimensional class because the X-complexes are 2Z/2-graded, the product of two classes in K1 lies a priori in K2 and can only be identified via the Bott periodicity theorem with a class in K0. The fact that the periodicity theorem (a deep transcendental result) is involved is responsible for the "period factor" 2~ri. []
Remark:
Tile theorem also justifies finally our choice of constants in the definition of the exterior product on cyclic cohomology. []
Proofi
After replacing all algebras by matrix algebras over them one may suppose all classes in K-theory to be represented by idempotent (invertible) elements rather than by idempotent (invertible) matrices. Also the trace Tr may be suppressed from the notation (This uses (Tr x ~) x (Tr x r = Tr x (~ x r which is true on the level of cochains). First case: i = j = 0 Let a = [el] 9 K0(A), b = [e2] 9 Ko(B) Then a x b = [el @ e2] and • r = (x,(~oRmoi))*(~|162
where X . (~r o R m o i) is given by
X o R ( A @ B) x:(Rmoi)> X o R ( R A | RB) x,:> Xo(RA | RB) = Xo(RA) | Xo(RB) Therefi)re
(ch(a • b), ~ x r
:
(F(~(el | e2)), (71 o R m o i)*(~ |
r
152
= ( ( 7 [ o R m o i ) F(~(e1@62)),9~ | r
=
(F((o(~l))
= ( F ( ( ~ - o / ~ m o / ) ( 0 ( e l @62))),(fl @ r
|
o(~)), ~
|
r
The sum defining F((o(el))| ) converges in RA(K,N)| K(A), (K', N ~) E /C(B) large enough. On the other hand
for (K, N) e
(6h( a ), ~ ) (ch(b), r ) = (F(Q(6~)), ~ } (r(0(e2)), r } = (F(Q(c,)) | F ( ~ ( ~ ) ) , ~ | 1 6 2 As
~A | ~ ( F ( ( 0 ( 6 1 ) )
| ~(e2))) : 61 | 6. : ~A | ~.(F(~(6~)) | F(~(62)))
we may conclude with the help of Lemma 5.10 and by applying the Cartan homotopy formula as in the proof of Theorem 6.19. Second case: i = 0, j = 1 Let a 6 Ko(A), b c K I ( B ) ~
Ko(SB). Then
by Proposition 8.21
= (ch(a • ~ ( b ) ) , ~ • (r • ~-0) by definition of 9 and the associativity of the exterior product on cohomology = (ch(~),~)(ch(a~(b)),r
• ~-~)
by the first case treated above = (ch(a),~)(6t~(b),r by Proposition 8.21 again. Third case: i = j = 1 Let a E K I ( A ) , b E KI(B). We find
(ch( a ), ~ } (ch( b ), r
= (ch((I)( a )), qz • T1} (ch(~( b )), r • rl}
by Proposition 8.21 = (ch(~(a)
• r
• ~1) • (r • ~1))
by the first case treated above
= (ch((a x b) x Bott),(~ x @) x (rl • T1)}
153
Here tile isomorphism A|174174
~_ A | 1 7 4 1 6 2
has been applied. Furthermore, the associativity of the exterior product and the identity
9 ( a ) • ~ ( b ) = (a • b) • Bott E K o ( S 2 ( A | have been used. The last expression equals
(ch(a • b),qo • ~/J>(ch(Bott),rl • rl> by the first case above so that it remains to calculate
Bott = (Tr x 7r).fl' where r~ x r~ 9 HC2(C~~ evaluate the pairing
T 1 X T 1 = (Tr X 7r)*(T I X T~)
~_ H2(T2,(g) equals the fundamental class.
To
(ch(Bott),rl x rl} =
fi 9 K~
= K0(C0(~2)) = Ko(SC~176
under a smooth degree one map f : T 2 -~ S 2. As the homological fundamental classes correspond to each other under such a map the above pairing can also be evaluated on S 2. On S 2 it is known finally that the Bott element is the image of the fundamental class [u] = [t + exp(27dt)] 9 KI(C~176 under ~P: K I ( C ~ 1 7 6
~ K0(SC~176
whereas the honmlogical fundamental class of C~1762) equals again T1 • r, when restricted to SC ~ (S 1). So we conclude by Proposition 8.21 that
= = Tl(~/,-ld?.t) --
I'
e x p ( - 27rit)dexp(27rit) = 2~ri
This terminates the proof of Theorem 8.22. []
154 8-4 Slant p r o d u c t s From the Kiinneth isomorphism in periodic cyclic homology one knows that for unital algebras A, B tile chain map
•
X , R ( A | B) -4 X, R A 6 X , RB
constructed in 8.7, which defines the exterior product, has a homotopy inverse, i.e. there exists a chain map
X.RAOX.f~B ~ X.R(A | B) that provides an inverse up to homotopy of x. (To verify this claim strictly we should first of all identify our product with Connes's product up to suitable normalization constants, which we have not done, but which seems probable in light of the compatibility of both products with the product in K-theory.) Such a chain map could be used to construct an exterior product on the bivariant X-complexes generalizing the composition product and the exterior product. As a special case we would then obtain a pairing
\ : X . R A ~ X * R ( A | B) ~ X*RB satisfying for a E XjRA, 3 E XJRA, ~/ 9 X k R B and by composing with the Chern character a pairing
\ : K , ( A ) @ H C * ( A Q B ) --4 HC*(B) While we have not constructed a homotopy inverse of the exterior product yet, the slant product with K-theory can be defined in a simple and direct manner. See also the remark at the end of the introduction of this chapter. T h e o r e m 8.23: Let A, B be admissible Fr~chet algebras and suppose that B is unital. a) There exists a natural homomorphism
\ : K,(B) ---4 HC*,c~(A,A| called the " s l a n t p r o d u c t " . b) The naturality can be expressed as follows:
B)
155
If f E [A,A'],mooth g 9 [B,B'] ..... th (resp. f 9 [A,A']a g 9 [B,B']~) are smooth (unital) homotopy classes of (asymptotic) morphisms, then, for x E K . ( B ) the cohomology classes
x\ E HC*,~(A, A |
B), (f | g). E HC~
.f. E HC~
|
B, A' |
') and g.(x)\ E HC*,~(A',A' |
B')
B')
satist~
(f | in HC~,~(A,A'|
o (x\)
(g,(x)\)
=
o
f,
i.e. the diagram "A"
x\
"A
B"
,-1 "A'"
1 g.(*)\
"A' |
>
B'"
commutes.
Proof: C o n s t r u c t i o n of t h e slant product: Even case: For unital B the group Ko(B) can be represented by formal differences of (piecewise smooth) homotopy classes of idempotent matrices over B. An i(tenlpotent matrix e E Mn (B) defines a homomrphism of algebras e,: Put
\:
A a
Ko(B)
--+
e
~
-+ A | Mn(B) ~ age HC~174 [IdA |
B)
• TrMnr o [e.]
It is clear that the map is well defined because HC*~ are smooth homotopy functors. Odd case: Let Vl E HC 1(Sift) the fundamental class of the circle and denote by (I) the isomorphism (I) : K1A --+ KoSA. Define then \ to be the composition
KI(B) ~+ KoSB ~ HC~
SA|
B)
(r~•
HC:,~(A,A|
B)
Naturality of the slant product: The naturality of the construction under continuous morphisms of admissible Fr6chet algebras is obvious. With respect to asymptotic morphisms, we consider the cases f = id and g = id separately from which the gcneral case follows. The last one poses no problems. So let g : B ~ B ' be a smooth asymptotic morphism and let e = e 2 E A be an idempotent in A. Then E
KoB'
156
can be represented by the idempotent e' := F(g(~.)) 9 B' where F is the power series of 1.20 (We suppress asymptotic parameters and suppose them always to be chosen such that the curvature of f at (e, e) is sufficiently small.) The slant product A Q ~ B ') g,([e])\ 9 HC~ is then induced by the morphism a --+ a | F(g(e)) whereas
g, o (e\) 9 HC~
Or B')
is induced by a -+ a | g(e). As these asymptotic morphisms are horn| linear homotopy the conclusion follows.
by
a
El
Theorem
8.24:
Let A, B, C be admissible Fr6chet algebras and suppose A to be unital. There exist natural bilinear pairings
K.A@HC*~(A| K.AQHC*,~(A|
B) --+ HC*,~(B) B,C) --+ HC*~(B,C)
HC~,~(B,C)~K.A -+ HC*~,(B,A|
C)
which are given by applying slant- and composition-products. Let x E gi(A), ~ C HC~,~(A), r E HCJ,~(B), (resp.r e HCJ,~(B,C)). Then the equality x\( v x r holds.
= (ch(x),7)) r
157
Proof." The existence of the pairings is clear as well as several naturality properties which we do not state explicitely but which are easily derived. It remains to check the compatibility of the slant- and the exterior product claimed in b). By definition of the exterior product the chain map x o e\ may be obtained by taking either of the two possible paths around the commutative diagram
4
$
X,n(A | r ~' $
--4
X,~(i)~,,X,~(~)
'~|
However the identity map on X-,7r
X,~(~)
-X, Ti(A e B) $ I~'
X,~(7.)6,,X,~(h)
is chain homotopic to the composition
q% x , ~ r
x.% x , r 1 6 2 1
-,
X,~(~)
-~
ch(e)
so that x o e\ may be represented as well by the chain map
X.7~(~)
--~ $ . 7 ~ ( 2 ) | 1 6 2 --~ X.n(i)~X.7~(h) ~
~ |
1
~
~ch(e)
from which the conclusion
readily follows. In odd dimensions
u\(~ x r = ~(~)\(~ • r
x ~1
by definition of the slant product = ~('~t)\~ • (~ • T1) = ~ @ h ( ~ ( u ) ) , r • 7"1} by the even case just treated
= ~(ch(~), r by Proposition 8.21. []
158
C h a p t e r 9: E x a c t s e q u e n c e s
The possibility of calculating asylnptotic cohomology groups relies essentially on its excision properties. The main drawback of the theory as it stands is that we cannot say anything about this question if we regard the category of all admissible Fr6chet algebras. It is only after suspension that we get positive "stable" results, but it remains desirable to obtain unstable excision theorems because suspension destroys the purely algebraic-analytic character of the theory. The most basic excision problem concerns the extension
O --+ S A -+ C A --+ A -+ O where a corresponding exact sequence for bivariant cohomology would provide natural HC-equivalences
f i e HC)~(A;SA)
a E HC~(SA, A)
This would imply a cohomological periodicity theorem
HC~(A, 13) ~_ HC~+I(SA, B) ~ HC*+I(A, SB) Whereas a natural candidate for the HC-equivalence a E HC~(SA, A) is provided by the exterior product with the fundamental class of the circle there seems at the moment to be no reasonable definition of fl E HC~ (A, SA). The case A = q~ is the only one where a can be shown to be an unstable HCequivalence, which proves the important fact that the coefficient groups HC* (Sr Sqd) of stable asymptotic cohomology are what they ought to be. The most straightforward approach to cohomological periodicity and the construction of fl is to derive them from Bott periodicity in K-theory. The suitable version of Bott-periodieity was developed by Connes and Higson [CH] who construct (nonlinear) asymptotic morphisms
fie : SA |
lC --4 S3A |
l~ OZE : S 3A |
lC --+ SA |
IC
which are homotopy inverse to each other and induce the Bott-periodicity map in K-theory. These morphisms do not exist unstably because it is essential that the source of an asymptotic morphism describing a Bott element is a "group up to homotopy", which is true for suspensions but not for general algebras. Using this approach it is possible to define an element
j3' E UC~
S3A)
and finally a stable Bott element
fl E HC~(SA, S2A) by taking the exterior product with the fundamental class of the circle. The construction of fl' is based on the observation that although nonlinear, the restriction of the Connes-Higson morphism fiE to the "smooth" subalgebra C~(]0, 1[, A) of
159
SA may be represented by a linear asymptotic morphism. The inclusion of this subalgebra into SA is an asymptotic HC-equivalence by the derivation lemma. The stable eohomologieal periodicity theorem
HC~(SA, B) ~ HC*+I(S2A, B)
HC*(A, SB) "~ HC(~+I(A, S2B)
follows by showing that the stable Bott element is an asymptotic HC-inverse of the Dirac element a ~ HC~(S2A, SA). The demonstration is based on Atiyah's rotation trick but rather cumbersome due to the necessity of descending to "smooth" subalgebras which are not preserved during the constructions in the proof. The general excision ttmorems in asymptotic cyclic cohomology can easily be derived from the stable periodicity theorem. A well known argument from algebraic topology showing that fibrations and cofibrations coincide in the stable homotopy category can be used to derive for each homomorphisnl
f:A--+B of admissible Fr~chet algebras two six term exact cofibration (Puppe)-sequences relating the stable asymptotic cohomology of f : A --4 B to that of the mapping cone Cf of f. If
O--+ J ~ A ~ B - + O is an extension of separable C*-algebras (i.e. if J possesses a bounded, positive, quasieentral approximate unit), then it was shown by Connes and Higson [CH] that the ideal J is stably asymptotically homotopy equivalent to the mapping cone C]. By descending to suitable "smooth" subalgebras to gain linearity of the considered asymptotic homotopy equivalences we show that the kernel J of f is stably asymptotically HC-equivalent to tim mapping cone of f if f possesses a bounded linear splitting. This provides six term exact sequences for the stable asymptotic cohomology of an extension of separable C*-algebras with bounded linear section. Finally some comment on the difficulties in constructing an unstable Bott element fl C HC~(A, SA) for arbitrary admissible Fr~chet algebras. There are several ways to obtain chain maps [3 : X , RA --+ X,+IR(SA) with the right algebraic properties: one can use either H-unitality and excision for ordinary cyclic homology following Wodzieki or can mimic the procedure of Elliott, Natsulne and Nest. They obtain a Bott element from the deformation of the crossed product of the algebra of Schwartz flmetions on the real line with the group acting by rescaling the Pontrjagin dual real line and derive Bott periodicity from Takai duality. Finally there have been attempts of Lott to define a bivariant Bott element by using the superconnection forlnalism of Quillen. In all these approaches a typical feature arises: the obtained chain map is such that the image of a tensor with entries belonging to a compact subset of A will be a tensor with entries in the nmltiplicative closure of the union of the given compact subset of A and the set of elements of a bounded, approximate unit of Sift, which does not form a compact set anymore. Therefore a theory allowing the entries
160 of tensors to belong only to compact sets is unlikely to possess a bivariant Bott element. But on the other hand it is not possible to modify the theory by allowing the entries of tensors over a nonunital algebra to belong to the union of a compact set and a bounded, approximate unit because approximate units are not preserved under homomorphisms of algebras. Some further study is necessary to find the way out of this dilemma. 9-1 T h e s t a b l e p e r i o d i c i t y t h e o r e m Construction of a stable Bott-element: The Bott class/3 generating the infinite cyclic group
Ko(S2fi2) := Ker(Ko(C(S2)) evat,> Ko((F) ) can be represented by the formal difference
Z = [~(,]- [~.1] of two idempotent matrices
1 (1~ 2 Z)
e 0 : = I+~W~ in
-
1
el:=
(~ 0) 0
M2(C($2)) -- M2(C(ff:U {oc})) which coincide at oc. Due to Connes-Higson there exists a corresponding class in E(r
S~r
:
[[Sr S3r |
~]]
the group of homotopy classes of (nonlinear) asymptotic morphisms from Sfl~ to ]~. Its image under E(ff:, $2r --+ E(r C($2)) coincides with (e0). - (el). I~ with a homomorphism of (where we have identified an idempotent in C(S ~) | to the latter algebra). We will adopt their construction to obtain a Bott class in bivariant asymptotic cohomology.
S3(I~C *
Definition 9.1: Let (ut)tE~+, 0 < ut < 1 be a continuous, bounded approximate unit for $ 2 r = C0(ffi~2). Then
-+ C(/~+, M4($3(I]))
"Bott" : S r f
sin~x
D~ :=
(cos~x)(1 - ,~)
+
~
(~o.~}~)(1
-
-sin~x
~,)
)
defines a linear, asymptotic morphism S r ~ M4 ($3r Here the third suspension coordinate corresponds to "x" and the first two coordinates correspond to $2~.
[]
161 For a proof see 10.2. L e m m a 9.2:
The diagram 2r
M4($3~ )
"Bott">
=
M4($3~ )
$i
$ eo | idsr V el | (dsr
(M~(C~(S~)) |
Sr ~
+
(M4(C~(S')) |
Sr
-~
M4(Co(~ • S=))
commutes up to homotopy. Here iTt : S g --+ S ~ denotes the canonical inversion corresponding to the degree -1 involution of the unit interval and V : g g -+ S g O S g is the splicing map (see 9.4.). Proof:
Consider the family of asymptotic morphisms
Fa:
Dt,~ :=
Sr
-+
M4(C0(~•
.f
~
f ( D t , ~ ) ( Oo
(1 - A)(cos~x)(1 - ut)
elO) -sin~x
(Dt,~ is constructed from Dt by replacing the approximate refit (ut) of $2~ by AIc(s2) + (1 - A)ut). Then Fo = i o " B o t t " and Fl : f --+
0
f(-sin~x)ei
is naturally hornotopic to e0 | idsr V el | idsr as x -+ sin-~x defines a homeomorphism of [-1, 1] which is isotopic to the identity. D e f i n i t i o n 9.3.a):
Let A be an adufissible Fr~chet algebra. Let rl E class of the unit interval. Consider the diagrams
SA
M4(sa(F) |
A
k'
~
SA M4(S3A)
"B~174 k"
<
H C 1 (,-g~)
be the fundamental
M4(Sar | M4(8r
|
A S2A
of (asymptotic) morphisms, where k, k" are asymptotic HC-equivalences by the derivation lemma (see Chapter 7). Put then f3SA := [
"9 ~ ' ] o Lr~.">' ( T r x vi) x za, ,,j o [k',] o ["Bott" |
zd ~] o [ < q - '
162 flSA C HC~(SA, S2A)
is called the " s t a b l e B o t t element".
