Group C*-Algebras and K-theory

¡

Nigel Higson and Erik Guentner ¢ Department of Mathematics, Pennsylvania State University University Park, PA 16802

[email protected] £ Department of Mathematical Sciences University of Hawaii, Manoa 2565 McCarthy Mall, Keller 401A Honolulu, HI 96822 [email protected]

Preface

These notes are about the formulation of the Baum-Connes conjecture in operator

algebra theory and the proofs of some cases of it. They are aimed at readers who ¤ have some prior familiarity with -theory for ¥§¦ -algebras (up to and including the Bott Periodicity theorem). I hope the notes will be suitable for a second course in operator ¤ -theory. The lectures begin by reviewing ¤ -theory and the Bott periodicity theorem. Much of the Baum-Connes theory has to do with broadening the periodicity theo- rem in one way or another, and for this reason quite some time is spent formulating and proving the theorem in a way which is suited to later extensions. Following that, the lectures turn to the machinery of bivariant ¤ -theory and the formulation of the Baum-Connes conjecture. The main objective of the notes is reached in Lec- ture 4, where the conjecture is proved for groups which act properly and isometrically on affine Euclidean spaces. The remaining lectures deal with partial results which are important in applications and with counterxamples to various overly optimistic strengthenings of the conjecture. Despite their length the notes are not complete in every detail, and the reader will have to turn to the references, or his own inner resources, to fill some gaps. In ad- dition the lectures contain no discussion of applications or connections to geometry, topology and harmonic analysis, nor do they cover the remarkable work of Vincent Lafforgue. For the former see [7]; for the latter see [62, 44]. The notes are based on joint work carried out over a period of many years now with many people: Paul Baum, Alain Connes, Erik Guentner, Gennadi Kasparov, Vincent Lafforgue, John Roe, Georges Skandalis and Jody Trout. It is a pleasure to thank them all. I am especially grateful to Erik Guentner for writing the first draft of these notes and for his valuable assistance throughout their creation. Both authors were partially supported by NSF grants during the preparation of this paper. Nigel Higson 138 Nigel Higson and Erik Guentner 1 K-Theory

In the first three lectures we shall be developing machinery needed to formulate the

Baum-Connes conjecture and prove some cases of it. We shall presume some prior ¤ familiarity with ¥¨¦ -algebra -theory, but we shall also develop a ‘spectral’ picture

of ¤ -theory from scratch. In Lecture 1 we shall prove the Bott periodicity theorem in ¤

¥¨¦ -algebra -theory in a way which will be suited to generalization in subsequent lectures.

1.1 Review of K-theory ¤

We begin by briefly reviewing the rudiments of ¥§¦ -algebra -theory, up to and ¤ including the Bott periodicity theorem. As the reader knows, ¥©¦ -algebra -theory is a development of the topological ¤ -theory of Atiyah and Hirzebruch [4]. But the

basic definition is completely algebraic in nature:

¤   Definition 1.1. Let be a ring with a multiplicative unit. The group is the

abelian group generated by the set of isomorphism classes of finitely generated and

     projective (unital, right) -modules, subject to the relations .

Remark 1.1. Functional analysts usually prefer to formulate the basic definition in

 terms of equivalence classes idempotents in the matrix rings "! . This is be- cause in several contexts idempotents arise more naturally than modules. We shall

use both definitions below, bearing in mind that they are related by associating to an

!

 %'#(

idempotent #%$& ! the projective module .

¤

The group  is functorial in since associated to a ring homomorphism

,)+-(. *

)+* there is an induction operation on modules, . ¤ 

Most of the elementary algebraic theory of the functor  is a consequence

of a structure theorem involving pull-back diagrams like this one:

/10

/

87 7 87 87

4

:9 ¡ ¡>=@? ?A¡ ¡

;$<  > AB

65 . 0

/32  

¡

4

/ * 2

Theorem 1.1. Assume that in the above diagram at least one of the two homomor-

¡

 

phisms into * is surjective. If and are finitely generated and projective mod-

¡ ?E ?A¡ ¡

DC  )  *

ules over and , and if is an isomorphism of modules,

¦ ¦

then the -module

8F F JF F

9 =

¡ I ¡ ¡

665 G$H   -KLM -'KNB

is finitely generated and projective. Moreover, up to isomorphism, every finitely gen-

OP erated and projective module over has this form. Group C*-Algebras and K-theory 139

This is proved in the first few pages of Milnor’s algebraic ¤ -theory book [49].

The theorem describes projective modules over in terms of projective modules

¡

over , projective modules over , and invertible maps between projective mod-

¤

*© ules over * . It leads very naturally to the definition of a group in terms of

invertible matrices, but at this point the purely algebraic and the ¥©¦ -algebraic theo-

ries diverge, as a result of an important homotopy invariance principle.

9

¥Q¦ Q R KS ¥Q¦

Definition 1.2. Let be a -algebra. Denote by the -algebra of contin-

9

R KT

uous functions from the unit interval  into .

JU

 ¥§¦

We shall similarly denote by the -algebra of continuous functions from

U

a compact space into a ¥Q¦ -algebra .

¥Q¦ 

Theorem 1.2. Let be a -algebra with unit. If is a finitely generated and pro-

9

R KT

jective module over § then the induced modules over obtained by evaluation

9 9

R KT K($X R KS OP at RV$W and are isomorphic to one another. As a result, ¤ -theory is a homotopy functor in the sense of the following defini-

tion: ¥§¦

Definition 1.3. A homotopy of Y -homomorphisms between -algebras is a family

87

9

 `<$a R KT `b,)_Zc[

of homomorphisms Z\[]C ^)_* ( ), for which the maps are

7

$d ¥§¦

continuous, for all . A functor  on the category of -algebras is a homotopy [

functor if all the homomorphisms Z in any homotopy induce one and the same map

[

Z ]Ce >)+ *§

 .

¤ ¤

We shall now define the -theory group  .

¦ ¦

¥ f!  ¥

Definition 1.4. Let be a -algebra with unit. Denote by the -algebra

I

g h(i;! 

of g matrices with entries in and denote by the group of invert-

 h(ij!  h(ij!lknm  ible elements in ! . View as a subgroup of each via the

embeddings

p

oR

o,)

Rrqsut

¤ w

 v h(i x

Denote by the direct limit of the component groups ! :

¤

>y{z}| v h(i>! x

~

)

t

h(i 

!

Remark 1.2. This is a group, thanks to the group structure in , and in fact an

€‚ € wn

„ƒ „ƒ



  abelian group since  is homotopic to , and hence to . Returning to our pullback diagram and Theorem 1.1, it is now straightforward to derive all but the dotted part of the following six-term ‘Mayer-Vietoris’ exact

sequence of ¤ -theory groups:

¤ w ¤ w ¤  ¤ 

¡

†  *§  / / (1)

O

¤ ¤ ¤

¤ 

¡

†  

*© o o t 140 Nigel Higson and Erik Guentner

The diagram is completed (along the dotted arrow) as follows. Consider first the

pullback diagram

ˆ‡ ˆ‡

  0

/ /

4

/ 0

2

ˆ‡

 ˆ‡ 

¡

 * 

4 /

2 involving algebras of functions on the circle ‡ . The Mayer-Vietoris sequence asso-

ciated to it,

¤ w ˆ‡ ¤ € 8‡ ¤  8‡ ¤Š  8‡

¡

x x‰ x * x / / (2)

O

8‡ 8‡ ¤ 8‡ ¤ ¤ ¤ 8‡

¡ 9

* x‰ x x

x o o

‡ K$ maps to the Mayer-Vietoris sequence (1) via the operation ‹ of evaluation at ,

and in fact this map is the projection onto a direct summand since ‹ has a one-sided

inverse consisting of the inclusion of the constant functions into the various algebras of functions on ‡ . The complementary summands are computed using the following

two results: ¥Q¦

Theorem 1.3. Let be a -algebra. The kernel of the evaluation homomorphism

¤  ˆ‡ ¤ w

‹\C ŒM) 

¤

 P is naturally isomorphic to . O

This is a simple application of the partial Mayer-Vietoris sequence (think of

8‡

9

 Q R KS

as assembled by a pullback operation from two copies of ). ¥Q¦

Theorem 1.4. Let be a -algebra. The kernel of the evaluation homomorphism

¤ ˆ‡ ¤

‹\C ŒM) 

¤ is naturally isomorphic to  .

This is much harder; it is one formulation of the Bott periodicity theorem.

But granting ourselves the result for a moment, we can complete the diagram ¤ (1) by the simple device of viewing its horizontal reflection (with the -groups

on the top) as a direct summand of the diagram (2). The required connecting

¤ ¤

*© ) 

map ŽC appears as a direct summand of the connecting map

¤ 8‡ ¤ 8‡

ŽC x>) x

* .

The full Mayer-Vietoris sequence is a powerful computational tool, especially

U ¤Š U

) x for commutative algebras. For example it implies that the functors , constitute a cohomology theory on compact spaces (as in algebraic topology). A simple consequence is the formula

Group C*-Algebras and K-theory 141

¡

¤ w 8‡ ¤  ¤ w

xM‘ ‰   which is a perhaps more familiar formulation of Bott periodicity. Let us conclude our review of ¤ -theory with a quick look at the proof of Theo-

rem 1.4. The launching point is the definition of a map

’

¤ w ¤ ˆ‡

C j) Œ



by associating to the class of an idempotent #$& f! the element

J• •

“E”

~

#– K #¨

M (3)

8‡

Πin h(i>! . The following argument (due to Atiyah and Bott [6]) then shows

that this Bott homomorphism is an isomorphism onto the kernel of the evaluation

’

¤ ˆ‡ ¤

‹cC  map x ) . The key step is to show is surjective; the proof in injectivity is a minor elaboration of the surjectivity argument3 and we shall not comment on it further. By an approximation argument involving trigonometric polynomials the proof of

surjectivity quickly reduces to showing that a polynomial loop of invertible matrices

J• • •› 

“

9 9

›

\˜—  — '™š™S™L — — $< u! 

which defines a element of the kernel of the evaluation map must lie in the image of

’

8• “

. By elementary row operations, the loop  is equivalent to the ‘linear’ loop

¤S¥

¥

›¡ ›

— — — —

•

tStSt

~

• J•

K R R

œ

9

tStSt

 f* MžŸ

¦

Ÿ

•

tLtštLtštLtLtštLtštLtLtšt¢tLtštLtšt

~

K R£R

tStSt

•

œ

* §K

for suitable matrices and . Evaluating at and bearing in mind that is in q

the kernel of the evaluation map we see that ¨"* is path connected to (in some

œ 

suitable h(ij© ) and so is equivalent to

¡

• • J•

ª

~

 *©M¥  q ¥( M "f*§ t

The final step of the argument is for our purposes the most interesting, since in in-

8•

ª 

volves in a crucial way the spectral theory of elements in ¥§¦ -algebras. Since is

• ‡ J•

¡

$ M

invertible for all the spectrum of ¥ contains no element on the line Re

# ¥

in « . If denotes the idempotent associated to the part of the spectrum of to the

8• ª right of this line (obtained from the Riesz functional calculus) then  is homo-

topic to the path

J• •

”

“

9

~

M#  K #¨ ’ the ¤ -theory class of which is of course in the image of . This concludes the proof. In the following sections we shall recast the definition of ¤ -theory and the proof of Bott periodicity in a way which brings spectral theory very much to prominence.

As we shall eventually see, this is an important first step toward our principal goal of ¤

computing -theory for group ¥§¦ -algebras. ¬ As Shmuel Weinberger puts it, uniqueness is a relative form of existence. 142 Nigel Higson and Erik Guentner

1.2 Graded C*-Algebras

To proceed further with ¤ -theory we shall find it convenient to work with graded

¥¨¦ -algebras, which are defined as follows.

¥Q¦ Y ­

Definition 1.5. Let be a -algebra. A grading on is a -automorphism of

¡

6K

satisfying ­ . Equivalently, a grading is a decomposition of as a direct sum of

Q° 

Y ®o f ¯8 Ž¯

two -linear subspaces, , with the property that k , where

87 7

±

93²

­ ©

$³´€µ . Elements of (for which ) are said to be of even grading-

J7 7

~

­ >

degree while elements of (for which ) are of odd grading-degree.

Y ­X

Example 1.1. The trivial grading on is defined by the -automorphism id, or

' R equivalently by setting and .

In fact, we shall require only a very small collection of non-trivially graded ¥©¦ - algebras, among which the following two are the most important.

Example 1.2. Let ¶ be a graded Hilbert space; that is, a Hilbert space equipped with

'¶ ¥¨¦ ¸ ¶d

an orthogonal decomposition ¶·„¶ . The -algebras of compact

¶d ¶

operators and ¹ of bounded operators on are graded. To describe the grading,

I

¶ µ µ think of an operator º on as a matrix of operators. We declare the diagonal

matrices to be even and the off-diagonal ones to be odd.

J¼

 ¥Q¦

Example 1.3. Let »˜¥ , the -algebra of continuous, complex-valued func- ¼

tions on which vanish at infinity, and define a grading on » by the decomposition

w 8¼

o5 B;–5 B

»¨¥ even functions odd functions

t

¾ ¾

~

j,)¿½ 

The grading operator is the automorphism ½ .

¤ w 8¼

¥  Warning: In -theory it is customary to introduce the ¥§¦ -algebra in con-

nection with the operation of ‘suspension’. But in what follows the algebra » will

play a quite different role.

Definition 1.6. A graded ¥Q¦ -algebra is inner-graded if there exists a self-adjoint

unitary ‹ in the multiplier algebra of which implements the grading automorphism

­ on :

J7 7 7

9

>'‹ ‹ $H

­ for all .

‹QÀK

Examples 1.5 The trivial grading on a ¥Q¦ -algebra is inner: take . In addi-

¶d ¹ ¶d ‹

tion the gradings on ¸ and are inner: take to be the operator which is

~

Á q Á » q on and on . However the grading on is not inner.

All the fundamental constructions on ¥§¦ -algebras have graded counterparts, and

we shall require below some familiarity with the notion of tensor product for graded ¥§¦ ¥¨¦ -algebras. As is the case with ungraded -algebras, tensor products of graded

¥¨¦ -algebras are defined as completions of the algebraic graded tensor product. And as is the case in the ungraded world, there is not usually a unique such completion.

Group C*-Algebras and K-theory 143 7

Let us introduce the symbol  defined by

7

9

7 $H

R if

 aÂ

7

9 $H

K if

t

7 7 7 7

$f $"  An element $f is homogeneous if or . Keep in mind that is

defined only when 7 is a homogeneous element.

Ä

* ¥Q¦ VÃ *

Definition 1.7. Let and be graded -algebras. Let be the algebraic ten- *

sor product of the linear spaces underlying and . Define a multiplication, involu-

Ä * tion and grading on à by means of the following formulas involving elementary

tensors:

J7 J7 7 7

Ä Ä Ä

¡ ¡ ¡ ¡€9

~

à —  à — M KLŒÅLÆ ÅLÇ Ã — —

0

2

J7 7

Ä Ä

~

¦ ¦ ¦

ÅLÇSÅLÆ

à —T  KL à —

J7 7

Ä

9 9

à —TM  — µ€

 mod

7 7 7

9 9 9 9

¡

¡

$f — — $f*

for all homogeneous elements and — . (The multiplication

Ä

*

and involution are extended by linearity to all of à .)

Ä *

The construction of VÃ satisfies the usual associativity and commutativity

Ä Ä

*ž)È*dà rules but with occasional twists. For example, an isomorphism ŠÃ is

defined by

7 7

Ä Ä

~ ~

à —G, )

KLŒÅLÇSÅLÆɗ€Ã (4) t

Definition 1.8. The graded commutator of elements in a graded ¥©¦ -algebra is given

by the formula

7 7 7

9 9

~ ~

 —ÉE — K¢ — Å¢ÇTÅ¢Æ

on homogeneous elements (this is extended by linearity to all elements).

¥Q¦ ZQCE Ê)È¥ ˊCE*¿)Ì¥ Lemma 1.1. If ¥ is a graded -algebra and if and are 4

graded Y -homomorphisms whose images graded-commute (meaning that all graded

87

9

Z  Ë —TÍ Y

commutators  are zero) then there is a unique graded -homomorphism

7 87

Ä Ä

¥ Z 1Ë —T OP * — Ã

from à into which maps to .

¶fÃ-Ž¶ Example 1.4. Let ¶ be a graded Hilbert space and denote by the ordinary Hilbert space tensor product, but considered as a graded Hilbert space. The con-

struction of the lemma produces a graded Y -homomorphism from the tensor product

Ä

¶dLÃ ¹ ¶d ¹ ¶o-H¶H

algebra ¹ into which takes the homogeneous elementary ten-

‡

Ä º

sor à to the operator

‡

œ ª œ ª

~

- ,) - K¢xÅLÎTÅLωº

t Ð

A Ñ -homomorphism is graded, or grading-preserving, if it maps homogeneous elements to homogeneous elements of the same grading-degree.

144 Nigel Higson and Erik Guentner

Ä

* ¥§¦ *

Definition 1.9. Let and be graded -algebras and let à be their algebraic

-*

tensor product. The maximal graded tensor product, which we will denote by à ,

Ä

›

- * * Ã

or occasionally by à , is the completion of in the norm

ÇÓÒ

Ô‰Õ Ô Ô‰Õ Ô

7 87

Ä

9

¯ —Ó¯ Öx×eØ Z ÍË —Ó¯3

à ¯

where the supremum is taken over graded-commuting pairs of graded Y -homo-

* ¥©¦ ¥

morphisms, mapping and into a common third graded -algebra . - Warning: Our use of the undecorated symbol à to denote the maximal tensor product (as opposed to the minimal one, which we shall define in a moment) runs

counter to ordinary ¥Q¦ -algebra usage. In situations where the choice of tensor prod-

› -

uct really is crucial we shall try to write à .

ÇÓÒ -

Remark 1.3. It is clear from the definition that the tensor product à is functorial: if

ËÀC*Ú)ÜÛ Y

Z®CE a)Ù¥ and are graded -homomorphisms then there is a unique

7 J7

-— Y Z -ËÀCÝ -*Þ)Ü¥ -Û Z  -Ë —S

à à à Ã

graded -homomorphism à mapping to , for

7 —$d*

all $d and .

* ¥©¦

Example 1.5. If one of or is inner-graded then the ungraded -algebra under- -*

lying the graded tensor product à is isomorphic to the usual tensor product of

*

the ungraded ¥Q¦ -algebras underlying and . If say is inner-graded then the -*%)¿ f-f*

isomorphism VÃ is defined by

7 7

~

-—G,à ) ‹€Å¢Æn-"— t

We also note that the graded tensor product of two inner-graded ¥©¦ -algebras is

itself inner-graded. Indeed

.

‹ -ߋlà–$Háà ©Ã-*§já f- *© t

For the most part we shall use the maximal tensor product of graded ¥©¦ -algebras,

but occasionally we shall work with the following ‘minimal’ product:

Ä

* ¥Q¦ VÃ *

Definition 1.10. Let and be graded -algebras and let be their alge- *

braic tensor product. The minimal graded tensor product of and is the com-

Ä

*

pletion of ©Ã in the representation obtained by first faithfully representing and

Ä

¹ ¶d * ¶d:à ¹ ¶d ¹ ¶XÃ-߶d * as graded subalgebras of , and then mapping to as above. The minimal tensor product is also functorial, but from our point of view it has

some serious shortcomings. These will be explained in the next lecture.

Ä

¸ ¶d

Exercise 1.6 Show that the minimal and maximal completions of ŠÃ and

Ä

»(Ã are the same.

»Ã-Ž» »

Exercise 1.1. Describe the tensor product ¥§¦ -algebra (note that although

»(Ã-Ž»

itself is a commutative ¥Q¦ -algebra, the tensor product is not).

