Asymptotic Cyclic Cohomology

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 _ 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~_

sup0_

SUPo suPo 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 ~ 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)
0 Re(z)

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( 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 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 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)
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 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 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 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


,,,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)
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
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

- k~, k-, < k~ We therefore find the estimate K t[ R f ( a ) [u,m 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 ~

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 _ 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

_
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

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(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) -~" 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 ---- { 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

----

"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 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) 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)


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 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 ) 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)

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]

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) 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~ 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

X0(r

= +<x> Xl(e[Yn]) :

=

0+>

+ (@~ ]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 +