[] If f : A --~ B is all (asymptotic) morphism the d i a g r a m
"SA"
~3s~, > ,,S2A ,,
s'* 1
Is2,.
" S B"
> " S2 B '' ~,q B
commutes as is i m m e d i a t e l y clear fiom the definitions.
Definition 9.3.b): Let A be an admissible Fr~chet algebra. Then
A -1 E HC~(SA, A) r~A := It1 x id A ] o[,,]k is called tile " D i r a c element".
[] The d i a g r a m "SA"
~A > "A"
"SB"
> "B" Ct B
c o m m u t e s as for the stable Bott element.
Theorem 9.4: (Stable periodicity theorem) Let A be an admissible Fr~ehet algebra. The stable Bott- and Dirac-elements flSA E HC~(SA, S2A)
O~SA E HC~(S2A, SA)
define a s y m p t o t i c HC-equivalences inverse to each other:
C~SA o flSA = id sA~., E HC~ flSA o OlSA = idS, 2A e HC~
SA)
S2A)
163
Proof."
First part: O:SA 0 flSA = ,d. SA E H C ~
SA)
By definition O~SA o flZA equals
[~ • ia.5~1 o [k2~] -~ o [ 1--==(T~ • ~ ) • id s~A] o r~:"l-~ o ,.~~
ldA]
[k'.] o [" Bo~" |
O
[k.] -~
First of all note that the diagram k111
>
M4(82SA) M4(S2SA)
=
k"
H
AI4(S2SA) $
-+
SA
=
M4($S~A) $
[ ( T r x rl)] x k SA
8SA
It2] x $
M4(S3A)
[(Tr
x
~'1)] x
S2A
SA
conmmtes where 1 ( T , ' x 7"5 X T1) e HC2(M4(82r
(Here we denote by S the smooth and by S the continuous suspensions.) Therefore aSA o flSA equals (*)
[T2 X idS,A] o [k"'] -1 o [k:] o ["Bott" |
Id A] o [k,] -1
The cohomology class ~-2 C HC2(M4(82r is known to be induced from a class T~ C HC2(M4(C~(S2))) which is in t~ct the flmdamental class of the two sphere. This gives rise by lemma 9.2. to the diagram
8A
G
(M2(C~(S2)) |
SA) 2
$ k I o ("Bott" | id A)
$
M4(C~($2)) |
$
M4(S3A)
-+
M 4 ( C o ( ~ x S 2, A))
~
I" M4(C~(S2)) Or SA
]~lll
M4(S2r |
SA
SA
"1-2•
which commutes up to homotopy, where
qa = (eo | id V el | id) o ]~:A From this we derive (~SA o flSA =
= IT2 x idS,A] o [k~,']-1 o [k',] o ["Bott" |
Id A] o [k,] -1
k'" k'"
164
= [7~ x idS, A] o [kZ'] -1 o [k~"] o [(eo @ i d v el @ id),] = [ ~ X i d ~ A] o [(~0 @ id V el |
i6/),]
= [4 x idS, ~] o [(~0 | ,:dsA).] + [g • irish1,~ o [(~1 O iRsA).]
(as will be verified at the end of the demonstration)
by definition of the slant product I
.- S A
= (ch(e0), 7;)[idS, A] + {ch(el), T2)[zd, ] by theorem 8.24
= l[idfA ] + ()[i~/~A] = [idfA] by the well known integrality properties of the ordinary Chern character (see [CO]). Second Part:
9, S 2 A ~ S A o O~SA = 7,a,
By definition flSA o O~SA equals [ ~x/ ( T rl x T1) ig s2A ] o [k',']-lo[k'.]o[("Bott"|
•
SA ]o[k.SA ]-1
Note that all bivariant classes involved are induced by exterior products or (asymptotic) morphisms. The naturality of the exterior product with respect to (asymptotic) morphisms implies then [k','] -1 o [k'.] o [("Bott" | = [T1 X
i d M4($qJ)|
o [Sk~'] -1
IDA),] o [k,A]-1 o IT1 • o [$ ]r,] o [8("Bott" |
A] = IDA),] o [SkA] -1
(S still denoting the smooth suspension). Inserting this into the formula for t3SA o yields
O~SA
f l S A o O:SA
= [@ g (Tr X T1) X id s~A] o ['rl • id M~(sr174 o[Sk',] o [S("Bott" |
IDA).]
"
o [Sk,A] -1 o
= [r~ x id s~A] o [$k'.'] -1 o [Sk',] o [S("Bott" |
H -1 o o [Sk.]
[k,S A ]--1
IDA),] o [SkA1-1 o [kSA] -1
165
Now there is a commutative diagram
$2 A "r
skA
-sw -~
S2 A ~2
S(SA) kSA
k' k"
k SA
S(SA)
$ S(SA) ; M4(S 3) | S A $ M4(S3$A) ? M4(SOJ) | S 2 S A
SW
t S(SA) $ S(M4(Saq~) | A) $ S(M4 (S3A)) -~ ~ 8 ( M 4 ( S r | S2A)
8 ( k A) r 8k' Sk"
where
*/, = "Bott" |
Id sA
~b' = S ( " B o t t " |
Id A)
and
sw : SeA --+ SeA S W : M4(8@) | S 2 S A N /144 (S4A)
~
S(M4(8(~) | S2A) A ---+ M4(S4A)
( 00 1)
are the isomorphisms given in suspension coordinates by the matrices
sw =
( 01 o l )
sw
01 0 00
0
0
0
1
respectively. Note that [sw.] = lid.] C H C ~
S2A), [SW.] = [id.] E HC~
S4A)
as both permutation matrices sw, S W are of positive determinant and thus connected by a continuous path to the identity matrix giving rise to a homotopy of the induced maps on suspensions connecting the switch maps to the identity. This implies the equality flSA o OISA
= [ ~ • id s~A] o [sk"] -~ o [ s < ] o
[s("Bott" v~
= [T2 x id s~A] o [Sk"] -1 o [$k'.] o [$('Bott" |
i d A ) . ] o [s
IDA).] o [Sk.A]-I o [kS.A]-1 o [sw.]
= [7~ • id s2A] o [SW,] o [k"J -1 o [k',] o [("Bott" |
IdeA).] o [kS, A] - I o [SkAJ -1
166 Consider the commutative diagram
k" i"
8k"
t $(M4($r
)
SW
|
S2A)
?
M4(,-S'2r | i'
8(M4(S3A))
M4(S38A) t M 4 ( 8 r | $28A SSA
?
M4(84A)
M4(84A)
)
SW'
)
M4(S4A)
M4(S4A)
SW~id
where all vertical maps induce asymptotic HC-equivalences by the derivation l e m m a and where [SW,] = [id,] 9 HC~ S4A) by what we said above. Note that
k" o i" = k m in tile notations of the f r s t part of the proof. From this we derive the equality [SW,] o [kt,t] -1 = [i,] o [SW:] o [it,]-I o [k:t'] -1 = [i,] o [ i ' , ] - ' o [k"'] -1 = [ ( , : d ' , ( ~ r
|
S k a ) , ] o [k'."] -1
which implies by the naturality of tile exterior product [r2 x id s2A] o [SW,] o [kt,'] -' = = [72 x idS,2A] o [(id M'(s2r
|
SkA),] o [k~,"] -'
= [ ( S k A ) , ] o [ ~ • i d ,s~A] o [k:"]-'
Inserting this into the expression for/3 o a one finds /3SA o OZSA -----
= [72 x id s~A] o [SW,] o [k~,r]- t o [k',] o [("Bott" |
= [(SkA),]
:'SSA~o o ([~ • .,,, j [k~."] -1 o [k'.] o = [(SkA),]
[("Bott" |
IdSA).] o [kS.A]-1 o [SkA] -1
IdSA),] o [k,Sn] -1) o [SkA] -1
o (~sA o ~sA) o [Sk)] -1
by the equality (*) of the first part of tile demonstration = [(sk%,]
o
lidS, sA] o [Sk,A] -~
by what we proved so far
= lidS, ~A]
167 It remains to verify the assertion about the action of i-d$A on asymptotic cohomology. It is well known that the suspension of an algebra becomes a group object in the homotopy category of algebras. The "addition up to homotopy" being given by the splicing map y(2t) 0 < t < 1 V : SA | SA --4 S A (f,.q) -+ . f V g : = g(2t-]) 89 The compositions SA
SA
(id~O)) S A | S A _(O~d) S A | S A
-Y+ S A Z+ S A
are homotopic to the identity whereas
SA
(~a~(d)
SA | SA
-~
SA
is nullhomotopic. From this we conclude
0 = [V,] o [(id 9 i3),] = [v,] o [i d,sAesA ] o [(id 9 i3),] = [v,] o ([(id 9 0),] + [(0 ~ id).]) o [(id 9 i~),l by the structure theorem 6.16 about the asymptotic cohomology of a direct sum = [V,] o [(id @ 0),] o [id,] + [V,] o [(0 @ id),] o [i~/,]
= [id,] + [~d.] which proves the claim.
[]
Proposition 9.5" HCt~ (Sr r defines an asymptotic HC-equivalence and
Tile Dirac element a r consequently H
* C~(Sr
=
{r
0
*=0 *= 1
Proofi 1) Construction of an HC-inverse/3r of ar Let eo, el E M2(C~(S~)) be the matrices of 9.2. defining the Bott class and let ~o, ~l : r ~
M2(C=(S2))
be the corresponding unital homomorphisms. Let f : S 1 • S 1 --+ S 2 be a smooth map of degree one and denote as before by T1 the cohomological fundamental class of the circle. Put
r
1 := [~*] O [ ~ / ( T r
x T1) • ia~,~(S')] o [M2(f),] o [~o,1 ,]
168
r
~
HomI(X.(T4~), X.(TgC(S1)))
where k : C~176l) -+ C(S 1) is the inclusion. X.(~) t h e y descend to chain m a p s r
9
As b o t h chain m a p s annihilate l v 9
Homl(X.(Tg~),X.(TgC(S1)))
: Horn 1(-X. (T4~), -X. (TiS-~)) = X: (r Sr As chain m a p s b o t h r t h a t one can define
and r
fie := [ r
are cocycles in the bivariant analytic X - c o m p l e x so
[r
HC)(r162
9
= HC~(r162
2) ~r o ~
= idr 9 H C ~ ( r r
By definition 0'~ 0 ; ~
= [T1 x idr o [k,r -1 o [~,] o [2~(Tr x rl) x ia~. ~(sl)] o [M2(f).] o ([e0.] - [el .]) -- [71 x idr o [ ~ i ( T r x 71) x i ~
(s )] o [M2(f).] o ([eo.] - [el.])
by definition of k r and 1
= [ ~ i ( T r x vl x T1) X idr
o
[M2(f).]
o
([e0.] - [el .])
= [g • i~.r o ([e0 . ] - [e~ ,]) in the n o t a t i o n s of the previous t h e o r e m = e0\[•
x
idr - el\[g x idr
= ((eh(eo), r;) - (eh(el), r~))idr
= idr
(see 9.4.)
3) fie
o
(~r = [id sr
E
HC~162162
By definition ~
= [k.] o = [k,.]o[
o 0~
1
[~-~i(Tr x T1) X ida. ~(s~)] o [M2(f).] o ([e0.] - [el.]) o IT1 >< idr o [k.r -1
@/
r
1
M
~ S2
(TrxT1)xid~. (s)]o[M2(f).]o[~'lxid. 2(c(
= [k.]o[~_~i(TrxT1)xi~(sl)]o[TlxidM2(c~(s
))]o([Seo.]-[$el.])o[k.]r
xS ))]o[M2(Sf).]o([Seo.]_[Sel.])o[kr
= [k.lo[~---~z(Trxr, x~-l)xid~.~(sl)]o[M2(i').]o[M2(Sf).]o([Seo.]-[Sel.])o[kr
-1
169
where
i':M2(SC~176 •
$1)) -~ M2(C~
1 • S 1 • S1))
is the inclusion. Now the diagrams Sr 1"
S~
S W o" B o t t " )
M4(Sar 4i
$
id s | eo V id s | el
k Hr
SM2(C~(S2)) 2
M 4 ( C o ( ~ • $2))
and M4 (S 3 r
S W -1
i k"
i -+
M4(Co(h~ x S2))
$ Mn(S~r $ M4(C0(S 2 • ~ ) ) j" M 4 ( C ~ ( S 2) |
S)
$
k'"
$ M4(SCec(S 1 • S1)) $
M4($f)
M~(SC~(SD)
M,(I|
M4(i')
M4(C~(S~ • S~ • S~))
commute up to homotopy as can be seen from 9.2., where S W : M4($3r --+ /l.14($3G) corresponds to the map induced by the cyclic permutation (1 2 3 ) of the suspension coordinates. S W is homotopic to the identity. As the involution (d sA induces -1 on asymptotic cohomology groups (see the proof of theorem 9.4) one may conclude that [M2(i').] o [M2(Sf),] o ([Seo,] - [Sel ,]) = = [M4(I | i'),] o [k',']-1 o [i,] o ["Bott",]
Thus / ~ o O~cll ----
= [L] o [2@/(T, • T, • T,) • ~ ( s , ) ]
= ILl o [~; •
o [M4(/oi,).]
o [k,,]_l o[i,] o [" B o t t ' , ] o [~,~] -1
i~ ~s~l] o [k"]-' o [i.] o ["~oW'.] o [k.~] -~
= It; x id c(s`)] o [<"]-~ o [<] o ["Bott",] o [k,~]-~
in the notations of the previous theorem ~- OIS~ J o ~ S r
by equation (*) of 9.4.
= [idf r by the stable periodicity theorem. []
170
9-2 P u p p e
s e q u e n c e s a n d t h e first e x c i s i o n t h e o r e m
Any asymptotic morphism f : A -4 /3 of admissible Fr6~chet algebras induces maps of complexes f.:
X,~(C, A) -4 X~,(C, t3) f* : X*(A, D) +- X~,(B, D)
Recall that the cone o f a m a p f : X. -+ Y. of c o m p l e x e s is given by the complex
Cone(f., X., Y.) :=
Oc~,,,e:=
f.
Ov,
The cohomology of the cone is then called the relative c o h o m o l o g y of the pair
f,:X,-4Y, and is related to the cohomology of X, and 1~ via a long exact sequence which in fact becomes a six term exact sequence for 23/2-graded complexes. Our aim in this chapter is to describe the relative asymptotic X-complexes
Cor~e(f., X•(C, A), X~,(C, B)) Cone(f*, X~,(A, D), X~,(B, D)) in suitable cases in a more direct manner. These results form the basis for calculations of asymptotic cohomology groups. If f : A --+ B is a morphism of algebras, the m a p p i n g c o n e by the cartesian square Cf
> CB
1 A
1 ~w~ > B
Cf of f is defined
f If f is continuous and if A, B are admissible Fr~chet algebras, then too.
CI is admissible,
171
T h e o r e m 9.6: (First Excision T h e o r e m ) Let A, B, C, D be admissible Fr~chet algebras and let
f:A-+ B be a continuous homomorphism. sequences
HCO(C, SA )
Then there are natural six term exact Puppe
sI.> HCO(C,SB )
HC~(C, SCf)
H C ~ (SCf,
HCI (C, SCI)
HC~(C, SB)
<
HCO(SA, D )
<sf* HCO(SB, D )
SI.