¸ ¶d -¸ ¶ â@>‘ ¸ ¶ -¶ âã

à à Exercise 1.7 Show that  . Group C*-Algebras and K-theory 145

1.3 Amplification

J¼

»®¥ 

The graded ¥¨¦ -algebra will play a special role for us. Using it we shall

¦ Y enrich, or ‘amplify’, the category of graded ¥ -algebras and -homomorphisms.

To do so we introduce two Y -homomorphisms, as follows:

ä

C»¨)+« åfC»¨)æ» -Ž»

and Ã

t

¤

ä

Rw ¥§¦

The first is defined by ½ç½ . In the world of ungraded -algebras and - ä

theory is not so interesting since it is homotopic to the zero Y -homomorphism. But

ä ¥Q¦ as Y -homomorphism of graded -algebras is definitely non-trivial, even at the level of ¤ -theory (which we will come to in the next section). The defining formula

for å ,

JU JU U

9

å Cݽ M,)¿½ -QKj'K]Ãà -  »

is explained as follows. Denote by »cè the quotient of consisting of functions on

9

~ßé é 

the interval  (the quotient map is the operation of restriction of functions)

U ¾ ¾

èÚ$»‰è ,)

and denote by the function . If ½§$%» then we can apply the

U U

è -QKjK - è $d»‰è -ß»‰è

à Ã

functional calculus to the self-adjoint element à to obtain

U U

½ è -QKjK - èN;$ »‰è -Ž»‰è

à Ã

an element à .

åfC»")滧Ã-»

Lemma 1.2. There is a unique graded Y -homomorphism whose com-

è è

-Ž»¨)æ» -Ž»Ã Y

position with the quotient map »QÃ is the -homomorphism

U U

9

åfCݽd,)ê½ è -§Kj'K - èN

à Ã

éë

OP

for every R .

è è

-Ž»f)+» -Ž»Ã Exercise 1.8 Show that the intersection of the kernels of the maps »©Ã

is zero. This proves the uniqueness part of the Lemma.

“ œ

Remark 1.4. If the self-adjoint homogeneous elements and in » are defined by

J¾ F¡ ¾ ¾AF¡

“ œ

9

2 2

Ò j Ò

> and t

then

“ “ “ œ “ œ œ “

å M - å > -  -

à Ã

à and

t

“ œ ä

» å

Since and generate the ¥Q¦ -algebra , formulas involving and can often be œ verified by checking them on “ and .

Remark 1.5. Another approach to the definition of å is to use the theory of un-

bounded multipliers. See the short appendix to this lecture.

ä

å » The Y -homomorphisms and provide with a sort of coalgebra structure: the diagrams

146 Nigel Higson and Erik Guentner

» ì »

» -Ž» / à (5)

C ð ð {{ CC {{ CC {{ CC

{{ CC Sí

î {} !

» »

ì ì ìjï Ca = CC {{

CC {{

í Sí

î î ñ ñ CC {{

  C  {{

» -Ž» -Ž» » -Ž» » -Ž»

à à à Ã

í

î /

ì

“ œ

commute, as is easily verified by considering the elements and $d» .

¦

¥

Definition 1.11. Let be a graded -algebra. The amplification of is the graded - tensor product »j –»¨Ã .

Definition 1.12. The amplified category of graded ¥§¦ -algebras is the category whose

*

objects are the graded ¥Q¦ -algebras and for which the morphisms from to are

»j * ZQCÝ )+* the graded Y -homomorphisms from to . Composition of morphisms

and ËbCA*ò)ó¥ in the amplified category is given by the following composition of

Y -homomorphisms:

í

î

¡

ì

»j »j* ù ¥ õôEö}÷wø

/ » / / t

Exercise 1.2. Using (5) verify that the composition law is associative and that the »j §)ú

Y -homomorphisms obtained by taking the tensor product of the augmen-

ä tation CA»§)û« with the identity map on serve as identity morphisms for this

composition law. ¦ Remark 1.6. Most features of the category of graded ¥ -algebras pass to the am-

plified category. One example is the tensor product operation: given amplified mor-

¡ ¡ ¡

CÝ )ü* Z CA )ú*

phisms from Z and there is a tensor product morphism

¡ ¡ ¡

-Ž -* -  * Y »

à Ã

from à to (in other words a -homomorphism from into

¡

-* * Y

à ) defined by the composition of -homomorphisms

í

î

í Sí

î î

¡

¡ ¡

¡ ¡ ¡

ì

» -Ž à  »¨Ã- -Ž Ã

0

‘ 2

» - à - à * -*à »j -ß»j Ã

 ÷ / ÷ /

(the formula incorporates the transposition isomorphism (4)).

Exercise 1.3. Show that the tensor product is functorial (compatible with composi- tion) and associative.

1.4 Stabilization

A second means of enriching the notion of Y -homomorphism is the process of sta-

bilization. This is of course very familiar in ¤ -theory: stabilization means replacing

VÃ-Ž¸ ¶d ¥Q¦ a ¥¨¦ -algebra with , its tensor product with the -algebra of compact operators.

Group C*-Algebras and K-theory 147

?

¥¨¦ ©-§¸ ¶d

If is a trivially graded -algebra with unit then each projection in

?

%-6¸ ¶dŒ determines a projective module over (namely with the obvious

right action of ) and in fact the set of isomorphism classes of finitely generated

-modules is identified in this way with the set of homotopy classes of projections

¤ ¶d in "-u¸ . For this reason stabilization is a central idea in -theory.

Let us now return to the graded situation. There are Y -homomorphisms

¶d ¸ ¶dšÃ-¸ ¶d>)^¸ ¶d

«–)+¸ and

F F ý

defined by mapping ýD$W« to , where is the projection onto a one-dimensional,

¶fÃ-Ž¶ ¶

grading-degree zero subspace of ¶ , and by identifying with by a grading- ä degree zero unitary isomorphism. These play a role similar to the maps and å

introduced in the previous section. There is no canonical choice of the projection

F

‘

¶uÃ-߶ ¶ or the isomorphism  , and for this reason we cannot ‘stabilize’ the

category of ¥¨¦ -algebras in quite the way we amplified it in the previous section. But

at the level of homotopy the situation is better: ¶Hâ

Lemma 1.3. Let ¶ and be graded Hilbert spaces. Any two grading-preserving

¶dâ Y ¸ ¶d ¸ ¶Hâ@

isometries from ¶ into induce graded -homomorphisms from to

OP

which are homotopic through graded Y -homomorphisms.

¸ ¶d

As as result there are canonical, up to homotopy, maps «þ) and

¸ ¶d -¸ ¶d") ¸ ¶d

à . We could therefore create a stabilized homotopy cate-

* Y

gory, in which the morphisms from to are the homotopy classes of graded -

*HÃ-¸ ¶d

homomorphisms from to . We could even stabilized and amplify simul-

taneously, and create the category in which the morphisms between ¥V¦ -algebras

Y »j *&Ã-¸ ¶d and * are the homotopy classes of graded -homomorphisms from to . We won’t exactly do this, but the reader will notice echoes of this construction in the following sections.

1.5 A Spectral Picture of K-Theory

We are going provide a ‘spectral’ description of ¤ -theory which is well adapted to Fredholm index theory and to an eventual bivariant generalization. Actually our definition is a back formation from the bivariant theory described in [13, 14, 27] (it is also closely related to various other approaches to ¤ -theory).

For the rest of this section we shall fix a graded Hilbert space ¶ whose even and odd grading-degree parts are both countably infinite-dimensional. Unless ex-

plicitly noted otherwise we shall be working with graded ¥§¦ -algebras and grading-

preserving Y -homomorphisms between them.

9

*¨

Definition 1.13. We shall denote by  the set of homotopy classes of grading-

¦

¥ * preserving Y -homomorphisms between the graded -algebras and .

With this notation in hand, our description of ¤ -theory is quite simple:

148 Nigel Higson and Erik Guentner ¥§¦

Definition 1.14. If is a graded -algebra then we define

¤"

9

M% » ©Ã-¸ ¶d3

t ¤f

For the moment  is just a set, although we will soon give it the structure of ¤" an abelian group. But first let us give two examples of classes in  to help justify

the definition. Û Example 1.6. Take ÿÿ« . Let be an unbounded self-adjoint operator on the

graded Hilbert space ¶ of the form

p

¡

RõÛ

Ûa

Û R"s

k Û (in other words Û is a grading-degree one operator) and assume that has compact

resolvent. (For example, Û might be a Dirac-type operator on a compact manifold.)

The functional calculus

Ë¡ CA½d,)¿½ Ûb

¤f

Y Ë C]»¨)^¸ ¶d «c

defines a graded -homomorphism and hence a class in . ¤

Example 1.7. Suppose that is unital and trivially graded, so that the -theory

¤

group  of Section 1.1 can be described in terms of equivalence classes of

? 9J?

-¸ ¶d

projections in VÃ . If are two such projections, acting on the even and

'¶ X¶

odd parts of the graded Hilbert space ¶ , then the formula

p

?

½ R R

~

4

Ë CݽH, )

?

R ½ R s

» ŠÃ-¸ ¶d defines a grading preserving Y -homomorphism from to .

The second example is related to the first as follows: if Û is a self-adjoint,

grading-degree one, compact resolvent operator on ¶ then the family

¥¤l¡ ¤

9 9

Ë£¢ACݽH,)¿½ Ûb $W R KT

¤

4

Y Ë çK Y Ë

is a homotopy from the -homomorphism at to the -homomorphism

¤

? ? ?

  Û at R , where is the projection onto the kernel of .

Before reading any further the reader may enjoy solving the following problem.

¤f

‘

³ «\ Y

Exercise 1.9 Prove that  in such a way that to the class of the -

£ ۊk

homomorphism Ë of Example 1.6 is associated the Fredholm index of . ¤f

Let us turn now to the operation of addition on  . This is given by the direct

¡

Ë Ë Y sum operation which associates to a pair of Y -homomorphisms and the -

homomorphism

¡

Ë  Ë C»")¿ VÃ-Ž¸ ¶§u¶d

t ¶ (One identifies ¶¨Š¶ with by some degree zero unitary isomorphism to complete the definition; at the level of homotopy any two such identifications are equivalent.) Group C*-Algebras and K-theory 149

The zero element is the class of the zero homomorphism. To prove the existence of additive inverses it is convenient to make the following preliminary observation which will be important for other purposes as well. The proof is a simple exercise

with the functional calculus.

¦

¥ ËbCÝ»Ú) Û

Lemma 1.4. Let Û be any graded -algebra and let be a grading-

» Û Ë preserving Y -homomorphism. Adjoin units to and , extend , and form the uni-

tary element

p

¾

±

~

¦

–Ë ¾

±

s



ù

¦

Z¨§

in the unitalization of Û . The correspondence is a bijection between the set

¦

ù ˊC»)õÛ

of Y -homomorphisms and the set of unitary elements in the unitaliza-

K Û

tion of Û which are equal to modulo and which are mapped to their adjoints by

©¦ ¦

> ¦ OP

the grading automorphism: ­ .

¦

¥ Û Definition 1.15. If Û is a graded -algebra then by a Cayley transform for we

shall mean a unitary in the unitalization of Û which is equal to the identity, modulo

Û , and which is switched to its adjoint by the grading automorphism.

¤f ¦

Returning to the question of additive inverses in  , if is the Cayley trans- Ë

form of Ë then it is tempting to say that the additive inverse to should be repre-

¦ ¦ ¦ sented by the Cayley transform ¦ . But this is not quite right; we must also view

opp opp

-¸ ¶  ¶ ¶ as a Cayley transform for VÃ , where is the Hilbert space but with

the grading reversed. The rotation homotopy

p

¦

Ö `x Öxz `x1q

¦

~

s

֌z `x1q Ö `x ¦



opp

-¸ ¶ò®¶   Ã

is –ƒthen a path of Cayley transforms for connecting to





ƒ> ¡  , which is in turn connected to the identity.

Remark 1.7. In terms of Y -homomorphisms rather than Cayley transforms, the addi- Y tive inverse of Ë is represented by the -homomorphism

opp opp

Ë 'Ë;­§C]»")¿ -߸ ¶ 

à » obtained by composing Ë with the grading automorphism on and also reversing

the grading on the Hilbert space ¶ .

Remark 1.8. In the next lecture we shall give an account of additive inverses using

the comultiplication map å we introduced in the previous section.

Proposition 1.10 On the category of trivially graded and unital ¥©¦ -algebras the

¤f ¤

functor  defined in this section is naturally isomorphic to the -theory functor ¤ 

 introduced at the beginning of this lecture.

150 Nigel Higson and Erik Guentner ¤"

Proof. We have already seen that  is the group of path components of the space

ƒ¨ ¶d

of Cayley transforms for <-<¸ (we can dispense with the graded tensor product



¦

ƒ

¡

 ‹©

here since is trivially graded). If is the grading operator and if is a

¦ §K

Cayley transform then ‹ is a self-adjoint unitary whose spectral projection,

¦

9

¡

#6 ‹ KL

ƒ>

j

#ß  ‹ ¨-(¸ ¶d

is equal to the §K spectral projection of , modulo . Conversely

# "- ¸ ¶d

if # is a projection which is equal to modulo then the formula

¦

~

'‹ µl# qw

¶d

defines a Cayley transform for '-–¸ . We therefore have a new description of ¤f

the new  , as the group of path components of the projections which are equal

 -¸ ¶d

to # , modulo . We leave it to the reader to determine that the formula

~



#ß,)  #  # 

 is an isomorphism from this new component space to the usual

¤ ¤

 (the argument involves the familiar stability property of -theory).

¡

¥Q¦ «ß©« ý Vý ,)

Exercise 1.11 Denote by  the -algebra with grading operator

¡

rý

ý (this is an example of a Clifford algebra — see Section 1.11). Show that if

¤ ¤f

‘

 Ã- 

is trivially graded and unital then  . *

Exercise 1.4. Show that if a graded ¥§¦ -algebra is the closure of the union of a

*

direct system of graded ¥Q¦ -subalgebras then the natural map

¤f ¤f

~

*  ) *§ y{z}|

~

)

-¸ ¶H

is an isomorphism. (Hint: Show that every Cayley transform for *&Ã is a limit

-¸Ã ¶H of Cayley transforms for the subalgebras * .)

1.6 Long Exact Sequences

Although it is not absolutely necessary we shall invoke some ideas of elementary homotopy theory to construct the ¤ -theory long exact sequences. For this purpose

let us introduce the following space:

¥Q¦ ! 

Definition 1.16. Let be a graded -algebra. Denote by the space of all

» Ã-¸ ¶d

graded Y -homomorphisms from into , equipped with the topology of

½M)^Ë ½ )^Ë Ë pointwise convergence (so that Ë iff in the norm topology, for

every ½r$b» ). Thus:

9

! Ž#"%$lØ » VÃ-¸ ¶dŒ

t

 Remark 1.9. As it happens, the space ! is a spectrum in the sense of homotopy theory—see for example [1]—but we shall not need the homotopy-theoretic notion of spectrum in these lectures.

Group C*-Algebras and K-theory 151



The space ! has a natural base-point, namely the zero homomorphism from

» -¸ ¶d

into à . It also has a more or less natural ‘direct sum’ operation

I

!  ! )&! 

¡

Ë Ë Y

which associates to a pair of Y -homomorphisms and the -homomorphism

¡

dZ ©Ã-¸ ¶–b¶d ¶–b¶ ¶ Z into . (One identifies with by some degree zero uni- tary isomorphism to complete the definition; at the level of homotopy any two such

identifications are equivalent.) It is of course this operation which gives the addition

¤" ]

! É operation on the groups ŠÚv . By a general principle in homotopy

theory the direct sum operation agrees with the group operations on the higher ho-

v ! É g'˜K

motopy groups ! , for .

v ! É As for the higher groups ! , they may be identified as follows. There is

an obvious homeomorphism of spaces

8¼

! !

‘

! ¥ ‰-f  ! 

)(

t

¼ J¼

! !

¥ †- 

Indeed by evaluation at points of we obtain from an element of !

¼

! 

a map from to ! which converges to the zero homomorphism at infinity, or in

‡ ¼

! ! 

other words a pointed map from the one-point compactification of into ! ,

!

 !

which is to say an element of ( . It follows that

¤f J¼

! !

vÝ! ! ŒŽv ! ÉŽ‘ ¥ ‰-f 



(

t

¤

¥Q¦

Definition 1.17. Let be a graded -algebra. The higher -theory groups of



are the homotopy groups of the space ! :

¤

9

>–v ! x g*'–R

! !

t

¤

 !  The space ! , and therefore also the groups , are clearly functorial in

. They are well adapted to the construction of long exact sequences, as the follow-

ing computation shows: ¥©¦

Lemma 1.5. If §)Ü* is a surjective homomorphism of graded -algebras then

 ! *§

the induced map from ! to is a fibration.

U ,+

Recall that a map ) is a (Serre) fibration if for every map from a cube (of

U ½ any finite dimension) into + , and for every lifting to of the restriction of to a

face of the cube, there is an extension to a lifting defined on the whole cube.

 ŠÃ-Ž¸ ¶d

Proof. Think of ! as the space of Cayley transforms for , and thus as

H)-! *§ a space of unitary elements. The proof that the map ! is a fibration is then only a small modification of the usual proof that the map of unitary groups

corresponding to a surjection of ¥§¦ -algebras is a fibration.

<).! *©

The fiber of the map ! (meaning the inverse image of the base-

0/ /  point) is of course ! where the ideal is the kernel of the surjection. So elemen- tary homotopy theory now provides us with long exact sequences

152 Nigel Higson and Erik Guentner

¤ ¤ ¤ 1/ ¤

 *©  *©

!lk ! !

/ !lk / / / /

tStSt tStšt ¤f

(ending at *§ ) as well as Mayer-Vietoris sequences

¤ ¤ ¤ ¤ ¤

¡

~ ~ ~ ~ ~

*© ) !  ) ! ‰ !  ) ! *© ) ) !lk

tštSt tStSt associated to pullback squares of the sort we considered in the first part of this lecture.

1.7 Products

A key feature of our spectral picture of ¤ -theory is that it is very well adapted

to products. Recall that in the realm of ungraded ¥§¦ -algebras there is a product

operation

¤ ¤ ¤

‰- *§M) "- *©

? ?

 -% 2L+ -#2L defined for unital ¥Q¦ -algebras by the prescription . This is the first in a sequence of more and more complicated, and more and more powerful, product operations, which culminates with the famous Kasparov product in bivariant ¤ -theory.

In our spectral picture the product is defined using the ‘comultiplication’ map å å that we introduced during our discussion of graded ¥§¦ -algebras. Using we obtain

a map of spaces

I

!  ! *©Ž)&! Ã-©*§

9

. ËNà;

by associating to a pair Ë the composition

í

î

 

~ ~Ó~ ~ ~e~

  ‘

» )æ»(Ã-Ž» ) ©Ã-߸ ¶d - à *dÃ-¸ ¶d ©Ã-*dÃ-¸ ¶d



ì

ù43 ù45

(in the last step we employ a transposition isomorphism and we also pick an isomor-

¶ -‚¶ ‘ ¶ Ã

phism  ). Taking homotopy groups we obtain pairings

¤ ¤V ¤ 

9

¯ ¯

‰- *§M) k ©Ã-(*§

as required.

¡

ç*Þç« Ë Ë

Example 1.8. Suppose that and that and are the functional cal-

¡ Û

culus homomorphisms associated to self-adjoint operators Û and , as in Exam-

¡ Ë

ple 1.6. Then the product of Ë and is the functional calculus homomorphism ¡

5

-q("qEÃÃ -Û for the self-adjoint operator Û . This type of formula is familiar from

index theory; in fact it is the standard construction of an operator whose Fredholm

¡ Û index is the product of the indices of Û and . It is this example which dictates

our use of the comultiplication å .

The various features of the product are summarized in the following two results. 6

To be accurate, the formula defines an essentially self-adjoint operator defined on the alge-

¢ £ 7 braic tensor product of the domains of 7 and .

Group C*-Algebras and K-theory 153 ¤ Proposition 1.1. The -theory product has the following properties:

(a) It is associative.

¾ ¤f ¤f

 8æ$ *©

(b) It is commutative, in the sense that if $ and , and if

¾ ¾

9 9

I I

C -*%)ê* - 8 M:8 Ã

à is the transposition isomorphism, then .

¦ ˊC*ÿ)_*§â

(c) It is functorial, in the sense that if Z§C ^)_ â and are graded

J¾ J¾

I I

Z -Ë; 8 MZ  Ë 8  Y OP

-homomorphisms then à .

¦ ¦ ¦

¾ ¤ ¤V€

¯

$  8b$ *§

Remark 1.10. In item (b), if we take and then the appropriate

J¾ ¾

¯

9

I I

~

M K¢ 8

formula is 8 .

¦

¤f «N

Proposition 1.2. Denote by K $ the class of the homomorphism which maps

½ Rx#%$d¸ ¶d #

the element ½&$ » to the element , where is the orthogonal projec-

tion onto a one-dimensional, grading-degree zero subspace of ¶ . If is any graded

¾ ¤f ¾

I

‘

¥¨¦ $ *§ «Ã-V* * K

-algebra and if then under the isomorphism  the class

¾ P corresponds to . O

1.8 Asymptotic Morphisms

We are now going to introduce a concept which can be used as a tool to compute ¤

-theory for ¥Q¦ -algebras. Other tools are available (for example Kasparov’s theory

or the theory of ¥Q¦ -algebra extensions) but we shall work almost exclusively with

asymptotic morphisms in these lectures.

* ¥§¦

Definition 1.18. Let and be graded -algebras. An asymptotic morphism from

9<;

* Z\[VCc )û* ` $6 K 

to is a family of functions , satisfying the continuity 7

condition that for all $<

J7

9<;

~

[

;Ce{K M)+* )+Z

`N, is bounded and continuous

7 7 7

9 9

¡ ý&$d«

and the asymptotic conditions that for all $H and

87 87 J7 7

= >

>

¡

¡

~

>?

[ [ [

 1Z  Z Z

J7 J7 J7 7

¡

¡

~ ~

[ [ [

  Z   Z Z

9 ;

>

)+R `M)

J7 7

> as

>

~

t

[ [

 ý  ýeZ Z

@

J7 87

~

¦ ¦

[ [

 Z  Z *

If and are graded we shall require that in addition

J7 J7

;'9

~

[ [

­ Œj)+R Œ Z Z `N) ­ as

where ­ denotes the grading automorphism. We shall denote an asymptotic mor-

BA ADC * phism with a dashed arrow, thus: Z C€ .

In short, an asymptotic morphism is a one-parameter family of maps from to Y * which are asymptotically -homomorphisms. We shall postpone for a little while the presentation of nontrivial examples of asymptotic morphisms (the main ones are given in Sections 1.12 and 2.6). As for

154 Nigel Higson and Erik Guentner

*

trivial examples, observe that each Y -homomorphism from to can be viewed as * a (constant) asymptotic morphism from to . It is usually convenient to work with equivalence classes of asymptotic mor-

phisms, as follows:

¡

Z C EA AFCd*

Definition 1.19. Two asymptotic morphisms Z , are (asymptotically) 7

equivalent if for all $H

¡

87 J7

~

y{z}| Z  Z  R

[ [

[HGJILK K

t

K K

MANADC'*

Up to equivalence, an asymptotic morphism Z§C† is exactly the same thing as a Y -homomorphism from into the following asymptotic algebra associated

to * .

¥§¦ z *© ¥Q¦

Definition 1.20. Let * be a graded -algebra. Denote by the -algebra of

9N;

 * z *§

bounded, continuous functions from {K into , and denote by the ideal

¦ *

comprised of functions which vanish at infinity. The asymptotic ¥ -algebra of is O

the quotient ¥Q¦ -algebra

O *§>'z *§Œ´:z *©

t

O

*© Y Z

If Z§Ce ) is a -homomorphism then by composing with a set-theoretic

*© *§

section of the quotient mapping from z to we obtain an asymptotic mor-

*

phism from to ; its equivalence class is independentO of the choice of section.

z *§

Conversely O an asymptotic morphism can be viewed as a function from into ,

Y

and by composing with the quotient map into *© we obtain a -homomorphism

*© from to which depends only on the asymptotic equivalence class of the asymptotic morphism.

Suppose now that we are given an asymptotic morphism

Z§Ce ©Ã-¸ ¶HPANADCV*dÃ-¸ ¶d

t

ˊC»¨)+ -¸ ¶d Y

If à is a graded -homomorphism then the composition

-¸ ¶d * -Ž¸ ¶d

à Ã

» ù

/ _ _÷ _/ (6)

» * -¸ ¶d

is an asymptotic morphism from into à .

¥§¦ Û

Lemma 1.6. Every asymptotic morphism from » into a graded -algebra is

» Û

asymptotic to a family of graded Y -homomorphisms from to .

» Û

Proof. We saw previously that a Y -homomorphism from to is the same thing as

Û Û K Û

Cayley transform for — a unitary O in the unitalization of (equal to modulo )

which is switched to its adjoint by the grading automorphism. In the same way, by

making use of the asymptotic algbra Û  we see that an asymptotic morphism from Û

» to is the same thing, up to equivalence, as a norm continuous family of elements

U

K Û [ in the unitalization, equal to modulo , which are asymptotically unitary and Group C*-Algebras and K-theory 155 asymptotically switched to their adjoints by the grading automorphism. But such an

‘asymptotic Cayley transform’ family can, for large ` , be altered to produce a family U

of actual Cayley transforms: first replace [ by

U U

K

¦

[ [

 ­ Œ + 

[ µ

(this ensures that the grading automorphism switches the element and its adjoint) and

then unitarize by forming

¡

¦

¦

[ [ [

Q+ + +  0

[

2

U ¦

[ [ [ `

(note that + is invertible for large ). Since and are asymptotic we have shown ¥§¦

that every asymptotic morphism from » into a -algebra is asymptotic to a family

¦ [

of Y -homomorphisms (corresponding to ), as required.

Z *

Definition 1.21. Two asymptotic morphisms Z and from to are homotopic

9

*b R KS Z Z

if there is an asymptotic morphism Z from to from which and can

9 9

K$W R KT be recovered by evaluation at R . Homotopy is an equivalence relation and

we shall use the notation

9

 *¨ 65 *çB

 homotopy classes of asymptotic morphisms from to

t

9 9

*(   *(  Y There is a natural map from  into since each -homomorphism can be regarded as a constant asymptotic morphism. It follows easily from the previous

lemma that: ¥Q¦

Proposition 1.3. If Û is any graded -algebra then the natural map

9 9

» ÛV   » ÛV 

 / P is an isomorphism. O

Returning to the composition (6), it gives rise to the following diagram:

composition with

9 9

 » -¸ ¶d3   » * -¸ ¶d3 

à Ã

÷ /

O

R

ð

9

 » *dÃ-¸ ¶d3

-߸ ¶dPA AFCŠ*dÃ-¸ ¶d

We arrive at the following conclusion: composition with ZQCA Ã

¤f ¤"

C M) *©

induces a homomorphism Z . ¦

1.9 Asymptotic Morphisms and Tensor Products

¤f ¤f

C >) *© The construction of maps Z from asymptotic morphisms has sev- eral elaborations which are quite¦ important. They rely on the following observation:

156 Nigel Higson and Erik Guentner

¥Q¦ ZQCe BASAFC *

Lemma 1.7. Let Û be a -algebra and let be an asymptotic morphism

¥¨¦ Z -§K†C -Û-ASAFC6* -Û

à à between -algebras. There is an asymptotic morphism Ã

such that, on elementary tensors,

7 87

[ [

ZßÃ-§KL C -UTà ©,)+Z šÃ-UT

t

Z -§K

Moreover this formula determines à uniquely, up to asymptotic equivalence.

* Û

Proof. Assume for simplicity that and areO unital (the general case, which can be

attacked by adjoining units, is left to the reader). There are graded Y -homomorphisms

Û *<Ã-Ûb

from and into the O asymptotic algebra , determined by the formulas

7 87

[

)ÿZ šÃ-§K Tb,)ÜK]Ã-T

, and . They graded commute and so determine a homomor-

ZßÃ-§K†Ce ©Ã-Û ) *dÃ-Ûb

phism . This in turn determines an asymptotic morphismO

-§K‰Ce ©Ã-ÛVANADC *dÃ-Û

Z‚Ã , as required. Two asymptotic morphisms which are asymp-

7

-UT Y *<Ã-Ûb totic on the elementary tensors à determine -homomorphisms into which are equal on elementary tensors, and hence equal everywhere. From this it follows that the two asymptotic morphisms are equivalent.

Remark 1.11. It is clear from the argument that it is crucial here to use the maximal tensor product.

Here then are the promised elaborations:

WASAFC§*

(a) An asymptotic morphism ZQCn determines an asymptotic morphism

¤

-¸ ¶H *HÃ-¸ ¶d

from ©Ã to by tensor product, and hence a -theory map

¤f ¤"

C M) *§

Z .

¦

XA ADCX*HÃ-¸ ¶d

(b) An asymptotic morphism Z§CA determines an asymptotic mor-

-¸ ¶H *dÃ-Ž¸ ¶dLÃ-¸ Ár

phism from VÃ to by tensor product. After identifying

¸ ¶H -¸ ¶H ¸ ¶H

à with we can apply the construction of the previous section

¤f ¤"

C M) *§

to obtain a map Z .

¦

Z§C » - YA AFCH*

(c) An asymptotic morphism à determines an asymptotic morphism

» - -¸ ¶d * -¸ ¶H -¸ ÁD ˊC»¨)+ -¸ ¶d

à à à Ã

from à to by tensor product. If ¤f

represents a class in  then by forming the composition

Sí í

î î

» - -¸ ¶d * -߸ ¶d

à à Ã

» ù

ì

»¨Ã-Ž»

/ / _÷ _ _/

¤" ¤ ¤" ¤f

Z C j) *§

we obtain a class in *© , and we obtain a -theory map .

¦

Z§C » - )Ü* -¸ ¶H Ã

(d) Combining (b) and (c), an asymptotic morphism à deter-

¤ ¤" ¤f

C j) *§

mines a -theory map Z . ¦

1.10 Bott Periodicity in the Spectral Picture

We are going to formulate and prove the Bott periodicity theorem using the spectral picture of ¤ -theory, products, and a line of argument which is due to Atiyah [5]. In the course of doing so we shall introduce many of the ideas which will feature in our later discussion of the Baum-Connes conjecture. Group C*-Algebras and K-theory 157

In this section present an abstract outline of the argument; in the next three sec- tions we shall fill in the details using the theory of Clifford algebras to construct

suitable ¤ -theory classes and asymptotic morphisms.

¦ *

Definition 1.22. Let us say that a graded ¥ -algebra has the rotation property if

¡ ¡

~

— -—à ,) K¢ — -—à ŢÆ

the automorphism Å¢Æ which interchanges the two factors

0

2

-* Y KG-

in the tensor product *HÃ is homotopic to a tensor product -homomorphism

Z

-*%)+*dÃ-*

Ce*HÃ .

¡

8¼

!

¥§¦ *ço¥ Example 1.9. The trivially graded -algebra  has this property (with

Z 6

K ). ¥Q¦

Theorem 1.12. Let * be a graded -algebra with the rotation property. Suppose

¤" *©

there exists a class —$ and an asymptotic morphism

­§C»(Ã-*%)+¸ ¶d

¤ ¤f ¤"

C *§V) «\

with the property that the induced -theory homomorphism ­

¦

K ¥Q¦

maps — to . Then for every -algebra the maps

’

¤f ¤" ¤" ¤f

­ C VÃ-*§M)  j) ©Ã-*§

and C

¦ ¦

¤ — induced by ­ and by multiplication by the -theory class are inverse to one an- other.

Proof. From our definitions it is clear that the diagram

[

¤" ¤"

¤f -theory product

¥(‰- -*§ ¥ - -*©

à Ã

à /

î





¤f ¤f

 ¤f 

¥(‰- 

[ -  / ¥ŠÃ

-theory product

¤" ¤f

C ©Ã-*©&) 

commutes. Let us express this by saying that the maps ­ ¦

are multiplicative. It follows directly from the multiplicative property that ­ is left-

’

¦

¤f ¤f

>) VÃ-*§

inverse to the map C :

¦

’

¾ ¾ ¾ ¾ ¾

I I I

­ Œ>­ —TM ­ —TM KŽ

¦ ¦ ¦ ¦

t ’

To prove that ­ is also left-inverse to we introduce the isomorphisms

¦ ¦

\

Ce ©Ã-*%)ê*dÃ-

and

9

C * - -*%)+* - -*

à à à Ã

which interchange the first and last factors in the tensor products. Note that

] £©gehE¢ i

So does ^`_bacedf , but Theorem 1.12 does not apply in the odd-dimensional case.

158 Nigel Higson and Erik Guentner

• ¤f • J• ¤f

\ 9

9 j 9

I I

$ *§ 8  8  8 $ -*§

Ã

¦ ¦

t

Z

9

-§K]ÃÃ -QK

Since * has the rotation property, is homotopic to the tensor product , where

• Z

is as in Definition 1.22. Therefore, setting ˜— above, we get

Z

\ 9

I I I

8  —‚ — 8 M —T 8

¦ ¦ ¦ t

Applying ­ we deduce that

¦

Z Z

\ \

I I I

8 >'­ 8 —TM­ —T 8M —T ­ 8 

¦ ¦ ¦ ¦ ¦ ¦ ¦

(the first and last inequalities follow from the multiplicative property of ­ ). Ap-

¦

Z

I

ua­ 8  —T

plying another flip isomorphism we conclude that 8 . This shows

¦ ¦

Z

­ ­

that multiplication by —T is left-inverse to . Therefore , being both left and

’

¦ ¦ right invertible, is invertible.¦ Moreover the left inverse is necessarily a two-sided

inverse. ¦

Z

Remark 1.12. It follows that —T<ê— . This fact can be checked in the example presented in the next section. ¦

1.11 Clifford Algebras

We begin by venturing a bit further into the realm of graded ¥©¦ -algebras. We are going to introduce the (complex) Clifford algebras, which are a familiar presence in ¤ -theory and index theory.

Definition 1.23. Let k be a finite-dimensional Euclidean vector space (that is, a real

vector space equipped with a positive-definite inner product). The complex Clifford

¥Q¦ k algebra of k is the graded complex -algebra generated by a linear copy of ,

whose elements are self-adjoint and of grading-degree one, subject to the relations

¡ ¡

Ô Ô

œ œ œ

™K $lk  for every .

Remark 1.13. The Clifford algebra can be concretely constructed from the complex-

k¨ º k¨

ified tensor algebra º be dividing the ideal generated by the elements

¡

Ô Ô

œ œ œ

~ ™€K

- .

F F

9 9

It follows immediately from the definition that if ! is an orthonormal

tStSt

mjy}zon k§

basis for k then regarded as members of these elements satisfy the relations

¡

F F FS FSSF

±qp



²

¯ ¯

K  

and 'R if

t

F F

± ±

lt t

4

¯ ™ ™S™š™e™ ¯or K%s mjy{zn k© s g

The monomials , where ™S™S™ span as a

0

jy{zn k§

complex linear space. In fact these monomials constitute a basis for m . The

F F

?

¯ ¯

µ r

monomial ™l™S™š™l™ has grading-degree (mod ).

0

J¼ F

¦

mjy{zn  «Š«

Example 1.10. The ¥ -algebra is isomorphic to , with correspond-

9

~

K « ing to KL . The grading automorphism transposes the two copies of .

Group C*-Algebras and K-theory 159

¡

8¼

¡

¥Q¦ mjy}zon «N

Example 1.11. The -algebra  is isomorphic to in such a way that

p p

±

F F

R K R

¡

± 

 and

~

t

KŽR€s Rls



F F

±

¡

¡

 

The (inner) grading is given by the grading operator ܬ .

¡

J¼

m

ƒ"

mjy{zn

Remark 1.14. More generally, each even Clifford algebra  is a matrix al-



F F

±

ƒ

m

¡

¡vu

¡



«\ ‹f

gebra , graded by m  ; each odd Clifford algebra

¡

J¼

tStSt

mTk

u u

¡ ¡

«\ «N:Š mjy{zn  is a direct sum , graded by the automorphism which switches the summands.

Definition 1.24. Let k by a finite-dimensional Euclidean vector space. Denote by

kQ ¥Q¦ k

 the graded -algebra of continuous functions, vanishing at infinity, from

jy{zn k§  kQ mjy}zon kQ

into m . (The grading on comes from alone—thus for example

jy}zn k§

an even function is a function which takes values in the even part of m .)

J¼ ] J¼ w 8¼

 ¥ o¥ 

Example 1.12. Thus  is isomorphic to (and the grading au-

¡

J¼

ƒf

¥§¦ 

tomorphism switches the summands) while the -algebra  is isomorphic to



¡

] J¼

ƒ

¡

¡



¥ x

, graded by . w

Suppose now that k and are finite-dimensional Euclidean vector spaces. Each

w kŠxw

of k and is of course a subspace of , and there are corresponding inclusions

jy}zn kQ mjy}zon w6 mjy}zon kyw6 Y

of m and into . They determine a -isomorphism

mjy}zn k© -mjy{zn w6>‘ mjy{zn k˜yw6

à 

(this can be checked either by computing with the standard linear bases for the Clif-

jy}zon k©:Ã-mjy{zn w6

ford algebras, or by checking that that the tensor product m has

jy{zn kzw6

the defining property of the Clifford algebra m ). w

Proposition 1.13 Let k and be finite-dimensional Euclidean spaces. The map

œ ª œ ª

¡ ¡

-½Ã ,)ÿ½ ½  G½ x½  ½ , where determines an isomorphism of graded

¥¨¦ -algebras

 kyw6M‘  k§ - w6

Ã



mjy}zn kQ -{mjy}zon w6 ‘ Ã

Proof. This follows easily by combining the isomorphism 

] w ]

mjy{zn k -|w6 ¥ k§‰-f¥ w6>‘ ¥ k˜yw6 Ã

above with the isomorphism  . ¥©¦

Proposition 1.4. Let k be a finite-dimensional Euclidean vector space. The -

kQ

algebra  has the rotation property.

¡

jCw )~w

Proof. Let } be an isometric isomorphism of finite-dimensional Eu-

Y } C|mjy}zon Š)

clidean vector spaces. There is a corresponding -isomorphism w

¦

¡

Y jy{zn w 

m and also a -isomorphism

¡

} C w M) w 

¦ ¦

160 Nigel Higson and Erik Guentner

¡

ª ª

¡ ¡

} ½ M:} ½ }

defined by x . Under the isomorphism

¦ ¦ ¦

 k§ - kQ>‘  kzkQ

à 

of Proposition 1.13 the flip isomorphism on the tensor product corresponds to the

9 9

 k§)k§

Y -automorphism of associated to the map which exchanges the

¦ ¦

9

k%€k

two copies of k in the direct sum . But is homotopic, through isometric

œ œ œ œ

9

9 9

¡ ¡

~

 ˜k ;,)

isomorphisms of k , to the map , and so is homotopic

¦ ¦

Z Z

~

K K - C‚k)Mk

to à , where is multiplication by .

¦É¦

k§

Of course, as we noted earlier, the algebra ¥ has the rotation property too.

kQ ¥ kQ The virtue of dealing with  rather than the plainer object is that with

Clifford algebras to hand we can present in a very concise fashion the following

¤f

k¨x

important element of the group  .