HC~(C, SA)
D)
gc~ (SCf, D) HC~(SB, D)
Sf"
> HC~(SA, D)
Moreover, the cone of the bivariant X-Complexes under Sf and the bivariant X-Complexes of the mapping cone of Sf are naturally quasiisomorphic:
X~(C, SCI) qis>Cone(S f,, X~(C, SA), X~(C, SB))[1] X~(SC I, D)
-+ S2Cf -+ S2A
s~I> S2B - ~ SC f - ~ SA sI> SB
and taking cohomology. The stable Bott periodicity theorem for asymptotic cohomology allows then to turn this sequence of cohomology groups into a periodic one. In order to show its exactness it suffices to prove that the cofibre sequence induces long exact sequences in asymptotic homology resp. cohomology. As a long
172
cofibre sequence consists (up to homotopyequivalences) of short cofibre sequences, it remains to show the exactness of
HC~(C, SCy) si.> HC~(C, SA)
sy.> HC~(C, SB)
and
H C ~* ( S C I , D ) <si* t t C ; ( S A , D) <sl* HC~(SB, . D) respectively. The composition of the two maps is zero ill both cases because tile composition f o i is nullhomotopic as the diagram
Cf
A
--4 C B
I
> B
shows. First case: The cohomological X-complex
HC*(SCI,D ) ~
H C t ( S A , D) <s]* H C * ( S B , D)
Let [~] E HC*(SA, D) be such that [Si*~] = 0 E H C ~ ( S C I , D ). So there exists r E X~9- 1 (SC I , D ) with 0 r = i * r is well defined up to a cocycle. It follows that
S j * r E X ; - I ( S 2 B , D) is a cocycle:
O(Sj*r
-- Sj* ( 0 r
~- Sj* Si* ~ = S(i o j)* ~o = 0
as i o j = 0. Its cohomology class is well defined up to the image of
Sj* : H C ~ - I ( S C I , D) ~ HC*-I(S2B, D) We put u := [Sj*r
* o flSB E HC~(SB,
D
)
and claim
S f * u = [~o] E HC~(SA, D) The demonstration proceeds in several steps. 1) The eohomology class of Sf*u is independent of all choices made: Any element of the indeterminacy group is killed under S f*: If [X] e H C ~ - I ( S C I , D) then
S f* ( ( S j * x ) o ~SB) = X o Sj, o flsB o S f . = X o S(j o S f ) . o flSA = 0 because j o S f is nullhomotopic.
173
2) Tile construction of S f * u is natural with respect to maps of cofibration sequences:
If
SZA
-+
SZB
+
-~
SCf
+
S2A t
~
S2B '
--~
$
SA
s f>
$
--+ SCf,
-+
SB
+
SA'
s['
> SB'
is induced f r o I n a commutative square A
A'
I
f,
> B
> B'
and *
u 9 HC,,(SB, D),
!
u' C H C ~ ( S B , D )
are classes associated to
~' 9 HC,*(SA',D),
:= $9" ~' 9 H C ~ ( S A , D)
by the procedure above, then
S f* v = Sg* (Sf'* u')
3) Applying 2) to the square
A
A
rd > A
I
>B
shows that only tile case of the cofibre sequence
$2 A --+ $2 A sj'> C S A eval|
SA ~ SA
has to be treated. 4) Consider the exact sequence 0 -+ $ r --+ C~(]0, 1]) := { f 9 C~([0, 1]), f(0) = 0} r where eval is given by evaluation at 1. Then To 9 X~ 7-1 9 XI(C~(]O, 1]))
To(l) = 1
7-1((fdg)tl) = f l f d g
r -+ 0
17"4
are related by
071 = eval* ro As we saw already, 9 HaI(Sr
[j'*~]
is the fundamental class of the circle. The fact t h a t the exterior product on cohomology was constructed via a m a p of complexes shows t h a t for any cocycle 99 E X~(SA, D) and eval | id:C~(]O, 1])|
S A -+ r | S A (evalNid)*99 = (eval | id)* (to x 99) = (eval* TO) X 99 = 071 X 99 = 0 ( r l x 99) with 7i • 99 E X ~ + I ( C ~ ( ] 0 , 1]) | SA, D). The inclusion k : C~0(]0,1]) | S A --+ C S A being an a s y m p t o t i c HC-equivalence (7.8), we see that the cochain
(T1 • 99) o (k.) - i E X ; + I ( C S A , D) satisfies 0 ( ( r l x 99) o (k.) - i ) = 99 o (eval|
o k , i = 99 o (eval|
= (evalNid)*99
So we can achieve the construction 1) by p u t t i n g /Y :----- ((T 1 X 99) o ( / g , ) - l )
o
Sj; o /3SA 9 Z~(SA, D)
and find id* [u] = [(7-1 X 99) o (j' | id)S, A o ( k , ) -1 o flSA] = [ ( j ' * v l X 99) o ( k , ) -1 o flSA] = [99 o aSA o flSA] = [99] by the periodicity theorem. Second case: the homological X-complex:
HC~(C, SCf)
si.> HC*(C, SA)
si.> HC;(C, SS)
Let [99] E HC~(C, SA) be such t h a t S f, [99] = 0. T h e n [99'] := /3SA o [99] e Hc~+i(c, S2A) satisfies $2f,[99 '] = S2f,
o
~SA o [99] = /TSB o S f , [99] = 0
So
S2I, ~' = 0 r for some
r e X~(C, S2B) which i8 well defined up to a cocycle again. Then
Sj, r
9 X~(C, SCs)
175
satisfies
O (Sj. r
= S j . (O g/) = S ( j o S f ) . So'
As j o S f is canonically nullhomotopic, there exists a natural element h, 9 X I ( S 2 A , S C I ) w i t h Oh, = S ( j o S f ) ,
given by the Cartan homotopy operator. We may then conclude that, ,/ := - S j .
o ~' + h o ~'
9 X g ( C , SCs)
is a cocycle whose cohomology class is well defined up t,o the image of
S j , : H C ; ( C , S " Z ~ ) - , H C ; ( C , SC~) We claim that Si. [u'] = [~] 9 H C ; ( C , S A ) To prove this calculate
Si.J
= -Si.
o Sj. otb+
Si. o h o
i = S i . o h o ~oI
as i o j = O. The commutative diagram
SA I! SA
2+
CA
e~at
A
-~ i
A
+ (eval, C f) joSf
>
Cf
II
and the fact that the canonical nullhomotopy of j o S f is given by the factorization over tile contractible cone C A in the diagram above show that
Si.
o h =
hI
where,
h' E X ~ ( S 2 A , SA) is the Cart, an homotopy operator associated to the canonical nullhomotopy of the composition S2A sj'> C S A eval| SA Note that h,' is in fact, a cocycle because the two evaluations at the endpoints coincide. Its cohomotogy class is well known L e m m a 9.7:
[h'] = (71 • idS. A) o k . 1 o Sj" = (3"* T1 x id sA) o k . 1 = aSA E H C ~ ( S 2 A , S A )
176
Proof:
The lemma asserts that
h' : rl • id A E HC~(SA, A) Let r E X ~
RA) be defined in even degrees by 1 times the composition XoR(SA) a~.> X~R(SA) f3 h~> Xo(RA)
i.e. by
1 fl n i r : @Wn --4 -2 Jo (wno + E w n - l - J flwJ O) dt 0 1 fl n-1
.
]o and in odd degrees by
~: owdo ~ -2 Then ~b e X ~
(oJ'do) dt
A) and h ' - rl x id A = 0 o r 1 6 2
as a lengthy but elementary calculation shows. []
Thus finally
[Si, u'] = Si, o h o [~'] = t{ o [~'] = ~SA o [~'] = OtSA o flSA o
[~] =
[~]
by the periodicity theorem again. The proof of exactness of the two six term exact sequences being achieved, we go on to define the desired quasiisomorphisms. The map
~ : x~(c, scs)
-~
Con4Sf,)[-1]
-+
((Si), ~,-(-1)d~avhs(ioi) o ~)
is easily seen to be a map of complexes. (Here again, hs(IoO c X ~ ( S C S, S B ) is the Cartan homotopy operator associated to the canonical nullhomotopy of S ( f o i).)
177
Moreover, this m a p of complexes fits into a d i a g r a m which commutes up to signs and up to homotopy:
X;(C, SA)[-1]
+
X;(C, SB)[-1]
~>
X~(C, SCI)
X:~(C, SA)[-1]
~
Xs
~
Co,~e(&f,)[-1]
SB)[-1]
i
x ; ( c , scs)
--,
X;(C, SA)
; ~
~
x;(C, SB)
tll
Cone(Sf.)[-1]
~
;11
X;(C, SA)
~
X;(C, SB)
Therefore a is a quasiisomorphism by the exactness of tile induced cohomology sequences and the five lemma. In tile case of tile cohomological X-complex we put
c/ : Cone(S f*, X~(SB, D),X;(SA, D)) (~, ~)
--+
X~(Cf, D)
-*
~ o hs(soO + (Si)* r
The same reasoning is then valid in this case, too, and is left to tile reader.
[]
9-3 Stable cohomology of C*-algebras and the second excision theorem W h e n does a short exact sequenee of admissible Fr~chet algebras give rise to a six term exact sequence on (stable) asymptotic cohomology ? A look at the, c,o m m u t a t i v e d i a g r a m A
;~ 0
-~
Cs
-~
--+
B
-~
0
II
Cyl s -+ B
-~ 0
and the first excision theorem show t h a t a necessary and sufficient condition is given by
178
D e f i n i t i o n 9.8: A unital epimorphism ,f : A ~ B of admissible Fr6ctlet algebras satisfies s t a b l e e x c i s i o n iff j :
Cf
J
+
(:~:,O)
is a stable asymptotic HC-equivalence.
[] In general one would expect excision to hold if f would turn out to be a cofibration on soille stable asymptotic tlomotopy category. This condition would then force j to be a stable homotopy equivalence. We have however not developed this point of view far enough to get any reasoimble conclusions. Anyway, an arbitrary epimorphism of admissible Fr6chet algebras shouht be far fl'om satisfying excision. To obtain a sufficient criterion for excision note, timt a stable asymptotic morphism strictly inverse to j would carry any positive, quasicentral, bounded approximate unit of C I to a similar approximate unit of J. This leads one to restrict attention to separable C*-algebras. which form essentially the largest category of Fr6chet algebras, for wtfich the kernel of any epimorphism possesses a poitive, quasieentral, bounded, approximate unit.
T h e o r e m 9.9: ( S e c o n d E x c i s i o n T h e o r e m ) Let 0 -~ J ~ A f~ B --+ 0 be a short exact sequence of separable C*-algebras with f unital. Suppose that f admits a bounded, linear section. Then f satisfies stable excision. Consequently, there are natural six term exact sequences
HC~(C. SA)
s~.> HCO(C, SB)
0 HC,~(C, S J)
HC'~(C, ss) HC~(C, SB)
HC~(C, SA)
<
Sf.
179
HC~
D)
(sI*
HC~~
D)
~/ HC~
%0 D)
HC~(SJ, D)
H C I ( S B , D)
Sf*
> HC,~(SA,D)
tbr any admissible Frfichet algebras C, D. They are natural in C, D under asymptotic nlorphisms and under maps of extensions 0
~
J
~
A
-~
B
-+
0
0
-+
jr
__+
A~
_+
B p
__+
0
Proof:
In Connes-Higson [CH] it has been shown that j is a stable asymptotic homotopy equivalence. As their notion of asymptotic morphism differs from ours, we repeat their argnment with the necessary modifications. We have to show that
Sj. E HC~
SCI)
is an asymptotic HC-equivalence. We will construct an HC-inverse [(9] of S j . explicitely. To do this choose a positive, quasicentral, bounded approximate unit
( ut ) C J, 0 <_ ut <_ 1 V t C ff~+ and fix a bounded, linear section of f
s: B --+ A f o s = idB Let ( v t ) E Co(]O, 1 D, 0 < vt < 1 Vt 9 ~+ be a bounded, positive, approximate unit for (7o(]0, 1D consisting of functions that differ from 1 only near the endpoints of the unit interval. To define the bivariant cocycle (9 the asymptotic connecting map associated to the extension
0 ~ S J ~ CA & Cf -+ 0 will be used. p also admits a bounded, linear section, for example
s' :
Cf (a,h(t))
-~
CA
--* x(t) ( a - s(h(O))) + s(h(t))
where X 9 S ~ equals 1 near 0 and vanishes near 1.
180
We put then 0t :
Sl~ @Tr C f
~
9 |
SJ
--+ 9('vt | ut) s'(x)
It is easily verified that 0 is an asymptotic morphisru so that we may finally define O. o (~.c~)-1 C
e ::
HC~
SJ)
with
k : S r 1 7 4 Cy ~ S C I given by tile obvious inclusion. First step: e o IS3.] = [id s J] C H C 2 ( S J , S J) O o [Sj.] = [0.] o [kc'] -1 o [Sj.] = [0.] o [(j |
idSr
o [k.] -1
by the naturality of k. However
0 o (j|
sr
~ k : Sr174
are homotopic, as one can see by considering the fanfily of asymptotic morphisms
F:, : Sr174 J
--+
g|
SJ
--+ g(vt |
where Fo = 0 o (j N~t id sr
((1-~)ut
+ A1A))s'(j(x))
and F1 is explicitely homotopic to k J. So we conclude
(9 o [ S j , ] = [(0 o (j |
i d ) ) , ] o [k.a} -1 = [k]] o [k,V] -1 =
[id s J]
Second step:
[&,]
o e
=
Fdsc'] c Hc~
sc,)
Again [Sj.] o 0 = [(Sj o 0).] o [kCJ] -1 where S j o 0 is homotopic to kC~: If
( w t ) E C I 0 <_ wt <_ 1 V t E ~ + is a bounded, approximate unit for Cf consisting of real valued functions that vanish around the vertex and are equal to 1 near the base of the mapping cone, then
G:~ : S r 1 7 4 CI g @y
--+ -+
SC I 9(vt |
( ( 1 - ) ~ ) j ( u t ) + )~wt))y
is a continuous family of asymptotic morphisms with Go = S j o 0 and G1 explicitely homotopic to k c~. Thus [ S j , ] o {9 =
[(Sj o 0),] o [k,C'] - ' = [k c'~] o [ k C S ] - i = [idC,']
181
As [Sj,] is now known to be an a s y m p t o t i c HC-equivalence, the assertion follows fl'om the six t e r m P u p p e exact sequence of tile first excision theorem. []
182
Chapter 10: KK-theory and asymptotic cohomology In this chapter we show that asymptotic cohomology is the target of a Chern character on Kasparovs bivariant K-theory. (The cyclic theories known so far do not achieve this goal due to their pathological behaviour for C*-algebras.) Our construction generalizes the well known cases of Chern characters for finitely summable and O-summable Fredholm modules treated ill [CO], [C02] and is the second main result of the paper. Morally, the Chern character is given by the composition ch :
KK*(A,B) -+ E*(A,B)" -+ " HC~(HA, SB)
of the Connes-Higson map [CH] and the evident "map" associating to a homotopy class of asymptotic morphisms its bivariant cohomology class. Due to the fact that E-theory is defined using nonlinear asymptotic morphisms the second "map" cannot be constructed rigorously. The Chern character can nevertheless be obtained with the help of the derivation lemma because the image of bivariant K-theory in E-theory can be described by asymptotic morphisms whose restriction to suitable "smooth" subalgebras is linear. The most flmdamental property of the bivariant Chern character is its compatibility with the Kasparov- resp. composition product, which can be viewed as a generalized Grothendieck-Riemann-Roch theorem. To derive the compatibility of the bivariant Chern character with the Kasparovresp. composition product (up to a period factor) we use Cuntzs description of bivariant K-theory which provides clean and simple fornmlas and give only afterwards a description in terms of linear asymptotic morphisms. The period factor 21ri has the same origin as the corresponding factor showing up for the ordinary Chern character (See Chapter 8). After comparing the bivariant with the ordinary Chern character of the previous chapters we show finally that the complexified bivariant Chern character yields an isomorphism
eh: KK*(A, B) |
(F ~ . HC~(SA, SB)
on a class of separable C*-algebras containing (~ and being closed under extensions with completely positive splitting and KK-equivalences. An explicit treatment of the most interesting special cases has still to be worked out.
183
10-1 T h e bivariant C h e r n character and t h e R i e m a n n - R o c h t h e o r e m T h e o r e m 10.1: ( G r o t h e n d i e c k - R i e m a n n - R o c h )
a) There exists a natural transformation of bifunctors on the category of separable C*-algebras
ch : K K * ( - , - )
-~ H C ; ( S - , S - )
(:ailed the bivariant C h e r n character b) For any separable, unital C.-algebras A, B, C the diagram
KKJ(A, B) @ KKZ(B, C)
- ~ > KKJ+I(A, C)
HCJ(SA, SB) | HC~(SB, SC)
§ 1
(2.i)J~
HC~+'(SA, SC)
o
colnIIlutes, where the upper horizontal map is tile Kasparov product and the lower horizoutal map is given by ~ 1 times the composition product. (See Theorem 8.22 for an explanation of the factor 2rri.) c) (Grothendieck-Riemann-Roch Theorem) Let. ch' := (2~ip ch : Kj --+ HQ~' be the normalised Chern character K-theory. Let r E KKI(A, B) and denote by Sr ~ KKZ(SA, SB) the corresponding stabilised elenmnt. Then the diagram
Kj(SA)
-|162
Kj+z(SB)
oh'1
l ch'
HC?(SA)
~ HC]+t(SB )
commutes. d) If
0--+ J ~ A --+ B ~ 0 is an extension that admits a completely positive lifting, ch is compatible with long exact sequences, i.e.
-+
KKJ(C, B)
$ ch
o -+
KKJ-I(C, J)
$ ch
-+
184 &II(t
+-
K K J + I ( B , D)
o ~--
K K J (d, D)
$ ch
+--
$ ch
+-- H c J + I ( S B , SD)
((2,iyo
H C J ( S J , SD)
+-
('.Onlluute.