œ œ

ƒk),mjy{zn k§ ¥ >

Definition 1.25. Denote by ¥WC the function which includes

mjy}zn kQ

k as a real linear subspace of self-adjoint elements in .

mjy}zon kQ

This is a continuous function on k into , but

¡ ¡

Ô Ô

œ œ

¥   ™lK

so ¥ does not vanish at infinity (far from it) and it is therefore not an element of

kQ ½<$d» ½ ¥(

 . However if then the function defined by

œ œ œ

9 9

,)ž½ ¥ Œ $lk

œ

¥ G$*mjy{zn k§

where ½ is applied to the element in the sense of the functional cal-

’

k§ Ce½H,)¿½ ¥( Y

culus, does belong to  and the assignment is a -homomorphism

 k§

from » to .

¤f ¤

 k¨Œ Y

Definition 1.26. The Bott element —¨$ is the -theory class of the -

’ ’

»f) kQ Ce½H,)¿½ ¥(

homomorphism C defined by . ¥V¦

Remark 1.15. The function ¥ is an example of an unbounded multiplier of the -

k§ algebra  . See the appendix. Example 1.13. Bearing in mind the isomorphisms of Examples 1.10 and 1.11, we

have

¾ ¾ ¼ ¾ J¾

9 9

~

 $ ¥ > p

and •

¡

J• • ¼

R

9

•

‘

¥ M „ $<«



s

t R

We can now formulate the Bott periodicity theorem.

Theorem 1.14. For every graded ¥§¦ -algebra and every finite-dimensional Eu-

clidean space k the Bott map

’

¤" ¤f

9

C M) - k§x

Ã

’

J¾ ¾

I — defined by > , is an isomorphism of abelian groups. Group C*-Algebras and K-theory 161

We shall prove the theorem in the next two sections by constructing a suitable

­ —SM6K

asymptotic morphism ­ and proving that . ¦

Remark 1.16. To relate the above theorem to more familiar formulations of Bott pe- †

riodicity we note, as we did earlier, that if gu%µ is even then the Clifford algebra

] J¼

!

¡

¥ Œ  ! is isomorphic to , from which it follows that if is trivially graded

then

¡ ¡

¤f J¼ ¤f J¼

m m

©Ã-q x>‘ "-f¥ x

 t

The ‘graded’ theorem above therefore implies the more familiar isomorphism

¡

¤f w J¼ ¤f

m

"-f¥ xM‘ 

 t

1.12 The Dirac Operator

We are going to construct an asymptotic morphism as in the following result. (The actual proof of the theorem will be carried out in the next section.)

Theorem 1.15. There exists an asymptotic morphism

­§C» - k¨‡ANADC©¸ ¶d

Ã

¤f ¤f

 k¨Œ>) «c

for which the induced homomorphism ­§C maps the Bott element

¤f ¤f

 kQŒ K($ «N —Ž$ to .

Definition 1.27. Let k be a finite-dimensional Euclidean vector space. Let us pro-

jy{zn k§

vide the finite-dimensional linear space underlying the algebra m with the

F F

¯ ¯ ™š™S™

Hilbert space structure for which the monomials r (associated to an or- 0

thonormal basis of k ) are orthonormal. The Hilbert space structure so obtained is

F F

9 9

¶ k§

independent of the choice of ! . Denote by the infinite-dimensional

tštSt

jy}zn kQ k

complex Hilbert space of square-integrable m -valued functions on , Thus:

¡

9

¶ k©Mi k mjy{zn k§x

t

kQ

The Hilbert space ¶ is a graded Hilbert space, with grading inherited from

jy{zn k§

m .

F

9 ½&$ Definition 1.28. Let k be a finite-dimensional Euclidean vector space and let

k . Define linear operators on the finite-dimensional graded Hilbert space underlying

jy{zn k§

m by the formulas

F] ¾ F ¾

j ™

¾ ¾

~

Ã

½ j ™š½ K¢

ÅLÒ

t

F

Umjy}zon kQŽ)ˆmjy{zn k§

Observe that the operator C is self-adjoint while the op-

Ã

‰mjy}zon kQM)Šmjy}zn kQ erator ½C is skew-adjoint.

162 Nigel Higson and Erik Guentner

F F

±

9 9

t t

k ™S™S™

Exercise 1.5. Let ! be an orthonormal basis for . Show that if

±

tStšt 4

then the ‘number’ operator ‹

!

Õ

F F

 ¯ ¯

qŒ

ð

¯

F F F F

?

~

¯ ¯ ¯ ¯

™S™š™ r ™š™S™ r g‰ jy{zn k§ µ

maps the monomial in m to .

0 0

Definition 1.29. Let k be a finite-dimensional Euclidean vector space. Denote by



§ ¶ kQ mjy{zn k§ k the dense subspace of comprised of Schwartz-class -valued

functions:



kQM jy}zn kQ

Schwartz-class m -valued functions

t



k§ Û ¶ kQ The Dirac operator of k is the unbounded operator on , with domain ,

defined by

!

Õ

F

½

œ œ

9

Œ Ûb½ ; ¯ ¾

Ã

¯



F F ¾ ¾

¢9 9 l9 9

k !

where ! is an orthonormal basis of and are the corresponding

tStSt tStšt

coordinates on k .

F ¯

Since the individual à are skew-adjoint and since they commute with the partial



k¨ derivatives we see that Û is formally self-adjoint on .

Lemma 1.8. Let k be a finite-dimensional Euclidean vector space. The Dirac opera-

½r$b» ŽH$ kQ z

tor on k is essentially self-adjoint. If , if and if is the operator of

¶ k© ½ ÛbŒ 

pointwise multiplication by Ž on the Hilbert space , then the product

kQ is a compact operator on ¶ .

Proof. The operator Û is a constant coefficient operator acting on a Schwartz space

¼

!

!

¯

k%‘ ÛaL‘ 

Å

ð 

of vector valued functions on . It has the form ¯ , where the

ÅLÒF’ ¯

matrices  are skew adjoint. Under the Fourier transform (a unitary isomorphism)

!

~

¯H•S¯

Û Û+”“ K 

‘ ð

corresponds to the multiplication operator ¯ , and from this

Œ Û

we see that Û , and hence , is essentially self-adjoint. Moreover from the formula

Œ

!

¡

¡ ¡



Õ Ô Ô

9

~

“

• •

Û – K Ž¯ ¯ 

Œ

ð

¯

¾ F ¼

¡

!

•

½ ; ½ $ Ûb 2

for all , it follows that if say ÇÓÒ then is pointwise multipli-

Œ

F

¡—¥˜F—

½ Ûb

cation by 2 , and therefore the inverse Fourier transform is convolution by

F

¡ — —

™

ŽX${ kQ

2 0

Ò (give or take a constant). It follows that is compactly supported

 Û x

then ½ is a Hilbert-Schmidt operator, and is therefore compact. The lemma



½ ÛbŒ Ž

follows from this since the set of ½D$H» for which is compact, for all , is

F

¡

» » 2 an ideal in , while the function Ò generates as an ideal.

We are almost ready to define our asymptotic morphism ­ .

Group C*-Algebras and K-theory 163

ŽW$* k§

Definition 1.30. Let k be a finite-dimensional Euclidean space. If and if

¡

œ œ

9<;

`j$X K  Že[>$š k§ Že[ >#Ž `

then denote by the function  . Û

Lemma 1.9. Let k be a finite-dimensional Euclidean space with Dirac operator .

ŽH$ k¨

For every ½r$ » and we have

¡

9 9

D©ž

y{z}| ½ ` Ûb R

[HGJILK K

Kœ› K



b $+¹ ¶ kQx ŽÝ[

where is the operator of pointwise multiplication by and

¡

` Ûb

½ is defined using the functional calculus of unbounded operators.

9  Remark 1.17. The commutator  here is the graded commutator of Definition 1.8.

Proof. By an approximation argument involving the Stone-Weierstrass theorem it

J¾ ¾ Ÿ

±

¡

½  Ž suffices to consider the cases where  and where is smooth and

compactly supported. We compute

¡ ¡ ¡ ¡ ¡ ¡ ¡

Ÿ Ÿ Ÿ

± ± ±

9 9 9

D D

` Û q] žH` ` Û q]  ÛV ` Û q]

Ô Ô

›

¡

9

 

` z Û b ÛV which has norm bounded by  . But the commutator of with is

the operator of pointwise multiplication by (minus) the function

!

Õ

¡ ¡

F †œŽ

œ œ

¾

¯

,)+` ` Œ

t

¯



Œ

ð

¯

¡

`  So its norm is ¡ , and the proof is complete.

Proposition 1.5. There is, up to equivalence, a unique asymptotic morphism

­n[C » - kQM)+¸ ¶ k©x Ã

for which, on elementary tensors,

¡

F [

½NÃ-|ŽÝM½ ` Ûbx ­

t

Ä

9<;

`j$X{K  ­c[jC:»  k§M)^¹ ¶ k§x

Proof. For define a linear map à by the formula

¡

F [

½NÃ-|ŽÝM½ ` Ûbx ­

t

O

Ä

[

»¨Ã  k§

Lemma 1.9 shows that the maps ­ define a homomorphism from into

¶ k¨xŒ - Ã

¹ . By the universal property of the tensor product this extends to

»(Ã- k¨

a Y -homomorphism defined on . Now, although neither of the operators

¡



½ ` Ûb y O

or O are compact it follows from elementary elliptic operator theory

Y » -‰ k¨

that their product is compact. So our -homomorphism actually maps à into

¶ k©xŒ ¢ ¹ ¶ kQxx the subalgebra ¸ . Therefore we obtain an asymptotic

morphism as required.

A[ Ž ­ Remark 1.18. The presence of Ž , instead of the plainer , in the definition of is

not at this stage very important. The ‘ ` ’ could be removed without any problem. But [

later on it will turn out to have been convenient to have used Ž .

/

Exercise 1.6. Show that if is an ideal in a ¥§¦ -algebra then there is a short exact

O O

sequence of asymptotic algebrasO

0/ /

  ´  R

R / / / / t 164 Nigel Higson and Erik Guentner

1.13 The Harmonic Oscillator

—Tß K In this section we shall verify that ­ , which will complete the proof of the Bott periodicity theorem. Actually we¦ shall make a more refined computation which will be required later on. We begin by taking a second look at the basic construction of Section 1.11.

Definition 1.31. Let k be a finite-dimensional Euclidean vector space. The Clifford



kQ k§ operator is the unbounded operator on ¶ , with domain the Schwartz space ,

which is given by the formula

!

Õ

¾ F

œ œ

9

¯ ¯

¥(½ > ½ x

ð

¯

¾ F

k ¯ k where ¯ are the coordinates on dual to the orthonormal basis of (the defini-

tion of ¥ is independent of the choice of basis).



§ ½r$b»

The Clifford operator is essentially self-adjoint on the domain k . So if

¥(;$H¹ ¶ k¨x

we may form the bounded operator ½ by the functional calculus.

’ k

Lemma 1.10. Let be a finite-dimensional Euclidean vector space and let C]Ȭ)

 kQ kQ

 be the homomorphism of Definition 1.26. If is represented on the Hilbert

kQ

space ¶ by pointwise multiplication operators then the composition

 kQ ¤ ¹ ¶ k§x £

» / /

½ ¥(;$d¹ ¶ k¨Œ OP

maps ½&$ » to .

—T

We shall compute the compostion ­ by analyzing the following operator: ¦

Definition 1.32. Let k be a finite-dimensional Euclidean vector space. Define an



¶ k© kQ

unbounded operator * on , with domain , by the formula

! !

Õ Õ

¾ F F

½

œ œ œ

¯ ¯ ¾ ¯

*©½ > ½ Œ‰ x

Ã

t

 ¯

¥ Û

Thus *%¥'fÛ , where is the Clifford operator and is the Dirac operator.

¼ 

Example 1.14. Suppose k . Then

p

¾ ¾

~

R T´¥T

9

¾ ¾

*%

bT´¥T R s

¡ ¡

8¼ 8¼

k§ i > i 

if we identify ¶ with in the way suggested by Example 1.10.



* Q

Observe that the operator maps the Schwartz space k into itself. So the

¡



kQ operator ÁÚ* is defined on . Group C*-Algebras and K-theory 165

Proposition 1.16 Let k be a finite-dimensional euclidean vector space of dimension



*%˜¥ ŠÛ kQ ¶ kQ

g , let as above. There exists within an orthonormal basis for ¡

consisting of eigenvectors for * such that (a) the eigenvalues are nonnegative integers, and each eigenvalue occurs with finite multiplicity, and

(b) the eigenvalue R occurs precisely once and the corresponding eigenfunction is

¡

Ô Ô

¦S§

œ

~

¡

Ø 

¼ 

Proof. Let us consider the case k first. Here,

¡

¾

~ª© ~

¡

2

K R

9

©

¡

* ¨

¾

Ò

2

~«©

2

R 'KS¬

©

Ò

2

¡

8¼  and so it suffices to prove that within the Schwartz subspace of i there is an

orthonormal basis of eigenfunctions for the operator

¡

¡

¾

T

9

~

¡

Áò ¾ T for which the eigenvalues are positive integers (with finite multiplicities) and for

which the eigenvalue K appears with multiplicity one. This is a well-known com-

¤ ¾ ¾

~

© ©

 i 

putation, and is done as follows. Define  and , and let

© ©

J¾ F

Ò Ò

¡

½ >

2 Ò

0 . Observe that

2

¤ ¤

~

Áò i q§'i  q

¤

ÙR Ár½ ó½ Á&iÿÙijÁ¿絀i

and that ½ , so that . It follows that and

! ! ! !

'i Á%Dµlg†i ½l!lk i ½ Ár½:!€k  µ:gQfKLx½l!lk

Á&i . So if we define then .

The functions ½l!lk are orthogonal (being eigenfunctions of the symmetric operator

¡

8¼

i 

Á with distinct eigenvalues), nonzero, and they span (since, by induction,

¡

g ½ i ½:!lk is a polynomial of degree times ). So after -normalization we obtain the required basis.

The general case follows from the (purely algebraic) calculation

‹

¡

! !

¡ ¡ ¡ ¡

Õ Õ

¾



? 9

~ ~

4

¡

¾

* ¥ fÛ     µ g‰ ¶ kQ

¯ on



¯

ð ð

‹

¯ ¯

4

k§

where is the number operator introduced in Exercise 1.5 and ¶ denotes the

kQ ka)Wmjy{zn k§

subspace of ¶ comprised of functions whose values are combi-

¡

F F

?

¯ ¯

™š™S™ * nations of the degree monomials r . From this an eigenbasis for may be found by separation of variables. 0

We shall use the following consequences of this computation:

* %*®­

Corollary 1.1. Let k be a finite-dimensional Euclidean vector space. Let

¶ kQ

be the Bott-Dirac operator of k , considered as an unbounded operator on

 Q with domain k . Then 166 Nigel Higson and Erik Guentner

(a) * is essentially self-adjoint

(b) * has compact resolvent.

¡

Ô Ô

¦ §

œ

~

Ø  (c) The kernel of * is one-dimensional and is generated by the function .

Theorem 1.17. Let k be a finite-dimensional Euclidean vector space. The composi-

tion

Sí

î

¸ ¶ k©x

»(Ã- kQ

» £

ì -»

/ »(Ã / _ _ _/

C]»QANAeC©¸ ¶d

is asymptotically equivalent to the asymptotic morphism ¯ defined by

¡

¯][ ½>½ ` *§ `°'oKL t

The idea of the proof is to check the equivalence of the asymptotic morphisms

’

 ¯

­ and on the generators

J¾ F¡ ¾ ¾AF¡

“ œ

9

2 2

Ò j Ò

> and

t »

of the ¥¨¦ -algebra . Since for example

’

F F F

¡ ¡ ¡

[²± [ [²³

“ “

2 2

][ M ­n[ Œ>

¯ and

F

¡ ²±

(the latter thanks to Lemma 1.10) we shall need to know that [ is asymptotic to

F F

¡ ¡

[ [²³

2 2 . For this purpose we invoke Mehler’s formula:

Proposition 1.6 (Mehler’s Formula). Let k be a finite-dimensional Euclidean space

¡ ¡

Û k Û ¥

and let ¥ and be the Clifford and Dirac operators for . The operators ,

¡ ¡

¤



ë

k§ ¨Û R and ¥ are essentially self-adjoint on the Schwartz space , and if

then

¡ ¡ ¡ ¡

F F F F

³ ³ ³

k ¢ ¢ ¢ ¢

9

2 2 2 2 2

0 0



0 0

2

ø ö

2 2

S

¤ ¤ ¤ ¤ ¤

¡

~

Öµ´ µ  K¢Œ´\֌z¶´ µ  ֌z¶´ µ Œ´lµ

where  and . In addition,

¡ ¡ ¡ ¡

F F F F

³ ³

k ¢ ¢ ¢ ¢

9

2 2 2 2 2

 0 0

0 0

2

ø ö

2 2

¤ ¤

¡ P for the same and . O

See for example [16]. Note that the second identity follows from the first upon

¡ ¡ ¡

8¼

 Û ¥

taking the Fourier transform on i , which interchanges the operators and . U Lemma 1.11. If is any unbounded self-adjoint operator then there are asymptotic

equivalences

   

¡ ¡ ¡ ¡

F F F F

[©¸ [©¸

9

2 2 2 2 2 2

0 0

‘ ‘

0

2

2e· 2 ·

and

   

¡ UrF¡ ¡ UDF€¡ ¡ UrF¡ ¡ UrF¡

[©¸ [©¸

9

2 2 2 2 2 2

` 0 ` ` ` 0

‘ ‘

0

2

· 2D· 2

¡ ¡ ¡

¡ ¡ ¡

9 9

¡

~

  KLÉ´\Öxz¶´ µ:` Œ´lµ Ö¹´ µ:` ֌z¶´ µl` where  and . Group C*-Algebras and K-theory 167

Remark 1.19. By ‘asymptotic equivalence’ we mean here that the differences be- tween the left and right hand sides in the above relations all converge to zero, in the

operator norm, as ` tends to infinity.

Proof (Proof of the Lemma). By the spectral theorem it suffices to consider the same ¾

problem with the self-adjoint operator U replaced by a real variable and the oper-

J¼ 

ator norm replaced by the supremum norm on ¥ . The lemma is then a simple

¡ ¡

¡ ¡

9 9

¢9 ¡

` yº `

calculus exercise, based on the Taylor series  .

J¼

9

}b$ »¨¥ 

Lemma 1.12. If ½ then

¡ ¡

9

y{z}| ½ ` ¥( } ` Ûb R

ž

K K

[HGxI

t

K K

›

¦

} $"» ¥

Proof. For any fixed ½'$"» , the set of for which the lemma holds is a -

] J¼ 

subalgebra of ¥ . So by the Stone-Weierstrass theorem it suffices to prove the

J¾Ÿ

±

¡ } lemma when is one of the resolvent functions  . It furthermore suffices

to consider the case where ½ is a smooth and compactly supported function. In this

case we have

Ÿ

¡ ¡ ¡ ¡ ¡

±

9 9

½ ` ` ¥(  s ½ ` ¥( ` Û

ž ž

K K K K

K K K K

› ›

by the commutator identity for resolvents. But then

¡

Ô¼»¾½ Ô

¡ ¡ ¡

9

½ ` ¥( ` Û s"` ™ ™ $¾¿ ½ ¥(Œ

ž constant

K K

t

K K ›

This proves the lemma. ‹

k§G)ê¶ kQ

Proof (Proof of Theorem 1.17). Denote by Ce¶ the ‘number opera-

? ?