[]
T h e o r e m 10.2: ter)[CH]
(Explicit d e s c r i p t i o n of t h e b i v a r i a n t C h e r n charac-
Let. A, B be separable C*-algebras. a) Let x E K K ~
B) be represented by a Kasparov bimodule of the form
Choose a bounded, approximate unit ( u t ) t ~ + , 0 < ut _< 1 in K: | B which is quasicentral with respect to the C*-subalgebra of L:(HB) generated by po(A)Upl(A). Then the linear map ~P~ :
{
sin~x
D~ = \ c o s ~ x ( 1 - u~)
8A
--+ M2S(K. |
co.qxO - u~)) -sin~x
_
B) ~- S ( K |
B)
x t h e coordinate function on] - 1, 1[
defines an asymptotic morphism and the bivariant Chern character of x may be represented by the cocycle oh(x) = F ~ , . ] -~ o [ e ~ . ]
where
denote the inclusions.
iSB : S B
~
kA :
-+
,SA
o [k.~] -~
S(g~| SA
B)
185
b) Let y E K K I ( A , B) be represented by the extension 0~]C| admitting a completely positive splitting s. Choose a bounded, approximate unit (vt)te~+, 0 < vt _< 1 of ~U| B which is quasicentral in E. Then the linear map q%ad :
SSA
--~ S(1C Qc. B)
f | a
-~
f(v,)s(a)
defines an asymptotic morphism and the bivariant Chern character of y may be represented by the cocycle eh(y) = [iSB,] -1 o [~odd,] o [k,SA]-1 o ZSA (Here suspensions correspond to functions on ]0, 1[.) []
Before coming to a proof of Theorems 10.1,10.2 we want to recall Cuntz's description of KK-theory [CU] and the modifications in odd dimensions due to Zekri [Z]. For any C*-algebra A, the free product A * A of A with itself carries a natural C*-norm. If the completion is denoted by QA there is a short exact sequence of C*-algebras 0 ~ qA --+ QA ~d,id A -~ 0 splitting naturally in two ways. The morphisms i d * O , O * i d : QA ~ A restrict, to maps ~o, ~:1 : qA -~ A These maps are KK-equivalences in fact. The interest in these algebras stems from the result of Cuntz that there is a natural isomorphism KK~
if- [qA |
~,qB |
~]
between the group of homotopy classes of homomorphisms of C*-algebras from qA | IC to qB | 1~ and K K ~ B).
186
The group 2E/22E acts on Q A and it,s ideal qA by switching the two factors of A. The crossed product by this action is denoted by
eA := qA • ~/22Z It fits into a short exact sequence with a completely positive splitting
0 - ~ eA ~ C*(A[P]) -~ A |
~ 0
where
C*(A[P]) ~- Q A • 2 ~ / 2 ~ is the universal C*-algebra generated by A and a selfadjoint involution P. For separable A the algebras C*(A[P]) (resp.eA) are KK-equivalent to A by an even (resp. odd) KK-equivalence. Zekri proved that the natural map
KKI(A,B)
~ - [eA |
IC, qB |
i~]
is an isomorphism providing thus a description of KK-theory in odd dimensions similar to that of Cuntz in the even case.
Proposition
10.3"
Let A be a separable C*-algebra. a) The maps
7to, 7rl : qA ~ A induce asymptotic HC-equivalences [STr0.], [$7ri.] E H C ~
SA)
b) There exist natural asynlptotic HC-equivalences
aA : E H C ~ ( S e A , SA), [3~ : e H C ~ ( S A , SeA) inverse to each other. Naturality means that for any homomorphism of C*-algebras f : A -+ B the equalities f. o
a A
= a B o (e f ) .
/~B o f,
=
(c f ) , o flA
hold, i.e. the diagrams "SEA"
"SeB" CO~lnlute.
~ > "SA ....
SA"
~ -+ "SEA"
> "SB ....
SB"
-+ " S e B "
a
187
Proof:
We repeat the arguments of Cuntz and Zekri: a) Consider the homoinorphisms (Tr0,1h) : A , A -+ A O A and
A| A
--+
M2(A * A)
(x,y)
-+
(i(X)O ~(Y)O)
with
i , ] : A -+ A * A the two inclusions. The compositions of them are homotopic to the canonical inclusions
A * A -~ M2(A * A) A@A ~ M~(A| which yield asymptotic HC-equivalences by (8.17). So both morphisms and their suspensions are asymptotic HC-equivalences and we find
H C * ( S Q A , - ) ~_ HC~(SA 9 S A , - ) ~_ H C ~ ( S A , - ) 9 H C ~ ( S A , - ) ~_ S(id 9 0)* H C * ( S A , - ) • S(O 9 id)* H C ~ ( S A , - ) If A is separable, the second excision theorem may be applied to the splitting extension
0 ~ qA -+ QA
id*id)
A -+ 0
and yields an isomorphism
H C * ( S q A , - ) ~ H C * ( S A , - ) ~- H C * ( S Q A , - ) ~~- S(id * O)*HC*(SA,-) | S(O * id)*HC~(Sm,-) which can be provided either by It0 or by ~h:
S~r;, S~r~ : H C ~ ( S A , - ) ~
HC~(SqA,-)
Treating the case HC* ( - , SA) similarly allows finally to establish the claim. b) The extension defining qA is equivariant under the involution switching the two factors of A and gives rise to an extension of crossed product algebras
0
-+ eA
-+ QA x W,,/27Z -+
0
--> eA
-+
C*(A[P])
-~
A x2Z/22Z
A|
-+
0
-+ 0
188
The crossed product QA • is easily seen to be isomorphic to the universal C*-algebra C* (A[P]) with relation (p2 _ p ) A = 0 where p
]+F 2
z
-
and the inner a u t o m o r p h i s m defined by F = F*,
(F ~ - 1)A = 0
corresponds to the action of the nontrivial element of 2~/22g on epimorphism (P0,Pl) :
C*(A[P]) -+ A • 2Z/22~ _~ A|162
-~ A |
QA. F i n a l l y the
(r
=
AoA
is given by p u t t i n g P = 0 and P = 1, respectively. The h o m o t o p y
C*(A[P]) ~ a
-§
P
~
M2(C*(A[P]))[O,1] 0 0 (PO)_ ut 0 0 ut 1
with
ut =
(cos2t -sin~t~ sin2 t cos~t ] C M2(~
allows to conclude t h a t the n a t u r a l inclusion
A --+ C*(A[P]) as well as its suspension is an asymptotic HC-equivalence inverse to P0:
C*(A[P]) --+ A | A ~
A
(resp. Spo). As the projection C*(A[P]) ~ A @A possesses an obvious completely positive lifting the second excision theorem may be applied in the case t h a t A is separable and yields the exact sequences
0 ~ H C~(SeA,-) -~ HC~+I(SA, _)2 (Spo,Spl)*} HC~+I(SA, _) ~ 0 * 0 -~ HC*(-,SA) (Svo,Sv,)* HC,(_,SA)2 o HC,+I(_SeA) -~ 0 from which the isomorphisms
HC~(SeA,-) ~
HC~+I(SA,-)
and
HC*(-,SA) A HC;+'( -, SeA)
189
are obtained. They are natural in A under morphisms of algebras (which give rise to a map of extensions defining cA) and under asymptotic morphisms in the free variable - . The Yoneda lemma shows then that these isomorphisms are induced from the composition product with canonical (:lasses
ozA e HC~(SeA, SA), /3A E HC~(SA, SeA) and the claimed equalities follow from the naturality of the boundary maps above. []
Proof of Theorem
10.1:
Construction of the Chern character
Define the Chern character in even dimensions by the composition
KK~
B) -% [S(qA|
1C),S(qB |
~+ HC~
K)] -+ HC~
SqB) ~+ HC~
|
K, S(qB) @c" K)
SB)
where the two isomorphisms in the lower line are given by
[iSqB]-1 o -- o [iSqA] : HC~
|
~, SqB |
t:) ~ HC~
SqB)
and [suB,] o - o [slrA.] - t : HC~
SqB) -4 HC~
SB)
respectively. Define the Chern character in odd dimensions to be the composition
KKt(A,B) -% [S(cA|
t:),S(qB| HC~
t:)] --+ HC~174
SqB) ~
9t~,S(qB)@c. ]C)
HC~(SA, SB)
where the two isomorphisms in the lower line are given by
[iSqS] -1 0 -- 0 [Z. r.SeA1] : HCO((SeA) @c* tG, (SqB) |
K) ~ HC~
and [S~rB] o - o ~Ar : HcO(SeA, SqB) + HC~
SB)
respectively. The Chern character is natural under morphisms of algebras and
ch( idK. K )
=
idHC~
SqB)
190
N a t u r a l i t y of the C h e r n character
First case: KK ~ | KK ~
) KK ~
hh H C ~ | It C[~
> H C2
Obvious from the definitions. Second case:
KK 1| KK ~ _
> KK 1
HC~ | HC ~
, HC~
Obvious from the definitions, as the Kasparov product is just given by the composition of the corresponding morphisms of universal algebras. Third case:
KK ~ | KK 1
) KK 1
ch| I
I ch
H C ~ @ HC~
> HC~
Let z E [qA|
K, q B |
1C], y E [eI?|
lC, q C |
]C]
The Kasparov product x @ y of x and y is given by the composition x O y : ~A O c . K
~(z)|
""% e(qA) @c" K
e(qB |
e(B)|
~(i)|
K) @c" K ~ e(qB) |
K|
K y|
qC |
1C |
1~) |
e(qA |
-~
IC @c" 1C ~(,~o)~ 1(2 ~_ qC |
K
(See [Z]). The morphism PA : eA |
K --+ e(qA) @c" lC
is a homotopy inverse of e(Tr0) up to stabilization by matrices: The composed morphisms eA |
K "% e(qA) |
e(qA) O c . K
~(~o)|
K e(rr~174 eA |
IC
eA @c. K~ ~a> e(qA) @c* K
are homotopic to the identity so that we may conclude
[Sp A] = [i.] o [Se(rr0).]-'
o [i.]-'
e HC~
|
The natural morphism k : e(C |
K) ~ ~C |
K
IC,Se(qA)|
K.)
191
is defined by the map of extensions
e(C |
lC) -> O(C |
IC) x 2ZI22Z
k;
e.C |
~
C |
IC (~ C |
~.
K
~
lC
4.
QC • 2z/22z |
lC
-~
C |
lC (~ C |
lC
which shows also, after repeating the argument proving the existence of the elements (~, (3~ that [S~:,] o f32 |
o i c = i,c o fig
9 HC~(SC, S4C ) |
lC)
Let, us calculate the Chern character of :~:| y: We find after suspending the sequence of morphisms defining x | y
~,t,,(.~, | y) = [S~0c,] o [i,~qC]-' o [i?,c| = [STrO0,] o [iS, qC] - ' o [isqc| =
[s~g,]
o [S(x | y),] o [i s~] o flA
- ' o [S(Ome(Tro)okoe.(x)oe(i))|
o [ i s l e ] - , o [is~c|
)
- ' o [s((y o 4 ~ 0 ) o k o ~(~ o i)) | id),]o
oG] o [&(~0).l-'
o [~.]-, o [.~.] o ;~2
= [s~c.] o [is~c] -, o [s(y o 4~0)).1 o [sk.] o [s~(~ o ~).] o [s4~0).]-' = [s,.o.]c o [~.-~o ] -, o[S(yo4~0)).]o :
[s~;.] o [if~c] -, o [S(yoi).]
= ([s~c.] o [~s~c]-,
([s ~,. ] o ~Z ~| o([S4~0).]
o [sy] o [~.] o/~y)
o f32
o [i. ]) o ([i. ] -' o [sx] o [,i. ] o [s~0. ] -' )
o ~.,.) o ([i.]-'
o[Sz] o [i.] o [s~0.]-')
o ([s~0.] o [~.]-' o [s~] o [~.] o [s~0,]-')
= oh(y) o ch(x) = ch(x) | ch(y) Betore proceeding further it is necessary to investigate the behaviour of the Chern chara(:ter with respect to boundary maps in tile long exact sequences in KK-theory and asymptotic cohomology. We treat the homological cast, the cohomological one being similar. Let
O-+ J - ~ A-+ B--+ O be an extension of separable C*-algebras admitting a completely positive splitting. If one considers the diagram
SA
s f)
SB
4
CI $ J
4
A
&B
the KK-theoretic connecting map 6 : KK*(C, B) --+ KK*+I(C, J) is given by the composition
6 : KK*(C,B) |
KK,+,(C, SB) J:+ K K , + I ( c , c f ) <~_ K K , + I ( c , j )
192
(The existence of a completely positive splitting is essential in showing that the canonical inclusion J -+ C I is a KK-equivalence). It is also a stable asymptotic HCequivalence by the second excision theorem (9.9) and the cohomological connecting map is defined similarly by the composition :
HCf~(SC,
|
HCf+,(SC, S2B) sj.
sj.> HC~+I(SC' SCI) ~-- HC~+I(sc, r So one obtains a diagram
6:
KK*(C, B) ch $ 6: HC~(SC, SB)
|
KK*+I(C, SB) ~ KK*+I(C, J) ch, $ ch $ HC~+I(SC,S2B ) -+ HC~+I(SC,SJ)
|
where the square on the right side is commutative by the naturality of the Chern character under algebra homomorphisms. So it remains to investigate the commutativity of the square on the left. The compatibility of the Chern character with the Kasparov- resp. composition-product already being established in the case where at least one factor is even-dimensional we see that the following diagrams commute
KK~ B) ch 4 HCO(SC, SB)
| |
KKI(C,B)
<|
ch $
HCI(SC, Sm
|
KKI(C, SB) ch 4 ~)> HC~(SC, S2B ) KKO(C,SB) ch 4
HCO(SC,S~B)
and consequently also the square
KKI(C, B) ch $ HC~(SC, SB)
| |
KK~ SB) ch $ HCO(SC,S~B )
The statement of Theorem 10.1.d) follows then from the L e m m a 10.4:
Let OLKK
C KKI(SB, B), [3KKE KKI(B, SB)
be the K-theoretical and
O~gc e HCI(S2B, SB), ~HC e HCI~(SB, S2B) the cohomological Dirac- resp. Bott-elements. Then c h ( / ~ g g ) = •HC
Ch(O~KK) = ~ 10~HC
[]
193
Assuming tile lemma for the moment we go on to establish the compatibility of the bivariant Chern character with products in the remaining fourth case: Fourth case:
KK 1| KK 1
ch|
> KK ~
~
~ ch
HC~ | HC~
>
HC ~
[x] E [r174
,U] <(~o|
[r q B |
]C] ~_ K K I ( A , B )
[y] E [eB, C |
K:] <(~o|
[(B, qC|
K] ~- K K I ( B , C )
Their Kasparov product x|
(see [z])
E KK~
(x | y): qA A+ r162174
k|
eB No" ]~ |
~ y|174
C) can be represented by the composition
;C ~(~)> r
C|
]~ |
|
| ]~- |
--+ ]~ ~ C @c* ]~
The element u C KK~ corresponding to v : qA --+ e(eA) | defined as follows: From the extension
0 --+ eA -+ QA x 2Z/22Z -+ A(~ A --+ 0 one obtains the connecting map
KKO(A,A|
o
K K I ( A , eA )
For a general separable C*-algebra D, the connecting map
KK*(D,A)
0oio.> K K . + I ( D , cA)
is induced by tile Kasparov product with the KK-equivalence flA. The KK-equivalence Iv] is then given by the square of tile connecting map: [v]:
KK~ id.
Oo~o.> K K I ( A , ~ A )
-+
flA
Ooio.> KKO(A,~(eA))
__+
flA | fl~A
;C is
194
The connecting mat)s being compatible with the Chern character, it is easy to calculate oh(v) by having a look at tile comlnutative diagram
id. E
KK~
$
Ooio.>
KKI(A,r )
oh, $
id. E HC~
Ooio.)