~

k§ µlg tor’ which multiplies the degree component of ¶ by . We observed in

the proof of Proposition 1.16 that ‹

¡ ¡ ¡

9

* ¥ fÛ 

‹

¡ ¡ Û

and let us observe now that the operator commutes with ¥ and . As a result,

F F F

[©¸ ³ [©¸ [©¸

© kÀ à

2 2 2 2 2 2



ø ö

and therefore, by Mehler’s formula,

F¡ F¡ F¡ F€¡ F€¡

[©¸ ³ ³ [©¸

à ©

9

2 2 2 2 2 2



0 0

0 0

2

2e· · 2F·

It follows from Lemma 1.11 that

F¡ F¡ F¡ F F¡

[©¸ [©¸ ³ [©¸ [©¸ ³ [©¸

à ©

9

2 2 2 2 2 2 2 2 2

0 0

‘

2 2

and hence from Lemma 1.12 that

F F F

¡ ¡ ¡

[©¸ [©¸ ³ [©¸

à

2 2 2 2 2 2 ‘

168 Nigel Higson ‹ and Erik Guentner

F

¡

[©¸ ©

(since the operator is bounded the operators 2 converge in norm to the

’

J¾ F

¡

“

C]»¨)æ»QÃ- k§ > 2

identity operator). Now the homomorphism maps Ò to

“ “

[

- ¥( ­

à , and applying we obtain

’

¡ ¡ F F¡

[©¸ [©¸ ³

“ “ “

9

2 2 2 2

[

­ x> ` ¥( ` ÛbM

’

F

¡

[©¸

à

“ “

[ [

¯ H ­ x 2

as we noted earlier. But 2 , and so we have shown that

¾

“ œ

[

 j

and ¯ are asymptotic to one another. A similar computation shows that if

’

¾ÝF

¡

œ œ “ œ

[ [

­ Œ ¯  2 Ò then and are asymptotic to one another. Since and generate

» , this completes the proof.

¤f ¤"

C  k¨x¨) «N —Š$

Corollary 1.2. The homomorphism ­ maps the element

¤f ¤" ¦

k¨x K($ «N

 to the element .

’

—S Y

Proof. The class ­ is represented by the composition of the -homomorphism ¦ with the asymptotic morphism ­ . By Theorem 1.17, this composition is asymptotic

to the asymptotic morphism

¡

[

¯ ½M˜½ ` *©

t

][ Y

But each map ¯ is actually a -homomorphism, and so the asymptotic morphism

?

Y ½6,) ½ *§ ¯ is homotopic to the single -homomorphism . Now denote by the

projection onto the kernel of * . The formula

>

Ĥ ¤

¡

9 9

Ûb $ R KS

½ if

Â

~ ?

>Ã

½H, ) Á

¤

½ Rw R

9 R

¨ if ,

R R

¬

—TMK

defines a homotopy proving that ­ . ¦

Appendix: Unbounded Multipliers

Any ¥¨¦ -algebra may be regarded as a right Hilbert module over itself (see the book [45] for an introduction to Hilbert modules). An unbounded (essentially self-

adjoint) multiplier of is then an essentially self-adjoint operator on the Hilbert

module , in the sense of the following definition:

¥©¦ Å

Definition 1.33. (Compare [45, Chapter 9].) Let be a -algebra and let be a

Å º

Hilbert -module. An essentially self-adjoint operator on is an -linear map

Å ¢yÅ Å

from a dense -submodule into with the following properties:

Ï

œ ªUÇ œ ª‰Ç œ ª

9 9 9

Ɛº )Æ º $šÅ

(a)  , for all .

¡ Ï

(b) The operator quº is densely defined and has dense range. º If º is essentially self-adjoint then the closure of (the graph of which is the

closure of the graph of º ) is self-adjoint and regular, which means that the operators

Group C*-Algebras and K-theory 169

Ÿ Ÿ

± ±

¡

º º Å º q] q] are bijections from the domain of to , and that the inverses

are adjoints of one another. See [45, Chapter 9] again. Y

If º is essentially self-adjoint then there is a functional calculus -homomorphism

] J¼ ¾ Ÿ

±

¡

»'®¥  Å

from into the bounded, adjoinable operators on . It maps  to

Ÿ

±

¡ º

q] .

Ÿ

±

¡

º X® qw In the case where Å , if the densely defined operators are given

by right multiplication with elements of , then the functional calculus homomor-

phism maps » into (acting on as right multiplication operators). If is graded,

º º

if the domain of is graded, and if has odd grading-degree (as a map from the

Ï

graded space into the graded space ) then the functional calculus homomor- Ï

phism is a graded Y -homomorphism.

U ¾ ¾ ¾

C½ Q,) ½  Example 1.15. If ʧ» then the operator , defined on say the

compactly supported functions, is essentially self-adjoint.

U U

¡

Lemma 1.13. If is an essentially self-adjoint multiplier of and if is essen-

U U

Ä

¡ ¡

 

-§KM¨K -

à Ã

tially self-adjoint multiplier of , then à , with domain , is

¡

0

2

- OP

an essentially self-adjoint multiplier of à .

JU

»")滧Ã-Ž» å ½>½ -QKeÃ

Example 1.16. Using the lemma we can define åfC by

U

-  K]Ã .

2 Bivariant K-Theory

We saw in the last section that asymptotic morphisms between ¥©¦ -algebras deter- mine maps between ¤ -theory groups. In this lecture we shall organize homotopy classes of asymptotic morphisms into a bivariant version of ¤ -theory, whose pur- pose is to streamline the computation of ¤ -theory groups via asymptotic morphisms. In doing so we shall be following the lead of Kasparov (see [39, 37, 38]), although the

theory we obtain, called  -theory [13, 14, 27], will in fact be a minor modification of Kasparov’s ¤D¤ -theory.

2.1 The E-Theory Groups

* ¥§¦

Definition 2.1. Let and be separable, graded -algebras. We shall denote by

9

*§ »§Ã- ©Ã-¸ ¶d

 the set of homotopy classes of asymptotic morphisms from to

-¸ ¶d

*dà . Thus:

9 9

 *§>%  »¨Ã- ©Ã-¸ ¶d *dÃ-¸ ¶d3 

t

Z *

Example 2.1. Each Y -homomorphism from to , or more generally from

9

» - -¸ ¶H * -Ž¸ ¶d  *§

à Ã

à to , determines an element of . This element de-

9

 Z‰n$H *§

pends only on the homotopy class of Z , and will be denoted .

9

*© The sets  come equipped with an operation of addition, given by direct sum of asymptotic morphisms, and the zero asymptotic morphism provides a zero element for this addition.

170 Nigel Higson and Erik Guentner

9

*©

Lemma 2.1. The abelian monoids  are in fact abelian groups.

ZQCÝ» - -¸ ¶dxANADCD* -¸ ¶H

à à Proof. Let à be an asymptotic morphism. Define an

asymptotic morphism

opp opp

Z C»(Ã- ©Ã-¸ ¶dPANAeCV*dÃ-¸ ¶ 

¾

opp J¾

Z ;'Zc[ ­ x ­

by the formula [ , where is the grading automorphism. We shall

opp 9

Z  *§

show that Z defines an additive inverse to in .

¤ ¨R

For a fixed scalar ' the formula

p p

¤ ¾

È ¾ ¾

[

R Z  R

¢

9

¤ J¾

Ce½MÃ- ,)¿½ $ »(Ã- ©Ã-¸ ¶H ½&$d»

[ opp

s s

R R Z 

[

È

¢ opp

» -Ž» - -¸ ¶d * -¸ ¶a"¶ 

à à Ã

defines an asymptotic morphism from à into .

È

¢

åfCA»a)û» -Ž» By composing with the comultiplication à we obtain asymptotic

morphisms

í

î

opp

ì

» - -¸ ¶H » -Ž» - -¸ ¶d-ÉƒÊ * -Ž¸ ¶çu¶ 

à à à à Ã

à / _ _ _/

¤ <;

9 opp

R  Z§

Remark 2.1. The above argument provides another proof that the ¤ -theory groups

described in the last lecture are in fact groups.

F ¶d

If is a rank-one projection in ¸ then by composing asymptotic morphisms

7

½jÃ- $®»(Ã-

with the Y -homomorphism which maps the element to the element

7 F

- - Ã $d»(Ã- VÃ-¸ ¶d

½NÃ we obtain a map (of sets, or in fact abelian groups)

9 9

~

  »(Ã- ©Ã-¸ ¶d *dÃ-¸ ¶H3  )·  »(Ã- *dÃ-¸ ¶d3  t

Lemma 2.2. The above map is a bijection.

¶d Proof. The inverse is given by tensor product with the identity on ¸ . Details are

left to the reader as an exercise.

9

*§

The groups  are contravariantly functorial in and covariantly functo- ¥Q¦

rial in * on the category of graded -algebras.

9

« *§ ¥©¦

Proposition 2.1. The functor  on the category of graded -algebras is nat- ¤" urally isomorphic to *§ .

Proof. This follows from Proposition 1.3 and Lemma 2.2. Group C*-Algebras and K-theory 171

2.2 Composition of Asymptotic Morphisms

The main feature of  -theory is the existence of a bilinear ‘composition law’

9 9 9

 *§‰-f * ¥(j)+ ¥(

which is associative in the sense that the two possible iterated pairings

9 9 9 9

 *§c-  * ¥(‰-f ¥ Ûb>)¿ Ûb

are equal, and which gathers the  -theory groups together into an additive category

*

(the objects are separable graded ¥§¦ -algebras, the morphisms from to are the

9

*§ elements of the abelian group  , and the above pairing is the composition

law). ¥©¦

The  -theory category plays an important role in the computation of -algebra

¤ ¤

-theory groups, as follows. To compute the -theory of a ¥©¦ -algebra one can,

9 9

*  *§  * 

on occasion, find a ¥Q¦ -algebra and elements of and whose com-

9 9

  * *§

positions are the identity morphisms in  and . Composition with

9 9

*§  * 

these two elements of  and now gives a pair of mutually inverse

9 9 9

«   « *©  « 

maps between  and . But as we noted in the last section

¤ ¤f ¤f

9

« *©  *§

and  are the -theory groups and . It therefore follows that

¤f ¤" ¤f ¤f

©‘ *§ *§ 

 . Therefore, assuming that can be computed, so can . ¤

This is the main strategy for computing the -theory of group ¥©¦ -algebras. In this section and the next we shall lay the groundwork for the construction of the composition pairing. The following sequence of definitions and lemmas presents a reasonably conceptual approach to the problem. The proofs are all very simple, and by and large they are omitted. Details can be found in the monograph [27].

We begin by repeating a definition from the last lecture.

¦ ¦

¥ z *§ ¥

Definition 2.2. Let * be a graded -algebra. Denote by the -algebra of

9N;

 * z *§

bounded, continuous functions from {K into , and denote by the ideal *

comprised of functions which vanish at infinity. The asymptotic ¥©¦ -algebra of is ¥Q¦

the quotient -algebra O

*§>'z *§Œ´:z *©

t

O

BANADCV*

Observe (as we did in the last section) that an asymptotic morphism ZQCA

Y Z§C‰ ·) *§

definesO a -homomorphism in the obvious manner and that two

* Y

asymptotic morphism from to define the O same -homomorphism from to

*§ O

precisely when they are asymptotically equivalent. O

) *© Y

The asymptotic algebra construction *æ, is a functor, since a -homo-

¥ Y *§ ¥(

morphism from * to induces a -homomorphism from to by compo-

O O

sition. O

9 9

*©>*

O O

Definition 2.3. The asymptoticO functors are defined by and

tStšt

¡

! !

*©x *©M t

172 Nigel Higson and Erik Guentner O

O

!

9

Z Z CE ) *§ g Y

Two Y -homomorphisms are -homotopic if there exists an -

È

!

9

*b R KT Y Z Z homomorphism Cw o) from which the -homomorphisms and

are recovered as the compositions

O O

!

! evaluate at ,

9

*b R KT *©

/ /

t O

Lemma 2.3. [27, Proposition 2.3] The relation of g -homotopy is an equivalence re-

!

*§ OP

lation on the set of Y -homomorphisms from to .

O

9

¦

* ¥   *¨ 

Definition 2.4. Let and be graded -algebras. Denote by ! the set of

!

O

Y *§

g -homotopy classes of -homomorphisms from to :

!

9

  *¨  o5fg Y *§GB

! -Homotopy classes of -homomorphisms from to

t

9

 *¨  Y

Example 2.2. Observe that  is the set of homotopy classes of - homomor-

9

 *( 

phisms and  is the set of homotopy classes of asymptotic morphisms. O

Remark 2.2. The relation of g -homotopy is not the same thing as homotopy: homo-

!

O O

*§ g

topic Y -homomorphisms into are -homotopic, but not vice-versa, in general.

O O O O

! !lk

*§ *§ O

There is a natural transformation of functors,O from to , defined

! ! !€k

*§D *§Œ *© O

by including O as constant functions in . A second

! !€k *©

and different natural transformation from *© to may be defined by

! *© including * into as constant functions, and then applying the functor to this inclusion. Both natural transformations are compatible with homotopy in the

sense that they define maps

9 9

~

  *( @! )ü  *( @!lk t

Lemma 2.4. [27, Proposition 2.8] The above natural transformations define the

9 9

~

  *¨  )ü  *(  OP !

same map !lk .

9

 *¨ !

With the above maps the sets  are organized into a directed system

9 9 ¡ 9

  *(  )ü  *(  )ü  *¨ ÌË)ü™S™S™

9

I

* ¥§¦   *( 

Definition 2.5. Let and be graded -algebras. Denote by the direct O

limit of the above directed system. O

›

!

*§ ˊCÝ*Ú) ¥(

Proposition 2.2. [27, Proposition 2.12] Let Z§CÝ a) and Y

be Y -homomorphisms. The class of the composite -homomorphism

O

O

›

!

!lk

*§‰ÍÏÎ

¥(

ù

ö ø

÷ / /

t

9 9

 ¥ ²I Z Ë   *( ÄI

in the set  depends only on the classes of and in the sets and

9

 * ¥ ÐI

 . The composition law

9 9 9

I

  *¨ ²I   * ¥ ÐI )ü  ¥ ²I P so defined is associative. O

Group C*-Algebras and K-theory 173

Exercise 2.1. Show that the identity Y -homomorphism from to determines an

9

 ‚ ²I element of  which serves as an identity morphism for the above composition law.

Thanks to Proposition 2.2 and the exercise we obtain a category:

Definition 2.6. The asymptotic category is the category whose objects are the graded

9

I

  *(  ¥¨¦ -algebras, whose are elements of the sets , and whose composition law

is the process described in Proposition 2.2. Y Observe that there is a functor from the category of graded ¥©¦ -algebras and -

homomorphisms into the asymptotic category (which is the identity on objects and

9

I

Z§Ce ˜)¿*   *¨ 

which assigns to a Y -homomorphism its class in ). ¤

Exercise 2.2. Show that -theory, thought of as a functor from graded ¥V¦ -algebras to abelian groups, factors through the asymptotic category.

2.3 Operations

We want to define tensor products, amplifications and other operations on the asymp-

totic category. For this purpose we introduce the following definitions. ¥©¦

Definition 2.7. Let  be a functor from the category of graded -algebras to itself.

9

¥Q¦ ½®$˜ *b R KT ½

If * is a graded -algebra and if then define a function from

Œ

9 9

R KT  *© `$¨ R KT ½

 into by assigning to the image of under the homomorphism

9

[ [

‹ ]Ce *  R KS@M)+ *§ ‹ ` 

 , where is evaluation at . The functor is continuous

9

½&$H *  R KS@ ½

if for every * and every the function is continuous.

Œ

ê,)þ -* Example 2.3. The tensor product functors à (for both the minimal and

maximal tensor product) are continuous. ¥§¦ Definition 2.8. A functor  from the category of graded -algebras to itself is exact

if for every short exact sequence

/

/

´

R R / / / /

the induced sequence

1/ /

     ´  R R / / / /

is also exact.

›

)+ ŠÃ- *

Exercise 2.3. The maximal tensor product functor , is exact.

ÇÓÒ

›

ò,) - ¯ * ! Remark 2.3. In contrast the minimal tensor product functor à is not

exact for every * . See [66] for examples (and also Lecture 6).

174 Nigel Higson and Erik Guentner

½ ½

If  is a continuous functor then the construction of from described in Defi- Œ

nition 2.7 determines a natural transformation

9 9

 *  R KTM)+ *§T R KT t

The same process also determines natural transformations

z *§xj)^z  *©  z *©x>)+z  *©x

 and

9N;

z *§ ¥¨¦ {K  O

(recall that is the -algebra of bounded and continuousO functions from



z *§ 

into * and is the ideal of functions vanishing at infinity). So if is in addition

*§Œ  *§Œ an exact functor then we obtain an induced map from  into , as

indicated in the following diagram: O

 z *§x  z *©x  *©x R R / / / /

O

  

z  *§x z  *©x  *©x R

/ / / / R

t 

Proposition 2.3. [27O , Theorem 3.5] Let be a continuous and exact functor on Y

the category of graded ¥Q¦ -algebras. The process which assigns to each -homo-

! *§

morphism ZQCÝ ) the composition

O O

! !

 Ñ  *©x  *©x

ö}÷wø / / P defines a functor on the asymptotic category. O

Applying this to the (maximal) tensor product functors we obtain the following

result.

› -

Proposition 2.1 [27, Theorem 4.6] There is a functorial tensor product à on

ÇÓÒ P the asymptotic category. O

With a tensor product operation in hand we can construct an amplified asymptotic

category in the same way we constructed the amplification of the category of ¥©¦ -

algebras and Y -homomorphisms in Definition 1.12.

Definition 2.9. The amplified asymptotic category is the category whose objects are

*

the graded ¥¨¦ -algebras and for which the morphisms from to are the elements

9

I

 »¨Ã- *¨  ZQCn ÿ) * ˊC†*Ù) ¥ of  . Composition of morphisms and in the amplified asymptotic category is given by the following composition of morphisms in

the asymptotic category:

Sí

î

í

î

ì

ù ¥

»(Ã- »Ã-Ž»(Ã- »(Ã-*

/ ÷ / / t Group C*-Algebras and K-theory 175

2.4 The E-Theory Category

The main technical theorem in  -theory is the following:

* ¥§¦

Theorem 2.2. [27, Theorem 2.16] Let and be graded -algebras and assume

9 9

I

  *(    *( 

that is separable. The natural map of into the direct limit is a bi- *

jection. Thus every morphism from to in the asymptotic category is represented

* OP by a unique homotopy class of asymptotic morphisms from to .

Unlike the results of the previous two sections, this is a little delicate. We refer

the reader to [27] for details.

9

*§ It follows from Theorem 2.2 and Definition 2.1 that the group  (for

separable) may be identified with the set of morphisms in the amplified asymptotic

-߸ ¶d *dÃ-¸ ¶d

category from VÃ to . As a result we obtain a pairing

9 9 9

~

 *§n-f * ¥( )+ ¥( from the composition law in the asymptotic category. We have now reached the main

objective of the lecture:

9

 *©

Theorem 2.3. The  -theory groups are the morphism groups in an addi- ¥§¦ tive category Ò whose objects are the separable graded -algebras. There is a

functor from the homotopy category of graded separable ¥©¦ -algebras and graded

Ò OP

Y -homomorphisms into which is the identity on objects.

Y ˊCA*VANAeCr¥

Remark 2.4. If Z§CE ç)Ü* is a -homomorphism and if is an asymp-

9 9

Ë  Zn‰$d *§  ËM‰$< * ¥(

totic morphism then Z and determine elements and .

bGZ ¥

In addition the (naive) composition Ë is an asymptotic morphism from to ,

9

Ë{>Zn‰$< ¥(  Ë*;ZnE ËM4Ž Zn and so defines an element  . We have that . The

same applies to compositions of Y -homomorphisms and asymptotic morphisms the other way round, and also to compositions in the amplified category.