KKO(A, c2A)
ch $
SA) id.
ch $
ao~o.> HCI,(SA, SeA)
(2~,:)aoio.> HCO(SA ,Se~A )
flA
-~
--~
(27ri)r A o ~A
So
ch,(u) = (27ri)fl~ A o / ~ C HC~
Sc2A)
With this we find
ch(x | y) = L[iSCl-l* j o [is,c|
-1 o [iz,c|174
o [ s ( (~ | id) o k o ~(x) ) o i~t.] o [ s , . ] = [ i f c ] - l o r i SL.~ |
j
o [S((yOid)oko4x)),]
o [ s . 0 . ] -~
o ( [y, A - 1 ]
o [s,.]o
[s~0.]-b
~- (2ffi)[iS, C] -1 0 [SB] O [~.s~. ,, j -1 o [ s k . ] o [sd.~:).] o Z; A o f ~
because [i.s~A]-,
o IS..]
o [s~0.]-'
:
,:1,(.) = (2~)/3:A
o z2
Thus ch(x @ y) equals (27ri)([i,sc]-i o [Sy])o ([iSeB]-I o [Sk,] o,~B|163 = (2~,:)(F.sc]-~
o [Sy]) o ([i.s~]-~
o ([SJ::] o ~A)
o ,:S~BI.., o 9~) o ([i.SB]-~ o [S d o /32)
because of :
(2~i)([iS, C] -1 o
[Sy] o ,~B) o ([iSB]-lo
[,~Sg] o [~4)
= (2~i)ch(y) o ct,(x) = (2~i)d,,(x) o ch(~) The demonstration of Theorem 10.1.a) is thus achieved. Theorem 10.1.c) follows fi'om Theorem 10.1%), L e m m a 10.4. and 10.5. and the stable periodicity theorem. []
195
P r o o f o f L e m m a 10.4:
Recall the extension O -+ eB -+ E I B -~ B -+ O
defined as the pullback 0
-+
eB
--+
E1B
il 0
-+
-+
B
.~
eB
--+
0
; i0
--+ C*B[P]
~
BOB
--+ 0
of the extension defining eB in 9.3.(see [Z]). The inclusion E 1 B --+ M 2 ( E 1 B ) is nullhomotopic which gives rise to maps of extensions 0
~
SB
-+
CB
$
-+
;
0
-+
M2(SB)
0
-+
eB
~
-~
M2(B)
-+
B
t --->
EIB
-+
0
~
0
-+
0
$
M2(CB)
st
B
t
The KK-theoretic connecting map of the upper extension being the K-theoretic Bott element /JKK C K K I ( B , S B ) we conclude that under the isomorphism
K K I(B, SB) & [~U, SB |
lC]
/;~KK corresponds to the homomorphisuls s. The Chern character ch(f3KK) can therefore be computed by composing S s , with the cohomologieal connecting map 2r E H C I ( S B , S e B ) which equals the cohomological connecting map of the extension O~ SB-*CB
~ B ~O
by naturality. A look at the definition of the cohomological connecting map shows that it coincides for the extension under discussion with the cohomologica] Bott element so that
ch(f~K~ ) = [3uc, which proves the first part of the lemma. To calculate the Chern character of the Dirac element OZKK we use the equation
/~.c = ch(gK K ) = a ~ ( ~ 9 ~
K) = ch(~: )ch(9~ )
which is true because the compatibility of the bivariant Chern character holds already if at least one class of even degree is involved. By the proof of Theorem 9.4 and Lemma 10.5. we know however that =
idB]), [(T 1 X 7-1) X i d ,SB ]) = (2~ri)[idS,B]
and consequently ~:h(9~K ) = ( 2 ~ i ) ~ C so that
196
and finally 1
ch(CtKK ) = 27r~OlHC []
Proof of Theorem
10.2:
We divide the demonstration into several steps. 1) q)~v and C1~odd
are
asymptotic morphisms.
It can be readily verified that q~v and r d define bounded families of continaous linear maps. To calculate their curvature note that
wq,~, = f(Dt)[g(Dt), (po~a)
pl(a)0 )] (PO(ob) pl0(b))
WeoeO = fg(vt)(s(ab) - s(a)s(b)) + f(vt)[g(vt), s(a)]s(b) We may suppose in both cases that f,g are holomorphic near the intervals [-1, 1] resp. [0, 1]. In the even case one finds
2,a,)': - 27ri.
g(A)(A-Dt)-'f(A-
Dr),
Po a) pl(a) 0
](A - Dt)-ld~
and
[(A- Dt)' ( P~ a) pl(a)O ) ] = 0 =
c o s 2 x ( ( 1 - ut)Po(a)- pi(a)(1 - ut))
cos~x((1 - u~)pl(a) - po(a)(1 - ~ ) ) ) 0
so that the estimate
II (1
-
ut)pl(a)
-
po(a)(1
-
ut)I1~1] (1 - ~,)(pl(a)
- po(a)) II + II [u,,po(a)]
]1
shows that the curvature of ~ev becomes arbitrarily small near infinity (note that pl(a) - po(a) C ~ @c* B). The same argument keeps track of the component f(vt)[g(vt), s(a)]s(b) of the curvature of r whereas the fact that s(ab)-s(a)s(b) | B, the identity f(vt)j = f(vt)j - f ( 1 ) j =
1;
= 2~ri
f(A)((A - vt) -1 - ()~ - 1)-l)jd)~
and the estimate I[ ( ( A - vt) -1 - ( A - 1 ) - l ) j [1-<1] ( A - v t ) - l ( A - 1) .7 1[1[(vt - 1)j [1 take care of the second component fg(vt)(s(ab) - s(a)s(b)) of the curvature of ~odd.
197
2) The evendimensional case Let
O-+ J ~ A P+B ~ O be a splitting (under s) extension of separable C*-algebras and let (ut) be a bounded, quasicentral, approximate unit in J. Then the linear m a p
X:
SA
-~
f | a
-+
M2S.] f(Dt)
(o ~
s(v(a) )
is an asymptotic morphism by what we just proved. Moreover, following the arguments in the proof of 9.2 shows that the cohomology class
: : [TI" X idf J] o IX,] o [kA] -1 ~ H C ~
S J)
satisfies (Si),r
9A ] - [S(.~ op),] 9 HCO(SA, SA) [S~d,
If we apply this to the extension
0 -+ qA ~ QA
id*id)
A ~ 0
with splitting il : A ~ QA, then ( o [S/o,] 9 HC~
SqA) satisfies
[S/0,] = [S(id*O),] o ([Sid,]- [S(sop),])o IS/0,] = [id,] E H C ~
[STr0,] or
so that we conclude o [S/0,] = [S7%,] -1 9 H C ~ Let now x 9 K K ~
SqA)
B) be represented by the Kasparov bimodule
Then there is a corresponding diagram of extensions
o
-~
IC|
~
f t 0
-+
qA
M(IC|
~
Q(IC|
t PO * Pl -+
QA
~
A
B
represents the class x under the Cuntz isomorphism
KK~
o
---+
0
t
and the h o m o m o r p h i s m
f : qA -+ lC |
~
B) ~-- [qA, B |
K]
SA)
198
By definition oh(x) = [iSB,] -1 o [Sf,] o [STr0,]-' = [isB,] -1 o [S f,] o g o [S/0,]
= [ i S B , ] - ' o [(Sf o X o Si0),] o [kA] - '
9 HC~
SB)
Moreover. the fact that the bounded, quasicentral approximate units in an ideal of a separable C*-algebra form a convex cone shows that the diagram S(/C Oc- B) < |176
SA
sf I
ls~0
S(qA)
S(QA)
( X
commutes up to homotopy which proves [ ( S f o X o S/0),] : [~.,,,] E HC~
S(/C |
B))
and establishes the claim. 3) The oddimensional case If
O -+ lC Oc. B -+ E --+ A -o O is art extension admitting a completely positive splitting then it can be shown using Kasparov's generalized Stinespring tl~eorem that there is a map of extensions 0
~
/C|
-+
gt 0
-~
eA
E
-+
A
t ~
ExA
-+
0
~
0
II -+
A
where E1A is the universal C*-algebra of 10.4 (see [Z]) and g is the homomorphism corresponding to y E K K I ( A , B) under the Zekri isomorphism
K K X ( A , B) ~ - [cA, IC Go* B] By definition of the bivariant Chern character ch(y) = [~s.,] -~ o [ & , ] o O)
where BA 9 K K I ( S A , SeA) is the cohoinological connecting map of the extension
O -+ eA + E~A --+ A + O The connecting map being natural we conclude ch(y) = [ i s B , ] - '
o 5 o [idS,A]
where a is the cohomological connecting nmp of the extension O ~ IC |
defining y.
B -+ E -+ A -+ O
199
Recall t h a t the connecting m a p was defined as follows: = e o [sj,] o #SA
ill terms of the nlorphisms ill tile d i a g r a m
SA
J+
0
-+
CI
-+
E
-~
E
~t
f--+ A
tl
~|
II --+
A
-+
0
where (9 = [0,] o
[k~'] -1 = [Si,] -1
(see the two excision theorems 9.6,9.9). Thus = [0,] o [~,c'j-~ o [ s j , ] o # S A = = [0,] O [ S ]i . o ~:SA]-I,. J OflSA
As the a s y m p t o t i c m o r p h i s m 0 o 8 j is readily seen to be homotopic to q~oda we conclude ch(y) = [iSB,] -1 o (~ -----[iSB,] -1 e [ff2odd,] o L[kSA]-I* J 0 t~SA as claimed. []
L e m m a 10.5: The normalised o r d i n a r y Chern character ch' (10.1.b)) and the bivariant Chern character chbiv (10.1.a)) coincide, i.e. for a separable C*-algebra A the d i a g r a m
Kj(A)
ch'
~-
KKJ(qJ, A)
$ HC~.(A) HCJ(qJ, A)
$ + - o~ A
o
--
chbiv
HC~(S(~,SA)
o ~r
coIinlnltes.
Proof: Tile d i a g r a m being n a t u r a l under algebra homomorphisms we m a y suppose A = q: in tile even and A = S r in the odd case. In the even case the c o m m u t a t i v i t y of the d i a g r a m is obvious. In the odd case, tile flmdamental class u of K K l ( ~ , S(~) ~- 2Z is represented by the extension 0 --+ Sq~ --+ C r -+ (IJ ~ 0 whose bivariant Chern character is given b y / ~ s r E HC~(S(~, S2ff~). Then
c~sr o el~bi.(u) o #r : ~ s r o # s r o #r = #r in H C ~ ( r 1 6 2
~_ r
200
On the other hand the ordinary Chern character of the fundamental class u E K1 (S~) satisfies from which the claim immediately follows. []
C o r o l l a r y 10.6: Let A, B be separable C*-algebras. Let
r E KK* (A, B) be a KK-equivalence. Then ch(r
9 HC;(SA, SB)
is an asymptotic HC-equivalence. Consequently
HC;(SA, C) ~- HC~+I~~
C)
HC;(D, SA) ~_ HC;+I~I(D, SB) []
T h e o r e m 10.7: Let C be the smallest class of separable C*-algebras satisfying
1)r 2) If in an extension
0--+ J ~ A-+ B - + 0 with completely positive splitting two algebras belong to C, then also the third one. 3) C is closed under KK-equivalence. Then for any A, B E C the bivariant Chern character yields an isomorphism
ch : KK*(A, B) |
ffJ ~+ HC•(SA, SB)
201
Proof."
Consider the class C' of separable C*-algebras A such that
ch : K K * ( A , r
|
~ -+ HC~(SA, Sr
is an isomorphism. By our calculation of HC~,(S~, S~) (9.5) we know that r E Cf. An extension of separable C*-algebras with completely positive lifting yields six term exact sequences in KK-theory and bivariant cohomology, compatible (up to multiplication by 27ri) under the Chern character: o
+-
K K * (g, ff~)
+--
K K * (A, q;)
$ ch
+--
K K * (B, (~)
$ ch.
o HC~(SJ, Sq;) +---
o
+-
$ ch
* A ,S~) +-- HC~(S
e-- H C * ( S B , S(~)
o 4--
The five lemma shows then that C' satisfies also condition 2) above. It is also clear that C' is closed under KK-equivalence as the diagram (commuting up to multiplication by constants)
K K * ( B , ~)
v|
>
HC~(SB, S(F) ch(~)|
KK.+I~OI(A, (F)
HC.+IvI(SA, Sq;)
shows. Thus
ch : KK*(A,(F) |
~ ~
HC*(SA, Sq;)
for any algebra A in C. Running the same argument for the class C~ of separable C*-algebras B such that
ch : K K * ( A , B) |
(F --+ HC*(SA, SB)
is an isomorphism completes the proof of the theorem. []
C o r o l l a r y 10.8:
Let A be a separable C*-algebra belonging to the class C. Then the Chern character defines isomorphisms
HC~(r 2A)
ch: K , ( A ) |
(~ -%,
ch: K*(A) |
(~ ~" HC*(SeA, q;)
between the complexified K-theory (K-homology) of A and the asymptotic cyclic homology (cohomology) of S2A. []
202
11 E x a m p l e s Finally two explicit calculations of asymptotic cyclic (co)homology groups are presented. The two examples are of a very different nature. In the first, the stable bivariant asymptotic cyclic homology of separable, comnmtative C*-algebras is computed. The arguments are exclusively based on the functorial, homotopy- and excision-properties of asymptotic cohomology developed hitherto. If A is a separable, commutative C*-algebra with associated locally compact Hausdorff space X, the asymptotic cyclic homology of A equals the (2~/22~periodic) sheaf cohomotogy of X with compact supports and coefficients in the constant sheaf ~:
HC~,(S2A) ~ ~
H~+2"(X,~)
This is in some sense the most natural answer one could hope for and again provides evidence that asymptotic eohomology yields a reasonable cohomology theory for Banach algebras. The second example illustrates, how asymptotic cyclic groups can be calculated by methods of homological algebra. We treat the case of the Banach group algeb r a / I ( F n ) of a free group on n generators. One obtains an isomorphism between asymptotic homology and group homology
HC~(ll(Fn)) = H,(Fn, r as in the case of the algebraic cyclic homologyof the ordinary group algebra. The result coincideswith that for the (stable) asymptotic homologyof the reduced group C*-algebra:
HC~,(S2C;(Fn)) = H,(Fn, r (This follows from the fact that the group C*-algebra is KK-equivalent to a commutative C*-algebra (whose homology is known by the first example) and from the existence and properties of the bivariant Chern character of chapter 10.) We emphasize however that it is not the result but rather the way to obtain it, which might be of some interest. The case treated here is particularly simple, but the calculation as such applies (in principle) to a larger class of algebras. Finally it should be mentioned that the calculations in the cohomological case are more involved. They do not yield the full bivariant asymptotic cohomology but closely related groups which will be studied elsewhere.
203 11-1 Asymptotic cyclic cohomology of commutative C*-algebras In this section the stable asymptotic (co)homology of separable, commutative C*-algebras will be computed. Recall that every commutative C*-algebra coincides with the C*-algebra of continuous functions on a compact Hausdorff space in the unital case and with the C*algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space in the nonunital case. The algebra is separable if and only if the corresponding compact space (the one point compaetification of the corresponding locally compact space) is metrisable. First of all it is shown that stable asymptotic (co)homology defnes a (co)homology theory in the sense of Eilenberg Steenrod on the category of separable, commutative C*-algebras (i.e. compact Hausdorff spaces).
Theorem 11.1: Let A, B be adnfissible Fr@het algebras. For any pair X D X ' of compact, metrisable spaces denote by Cv the mapping cone of the natural suriection p: C(X) --+ C(X'). Then the flmctors
* " ') HA(X,X
:= HC,*~(SA,SCp)
H.B(X,X ') := HC:(SCp, SB) define generalised, 2Z/22Z-periodic cohomology (homology) theories on the category of pairs of compact, metrisable spaces.
Proofi By definition, tile flmctors H~,, H,u are 2Z/22Z-graded. It follows from the homotopy invariance of bivariant, asymptotic cyclic cohomology (Theorem 6.15) that H~, H,u are homotopy flmctors. If (X, X') is a pair of compact, metrisable spaces, then the first excision theorem (9.6.), applied to the homomorphism p : C(X) --+ C(X'), yields the six term exact sequenes
H~(X')
+--
H~(X)
+-- H~(X,X')
o+ H~+I(X, XO
Hp(X')
--+ H*A+I(X)
--+
to H~+l(x ,)
-+
-+
H,B(X,X ')
+-- H L I ( X )
+--
H B I ( X ')
ot H,B+I(X,X ')
;o
The same holds if one starts with a triad of spaces.
204
Consider the extension of C*-algebras
0 +
C(X,X')
-4
C(X)
~
C(X')
-+ 0
where C(X, X') is the algebra of continuous functions on X vanishing along X'. The second excision theorem in asymptotic cohomology (9.9) implies
H~(X,X')
=
H,B(X,X ') =
HC*(SA, SCv)
"~ HC~(SA, S C ( X , X ' ) )
HC*(SCp,SB)
~- HC~(SC(X,X'),SB)
Froln this it is clear that the fllnctors H~, H ,B satisfy the following strong version of the excision axiom:
Strong excision axiom: If f : (X, X') -+ (Y, Y') is a map of pairs of compact, metrisable spaces which induces a homeomorphism of X - X ' onto Y - Y' then f * : H*(X,X') 'fi- H*(Y,Y')
f. are isomorphisms.
[] To identify the (co)homology theories occuring in this way the following special case has to be considered first. T h e o r e m 11.2: Let X, Y be finite CW-complexes. Then (in the notations of ll.1)
H (X)
~_
(~n~=OH ~ (X, q3) | H~ -n (pt)
~- I]n~176Horn(Ha(Y, (U),H,B_,~(pt)) Especially
where the grading is such that the components (Pn,, of an element 9 of the right hand side vanish if n - m :# 9 rood (2).
205
Proof:
The groups above are well defined because every finite CW-complex is compact and metrisable. It is clear that the last statement follows from the two previous ones, applied to the cases A = (l; and B = C(Y) respectively. Consider the contravariant functor H~ on the category of finite CW-complexes. By Theorem 11.1, it is a tmmotopy functor taking values in abelian groups, taking cofibration sequences of spaces into exact sequences of abelian groups and satisfying the weak wedge axiom. The Brown representation theorem tells then that there exist CW-complexes En and a natural equivalence of functors
[-, En] Moreover, as the fnnctors under consideration are group valued, the complexes E~ are actually H-spaces and consequently nilpotent spaces. The homotopy groups
~rk(En) ~+ H~(S k) being Q-vector spaces and E~ being nilpotent, the complexes En are Q-local, i.e. coincide with their Q-localisations. As the k-invariants of H-spaces vanish rationally, every Q-local H-space is a product of Eilenberg-MacLane spaces and a check of homotopy groups shows oo
En ~- H K(H~(Sk)' k) k=0
Thus
H~(X) ~
[X,E~] ~_
l l Hk(X,H~(Sk)) k=O
oo
~_ ( ~ Hk(X, (~) | H~-k(pt)) k=0
Due to the stable periodicity theorem (9.4) the functor H,B is not only defined on the homotopy category of finite CW-complexes but extends to a functor on the homotopy category of finite spectra. For a finite spectrmn Y let Y* be its SpanierWhitehead dual. Y* is again a finite spectrum and (Y*)* ~- Y. For any finite spectrum put F~(Y) := HB_m(Y*) This defines a contravariant homotopy flmctor on the category of finite spectra. As Spanier-Whitehead duality turns cofibration sequences into cofibration sequences, the Brown representation theorem can be applied again. Repeating the arguments above one finds
H~(Y) ~_ r ~ ' ( r *) ~ 0
Hl(Y *, (~)| Hum(S-I)
l:--o0
~- 1-[ H~ l=0
•)' ~) | Hum-l(Pt)
206
U H~
~)' H~-I(Pt))
/=0
[]
The following well known lemma is the key to extend the results obtained so far to general compact spaces. The presentation follows the article [M] of Milnor. L e m m a 11.3: Every compact, metrisable space is homeomorphic to an inverse limit (indexed by 2 + ) of finite, simplicial complexes. P r o o f ( S k e t c h ) [M]: Let X be a compact, metric space. Let (/An, n c 2 + ) be a sequence of finite open covers of X satisfying the following conditions. a) The diameters of the open sets of the cover b/~ tend to zero as n approaches 0(2).
b) For n < n' the cover Un, is a refinement o f / ~ To every cover U~ there is an associated simplicial set X , , its nerve, which captures the combinatorial data of tim cover. As all covers L/,. are finite, the geometric realisations IX,~I of its nerves are finite simplicial complexes. A refinement b/I of a cover b/ gives rise to a map of geometric realisations of the nerves IXu, t -4 IXul which is well defined up to homotopy. Choosing representatives of these maps one can form X := lim~_ IX,d. Because the diameters of the open sets in the covers /.4, tend to zero it is possible to identify the inverse limit over the nerves with the original space homeomorphically. [] This allows to extend the calculation of the (co)homology from finite CW-complexes to arbitrary compact, metrisable spaces, provided the considered cohomology theory behaves well with respect to inverse limits. The relevant conditions are as follows.
Lemma 11.4: [M] Let H* (resp. H . ) be a cohomology (homology) theory on the category of compact, metrisable spaces. Assume that H* (resp. H.) satisfies the strong excision property (11.1). 1) In the cohomological case the following assertions are equivalent.
207
a) If (X~, n C 2E+) is an inverse system of compact, metrisable spaces, then
H*(limXn) ~- limH*(Xn) oo b) If Y = Vi=IY/ is an infinite union of compact, metric spaces with diameters tending to zero which intersect pairwise in a single point Y0, then oo
H*(Y, yo) -%, (~ H*(Yi,yo) i:l
2) In the homological case the following assertions are equivalent.
a)
H,(lhnX,~) ~ Rli~2H*(X, ) where R lim~ denotes the total right derived functor of the inverse limit functor. b)
H,(Y, yo) ~-~ f i H,(Y~,yo) i=1
P r o o f : [M] The implications a) ~ b) are clear because the infinite wedge sum Y is the inverse limit Y ~ V ~ I Y / u n d e r the obvious contractions. hnplication b) =v a). To begin with, we add a one point space X0 = pt to the given inverse system which does not change the projective limit. Consider the mapping telescopes Z,~ of the finite sequences Xo ~- X1 6- -.. ~- X,~ and let Z := lim~ Z,~ be their projective limit. The space Z contains X = lime- Xn as compact subspace and the complement can be identified with the union of tile finite mapping telescopes
Z - X ~- ~_JZn n~O
The point Z0 = X0 =
pt i8 taken
as base point of Z. It does not belong to X c Z.
Claim: Z i8 contractible: A contraction e : Z • [0, 1] ~ Z is defined as follows. Let c(z, O) := z and let c(z, !n ) denote the image of z under the projection map Z --+ Zn-1 C Z. The deformation retraction of Z,, onto Zn-1 is used to define e(z,t) for 1 As Zo = pt the map c ( - , 1) is constant.
208 The complement Z - X of X ill Z can be decomposed as union of two subspaces made up by the even (resp. odd) parts of the mapping telescopes. Z - X = y , U Y" Y' ~ IIX2~
Y" ~ IIX2i+1
Y' A Y" = UXi (~ denotes "homotopy equivalent"). For any locally compact space U denote by U its one point compactification. The long exact cohomology sequence 0 = H * ( Z , pt) -4 H * ( X U p t , pt) ~ H*+I(Z, X U p t )
-4 H*+I(Z, pt) = 0
for the pair (Z, X) shows that H* ( X ) ~" H* ( X U pt, pt) ~- H *+1 (Z, X U pt) ~- H *+I(Z - X , co U pt)
The eohomology sequence -4 H * + I ( Z - X , oc Upt) --+ H *+l(~v, cc Upt) | H * + I ( Yw, co) -4 H * + I ( Y ' N Y " ) -+
for the triad (Z - X, Y', Y") provides a long exact sequence oo oo -4 H * (Vn=I(X~, U oo), oo) --+ H * (Vn=l(Xn U CO), OO) --+ H * ( X ) -4
A word about the metrics of the spaces involved. If Xn is an inverse system of oo X ,, is compact and metrisable again and a metric compact, metric spaces, then [I,~=~ on the product space is obtained in an evident way from metrics on the individual factors, provided that the diameters of the factors tend to zero: lim,~oo d i a m ( X n ) = 0. The restriction of this metric to the inverse limit lira+__Xn C l-I,, X n defines the inverse limit topology. From this remark it is clear that the diameters of the wedge summands Xn U oc above tend to zero. Therefore the assumed continuity property b) leads to the exact sequence OO
CxTO
-4 @H*(X.)
@
H*(X) -4
It is not difficult to identify the map j in the above sequence as j(...,an,...)
= ( . . . . a~
--
f ~* , (a n - i )
....
)
which allows finally to deduce H * ( l i m X . ) = H * ( X ) ~- l i m H * ( X . ) 4-
-+
The reasoning in the homological case is analogous. []
Some of the considered cohomotogy theories possess the continuity property described above.
209
Lemma
11.5:
oo Let X = Vu=IX n be an infinite wedge sum of compact, metrisable spaces with diameters tending to zero. Then cx)
HC;(r SC(V~.<Xn)) ~ G HC;(r SC(Xn)) n=l
Proof: OO n Let p , : gi=lX i ~ gi=lX i be the natural contraction. The induced homomormaps C(V~:=lXi) isomorphically onto the algephism p~, : C(Vin=lXi) --} C ( V i ~ 1 7 6 bra of continuous functions on v~~176 which are constant on the subspace Vi~ Let/C := {(K, N)} be as in 5.5, 5.6 where K runs through the family of compact and let Kn be the corresponding families subsets of the open unit ball in C(ViO~ for the algebras Pn(C(Vn=lXi)) C C ( g i ~ 1 7 6 Then by definition of the asymptotic, resp. analytic cyclic homology one has
_ e oo 1X i)) ~ HC~(C(V~~ H e , (SC(VC~O=lXi)) ~ H C,(SC(Vi= :
H,(lin~X.(RC(VC~=lSXi)(K,N))
~_
=
Hn~H.(X.(RC(V~_ISXi)(Kjv)) )
We claim that the natural map lira lira H.(X.(RC(Vi~ISXi)(K,N)) ) -+ l i m H . ( X . ( R C ( V ~ I S X i ) ( K , N ) ) ) - - + r ~ --4- KZ n
--+K~
is an isomorphism. In fact it is immediately clear that the selfmaps fn : X ~ X,~ 2::+ X give rise to an asymptotic morphism ~ot : C ( X ) -+ C ( X ) such that ~on := f,*~ and ~ot for noninteger values of t is defined by linear interpolation. This asymptotic morphism is naturally homotopic to the identity. Let now (K, N) E IC. If n is choosen large enough (so that the curvature of ~ot becomes very small on the multiplicative closure K 00 of K for t _> n) one can find (K', N ' ) E /C by (5.12) so that the composition
(~")'=*:) X.(RC(SX)(v;K.N)) -+ X.(RC(SX)(K,W))
X.(RC(SX)(K,N))
is chain homotopic to the identity 9 This shows that the map considered in the claim is surjective and the injectivity follows from a similar argument. From this one obtains by definition
H C . ( S C ( X )) ~_ l i r n H . ( X . ( R C ( S X ) ) ) ~_ lim lira H . ( X . ( R C ( S X ) ) ) --+K.
---+n --+lC n
n ~_ h m H C . ( S C ( V i = I X i ) ) ~_ limHC, (SC( V i=lXi) 9
~
n
--+tt
ol
---~T~
which, by the second excision theorem (9.9), equals lira
(6 \i=1
HC~. (SC(Xi
,,) = 6
~ x
HC, (SC( n))
n=l
[]
210
Lemma 11.6:
Let X = V,~=tX~ be an infinite wedge sum of compact, metric spaces with diameters tending to zero. Then the sheaf cohomology with coefficients in the constant sheaf ~ satisfies
H*(X,C) ~_ ~ H * ( X . , ~ ) 77~1
Proof: Let X be a compact Hausdorff space and let (~,~,n E 7Z+) be an inductive system of sheaves of abelian groups on X. The sheaf associated to the presheaf U -~ l i m ~ , r is called the direct limit lim_+,~ P,~ of the sheaves 5c,~. The stalks of a direct linfit are the direct limits of the stalks of the individual sheaves:
(limY,~l
= lim(.T,dx
Consequently the functor lim_~ : Sh~ + -4 Shx is exact.. The obvious h o m o m o r phisms of sheaves 9rk --+ lim_+,~ ~,~ give rise to a natural transformation of left exact flmctors lirn r(Jt-n) -+ r ( l i m f'n) If X is compact and Hausdorff, this is actually a n a t u r a l equivalence. In this case the derived fimctors of these functors also coincide. Therefore
~nH*(X, jz ) ~ H*(X, li2}:Yn if X is c o m p a c t and Hausdorff. Let now X = V~=IXi be as above and denote by i,~ vn=lxi --+ X the n a t u r a l inclusion which maps V~=: Xi homeomorphically onto a compact subspace of X. The direct limit l i m ~ n ( i n , ~ ) over the direct image sheaves in,~ is then easily identified with the constant sheaf (1J over X: ~ x -~ lin~in,(l?v,~ ,x~ For the cohomology one finds therefore
H*(X,(~) ~- H* ( X, limi,~.r -.,,
imH*(X, i~.r , = . u ,) ~ l-~,,
-~
-~ hm H (Vi_IXi , IV) ~ M.] H* (Xi, II?) --+n
'--
[] W h a t has been obtained so far can be smnmarised in the
211
T h e o r e m 11.7:
Let. A be a separable, commutative C*-algebra and let. X be the associated locally compact space. Then there is a natural isomorphism
~_ HC~,(S2A) ~_ H C.(S2A) ~
H~*+2n (X, qJ)
where H~ denotes sheaf cohomology with compact supports. Proofi
If A is unital, then X is compact and the compact support condition is empty for the cohomology of X: H * ( X , - ) ~_ H * ( X , - ) . If A is not unital, the corresponding space X is only locally compact. Denote by X its one point compactification. Then there are exact sequences
0 ~ HC~,(S2Co(X)) --+ HC~,~(S2C(X)) -+ HC~,(S2(~) = HC~,((~) --+ 0 0 ~ H* (X, (1J) --+ H* (X, ~) -+ H* (pt, (IJ) --+ 0 which shows that one can assume A = C(X) unital and X compact, the separability of A implies that X (resp. X) is metrisable. By Lelnnm 11.3 X can be identified with the inverse limit of the nerves of finer and finer open finite covers: X = lira+_ Xn, X , finite simplicial complexes. Thus
ttC~.~(S2C(X)) = HC~. (S2C(I~I X . )) ~- li2}~HC~ (S2C(X,~)) by Lemma 11.4 and 11.5 lim ( ~ -+n
H*+~k(xn,~)
k=-oo
by Theorem 11.2 OG
-- O k=-oo
k=-o~
by Lennna 11.6. The naturality needs some argmnents but is not difficult to show. [] The calculation of the asymptotic cyclic cohomology of a commutative C*-algebra turns out to be more complicated however. As the eohomology of an inverse limit of complexes is not related to the cohomology of the individual complexes in general one cannot hope to get a closed expression for the asymptotic (bivariant) cohomology groups. It is however possible to introduce closely related groups which will in fact turn out to be computable. As tiffs should be treated elsewhere we will be brief and content ourselves with some remarks.
212
L o c a l cyclic c o h o m o l o g y w i t h c o m p a c t s u p p o r t s D e f i n i t i o n 11.8:
Let A, B be admissible Fr4chet algebras and let ](~A, ] ~ B be the fanfilies of compact subsets of the open unit balls as in 5.5, 5.6. The bivariant local cyclic cohomology with compact supports of the pair (A, B) is defined as
HCz%(A, B) := R lira lira Hom~ont(X.RA(K N), X.RB(K, N,)) +-K: A --4K; B
-
where R lira+_ denotes the total derived flmctor of the inverse linfit functor and both sides are viewed as objects in the derived category of the category of complex vector spaces. []
R e m a r k 11.9: a) There exist natural transformations of functors
HP*(-)
--~
HC:(-)
-+
HC*(-)
-~
HC?c(- )
UP,(-) +-- HC:(-) ~+ HC~, (-) ~, gcZ, c(-) HC~(-,-) -+ HC*(-,-) --+ H C ~ ( - , - ) b) There exists a composition product in bivariant local cyclic cohomology with compact supports such that the diagram
HC~(-,-)
|
HC*(-,-)
--+ HC*(-,-)
HCI*~(-,- )
|
HC~,(-,-)
-~
HCI*~(-,- )
commutes.
[]
Contrary to the asymptotic case, the bivariant local cyclic cohomology with compact supports of separable, commutative C*-algebras can be computed. T h e o r e m 11.10: Let A, B be separable, commutative C*-algebras with corresponding locally compact spaces X, Y. Then
where the grading is such that the components ~T~m of an element 9 of the right hand side vanish if n - m r 9 rood(2). []
213
Before we come to the proof a few more properties of local cyclic cohomology are needed. R e m a r k 11.11: a) The first, and second excision theorems (9.6,9.9) hold for
HC~c(- , -).
b) The natural transformation HC~ ( - , - ) --+HC{e(- , -) commutes with Puppe sequences and boundary maps in long exact cohomology sequences. []
C o r o l l a r y 11.12: a) The bifunctor (X, Y) --+HC~c(SC(X), SC(Y)) defines a generalised, bivariant cohomology theory on the category of pairs of compact, metrisable spaces. b) If X, Y are finite CW-complexes the natural map
HC*(SC(X), SC(Y)) ~+ HCtc(SC(X), SC(Y)) is an isomorphism. Proofi This follows from Theorems 11.1, 11.2 and the preceding remark. O The generalised homology theories obtained from the local cyclic theory satisfy the same continuity property as the homology considered before. L e m m a 11.13: Let X = V~=0Xn be an infinite wedge sum of compact metric spaces whose diameters tend to zero. Then for any admissible Fr~chet algebra B the natural map oo H
* Cl~(SC(Vn=oXn), SB) -% [ I HC,~(SC(Xn), SB) *
co
is an isomorphism. Proof:
Let (I)t : C(X) ~ C(X) be the asymptotic morphism defined by the retraction of X onto successively larger finite wedge sums (see the proof of 11.5). Let, in the notations of 11.5, be/Cn = p'K: C K: be the family of compact sets of continuous oo functions of norm smaller than one on X which are constant on Yi=n+lXi. It is then not difficult to establish the following facts:
214
a) (~t) gives rise to a bivariant eocycle g2. C Rlim lira Hom~ +-K --4 UK,,
(For n-tuples ((K1, N1),..., (Kk, N~:)) tile necessary higher chain honmtopies between the individual chain maps arc provided by tile evident linear homotopies between p . . . . . . . . pn, : C(X) -4 C(X). b) Let i. E R lira lira Hom~
X.(RAt:))
+- U/C,, ~'KZ
be the obvious inclusion. Then
q?. o i. = Id C R lira lira Itom~ 4-- UiC,, --,yIC.