The tensor product functor on the asymptotic category extends to the amplified

asymptotic category (compare Remark 1.6), and we obtain a tensor product in  -

theory:

›

- 

Theorem 2.4. There is a functorial tensor product à on the -theory category ÇÓÒ

which is compatible with the tensor product on ¥§¦ -algebras via the functor from the Y

category of graded separable ¥Q¦ -algebras and graded -homomorphisms into the

OP  -theory category.

The minimal tensor product does not carry over to  -theory, but we have at least

a partial result. First, here is some standard ¥§¦ -algebra terminology. *

Definition 2.10. A (graded) ¥Q¦ -algebra is exact if, for every short exact sequence ¦

of graded ¥ -algebras

/

/

´

R R / / / / 176 Nigel Higson and Erik Guentner

the sequence of minimal tensor products

/

/

›

› ›

¯

´ - Ã ! *

R R

¯ ¯

- Ã ! * - ! * / / ©Ã / /

is exact.

›

¯

,)+ ŠÃ- ! *

In other words, * is exact if and only if the functor is exact.

¦ ¥

Theorem 2.5. Let * be a separable, graded and exact -algebra. There is a functor

¡

›

,)+ - ¯ *  !

à on the -theory category. In particular, if and are isomorphic

¡

› ›

 - ¯ * - ¯ *

! Ã !

in the -theory category then à and are isomorphic there too.

OP

We shall return to the topic of minimal tensor products in Lecture 6.

2.5 Bott Periodicity

Our proof of Bott periodicity in Lecture 1 may be recast as a computation in  - theory, as follows.

Definition 2.11. Let k be a finite-dimensional Euclidean vector space. Denote by

’ ’

9

 Y «  k x CE»æ)~ kQ

$§ the -theory class of the -homomorphism in-

9

 kQ «N 

troduced in Definition 1.26. Denote by ­%$' the -theory class of the

­§C» - k§°ANADC©¸ ¶ k©x asymptotic morphism à introduced in Proposition 1.5.

Proposition 2.4. The composition

 kQ

£ «

« / / «')+« in the  -theory category is the identity morphism .

Proof. This follows from Remark 2.4 and Theorem 1.17, as in the proof of Corol- lary 1.2.

A small variation on the rotation argument we discussed in Section 1.10 now

proves the following basic result:

’

­§CÏ k¨ ) «  ). k¨

Theorem 2.6. The morphisms and CE« in the -theory P category are mutual inverses. O

2.6 Excision

The purpose of this section is to discuss the construction of Ó -term exact sequences

in  -theory. First, we need a simple definition.

¥Q¦ ¥§¦

Definition 2.12. Let be a -algebra. The suspension of is the -algebra

Ô

9

®5\½<$H Q R KS¨C>½ Rw\˜½ K¢M'RGB t

Group C*-Algebras and K-theory 177

Ô Ô ]

9

¥ R KL

In other words is the tensor product of with . If is graded

Ô Ô then so is (the algebra itself is given the trivial grading).

Theorem 2.7. The suspension map

Ô ¥Ô

9 9

~

*§  *§ )¿

is an isomorphism. Moreover there are natural isomorphisms

¡ ¡

Ô ¥Ô

9 9 9 9

 *©>‘   *§>‘  *§ *© 

 and

t

¡

Ô  Proof. It follows from Bott periodicity that is isomorphic to « in the -theory category, and this proves the second part of the theorem. With the periodicity isomor- phisms available, we obtain an inverse to the suspension map by simply suspending a second time.

Here then are the main theorems in the section:

¥§¦ q

Theorem 2.8. Let * be a graded -algebra and let be an ideal in a separable

¦

¥ -algebra . There is a functorial six-term exact sequence

9 9 9

´:q *©  *©  q *§  / /

O

Ô Ô Ô



9 9 9

q *§  *©  ´:q *©

 o o

/ ¥§¦

Theorem 2.9. Let be a graded -algebra and let be an ideal in a separable *

¥¨¦ -algebra . There is a functorial six-term exact sequence

/ /

9 9 9

  *©  *V´   / /

O

Ô / Ô Ô /



9 9 9

*©´   *©    o o

For simplicity we shall discuss only the second of these two theorems (the proofs of the two theorems are similar, although the second is a little easier in some re- spects). For a full account of both see [27, Chapters 5 and 6]. The proof of Theorem 2.9 has two parts. The first is a construction borrowed

from elementary homotopy theory, involving the following notion:

Y ¥§¦

Definition 2.13. Let vC*·)þ¥ be a -homomorphism of (graded) -algebras. ¥Q¦

The mapping cone of v is the -algebra

9

°ÕŠ)Ö;—Nf½&$H*"¥V R KS¨CNv —TM˜½ Rw ½ K¢M'R‰×

¥ and

t

¥ Y ¥©¦ Proposition 2.5. Let vC*ž) be a -homomorphism. For every -algebra

there is a long exact sequence of pointed sets

Ô Ô

9 9 9 9 9

      ¥     *(    ¥  *(  ¥ 

/ / / Õ / /

tštSt t 178 Nigel Higson and Erik Guentner

The proposition may be formulated for homotopy classes of asymptotic mor-

phisms, as above, or for homotopy classes of ordinary Y -homomorphisms (compare

[59]). The proofs are the same in both cases. There are Y -homomorphisms

Ô Ô

9

™S™S™

¥

¥°Õ

¥ * / * / / / / which supply the maps in the proposition, and since the composition of any two

successive Y -homomorphisms in this sequence is null-homotopic, the composition

of any two successive maps of the sequence in the proposition is trivial. Let us prove

9

 *¨ 

exactness at the  term. If the composition

Õ

* ¥

_ _÷ _/ /

is null homotopic then a null homotopy gives an asymptotic morphism from

È È

9

ØANADC¨¥V R K¢ Z

C . The pair comprised of and now determines an asymptotic

¥PÕ

morphism from into , as required. For more details see [27, Chapter 5].

Y ¥©¦ Corollary 2.1. Let vCÝ*%)ê¥ be a -homomorphism. For every -algebra there

is a functorial six-term exact sequence

9 9 9

¥°Õ   *©  ¥(  / /

O

Ô Ô Ô



9 9 9

 ¥(  *©  ¥  o o Õ

This follows from Proposition 2.5 and Theorem 2.7. To prove Theorem 2.9 it

/

PÕ vCÝ*%)¿¥

remains to replace ¥ with in the above corollary, in the case where is

/

² ²

) "R

a surjection with kernel . To this end, observe that there is an inclusion ,

/ °Õ of into ¥ . Using the following construction one can show that this inclusion is an

isomorphism in the  -theory category. /

Theorem 2.10. ([27, Chapter 5].) Let be an ideal in a separable graded ¥©¦ -

“

[

5 B

©Ù4Ú I

algebra . There is a norm-continuous family [ of degree-zero elements

/ ï

in such that ø

“

Ûs [Üs˜K `

(a) R for all ,

Ô Ô

/

“

² ² ²

~

HGJI [ 'R $

(b) y{z}|V[ , for all , and

Ô Ô

7 7 7

“ “

~

$H HGJI [ [ R

(c) y{z}|V[ , for all .

¤ / )+

If Ce ´ is any set-theoretic section of the quotient mapping then the formula

¾ ¤w ¾

“

[ [

½©- j½   Z

Ô / /

OP

defines an asymptotic morphism from ´ into . /

Theorem 2.11. ([27, Proposition 5.14].) Let be an ideal in a separable, graded

¥¨¦ -algebra . The asymptotic morphism associated to the extension

Group C*-Algebras and K-theory 179

Ô /

9

Q R KL

¥

Õ R

R / / / /

Ô / ©Ô

9

 ¥PÕ

determines an element of  which is inverse to the element of

¥Ô / Ô /

9

¥ÝÕ  ¥‡Õ OP

 which is determined by the inclusion of into .

/

‘

«Õ  In view of Theorem 2.7 it now follows that  in the -theory category, and the proof of Theorem 2.9 is complete.

2.7 Equivariant Theory

We are now going to define an equivariant version of  -theory which will be partic- ¤ ularly useful for computing the -theory of group ¥§¦ -algebras. To keep matters as simple as possible we shall work here with countable and discrete groups, although it is possible to consider arbitrary second countable, locally compact groups.

The following definition provides the main idea behind the equivariant theory:

*

Definition 2.14. Let h be a countable discrete group and let and be graded

¥¨¦ ¥Q¦ h

h - -algebras (that is, graded -algebras equipped with actions of by grading-

*

preserving Y -automorphisms). An equivariant asymptotic morphism from to is

ÞANADCV*

an asymptotic morphism ZQCÝ such that

7 J7

9 ;'9

~

}Q™  }Q™ Zc[ ŒM)+R `N)

Zc[ as

7

}b$dh for all $H and all .

Homotopy is defined just as in the non-equivariant case, and we set

9

O

 *( Ðßfo5 *–B

 Homotopy classes of asymptotic morphisms from to

t

* h ¥¨¦ *§

If is a - -algebra then so is the asymptoticO algebra , and an equivari- * ant asymptotic morphism from to is the same thing, up to equivalence, as an

7

*§ Y

equivariant -homomorphism from to . Thanks to this observO ation it is a

straightforward matter to define an equivariant version of the asymptotic category

!

*§ h

that we constructed in SectionO 2.2. The higher asymptotic algebras are -

9

¥¨¦   *¨  g ß

-algebras; we define ! to be the set of -homotopy classes of equivariant

!

*©

Y -homomorphism from to ; and we define

9 9

ß ß

  *(  –y}z{|   *( 

I

!

~

) t These are the morphism sets of a category, using the composition law described in

Proposition 2.2, and this category may be ‘amplified’, as in Section 1.3. Finally, if h is separable (and assuming, as we shall throughout, that is countable) then the

canonical map gives an isomorphism

9 9

  *(    *( 

R

ß ß

I

ð /

t

à á

This is one place where our assumption that á is discrete is helpful: if is not discrete

i â‡d²^ then the action of á on is not necessarily continuous. 180 Nigel Higson and Erik Guentner

See [27] for details.

To define the equivariant  -theory groups it remains to introduce a stabilization

operation which is appropriate to the equivariant context. h Definition 2.15. Let h be a countable discrete group. The standard -Hilbert space

¶ is the infinite Hilbert space direct sum

ß

¡

ã

I

9

ð

¶ ˜ h¨

! ß

equipped with the regular representation of h on each summand and graded so the even numbered summands are even and the odd numbered summands are odd.

The standard h -Hilbert space has the following universal property:

8 h

Lemma 2.5. If ¶ is any separable graded -Hilbert space then the tensor product

¶ h

Hilbert space ¶„-"¶ is unitarily equivalent to via a grading-preserving, - ß

equivariant unitary isomorphismß of Hilbert spaces.

¶ h

Proof. Denote by ¶ the Hilbert space equipped with the trivial -action. The

¡

¡

ã

œ œ

-o }wn,)Š} ¶ç- h¨ - }w

formula ™ defines a unitary isomorphism from to

¡

ã

- h¨

¶ , and from it we obtain a unitary isomorphism

R

¶ -Ž¶ ¶ -Ž¶

à Ã

ð /

ß ß

t

¶ -‚¶ ¶ ¶ -Ž¶ ‘ ¶

à Ã

Since is just a direct sum of copies of it is clear that  .

ß ß ß ß

¶XÃ-߶ ‘ ¶

Hence  , as required.

ß ß

*

Definition 2.16. Let h be a countable discrete group and let and be graded, sep-

9

¥¨¦  *©

arable h - -algebras. Denote by the set of homotopy classes of equiv-

ß

- VÃ-¸ ¶  *dÃ-¸ ¶ 

ariant asymptotic morphisms from »¨Ã to ,

ß ß

9 9

ß

 *©>6  » - -߸ ¶  * -¸ ¶ 3 

à à Ã

ß ß ß t

Remark 2.5. The virtue of working with the Hilbert space ¶ , as in the above def-

ß

h ZQCE»§Ã- ,A AFC

inition, is that if ¶ is any separable graded -Hilbert space and if

-¸ ¶d Z

*dà is an equivariant asymptotic morphism then determines an element of

9

*© Z ¸ ¶ 

 . To see this, simply tensor by and apply Lemma 2.5.

ß ß Remark 2.6. The construction described in the previous remark has a generalization

which will be important in Lecture 4. Suppose that ¶ is a separable, graded Hilbert

space which is equipped with a continuous family of unitary h -actions, parametrized

9<;

K 

by `G$X . The continuity requirement here is pointwise strong continuity, so that

Š$

if } and then is norm-continuous in . Suppose now that and

h ¥¨¦

* are - -algebras and that ä

A graded á -Hilbert space is a graded Hilbert space equipped with unitary representations

of á on its even and odd grading-degree summands.

Group C*-Algebras and K-theory 181

ZQC» - ÞANADCV* -¸ ¶d

à à is an asymptotic morphism which is equivariant with respect to the given family of

h -actions, in the sense that

Ô Ô

¾ J¾

9

~

y{z}| Zc[ }Q™  }Q™ [ Zn[ Œ R

[HGxI

¾

9

Š$

for all } and . Then too determines an element of . Indeed,

ß

¶ 

after we tensor with ¸ and apply the procedure in the proof of Lemma 2.5 we

ß

-Ž¸ ¶ -߶à 

obtain an asymptotic morphism into *<Ã which is equivariant in the

ß

¶ -Ž¶Ã

usual sense for the single, fixed representation of h on . ß

Remark 2.7. One final comment: it is essential that in Definition 2.16 we include a

¶  factor of ¸ in both arguments. If we were to leave one out then we would obtain

a quite differentß (and not very useful) object.

9

*§ By comparing the definition of  to the construction of the equivariant,

amplified asymptotic category we immediatelyß obtain the following result:

9

 *§

Theorem 2.12. The  -theory groups are the morphism sets of an ad-

ß ß ¥©¦

ditive category whose objects are the separable graded h - -algebras. There is

¥©¦ h

a functor from the homotopy category of graded h - -algebras and graded - 

equivariant Y -homomorphisms into the equivariant -theory category which is the P

identity on objects. O

›

- Ã

The equivariant  theory category has a tensor product . Moreover there ÇÓÒ

are six-term exact sequences of  -theory groups associated to short exact sequences ¥¨¦ of h - -algebras. The precise statements and proofs are only minor modifications of what we saw in the non-equivariant case, and we shall omit them here. See [27].

2.8 Crossed Products and Descent ¥©¦ In order to apply equivariant  -theory to the problem of computing -algebra ¤ -theory one must first apply a descent operation which transfers computations in

equivariant  -theory to computations in the nonequivariant theory. This involves the

notion of crossed product ¥Q¦ -algebra, and we begin with a rapid review of the basic

definitions (see [53]) for more details).

h ¥§¦

Definition 2.17. Let h be a discrete group and let be a - -algebra. A co-

9

Z v‰ ¥Q¦ * Y

variant representation of in a -algebra is a pair consisting of a -

¥Q¦ * v

homomorphism Z from into a -algebra and a group homomorphism from * h into the unitary group of the multiplier algebra of which are related by the

formulas

7 J7 ¡ 7

9

j'Z }©™  } 1Z Ív } $< }Š$

v for all , t

182 Nigel Higson and Erik Guentner

h ¥§¦

Definition 2.18. Let h be a discrete group and let be a - -algebra. The lin-

9

På h  h ear space ¥ of finitely-supported, -valued functions on is an involutive

algebra with respect to the convolution multiplication and involution defined by

Õ

¡

¡

æ ¡

Ž } xx ŽA ŽV™ ½ } M ½ ½ ½

çÙ

ß

¡

¦ ¦

½ } M:}Q™ ½ }  

¥§¦ *

Observe that a covariant representation of in a -algebra determines a

9

I

å

Z v ¥ h  *

Y -homomorphism from into by the formula

Õ

9

I

å

Z v‰Œ½< Z ½ }ŒÍv }  h 

for all ½r$&¥ .

è

Ù

ß

9

¥¨¦ h 

Definition 2.19. The full crossed product ¥§¦ -algebra is the completion

9

å

¥ h  ¥Q¦

of the Y -algebra in the smallest -algebra norm which makes all the

I

Z v

Y -homomorphisms continuous.

¥©¦ ¥¨¦ h¨

Example 2.4. Setting « we obtain the full group -algebra . h

If is graded, and if acts by grading - preserving automorphisms, then

9

h 

¥¨¦ has a natural grading too (the grading automorphism acts pointwise on

9

På h 

functions in ¥ ).

9

¥Q¦ h 

Remark 2.8. The ¥Q¦ -algebra contains a copy of and the multiplier alge-

9 9

å

h  h  h ¥

bra of ¥¨¦ contains a copy of within its unitary group. Elements of

7 7 7

è è è

è

‘ ™Ä} $< R } can be written as finite sums Ù , where and for almost all . It will usually be convenient to useß this means of representing elements. For example

the grading automorphism is

Õ Õ

7 J7

è è

™N}V,) ­ \™N}

t

è è

Ù Ù

ß ß

¦ ¦

¥ ¥ The full crossed product is a functor from h - -algebras to -algebras which is (extending the terminology of Section 2.3 in the obvious way) both continuous and exact. As a result, there is a descent functor from the equivariant asymptotic category

to the asymptotic category,

9 9 9 9

~

¦ ¦

I ß

h *§Í  h  ¥   *¨  )ü  ¥

I t

In order to obtain a corresponding functor in  -theory we need the following com-

putation:

* h ¥©¦ ¶

Lemma 2.6. Let h be a discrete group, let be a - -algebra and let be a

}˜$h

h -Hilbert space on which the group element acts as the unitary operator

¦ è

C]¶„)^¶ . The formula

Õ Õ

¦

è è è è è

— -| à c™S}V,) — ™N} šÃ-|

è è

Ù Ù

ß ß Group C*-Algebras and K-theory 183

determines an isomorphism of ¥Q¦ -algebras

9 9

¥¨¦ h * -¸ ¶dŒ R ¥¨¦ h *§ -¸ ¶d

à Ã

ð /

t

Ä

9

å

¥ h *Hà ¸ ¶dŒ

Proof. The formula defines an algebraic Y -isomorphism from to

Ä

9

å

h *§LÃ ¸ ¶d

¥ . Examining the definitions of the norms for the max tensor product Y and full crossed product we see that the Y -isomorphism extends to a -isomorphism

of ¥Q¦ -algebras. Combining the lemma with the descent functor between asymptotic categories we obtain the following result:

Theorem 2.13. There is a descent functor from the equivariant  -theory category to

¦

h ¥

the  -theory category which maps the - -algebra to the full crossed product

9

¥¨¦ h   h Y

¥¨¦ -algebra , and which maps the -theory class of a -equivariant -

 Y

homomorphism Z§CA a)Ù* to the -theory class of the induced -homomorphism

9 9

h  ¥¨¦ h *§ OP

from ¥¨¦ to .

*

Corollary 2.2. Let h be a countable discrete group. Suppose that and are sep-

¥¨¦ *

arable h - -algebras and that and are isomorphic objects in the equivariant

¤f ¤"

9 9

¥Q¦ h x ¥Q¦ h *©x OP  -theory category. Then is isomorphic to .

2.9 Reduced Crossed Products ¤

We also wish to apply equivariant  -theory to the computation of -theory for re- duced crossed products. Here the operation of descent works smoothly for a large class of groups, as the following discussion shows, but not so well for all groups, as we shall see in Lecture 6.9 In the following definition we shall use, in a very modest way, the notion of

Hilbert module. See [45] for a treatment of this subject.