(even on the "chain level") i, o
= m
HC~
C(X))
because the asymptotic morphism q~ : C(X) -4 C(X) is naturally homotopic to the identity. This implies (by using the composition product) that the natural map
HCtc(C(X),
B) = Rlim
lira
+-,'C ~ K . B
Hom*(X,(RC(X)~c), X,(RB~c,,)) -4
lim Hom*(X.(RC(X)tc.),X.(RBtcB))
-4 R lim
+ - U / C . --~Ku
is a quasiisomorphism. The latter complex can however be identified as R lira
lira Hom*(X.(RC(X)~,.),X.(RBt:o)) qi~)
+--uK:. --+ K:B
qi~ R(limo lira) lira Hom*(X.(RC(X)r. ),X.(RBIc~)) qi~> t---n
~---~ n
-+K:B
n
qi,> Rlim~,~( R ~-tc~ --~tc~limHom*(X.(RC(X)~c.),X.(RBtc~))) = = R hm U C t c ( C ( V i = l X i ) , B) +--n 9
*
T~,
A similar sequence holds after suspension. Then the second excision theorem may be applied by Remark 11.11 and yields
HCt~(SC(X), SB) ": RIImHCI~(SC(V~=,X~), SB) ~- J . l HC~(SC(Xn), SB) ~---n rt=l
[]
215
Note that as a consequence of the foregoing lemma and 11.4, one obtains for an inverse limit of compact, metrisable spaces a quasiisonmrphism
HC[~(SC(limXn), SB) 4---n
Proof of Theorem
qi,>RlimHC[~(SC(Xn), SB) "~--n
11.10:
Because excision holds for both sheaf cohomology and local cyclic cohoinology, one may assume A and B to be unitah A = C(X),B = C(Y), X,Y compact, metrisable. Realise X as inverse limit of finite, simplicial complexes (Lemma 11.3): X -~ lim~ k Xk. Then
HCT~(SA, SB) ~- HCt~(SC(limXk), SB) ~_R lira HC[~(SC(Xk), SB) +---k
4--'k
Now for a finite, simplicial complex the arguments in the proof of Theorem 11.2 carry over from asymptotic to local cyclic cohomology with compact supports and show oo
HCt~(SC(Xk), SB)
~_
H H~
(F),HC'.~_,~(S2B)
As the three homology theories H(7,~ H C a H(7,lc coincide by 11.9, Theorem 11.7 shows fltrther that
HC;c(SC(Xk), SC(Y)) = Horn* naturally. Consequently
H~(Xk, ~), ~
HCz~(SC(X), SC(Y)) ~- R limHom* ~--k
On the category exact flmctors
Vectz~+ of inductive (Ca)
~
--
Hm(Y,•))
m--~O
systems of complex vector spaces tbe two left lim~.
Horn(C,D)
(C~) --+ Hom(lim~ C,~,D) (D a fixed complex vector space) are naturally equivalent. Therefore their right derived functors are also naturally equivalent:
RlimHom(C~,, D,) +~- Hom(limC~,,D,) +--n
-+ n
Thus
HG~(SC(X),SC(Y)) ~ RlimHom* t-.- k
~Hom*
Hn(Xk,~), n~O
lim (2~ g ~ (Xk, (IJ),
Hm(Y,(F) ~m=O
Hm(K ~)
216
~_
Ho,,~*
~ n=O ~kn~O
lira Hn(x~, -~"k
r 0 H'~(r'r m=O
+--k
111,=0
[]
217
11-2 E x p l i c i t c a l c u l a t i o n o f a s y m p t o t i c c o h o m o l o g y g r o u p s Contrary to K-theory, cyclic homology theories are defined by natural chain complexes. This should enable one, at least ill principle, to calculate cyclic (co)homology groups with the tools of homological algebra. In this paragraph we will illustrate a rather general scheme for tile calculation of asymptotic (local) cyclic groups by an example. We will treat the case of the convolution Banach algebra of summable flmctions oil a free group. Although the cohomology groups are known (stably) by the general excision properties of the cyclic theories their determination will be quite different now as it is based on a purely homologieal calculation. Local cyclic eohomology and the approximation
property
From its definition it is clear that analytic cyclic homology is a dirct limit of periodic cyclic homology groups
HCZ,~(A) = lira liP,
(RA(K,N))
whereas local cyclic cohomology is the linfit of a convergent spectral sequence
EPq = RP ~I~:(HPq(RA(K,N)) =:>GrPHC~,+q(A) To be able to use this for computations one has to find criteria for cutting down the the direct limit, resp. the spectral sequence to a controllable size. D e f i n i t i o n 11.14:
Let A be a eountably generated, normed algebra and let A be its completion. (If "4 is infinite dimensioal it will be of countable dimension and thus never complete.) Let {xl,. 9 xk . . . . } be a sequence of generators and let KN C "4 C A be the set of elements x ~ A satisfying a) n
ill
:c belongs to the (finite dimensional) span of all mononfials of length at most a:l,---,Xn.
b) II x
ll~
'~ []
Then K,, is a compact subset of the unit ball of A. D e f i n i t i o n 11.15:
Let ,4 be a countably generated normed algebra and let A be its completion. The local cyclic (co)homology with finite supports of the pair (A,.4) is defined as
HCI.f(A,A) := H.(~nX.(RA(K~,n))) HCI*I (A, .4) :-- H* (R li_n2 X* (RA(K. ,,~))) This definition does not depend on the choice of generators of M. []
218
It is clear from the definition that
HcIf (A, .4)~_ ~nHP.(RA(g.~,,~)) and that the corresponding spectral sequence in the cohomotogical case collapses to a short exact sequence
0 ---+m il H l p*-I(RA(Kn,.)_,_~,
-+ HCTI(A,A ) -4 limHP*(RA~,
(K.,,~)) ~ 0
There is an important class of algebras for which the local cohomologies with finite respectively compact supports coincide. To describe the class recall the D e f i n i t i o n 11.16:
Let E be a Banach space. Then E has the Grothendieck approximation property if, given any compact subset K C E and any e > 0 there exists a (bounded) linear l[< e In other words, selfmap r : E -+ E of finite rank such that supz.E K [[ x -- r the identity belongs to the closure of the finite rank operators in the topology of compact convergence. []
Typical examples of Banach spaces satisfying the Grothendieck approximation property are all kinds of LP-spaces. Examples of C*-algebras having the approximation property are all nucear C*-algebras and also the reduced group C*-algebras of the free groups and of discrete, cocompact subgroups of simple Lie groups of real rank one. For the class of algebras with approximation property one has R e m a r k 11.17: Let A be a Banach algebra and suppose that the underlying Banach space satisfies the Grothendieck approximation property. Let N C A be any dense, countably generated subalgebra. Then the natural maps between the local cyclic (co)homology groups with compact, resp. finite supports
HCI,C(A) ~-- HC;I(A,A ) HC(c(A ) 2+ HCt,I(A, A) are isomorphisms. []
Therefore the calculation of the asymptotic homology (local cyclic cohomology) of a Banach algebra is reduced to the problem of calculating the (co)homology of the complexes X,(RA(K~,,~)), i.e. of the periodic cyclic (co)homology of some completions of finitely generated subalgebras of the tensor algebra over A C A. This can be done (in principle) if A possesses a dense subalgebra ,4 of finite cohomological dimension.
219
Algebras of finite cohomological dimension and connections In this paragraph we will collect some facts (taken from Cuntz,Quillen [CQ] and Khalkhali [K]) about algebras of finite Hochschild cohomological dimension. All algebras are viewed as abstract algebras (not equipped with any topology). Let A be a complex algebra. The category of A-bimodules is an abelian category with enough injective and enough projective objects. The Hochschild (co)homology groups of the pair (M, N) of A-bimodules are defined as
H H A ( M , N ) := TorA|176
H H ~ ( M , N ) := Ext*A|
)
Definition and Proposition 11.18: ([CQ]) For an algebra A the following conditions are equivalent. a) The A bimodule A possesses a resolution by projective A bimodules of length b) The A bimodnle f~'~A of formal differential forms of degree n is projective. c) There exists a connection V : f~nA --+ f ~ + l A i.e. a linear map V satisfying V(aw) = aV(co)
V(wa) = V(co)a + (-1)l~lwda
Va 9 A, a~ 9 a'~A
For a given algebra A, its cohomological dimension is the smallest integer satisfying the conditions above. If the conditions are not satisfied for any integer, the cohomologieal dimension of A is defined to be infinity.
Proof: a) r
b): A possesses a standard resolution by free A-bimodules given by A ~-- po := A |
~...
b~
+-~ pk := A | 1 7 4 1 7 4
b'
+--...
m(a | a') := aa' k-1
b'(a ~ 1 7 4 1 7 4 1 7 4 1 7 4
k) = ~ ( - 1 ) J a
~ 1 7 4 1 7 4a J a J + l |
j=0
Tiffs resolution can be written as
A +-- f t ~
~- ... o f t k A |
with differential 0 := j o i n
o ...
|
ak
220 where m :
~"A
|
A
w | a
-+
~nA
-+
wa
is the (right module) multiplication and j :
f~'~A
--+
wn-lda
-+
lt~-lA | A (-1)n-1(w~-la|
n-l|
identifies ~ ' ~ A with the kernel of 0 : 1 2 ' ~ - l A | --+ ~ ' ~ - 2 A | As A has a projective resolution of length n iff the kernel K in any projective resolution A e - Po e - . . . e - P,~_ l v - K +- O
is projective itself, we are done. b) <=~ c): If ~ n A is projective as A-bimodule there exists an A-bimodule splitting s of the multiplication m : ~ ' " A O A -~ ~ ' ~ A
Then
~7 :
~n A
A~ w|
-~
(-1)nwda
is a connection on s Conversely, let V be a connection on ~]~A. Then s:
12'~A
~
~2'~A | A
w
-~
-j(Vw) + w | 1
is a bimodule splitting of the multiplication which shows that ~ n A is projective. [] Note that if V is a connection on ~ n A , then V':
~n+kA
~
aOdal...dan+ k
~
~]'~+k+lA V(a~
1 . . . d a n ) d a n + l . . . d a n+k
is a connection on [~'~+kA. This induced connection will be denoted in the sequel by tile same letter V. From this one easily obtains contracting homotopies of the standard projective resolution of A in degree larger than the cohomological dimension of A. Recall (3.10, 3.11) that the periodic cyclic homology H P , (A) of a unital algebra A can be calculated by any of the three following complexes: A
a) Tile X-complex of the I-adic completion R A of R A .
221
b) The periodic de-Rham complex or normalised (b, B)-bicomplex of A c) The full (b, B)-bicomplex of A. The three complexes are related by quasiisomorphisms
X.(R---A) q~> tiP.tin(A)(qi,~ c~p~, o . (A) which are actually deformation retractions. All three complexes can be identified with complexes of differential fornls. After this identification homotopy inverses of both quasiisomorphisnls in the diagram above are provided by the Cuntz-Quillen projection (see Chapter 3). Similarly, these chain nlaps yield (according to 5.25-5.27) quasiisomort)hisms (homotopy inverse to each other) in the topological context
X.(RA(K,N)) qi.~>CC.(A)(K,N) qis>X.(RA(K,N)) Recall that the norm of a homogeneous differential form in ~'~(A)(~t 2"+1) C CC.(A)(K,N) is given by
inf I[ c~
IIN.,,~=
~ I~r31(1+ '~)"~N-"('~!) -x
inf
(o(
i
2~+1
~
jAr,l(1 + n ) ' ~ N - n ( n ! ) - I
o J = ~ Z At~ tr~ ta[~ .. ,da B
Denote by At( c A the quotient algebra
AK
:=
RA(K,N)/IA(K N)
It is an admissible Fr~chet algebra. The subeomplexes F n generated by differential forms of degree at least n provide a natural filtration (Hodge filtration) of these complexes. If A is an algebra of Hochschild cohomological dimension at most n, the subcom~per plex F"+~C-CVe~(A) of CC. (A) becomes contractible and allows to calculate the periodic cyclic homology of A by a much smaller complex. Using tile quasiisomorphisms above, the corresponding X-complexes can be simplified in a similar manner. As we want to do this in a topological setting explicit formulas for the contracting homotopies are needed. Let A be an algebra of coholnological dimension n. Let V be a connection on ~Y~A, denote the associated connections on ~ : A , k > n by the same letter. Then there is a particularly simpte contracting homotopy in degrees above n for the standard complex calculating the Hochschild homology of A, discovered by M.Khalkhali:
222
Lemma ll.19:([K]) Let C . A : 0 ~- A b
~ 1 A ~-- . . . b
f~k A +-- . . .
be the standard complex calculating H H , (A, A). Let V be a connection o n f~kA, k >_ n. Then V o b + b o V = I d on ~ J A j > n
Proof:
(V o b + b o V)(wda) = V((-1)l~t[co, a]) + b(V(a~da)) = (-1)l~'[Va~, a]) + wda + (-1)l~l+*[Vw, a]) [] From the contracting homotopy of the Hochschild complex one can derive contracting homotopies of the cyclic bicomplexes. L e m m a 11.20: Let A be of cohomological dimension n. Let V be a connection on flJA, j > n. Consider the exact sequence of complexes 0 ~ F n + I c c ~ . ' e r A ~ CCP.r
Y+ C C t . ' e r A / F n + I c c P . erA --+ 0
a) The operator oo
h::
}2(-vm
v : Fn+tCCV.~" +
f n + 1~ {7.(Tper +~
k=O
defines a nullhomotopy of F n + l : h o (b + B ) + ( b + B ) o h = IdF,,+,CCP~.A
b) The map S' : C C P . ~ A / F " + I c c p . e"A -+ Ccp, e~A s' := ~ I d - b V
(
Id
on ~ n A / [ ~ A , A ] on ~ < ~ A
defines a linear section of p: p o s' = I d
c) The map s : CcP, e " A / F n + I c c P , e"A -+ CCP~"A
s :=
Id
on f~
Id + y~=o(-VB)k(Vb)(-VB)
on ~ n - l A
E~=o(--VB)k(Id
-- bV)
on ~Y~A/[anA, A]
223 defines a chain map splitting p:
p o s = Idcc,:~,A/F.+,CCK.A d) The chain map s o p : CCP~A ~ CCP,~A is chain homotopic to the identity. An explicit homotopy is provided by h':=
h' =
ho(Id-sop)
0
on~
E~,I=o(-VB)kv(-VB)I(Vb)(VB)
on a n - l A
E ~ = o ( - V B ) k V ( I d - E~=o( VB)~( h t - bY))
onft"d
V"o0 (_VB~kV z-~k=0~ J
~
h'o(b+B)+(b+B)oh'
= Id-sop []
The calculation of the local cyclic (co)homology of a Banach algebra A can now be done as follows: a) Check that A possesses the Grothendieck approximation property.
b) Find a dense, countably generated subalgebra A c A which is of finite cohomological dimension (as an abstract algebra). ( H C, (A) m lim_~,~ H,(X,(RA(K.,,,))), HC~(A) ~ Rlim~-~ H*(X*(RA(K.,n)))) c) Construct a sufficiently well behaved connection V on ftA which provides natural retractions of X , (RA(K. ,,) ). (Then HC] ~_ lim_~,~ HP,(AK,,) and HC[~(A) ~- Rlim~n HP*(AK~).) d) Calculate l i m ~ HP, (~4K~) resp. R lim~-n HP* (.,4K,). We will carry this out in a particularly simple case, namely for the completed group algebra P(Fn) of the free group on n generators.
Example: Cohomology of/l(Fn) Let F,, be the free group on n generators. Let A be a Banach algebra which contains the group algebra qJ[Fn] as dense subatgebra and satisfies the Grothendieck approximation property. Examples of such algebras are the Banach algebra of summable functions It(F,~) and the reduced group C*-algebra C~(F~), the norm completion of ~[Fn] acting as linear operators on In this situation the calculation scheme described before applies because the group algebra elF,,] is of cohomological dimension one. This follows from the
224
L e m m a 11.21: Let Fn be the free group with generators t l , . . 9 t,~ and let tl, 9 tn, t~-1, . . . , t~ 1 be the corresponding set of generators of the group algebra ~[F,~]. The associated word length flmction will be denoted by I - I. a) There exists a connection
v : ~'r
~ ~r
which is uniquely characterised by the property
V(dti) Therefore
r
=0
i = 1. . . . . n
is of cohomological dimension at most one.
b) Let N be a positive integer and consider the compact set KN C ff~[Fn] correslmnding to the inclusion (g[Fn] C A (11.14). T h e n there exists a constant
C(N)
> 1 such t h a t for all
V(da) = ~
a E KN
0 1 2 b~db~dba
c~
where ~ E
KN
and the number of s u m m a n d s is bounded by
C(N).
Proof: a) It is easily seen t h a t ~ ' ~ I ( ~ [ f n ] is a free bimodule over (1J[Fn] with generators dtl,..., dt,~. Thus giving a connection is equivalent to specifying a set of differential forms V ( d t l ) , . . . , V(dtn) E ~2. b) Each element
a E KN
{ r l i =kl
of monomials For such a monomial
t i+ 1
(
k
gilt
is contained in the linear span of the finite set k _< N}. k j-1
4-1
i=1
and
i=1 i=1
j-1
V
k
ti4-I )dtj• ( H ti4-1)
i )= E(I~
k
/=j+l
x
j-1
)
k
( H t ~ l ) d t j ( 1-I tl+1) = - ( I l t~l)dtjd( 1-I t~l) i=1 i--1
V
/=j+l k
t i3=1)dtj--1 ( 1-I tt• ) i=1
/=j+l
'~
)
i=1
/=j+l
j-1
k
=([Itfl)t;ldtjd(t; 1 H
tfl)
l=j-]-i
i=1 k
4-1
As connections are linear maps this shows t h a t V(I-L= 1 t i the form claimed in the assertion.
)
and also V ( d a ) are of
225
T h a t there is a b o u n d on the number of s u m m a n d s follows from the fact t h a t every linear map between normed, finite dimensional vector spaces is bounded. Of course there is no control of the size of the constant C ( N ) as it depends in general strongly on the choosen norm and connection on the considered algebra. L e m m a 11,22: Let N 2 1 be an integer. Then there exists M > N such that the n a t u r a l map oo
o k=0
extends to a bounded, linear operator oo
E(-V
o B ) k : CC.((~[FI])(KN,N) --+ CC.((~[FI])(K.,M)
k=O
Proof: Let w E ~2n((~[Fl]N ) and choose a presentation
w = E '
Aza~da~ "" .da~'
rt~i
with a~ = [ I j = i aj~, ~ aj~ E K N such that
I
lN, n
_
I1 ,0
Z ~-~2n m i which of course depends on the presentation of w. Let, m := z_~i=~
Now
9n
-VB(a~
" " da2'*) = - E
V(dai)dai+l " " d a i - i
i=0
and
mi
V ( d a i) = V ( E a ~ .
a ij _ l d ( a j i) a j +i l . . . a m , )i
=
j=l
=Eel
i
.
i i i a.j - l V.( d ( a. j ) ) a. j + l .
j=l
. .am, .
__ E a i l
i a ji _ i d ( a j ~) d ( a j +i i . . . am, )
j=l
Thus - V B ( a ~
") can be written (Lemma 11.21) as a sum of at most i
4C(N)m
terms of the form aoda ' .. . da~2n+2 with a~' = Nj~lm aij' , aij' E K N a n d such
~-~2n+2 that m __ < m ' := A--Ji=I mi! <--
7/}, -t- 2.
226 I t e r a t i n g the calculation shows t h a t
(-V o B)k(a~
da 2~) = Z
d~
da'2"~+2k
with a it ! : I]m' t, aijtt E KN such that m _< ~ t / t := ~,2n+2k j = t aij, A..~i=l ?7t,,i < m + 2k. and the n u m b e r of s u m m a n d s is bounded by ( 4 C ( N ) ) k m ( m + 2 ) . . . (m + 2k - 2). "
Choice of M : Choose first a number M ' > 2 large enough t h a t 2 ~-~ < "(~__A)89 ""' and let M be an integer such t h a t M > 32C(N)M'. For the norms one finds
I[ ( - V B ) k ( a ~ 99
<_ ( 4 C ( N ) ) k m ( m + 2)
llM,r ~--
.(m+2k
H l/ l! 2)max ]l aodal ... d a2n+2k i[ux
Now
re(m+2) .
(m+2k-2)=M . . .
.
m + 2k
'k~.
2
Mt
< M, k m
~ ( ~ m +1)...(~m +k- 1)
_
<M'k(~";k)k!<M'k2~+kk,<_(2M')k(N;l)~k,-by the choice of M . II II 11 Considering the norm of aoda 1 ... da2n+2 k note t h a t
,, II
~
[2_~_ I/
ai ~- l l aij ~j=l and t h a t
II
\ l=l
l!
b, :=
f/
ai(21_l)ai(21 )
II
ai(2t_l)aff20
In a"i(2/-1) a"i(2/) ira N
N + 1
a
Ill
E KM
because M > 2 N (the elements under consideration are linear combinations of monomials of word length at most 2 N < M ) . Thus H
l!
l! 9
If
aoda I ..da2n+zk i]M,~=
tl
z~+2k [-~-]
Ill] i----0 I=1
,, ,, N +1 ,, II ai(2L-1)ai(20 HA ~ II b~dbJ~... db2,~+2k IIM,~
f! with b~' = [ I ba, b.II E KM. Therefore
< (1 + n + k ) " M -('~+k) (n +1 k)! II b~db~'... db"2~+2k HM,r-
227
and one concludes that tl ,,
,,
(
,,
N
~zL~l
II aodal ...da2n+2k IIM,~_< \ ~ - - ~ ]
(1 + n + k)rM - ( n + k ) (n + k)!
Taking into account that 2n-I-2k
Z
/!
l!
m
@ ] -> ~2 V "~' -(2,~ + 2k + ,) -> 7
- (2n + 2k + 1)
i=0
and putting together the estimates obtained so far yields II ( - V B ) k ( a~
da2n IIM,r~
< (4C(N))k(2M,)k (_~_.~_) (2n+2k+U (1 + n + k)"M -('~+k) - -k~
(n + k)!
=(l+n+k)
r S C ( N ) ( N~ +) I
2
M ~, ]1 ~ k ( N ~ )+ I
_< C'(1 + n + k)"Ck4"M -n
2n+1 M - n
< k! (n + k)! -
k!
(n + k)!
for some constants C, C', C < 1. Thus finally
II ( - V B ) k~ IIM,r~ ~ I~fil II a~ (~-~ I~1)sup II a~ fl
fl
''' da2~n IIM,~
.. . d a p IIM,r~ k~
Nnn!(ll ~ IIg,o +,)C'(1 + n + k)rCk4nM -n (n + k)~ < _< C'(1 + n + k)~C k
(11~ IIN,0 + d
and the norm of the total sum can be bounded by
I[ ~ _ , ( - V B ) kw IIM,~< - C'
(1 + n + HrC k
(ll ~ IIN,o + d
k=O
-<
(} + 17c k c 'n~
(II ~ llN,o+d -< C"(ll ~ lJN,o+~) []
228
L e m m a 11.23: The notations are those of 11.14 and the remarks after 11.18. The natural projections qis) lim X,((F[Fn]K~ ) lim X,(Rr --+N
R lira +-N
---~N
X*(RgA[F,~](K~,N))
are quasiisomorphisms. Proof: Consider the comutative diagram q~s
lim_~NX. (Rr
lim~ N X . (~[Fn] KN )
)
)
lim_+N CC, (g[F,])(KN.N)
lim_+N CC,/F2CC,(r
qis
The horizontal maps are quasiisomorphisms by the remarks after 11.18. It suffices to show that 7r' is a quasiisomorphism which follows from 11.20 and 11.22. The cohomological case is similar. []
C o r o l l a r y 11.24: Let A be one of the algebras families of compact subsets of r
ll(Fn),C~(Fn).
Let KN be the corresponding (see 11.14). Then there are isomorphisms
HCl,C(ll(Fn)) ~_ ~mH,(X,(r * HC,lc (C~(Fn)) ~- limH,(X,(r -+N
and in the cohomological case short exact sequences
H * - - 1 (X * (r
-'+ HC~c(II(F,~)) -+ lim H * (X * (qA[Fn]KN(Z')))+ 0
0 -+ lim
1
0 - + lira
1H*-I(X*(gA[F,~]KN(C.))) -+ HCi* (C:(F,~)) ~ lim g*(x*(r
+..-N
~--N
Proof: This follows from 11.17 and 11.23.
+--N
+--N
--+ 0
229
Recall the cyclic cohomology of group rings. For a (torsion free) group F, the cyclic cohomology of g[F] decomposes as a direct product HC*(r
-~ H*(F,r
x l ~ HC*(r
of the group cohomology of F and of groups labeled by the (nontrivial) conjugacy classes {x} of F. For x E F let (x) be the infinite cyclic subgroup generated by x. Let, N. be the centraliser of (x) in F and let 1 - + (x) -+ N= -+ S~ -+ 1
be tile associated central extension with corresponding class ~ E H2(S~, 2~). Then is a module under H*(Sz, 2E) | (F and the multiplication the group HC*(r with ~ corresponds to the S-operation on cyclic cohomology under the isomorphism above. If F = F,, is a free group, the groups Sx associated to the nontrivial conjugacy classes are all finite cyclic, so that the action of the corresponding class ~x on HC* (~[F])(x) is zero. Thus all contributions to the cyclic cohomology of ~[F,] from nontrivial conjugacy classes are annihilated by the S-operation and therefore YP*(qJ[Fn]) ~- H*(X*(~[Fn])) ~- H*(F,~, q~)
This will now be carried over to the topological setting. Proposition
11.25:
The local cyclic (co)homology with compact supports of the Banach algebra P(Fn) of the free group on n generators is given by HCl*C(ll(yn)) ~-
{~ *=0 Cn , = 1
H C ~ ( l l ( F n ) ) ~-
{~ ~
*=0 *= 1
Proofi
For g E F,, denote by 19] its word length with respect to the generators Q , . . . , tn, t~-l,..., tn 1. Consider the X-complex X,(r X0(ff~[Fn]) = r
XI(r
Here
= (~=1 6J[F,,]dti
This complex decomposes according to the different conjugacy classes of F,: X0(r
=
+ + > X0(r
= +<x> Xl(e[Yn])<=> :
=
0+>
+<=> (@~<=) Cz)
51Czt -ldt )
230
For z E F~, z r e let X l , . . . , X r n be the (finite) set of elements of minimalword length in the conjugacy class ofz. For fixed i, 1 < i < m the set Y/(z) := {y[yxiy -1 = z} contains a unique element Yi of minimal word length. Define an operator X* : X * ( ( ~ [ F n ] )
--~ X * + l ( ( [ ~ [ i b ~ n ] )
by
Xo(z) := - - -
m
:riy[-ldyi i~1
I~@.,ati I ,I
]at~[ > ]a[
0
a = t[ 1
xl(adti) := { Then it is easily verified that
XOOx, + O x . o x = I d - P where P is the projection onto the complex X,(~[F,~])<e> associated to the trivial conjugacy class under the decomposition above. Thus the operator X provides an explicit chain homotopy annihilating the contributions from the nontrivial conjugacy classes to the cohomology of (~[F,~]. The complex X,(~[Fn])(e) is easily identified: Xo((IJ[F,~])<e ) = Ce and Xl(qJ[Fn])<e> = (~i~1 6Jt•ldti 9 The differentials in this complex are zero. To carry out the analogous calculation in the topological context, the algebras 9 [F,~]KN(t, ) C ll(F~) have to be identified. Recalling that
KN(I1) = { Z a g u 9 1 1 g I _< N, E l a g I }- N +
9
it is not difficult to show that ~---
- 9
~ O()}
g
and that the norm on this algebra is equivalent to "
a
"N'•"
E
ag
(N;1) - -
II '
9
To check the continuity of the homotopy operator ~ it suffices, as one works with /Lalgebras, to consider elements of the form a = ug G Xo, adb = ug, dut~ 6 X1. As the operator X1 increases the word length of such an element by at most one and all its coefficients ilg'tl -I are bounded, it extends to a bounded operator Xa : Xa(~[F,~]Ku) ~ X0(~[F,~]Ku) for all N. Concerning the operator X0 let for z E Fn be z = yixiYi -1, i m 1 , . . . , m be presentations as considered before. Then
231
one finds some i, 1 < i < m such that z = yixiy~ 1 is a minimal presentation of z, i.e. Izl = IXil + 21yil. Therefore 21yil < Izl and for the other elements yj,j ~ i one easily verifies ]yj[ <_ [Yi[+ [xi[ so that 21yjl ~_ 3[z[. Thus [] Xo(z)
t[M<_~ 1 ][ xiy~_ldy i [[M_< maxi [I xiy[ldyi [[M~ i=1
< mazilyi[ ;rovided that
< 21z[ <
Therefore the operator
x: x , ( e [ G ] ~ . ) --+ X.+~(r is continuous and the homotopy formula for X implies that not the individual complexes X.(r but their direct limit is quasiisomorphic to the finite dimensional complex X . (r162 lira X,(r
---~N
qi~> lira X,(r --4N
= X,(r
From this the conclusion follows. []
Index of Notations: admissible Fr4chet algebras
1.13
analytic cyclic (co)homology asymptotic cyclic cohomology asymptotic morphism asymptotic parameter space
5.16 6.4 1.5 1.1
bivariant analytic cyclic cohomology bivariant analytic X-complex bivariant asymptotic cyclic cohomology bivariant asymptotic X-complex bivariant Chern character bivariant X-complex bivariant differential graded X-complex Bott element Bianchi identity
5.16 5.15 6.4 6.4
Cartan homotopy formula Chern character cohomological analytic X-complex cohomological asymptotic X-complex cohomological X-complex compact supports composition product cone of a chain map connection Cuntz-Quillen projection PCQ curvature derivation lemma DG-modules differential graded X-complex Dirac element exterior product
10.1,10.2,10.5 2.2 2.5 8.22 1.4 4.3, 4.4, 4.10, 4.11, 5.21,6.12 2.8, 5.29, 6.17, 6.19 5.15 6.4 2.2 11.8 2.3, 5.15, 6.5 9.6 11.18 3.11, 5.25 1.3 7.1 2.4 2.5 9.3 8.7, 8.14, 8.15
233
Fedosov product finite supports first excision theorem formal inductive limit
3.6 11.15 9.6 5.4
Grothendieck approximation property
11.16
HC-invertible HC-equivalence homological analytic X-complex homotopy of asymptotic morphisms homotopy invariance I-adic filtration linear asymptotic category linear asymptotic homotopy category local cyclic cohomology
6.7 6.7 5.15 1.10 5.20,6.15 3.4 1.11 1.11 11.8, 11.15, 11 - 2
locally entire cochain
5.26
mapping cone multiplicative closure
9.6 1.12
periodic de Rham complex
2.1
Puppe exact sequences
9.6
reduced bivariant X-complex reduced cohomological X-complex reduced X-complex
2.2 2.2 2.2
second excision theorem slant product
9.9 8.23, 8.24
smooth linear asymptotic category
1.11
smooth linear asymptotic homotopy category
1.11 1.13
"small"
234
stable B o t t element
9.3
stable excision
9.8
stable periodicity tl~eorem strong excision axiom X-complex
9.4 11.2 2.2
Index of Symbols:
[A,B]~ " lim'Ai | I
s
Alg ~
1.11.
f, E X~
6.11
6.2
,f', (X)
9.10
2.7
)r* (X)
9.10
HC* HC~ HC~e HC~f
..4k
7.5
.A~
7.5
bs " Bott" CC~,~oe(A) ch ch(e) ch(u) (ch(e), ~} (ch(u), r C~(M, OM)
2.1 9.1
5.16 6.4 11.8 11.15
5.26
h~
4.3, 4.10
10.1, 10.2, 10.5
hy
4.11
3.13
i : RA --~ RRA
3.2, 5.11
3.13
i :A ~ A|
6.19
io
4.1
6.19
j
6.14, 8.16
7.7
k
6.14, 8.16
C(~) C ~ ( t~ + )o~
1.1
K cc c A
1.12
1.1
/C(A,U)
5.5
Cr162
6.3
s
4.1
C(X," lilm"Ai)
6.2
s
4.12, 4.13
C~(X," li~n"Ai)
6.2
m: R(A | B) -+ RA @ RB
8.10
Cr
6.2
N
2.1
|
li~n"Ai)
/(:(7-/)
8.18
T)
1.11
V
11.18
/)~
1.11
p
4.5
Dhomot
1.11
10.3
~:)h~
Dt
1.11 9.1
qA QA
eo, el
9.1
s
2.6
s F
ot
R
10.3 3.2 3.2
6.3
RA RAK RA(K,N )
1.20
7~A
5.6
5.6 5.6
236
R(~A)K R(~A)(K,N) Ti(~A)
5.12
~A
10.3
6.1
0
8.5
6.1
0
9.9
6.1
/~s
3.11
1.1 1.1 8
4.5
81,2
4.5
SA 5t X,A X*A
7.8
5.1 8.3
P pI
8.4, 8.13, 8.14
~r:RA~ A Q
6.3
3.2 3.2 8.2
2.2
T1
8.20,9.6
2.2
,F
3.11, 5.27
X*A
2.2
~1,2
4.5
X*(A,B)
2.2
~2 : K1A ~-~ KoSA
8.20
X*(A, B)
2.2
r
2.7
XSc(A) X~)o(A, B)
2.6 2.6
O30
1.3
,v.(RA)
3.10
~d
3.5
X,(RA)
3.12 3.12
X~,(A) X*(A) X* (A, B) X*(A) X~(A, B)
5.15 5.15 5.15 6.4
3.11, 5.27
5.1
~2A gt,PdR (A) f~IRA~
2.1,2.3
~(RAK)b,N f~RA
5.13
-IlN,m
5.6
6.4
2.1 3.7 6.1
9.3
)<
~e
10.3.
\
8.23, 8.24
~JSA
9.3
v
9.2,9.4
O~A
10.3.
8.7, 8.8, 8.14, 8.15
9.2
Bibliography:
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[co]
A.Connes, Noncommutative Geometry, Academic Press, (1995)
[co]
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[co2]
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[CH]
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[CM]
A.Connes, H.Moscovici, Cyclic cohomology, the Novikov Conjecture and Hyperbolic Groups, Topology 29, (1990), 345-388
[CQ]
J.Cuntz, D.Quillen, Algebra Extensions and Nonsingularity, Journal of the AMS 8(2), (1995), 251-289
[CQ]
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238
[D]
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[GO]
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[K]
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[M]
J.Milnor, On the Steenrod Homology Theory, in Ferry, Ranicki, Rosenberg, Novikov Conjectures, Index Theorems and Rigidity, LMS Lecture Notes 226, (1995), 79-96
[p]
M.Puschnigg, Explicit Product Structures in Cyclic Homology Theories, Submitted to Journal of K-theory
[z]
R.Zekri, Abstract Bott Periodicity in KK-Theory, Journal of K-Theory 3, (1990), 543-561