¡

ã

9

h  h ¥Q¦

Definition 2.20. Let be a - -algebra and denote by the Hilbert -

• • è•

}  } Œ¦ C†hÿ)

module comprised of functions for which the series ‘ is

norm-convergent in . The regular representation of is the covariant representa-

¡

ã

9 9

v‰ h 

tion Z into the bounded, adjoinable operators on given by the formulas

¡

¡

J7 7

ã

9 9 9

• • •

ŽA Z   ŽAj Ž ™  $ h 

and

¡

¡

ã

9 9

• • •

} ŽÝ v }  ŽA> $ h  t

The regular representation determines a Y -homomorphism from the crossed prod-

9

h  ¥Q¦

uct algebra ¥¨¦ into the -algebra of bounded, adjoinable operators on

¡

ã

9 

h . é

It should be pointed out here that Kasparov’s ê ê -theory has no such limitation in this

respect. However it has other shortcomings. Indeed as we shall see in Lecture 6 there is no aë ideal bivariant ê -theory for -algebras.

184 Nigel Higson and Erik Guentner

h ¥Q¦

Definition 2.21. Let be a - -algebra. The reduced crossed product algebra

9 9

ì

h  ¥¨¦ h 

¥¨¦ is the image of in the regular representation.

ì

¥§¦ ¥¨¦ h¨ Example 2.5. Setting « we obtain the reduced group -algebra .

Like the full crossed product, the reduced crossed product is a functor from

¥¨¦ ¥Q¦ (graded) h - -algebras to (graded) -algebras. However unlike the full crossed

product the reduced crossed product is not exact for every h (although inexact ex- amples are hard to come by — see Lecture 6). This prompts us to make the following

definition:

9

ì

§,)Ù¥§¦ h  Definition 2.22. A discrete group h is exact if the functor is exact in the sense of Definition 2.8.

There is a very simple and beautiful characterization of exact groups, due to

Kirchberg and Wassermann [43]. ¥©¦

Proposition 2.6. A discrete group h is exact if and only if its reduced group -

ì h¨

algebra ¥¨¦ is exact.

ì

h¨ h

Proof (Proof (sketch)). Exactness of ¥Q¦ is implied by exactness of since in

9

ì

¥©¦ h 

the case of trivial h -actions the reduced crossed product is the same thing

ì ì

› › ›

¯ ¯ ¯

h¨ u- h¨ ¥¨¦ - !¥¨¦ !VçÃ-

as (note that is trivially graded, so ! here). The

ì

¦ h¨

reverse implication is argued as follows. If ¥ is exact then the sequence

/

9 9

ì ì

› ›

¥¨¦ h ‰- ¯ ¥¨¦ h¨ ¥¨¦ h n- ¯ ¥¨¦ h¨

! !

R / /

/

9

ì

›

¦ ¦

h ´ ‰- ¯ ¥ h¨ ¥ !

/ / R

¦

¥ Û

is exact. But for any h - -algebra there is a functorial embedding

9 9

~

ì ì

›

¦ ¦ ¦

¯

h Ûb )ê¥ h Ûb‰- h¨ ¥ !¥

,)í}Š-#} T–,)îT -ÀK

defined by the formulas } and , and moreover a functorial,

-K,)MT }-b}V,)} }(-zŽb,)êR

continuous and linear left-inverse defined by T , and

p

€Ž

if } . It follows that the sequence

/ /

9 9 9

ì ì ì

¦ ¦ ¦

¥ h  ¥ h  ¥ h ´  R R / / / / is a direct summand of the minimal tensor product exact sequence above, and is

therefore exact itself. For more details see Section 5 of [43]

¥Q¦ * ¥¨¦ * *

Exercise 2.4. If * is an exact -algebra and if is a -subalgebra of then

then * is also exact.

Thanks to the exercise and to Proposition 2.6 it is possible to show that many classes of groups are exact. For example all discrete subgroups of connected Lie groups are exact and all hyperbolic groups (these will be discussed in Lecture 5) Group C*-Algebras and K-theory 185 are exact too. Every amenable group is exact since in this case the reduced and full crossed product functors are one and the same. For more information on exactness see for example [67]. We shall also return to the subject in Section 4.5. By retracing the steps we took in the previous section we arrive at the following result:

Theorem 2.14. Let h be an exact, countable, discrete group. There is a descent func- 

tor from the equivariant  -theory category to the -theory category which maps a

9

ì

¥¨¦ ¥§¦ ¥Q¦ h 

h - -algebra to the reduced crossed product -algebra , and which

Y Z§CE ç) *

maps the class of a h -equivariant -homomorphism to the class of the

9 9

ì ì

¥Q¦ h  ¥¨¦ h *§ OP

induced Y -homomorphism from to .

Corollary 2.3. Let h be an exact, countable, discrete group. Suppose that and

¦

h ¥ *

* are separable - -algebras and that and are isomorphic objects in the

¤f ¤f

9 9

ì ì

¥§¦ h Œ ¥¨¦ h *§Œ

equivariant  -theory category. Then is isomorphic to .

OP

2.10 The Baum-Connes Conjecture

In this lecture we shall formulate the Baum-Connes conjecture and prove it in some

simple cases, for example for finite groups and free abelian groups. We shall also ¦ sketch the proof of the conjecture for so-called ‘proper’ coefficient ¥ -algebras. This result will play an important role in the next chapter. The proof for proper algebras is not difficult, but it is a little long-winded, and we shall refer the reader to the monograph [27] for the details. We shall continue to work exclusively with discrete groups. Our formulation of

the conjecture, which uses  -theory, is equivalent to the formulation in [7] which

¤D¤ ¤r¤ uses -theory. Indeed there is a natural transformation from to  which determines an isomorphism from the ¤r¤ -theoretic ‘left-hand side’ of the Baum-

Connes conjecture to its  -theoretic counterpart. The isomorphism can be proved either by a Mayer-Vietoris type of argument (see for example Lecture 5) or by di-

rectly constructing an inverse. See also the discussion in Section 4.6 which in many ¤ cases reduces the conjecture to a statement in -theory, independent of both  - ¤D¤ 10 theory and -theory. Our treatment using  -theory is quite well suited to the theorems we shall formulate and prove in Lecture 4. However a major drawback of

 -theory is that it is not well suited to dealing with inexact groups. In any case, the ¤r¤

 -theoretic and -theoretic developments of the Baum-Connes theory are very

similar, and having studied by himself the basics of ¤r¤ -theory the reader could de-

¤r¤ ¤r¤ velop the Baum-Connes conjecture in -theory simply by replacing  with

throughout this lecture. ¢ c In fact the argument of Section 4.6 can be made to apply to any discrete group, but we shall not go into this here. 186 Nigel Higson and Erik Guentner

2.11 Proper G-Spaces

Let h be a countable discrete group. Throughout this lecture we shall be dealing U with Hausdorff and paracompact topological spaces equipped with actions of h

by homeomorphisms.

U ¾ U

$ h

Definition 2.23. A h -space is proper if for every there is a -invariant

¦ U ¾

¢ h h

open subset containing , a finite subgroup Á of , and a -equivariant ¦

map from to hQ´:Á .

U hQ´lÁ

The definition says that locally the orbits of h in look like . h

Example 2.6. If Á is a finite subgroup of then the discrete homogeneous space +

hQ´:Á is proper. Moreover if is any (Hausdorff and paracompact) space with an

U

I I

¿h ±B+ h +

Á -action then the induced space (the quotient of by the

Á h diagonal action of Á , with acting on by right multiplication) is proper.

In fact every proper h -space is locally induced from a finite group action:

U ¾ U

$ h

Lemma 2.7. A h -space is proper if and only if for every there is a -

¦ U U

¢ h Á

invariant open subset containing , a finite subgroup Á of , an -space

¦

I

±

h h + OP + , and a -equivariant homeomorphism from to .

Many proofs involving proper spaces proceed by reducing the case of a general

I

h ±:w proper h -space to the case of the local models , and hence to the case of

finite group actions, using the lemma. U

Lemma 2.8. A locally compact h -space is proper if and only if the map from

U U U ¾ ¾ ¾

9 9

I I

}  } 

h to which takes to is a proper map of locally compact P

spaces (meaning that the inverse image of every compact set is compact). O

¤ i

Example 2.7. If h is a discrete subgroup of a Lie group , and if is a compact

¤

i;´ h subgroup of i , then the quotient space is a proper -space.

2.12 Universal Proper G-Spaces

U

h +

Definition 2.24. A proper h -space is universal if for every proper -space

U

+Ê) there exists a h -equivariant continuous map , and if moreover this map is

unique up to h -equivariant homotopy. h

It is clear from the definition that any two universal proper h -spaces are - ¦

equivariantly homotopy equivalent. For this reason let us introduce the notation h

¦ h for a universal proper -space (with the understanding that different models for h will agree up to equivariant homotopy).

Proposition 2.15 Let h be a countable discrete group. There exists a universal

OP proper h -space.

Group C*-Algebras and K-theory 187 U

Here is one simple construction (due to Kasparov and Skandalis [36]). Let K be the space of (countably additive) measures on h with total mass or less. This

is a compact space in the topology of pointwise convergence. Let U be the closed

0

U

2

subspace of consisting of measures of total mass ¡ or less. The set-theoretic

U U U

£ï h

difference  is a locally compact proper -space which is universal. 0

In examples one can2 usually provide a much more concrete model. See [7] for examples (and see also Lectures 4 and 5 below). The following result, which we shall not prove, gives the general flavor of these constructions.

Proposition 2.7. Let be a complete and simply connected Riemannian manifold

of nonpositive sectional curvature. If a discrete group h acts properly and isometri-

h OP cally on then is a universal -space.

Remark 2.9. The manifold here could be infinite-dimensional.

2.13 G-Compact Spaces

U ¤

h ¢

Definition 2.25. A proper h -space is -compact if there is a compact subset

U U

whose translates under the h -action cover .

U U h

If is a h -compact proper -space then is locally compact and the quotient U

´lh is compact.

U U h

Definition 2.26. Let be a h -compact proper -space. A cutoff function for is

U

9

>C )· R KT

a continuous function ð such that

Ø ðw

(a) Öx×eØ is compact, and

¡

¾ U ¾

è

$ ð }§™ MK ‘

(b) Ù , for all .

ß h Observe that the sum in (b) is locally finite. Every h -compact proper -space ad-

mits a cutoff function. Moreover any two cutoff functions are, in a sense, homotopic:

ð

if ð and are cutoff functions then the functions

¡ ¡

9 9

~

ð¢[\òñ `0ð  K `x1ð `j$u R KS

are all cutoff functions.

U

h h Lemma 2.9. Let ð be a cutoff function for the -compact proper -space . The

formula

¡

¾ ¾ ¾

?

ð  } j:ð }

t

JU U ¤

9 9

å å

¥ ¥ ¥¨¦ x h  defines a projection in h , and hence in . The -theory class

of this projection is independent of the choice of cutoff function.

U

9

¥§¦ h

Remark 2.10. We are using here the streamlined notation  in place of

U

9

h ¥ Œ ¥¨¦ . 188 Nigel Higson and Erik Guentner

Proof (Proof of the Lemma). A computation shows that ? is a projection (note that

?

ð

the sum involved in the definition of is in fact finite). If ð and are cutoff func- l[

tions then associated to the homotopy of cutoff functions ð defined above there is a

¤

?

ð ð homotopy of projections [ , and therefore and give rise to the same -theory

class, as required. ¤ Definition 2.27. We will call the unique -theory class of projections associated to

cutoff functions the unit class:

U U ¤"

9 9 ? 9

¦ ¦

Œ x>‘  « ¥ h  n$ ¥ h



t U

Exercise 2.16 (See [54, Thm 6.1].) Let be a proper h -space. Show that the full

U U

9 9

ì

  h ¥Q¦ h

and reduced crossed products ¥Q¦ and are isomorphic.

U

9

qå h ¥På x KAd½

Hint: One approach is to show that if ½<$&¥ , and if is invertible in

U U

9 9

ì

 h K>f¥qå h ¥På x Kj"½

¥¨¦ , then the inverse actually lies in . It follows that

U U U

ì

9 9 9

  Š) ¥ h h ¥§¦ h is invertible in ¥Q¦ too, and therefore, the map is spectrum-preserving, and hence isometric.

Remark 2.11. As a result of the exercise, we can obviously define a unit class in

U ¤f

9

ì

x h ¥¨¦ too.

2.14 The Assembly Map

U

9 Ûb

In this section we shall further streamline our notation and write  in place

U U

ß

9 9

¥  Ûb  Ûb

of  . Observe that is covariantly functorial on the category

U

ß ß h

of h -compact proper -spaces .

Û h Definition 2.28. Let h be a countable discrete group and let be a separable -

¥¨¦ -algebra. The assembly map

U ¤"

ó

9 9

¦

Ce ÛbM) ¥ h Ûbx ß

is the composition

4<ô

Ú

U

U descent

9 9 9 9 9 9

¦ ¦ ¦

h ÛbŒ h  ¥ h Ûbx Ûb  ¥  « ¥

 / / ß

where the first map is the descent homomorphism of Section 2.8 and the second is

U

? 9 9

‰$< « ¥Q¦ h x

composition with the unit class  .

Û h ¥©¦

Definition 2.29. Let h be a countable group and let be a - -algebra. The

¤

h ¥§¦ Û

topological -theory of h with coefficients in a - -algebra is defined by

4

U ¤

[Hõ

9 9 9

h Û > y}z}| Ûb 

~

)

ß

‰öÏ÷

-inv, ß -cpt

ß ß h

where the limit is taken over the collection of h -invariant and -compact subspaces

U

¦ h ¢ , directed by inclusion.

Group C*-Algebras and K-theory 189

U

¦

h h h

To explain the limit, note that if ¢ø+ù¢ are -compact proper -

U h

spaces then is a closed subset of + and restriction of functions defines a -

  JU

¥ +© ¥ 

equivariant Y -homomorphism from to . This induces a homomorphism

JU

9 9

  + Ûb

from Û  to .

U

ß ß

¦

h h h

If ¢:+”¢ are -compact proper -spaces then under the restriction map

U

9 9 9 9

Œ « ¥¨¦ h +Vx  « ¥¨¦ h +

from  to the unit class for maps to the unit class

U

¦ h for ; consequently the assembly maps for the various h -compact subsets of are compatible and pass to the direct limit:

Definition 2.30. The (full) Baum-Connes assembly map with coefficients in a sepa-

¥¨¦ Û

rable h - -algebra is the map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h Ûb>) ¥ h Û x

which is obtained as the limit of the assembly maps of Definition 2.28 for h -compact

U

° ¦ subspaces h .

Definition 2.31. The reduced Baum-Connes assembly map with coefficients in a sep-

¥¨¦ Û

arable h - -algebra is the map

4

¤f ¤

[Hõ

ó

ì

9 9

ì

¦

h Ûb>) ¥ h Ûbx C

obtained by composing the full Baum-Connes assembly map ó with the map from

¤f ¤"

9 9 9

ì

¥¨¦ h Ûbx ¥¨¦ h Û x h Ûb

to induced from the quotient mapping from ¥§¦

9

ì

h Ûb

onto ¥¨¦ .

U

h h Remark 2.12. If h is exact and if is a -compact proper -space then there is a

reduced assembly map

] U ¤f

ó

9 9 9

ì

¦

h Ûbx CÝ ¥  ÛbM) ¥ ß

defined by means of a composition

4Nô

Ú

U U

descent

9 9 9 9 9

9

ì ì ì

 ¥¨¦ h  ¥¨¦ h Ûbx  « ¥¨¦ h ÛbŒ

¥  Û 

 / / ß

involving the reduced descent functor of Section 2.9. The Baum-Connes assembly ó map ì may then be equivalently defined as a direct limit of such maps.

2.15 Baum-Connes Conjecture

The following is known as the Baum-Connes Conjecture with coefficients (the ‘coef- Û ficients’ being of course the auxiliary ¥Q¦ -algebra ).

Conjecture 2.1. Let h be a countable discrete group. The Baum-Connes assembly

map

4

¤ ¤"

[Hõ

ó

ì

9 9

ì

¦

C h Ûb>) ¥ h Ûbx

t

¥§¦ Û is an isomorphism for every separable h - -algebra . 190 Nigel Higson and Erik Guentner

Not a great deal is known about this conjecture. We shall prove one of the main results (which covers, for example, amenable groups) in the next section. Unfortu- nately, thanks to some recent constructions of Gromov, the Baum-Connes conjecture with coefficients appears to be false, in general. See Lecture 6.

In the next conjecture, which is the official Baum-Connes conjecture for dis-

9

Û À« Û À¥ R KL

crete groups, the coefficient algebra * is specialized to and .

4

¤ ¤

[Hõ

ì

h¨ ¥¨¦ h¨x

We shall use the notations and to denote topological and ¥Q¦ -

¦ ¦ ¤ algebra ¤ -theory in these two cases (this of course is customary usage in -theory).

Conjecture 2.2. Let h be a countable discrete group. The Baum-Connes assembly

map

4

¤ ¤

[Hõ

ó

ì

ì

¦

h¨>) ¥ h¨Œ C

¦

t ¦ is an isomorphism.

Somewhat more is known about this conjecture, thanks largely to the remarkable work of Lafforgue [44, 62]. For example, the conjecture is proved for all hyperbolic groups (we shall define these in Lecture 5). What is especially interesting is that, going beyond discrete groups, the Baum-Connes conjecture has now been proved for all reductive Lie and ? -adic groups (this is part of what Lafforgue accomplished using his Banach algebra version of bivariant ¤ -theory, although by invoking a good deal of representation theory many cases here had been confirmed prior to Lafforgue’s work). Unfortunately we shall not have the time to discuss either Lafforgue’s work or the topic of ¤ -theory for non-discrete groups. At the present time, the major open question seems to be whether or not the Baum-Connes conjecture (with or without coefficients, according to one’s degree of optimism) is true for discrete subgroups of connected Lie groups. Even the case of uniform lattices in semisimple groups remains open. Considerably more is known about the injectivity of the Baum-Connes assem- bly map, and fortunately this is all that is required in some of the key applications of the conjecture to geometry and topology. We shall say more about injectivity in

Lecture 5.

9

h Ûb

Remark 2.13. We shall discuss in Lecture 6 the reason for working with ¥©¦

9

ì

h Û  in place of ¥Q¦ .

2.16 The Conjecture for Finite Groups

The reader can check for himself that the Baum-Connes conjecture is true (in fact it is a tautology) for the trivial, one-element group. Next come the finite groups. Here

the conjecture is a theorem, and it is basically equivalent to a well-known result of ¤ Green and Julg which identifies equivariant ¤ -theory and the -theory of crossed product algebras in the case of finite groups. See [23, 35]. What follows is a brief account of this.

Group C*-Algebras and K-theory 191

Û h ¥©¦ Theorem 2.17 (Green-Julg). Let h be a finite group and let be a - -algebra.

The Baum-Connes assembly map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h Ûb>) ¥ h Û x

¥§¦ Û

is an isomorphism for every h - -algebra .

9 9

ì

¦ ¦

¥ h Ûb>¥ h Ûb Û

Remark 2.14. If h is finite then for every .

¦ h If h is a finite group then can be taken to be the one point space. So the

theorem provides an isomorphism

9 9 9

 « Ûb  « ¥¨¦ h Ûbx

R ú

ð /

ß

t

? h¨

The unit projection $%¥¨¦ which is described in Lemma 2.9 is the function

= = ?

} ‚çK¢´ h h¨

, which is the central projection in ¥Q¦ corresponding to the trivial h representation of h (it acts as the orthogonal projection onto the -fixed vectors in

any unitary representation of h ).

Theorem 2.17 is proved by defining an inverse to the assembly map ó . For this

9

h Ûb

purpose we note that ¥Q¦ may be identified with a fixed point algebra,

¡

ã

9

9

¥¨¦ h Ûb

R

 Û-bû܃¿ h¨Œ3

ß

ð /

? ?

è è

è

T }™

by mapping to Ù (where is the projection onto the functions sup-

ß

S}AB } Kj-yü }  ü ported on 5 ) and by mapping to , where is the right regular represen-

tation (the fixed point algebra is computed using the left regular representation). The Y

displayed Y -homomorphism can be thought of as an equivariant -homomorphism

¡

ã

9

h Ûb h ÛÚ-Bû£ƒ¿ h¨x from ¥¨¦ , equipped with the trivial action of , into . It

induces a homomorphism

¡

ã

9 9

9

 « ¥¨¦ h Û x

« ۄ-bû܃¿ h¨Œx

/ 

ß

t

¤f ¤

4

[Hõ

9 9

h ÛbŒ h Ûb But the left hand side here is ¥§¦ and the right hand side is , and it is not difficult to check that the above map inverts the assembly map ó , as required. For details see [27, Thm. 11.1].

2.17 Proper Algebras

Theorem 2.17 has an important extension to the realm of infinite groups, involving

the following notion:

¦

¥ *

Definition 2.32. A h - -algebra is proper if there exists a locally compact proper

U U

Y Z ¥ 

h -space and an equivariant -homomorphism from into the grading-

JU

ZG ¥ 3A™S* degree zero part of the center of the multiplier algebra of * such that

is norm-dense in * .

192 Nigel Higson and Erik Guentner ý Remark 2.15. We shall say that * , as in the definition, is proper over . Throughout the lecture we shall deal with proper algebras which are separable.

The notion of proper algebra is due essentially to Kasparov [38], in whose work ¤r¤

proper algebras appear in connection with é -theory, a useful elaboration of

¤r¤ ¤r¤ é

-theory. We shall not develop here, or even its  -theoretic counterpart. While this limits the amount of machinery we must introduce, it will also make some

of the arguments in this and later lectures a little clumsier than they need be.

h ¥Q¦

Examples 2.18 If h is finite every - -algebra is proper over the one point space.

h ¥ ý h ¥Q¦ *

If ý is a proper -space then is a proper - -algebra. If is proper over

h ¥Q¦ Û *<Ã-Û

ý then, for every - -algebra , the tensor product is also proper.

9 9

ì

¦ ¦

h *§>¥ h *© ¥ Exercise 2.5. Prove that if * is proper then .

A guiding principle is that the action of a group on a proper algebra is more or ¦ less the same thing as the action of a finite group on a ¥ -algebra. With this in mind

the following theorem should not be surprising. *

Theorem 2.19. [27, Theorem 13.1] Let h be a countable discrete group and let ¥Q¦

be a proper h - -algebra. The Baum-Connes assembly map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h *©>) ¥ h *©x

is an isomorphism.

¤"

ó

ì

9

ì

h *§x Remark 2.16. Thanks to Exercise 2.5, the assembly map into ¥¨¦ is an isomorphism as well.

The proof of Theorem 2.19 is not difficult, but with the tools we have to hand it is rather long. So we shall just give a quick outline. The following computation is key not just to the proof of Theorem 2.19 but also to a number of results in Lecture 5.

Proposition 2.8. [27, Lemma 12.11] Let Á be a finite subgroup of a countable group

w Á

h and let be a locally compact space equipped with an action of by homeo-

h ¥Q¦

morphisms. If Û is any - -algebra there is a natural isomorphism

9 9 9

I

‘ ± ±

¥ w6 Ûb  ¥ h w6 Û  



ß

Á ¥§¦ where on the left hand side Û is viewed as an - -algebra by restriction of the

h -action.

F

I I

h ±Qw 5 B w

Proof. The space w is included into as the open set , and as a

] 

I

Á ¥ ¥ 6 h ±yw6 result there is an -equivariant map from w into . Composition

with this map defines a ‘restriction’ homomorphism

÷0ÿ

9 9

I

± ±

¥ h w6 Ûbþ  ¥ w6 Ûb

 / ß Group C*-Algebras and K-theory 193

To construct an inverse, the important observation to make is that every Á -equi-

]

w6 Û h

variant asymptotic morphism from ¥ into extends uniquely to a -equi-

¡



ã

I

h ± w6 Ûu-V¸ hQ´:ÁrŒ

variant asymptotic morphism from ¥ into . Decorating

¸ ¶H

this construction with copies of » and we obtain an inverse map

9 9

I

± ±

¥ w6 Ûb  ¥ h w6 Ûb

 / ß as required.

Proposition 2.8 has the following immediate application:

Lemma 2.10. Let h be a countable group. If the assembly map

4

¤" ¤

[Hõ

ó

9 9

¦

h *©>) ¥ h *§x C

¥Q¦ * h

is an isomorphism for every h - -algebra which is proper over a -compact

h ¥§¦

space ý , then it is an isomorphism for every - -algebra. ¥§¦

Proof. Every proper algebra is a direct limit of h - -algebras which are proper ¤ over h -compact spaces. Since -theory commutes with direct limits (see Exer-

cise 1.4), as does the crossed product functor, to prove the lemma it suffices to prove

¤

4

[Hõ

9

) h Ûb

that the same is true for the functor ÛÜ, . In view of the definition

¤

4

[Hõ

9

h Ûb h h

of it suffices to prove that if ý is a -compact proper -simplicial

9

¥ ýŽ Ûb

complex then the functor  commutes with direct limits. By a Mayer- ß

Vietoris argument the proof of this reduces to the case where ý is a proper homoge-

neous space hQ´:Á . But here we have a sequence of isomorphisms

 ¤"

9 9 9 9

¦

 ¥ hQ´lÁr Ûb>‘ q± « Ûb‘ ¥ h Û x

  ß the first by Proposition 2.8 and the second by Theorem 2.17. Since ¤ -theory com- mutes with direct limits the lemma is proved.

Lemma 2.11. Let h be a countable group. If the assembly map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h *©>) ¥ h *§x

¥Q¦ *

is an isomorphism for every h - -algebra which is proper over a proper homo-

¦

6hQ´:Á h ¥

geneous space ý then it is an isomorphism for every - -algebra which h is proper over a h -compact proper -space.

Proof. This is another Mayer-Vietoris argument, this time in the * -variable. Observe

¦

ý h ý

that if * is proper over then to each -invariant open set in there corresponds

/ w ©¦

¥ ;™€* * an ideal  of . Using this, together with the long exact sequences

in  -theory and the five lemma, an induction argument can be constructed on the

ý h number of h -invariant open sets needed to cover , each of which admits a -map to a proper homogeneous space. 194 Nigel Higson and Erik Guentner

The proof of Theorem 2.19 therefore reduces to the case where * is proper over

* hQ´lÁ

some proper homogeneous space hQ´lÁ . Observe now that if is proper over

hQ´:Á *¡

then * is a direct sum of ideals corresponding to the points of , and the ideal

F

Á ¥§¦ corresponding to Á is an - -algebra . The proof is completed by developing a

variant of the isomorphism in Proposition 2.8, and producing a commuting diagram

¤f

¤

4

[Hõ

9

9

¥¨¦ h *©x

h *©

ú /

R R

ð ð

¤f

¤

4  

[Hõ

9

9

¥¨¦ Á *§Œ

Á *¢ S

R

ú

ð / t

See [27, Chapter 12] for details.

2.18 Proper Algebras and the General Conjecture

The following simple theorem provides a strategy for attacking the Baum-Connes conjecture for general coefficient algebras. The theorem, or its extensions and rel- atives, is invoked in nearly all approaches to the Baum-Connes conjecture. As we shall see in Lecture 5 the theorem is particularly useful as a tool to prove results about the injectivity of the Baum-Connes map.

Theorem 2.20. Let h be a countable discrete group. Suppose there exists a proper

’

9 9

¥¨¦ * $" « *§ ­o$f * «N

h - -algebra and morphisms and in the equiv-

ß ß

ariant  -theory category such that

’

9

­ K$< « «ß

ß

t

¤ ¤f

4

[Hõ

ó

9 9

h Ûbu) ¥¨¦ h Û x

Then the Baum-Connes assembly map C is an

¥Q¦ Û h isomorphism for every separable h - -algebra . If in addition is exact then

the reduced Baum-Connes assembly map

4

¤f ¤

[Hõ

ó

ì

9 9

ì

¦

h Ûb>) ¥ h Ûbx C

is an isomorphism.

¦

Û h ¥

Proof. Let h be a countable discrete group, let be a separable - -algebra and ’

let ­ and be as in the statement of the theorem. Consider the following diagram:

¤ ¤f

4

[Hõ

9 9

h «‚Ã- Ûb ¥¨¦ h «ŽÃ-©ÛbŒ

ú /

 

£ £

¤ ¤"

4  

[Hõ

9 9

h * -Ûb R ¥¨¦ h * -ÛbŒ

à Ã

ú

ð /

 

¤ ¤f

4  

[Hõ

9 9

¥¨¦ h «ßÃ-VÛbŒ «‚Ã- Ûb

h /

t ú Group C*-Algebras and K-theory 195

The horizontal maps are the assembly maps; the vertical maps are induced from  -

’

9 9

-ÀKf$À « -©Û * -Ûb ­ -ÀK¨$% * -Û « -©Ûb

à à Ã

theory classes à and . The

ß ß

¦ *

diagram is commutative. Since the ¥ -algebra is proper, so is the tensor product

* -Û Ã and therefore by the Theorem 2.19 the middle horizontal map is an isomor- phism. By assumption, the compositions of the vertical maps on the left, and hence also on the right hand side are the identity. It follows that the top horizontal map is an isomorphism too. The statement concerning reduced crossed products is proved in exactly the same way.

2.19 Crossed Products by the Integers

In this section we shall apply the approach outlined in the previous section to just

about the simplest example possible beyond finite groups: the free abelian group ! hÊ„³ . What follows will serve as a model for the more elaborate constructions in the next lecture. For this reason it might be worth the reader’s while to study the

present case quite carefully.

¼

! h

Let h act by translations on in the usual way and then let act on the graded

8¼

!

œ œ

}§™š½ >˜½ }§™   

¥¨¦ -algebra that we introduced in Lecture 1 by .

8¼

! !

¥¨¦   Exercise 2.6. With this action of the free abelian group ³ , the -algebra is proper.

We are going to produce a factorization

J¼

! 

 

« «

/ /

£

’

!

 ­ in ³ -equivariant -theory. The elements and are very small modifications of

the objects we defined in Lecture 1 while studying Bott Periodicity.

’

8¼

!

  Y

Definition 2.33. Denote by C†»È) the -homomorphism that was in-

’

J¼

!

[

€' K C‰»ú)   Y

troduced in Definition 1.26, and for ` denote by the -

’ ’

¾ ¾

¡

[ [ [

½  ½M >½ ` 

homomorphism , where ½ .

’

¡

½M½ ` ¥( ¥

Thus [ , where is the Clifford operator introduced in Lecture 1.

’

J¼

!

QA AFCÛ 

Lemma 2.12. The asymptotic morphism C]» given by the above family

’

J¼

! !

Y ³ Ø 

of -homomorphisms []C]Ȭ) is -equivariant.

!

}Š$<³

Proof. We must show that if ½&$ » and then

Ô Ô

¡ ¡

~

y}z{| ½ ¥( } ¥(xx R ` ½ `

[HGJI

t

} ¥§¦ »

Since the set of all ½X$D» for which this holds (for all ) is a -subalgebra of it

J¾ Ÿ

±

¡

 » suffices to prove the limit formula for the generators ½r of . For these we have

196 Nigel Higson and Erik Guentner

Ô Ô Ô Ô

¡ ¡ ¡ Ÿ ¡ ¡ Ÿ ¡

± ±

~ ~

½ ` ¥( } ½ ` ¥(x  ` ¥  ` } ¥( 

Ô Ô

¡

~

sf` ¥ } ¥(

~

} ¥(

by the resolvent identity. Since the Clifford algebra-valued function ¥ is ¼

bounded on ! the lemma is proved.

’

J¼

!

9

¤£ «  x

Definition 2.34. Denote by $À the class of the asymptotic mor-

’

J¼

!

Î

QASAFCÛ 

phism C]» .

¼ ¤

! !

œ œ

9

}X$ ³ $ $ R KT } ™

Definition 2.35. If and , and if , then denote by ¢ the

¤ ¼

! !

œ

}X$ } ™ ½ }X$ ³

translation of by . Denote by ¢ the corresponding action of

8¼

!

 

on elements of the ¥Q¦ -algebra and also on operators on the Hilbert space

J¼

! 

¶ that was introduced in Definition 1.27.

8¼

!

9

¥£   «\

To define the class ­f$r that we require we shall use the asymp-

J¼ 8¼

! !

Î

- °ANAeCV¸ ¶ x totic morphism ­§C»(Ã that we defined in Proposition 1.5, but

we shall interpret it as an equivariant asymptotic morphism in the following way:

8¼

! !

9<;

-‰Ž<$ »Ã-  }Š$H³ `;$X K 

Lemma 2.13. If ½MÃ , , and then

Ô Ô

~

[ [

y}z}| ­ ½MÃ-q}c™ÐŽÝ ½NÃ-‰ŽA R }Q™ ­

[

¸

[HGJI

t

0

¡

Û }D™ ½ ` Ûb– [

Proof. The Dirac operator is translation invariant, and so ¸

¡

0

触



½ ` Ûb ` }d™ ` b ^

[  

for all . But ¸ for all . The lemma therefore fol-

0 ö

lows from the formula ø

¡



­n[ ½ -UŽÝM˜½ ` ÛbŒ b Ã

for the asymptotic morphism ­ .

J¼

!

9

  «N ¤£ 

Definition 2.36. Denote by ­Ê$À the -theory class of the equiv-

J¼ J¼ J¼

! ! !

Î

-‰ zA AFCç¸ ¶ Œ ¸ ¶ x

ariant asymptotic morphism ­§CE»QÃ , where is

9

} eM,)Ø}Q™ [

equipped with the family of actions ¸ (compare Remark 2.6).

0

’

9

­ 6K$< « «ß

Proposition 2.21 Continuing with the notation above, £ .

Î

¤ 8¼ J¼

! !

9

¦

$Ê R KS  ¥   

Proof. Let and denote by ¢ the -algebra , but with the

J¼

! ! !

9

³ } ŽAM,)Ø}\™ Ž   ³ ¢ scaled -action ¢ . The algebras form a continuous field of -

¥¨¦ -algebras over the unit interval (since the algebras are all the same this just means

J¼

! ! ©

ô

 ³  ³ ¥¨¦

that the -actions vary continuously). Denote by Ú the - -algebra of

ï

9

R KT

continuous sections of this field (namely the continuous functions from  into

¥¤ Ĥ 8¼

! ©

}Q™DŽA >`}§™ ¢Ž   ³

 , equipped with the -action ) . In a similar way, form

J¼ J¼

! © !

ô

¶ x ³ ¥¨¦ ¸ ¢ ¶ Œ ¸

the continuous field of - -algebras and denote by Ú

©

ï ¥¨¦ the ³ - -algebra of continuous sections. With this notation, what we want to prove

is that the composition

J¼

!

 

£ « « / /

is the identity in equivariant  -theory.

Group C*-Algebras and K-theory 197

J¼

J¼

!

!

¸ ¶ x

­§C» -  The asymptotic morphism à _ _ _/ induces an asymp-

totic morphism

J¼

8¼

„

!

!

9

ô

ô

¶ Œ ¸

Ú

 ­§C]» - Ã

Ú _ _ _/

ï

ï

’

8¼

!

  and similarly the asymptotic morphism C]» _ _ _/ determines an asymp-

totic morphism

„

8¼

’

!

ô

  Ú

C» _ _ _/

ï

’

9

R KS

by forming the tensor product of with the identity on ¥V and then composing

9

Y¢®» R KS with the inclusion » as constant functions. Consider then the diagram of

equivariant  -theory morphisms

¨

¨

J¼

!

9 9

ô

 

Ú

¥V R KT £

« / /

ï

ð

 

Ê Ê

J¼ 

 ! 

Ģ 

« «

/ /

£

¤

9

¾¢ $ç R KT

where ‹ denotes the element induced from evaluation at . Observe that

¤

ç¢  ‹¢

‹ is an isomorphism in equivariant -theory, for every (indeed, , considered

¤

 R

as a Y -homomorphism, is an equivariant homotopy equivalence). Set . In

9

 « «ß

this case the bottom composition is the identity element of £ . This is be-

¤ ¼

! !

Î

 R

cause when the action of ³ on is trivial and the asymptotic morphism

’

 8¼

!

C]»  Y

_ _ _/  is homotopic to the (trivially equivariant) -homomorphism

’ ’

J¼

!

­  ÀK Š  C»–) of Definition 1.26. So the required formula follows from Proposition 2.4. Since the bottom composition in the diagram is the identity it fol- 11 ¤ lows that the top composition is an isomorphism too. Now set %K . Since, as we just showed, the top composition in the diagram is the identity, it follows that the bottom composition is the identity too. The proposition is proved.

3 Groups with the Haagerup Property

3.1 Affine Euclidean Spaces Recall that we are using the term Euclidean vector space to refer to a real vector space equipped with a positive-definite inner product. In this lecture we shall be studying Euclidean spaces of possibly infinite dimension.

Definition 3.1. An affine Euclidean space is a set  equipped with a simply-transitive

action of the additive group underlying a Euclidean vector space k . An affine sub-

U

 k  

space of  is an orbit in of a vector subspace of . A subset of generates

U 

if the smallest affine subspace of  which contains is itself.

¢Í¢

©  

 

It is the identity once a is identified with via evaluation at any , or equivalently

©   ©   

  a once a is identified with via the inclusion of into as constants. 198 Nigel Higson and Erik Guentner

Remark 3.1. Note that even if  is infinite-dimensional we are not assuming any completeness here (and moreover affine subspaces need not be closed).

Example 3.1. Every Euclidean vector space is of course an affine Euclidean space (over itself).

The following prescription makes  into a metric space.

Definition 3.2. Let  be an affine Euclidean space over the Euclidean vector space

F F F F

œ œ

:9 ¡ ¡

k k  

. If $u , and if is the unique vector in such that , then we

Ô Ô

F F F JF

œ

9

¡ ¡ j

define the distance between and to be T .

¼

I

 —ACý ý )

Let ý be a subset of an affine Euclidean space and let be

¡

• 8• • J•

9 9

¡

¡

 HT the square of the distance function: — . This function has the

following properties:

8• • •

9

NR $*ý

(a) — , for all ,

8• • 8• • 8• •

¢9 ¡ ¢9 ¡ ¡l9

I

— G$lý ý M—

(b)  , for all , and

• • 7 7 ¼ 7

!

:9 9 :9 9

¯

! $%ý ! $ g 'R

‘ ð

(c) for all , all , and all such that ¯ ,

tštSt tStšt

!

Õ

7 J• •S 7€

9

¯ ¯

 s–R —



t

ð

¯

ï k

(To prove the inequality, identify  with and identify the sum with the quantity

¡

Ô Ô

7 •

!

~

¯ ¯

µ

ð ‘

¯ .)

¼

I

—AC‚ý ý)

Proposition 3.1. Let ý be a set and let be a function with the above

È

œý')+ ý

three properties. There is a map C of into an affine Euclidean space such 

that the image of ½ generates and such that

¡

8• • HȂ 8• Ȃ J•

:9 ¡ 9 ¡ 9

— >QT  x

• • È

9

¡

âšCÏý)+Qâ *ý

for all $ . If is another such map into another Euclidean space

HȂ 8• È 8•

‚CÝ )£Q⠎ ŒQ â 

then there is a unique isometry Ž such that , for every

• %ý

$ .

¼\

ý‚ Proof. Denote by  the vector space of finitely supported, real-valued functions

on ý which sum to zero:

Õ

¼ ¼ 8•

 ý‚Eo5c½&$  ý‚ŠC ½ NRGB

¼c

ý‚

If we equip  with the positive semidefinite form

Õ

K J• 8• • J•

Ç

¢9 ¡ :9 ¡ ¡

~

ƈ½ ½  ½ x— Œ½ 

µ

 

Ù

£

ï

0

2

¼ ¼

Ç

9

 ý‚ Æ8½ ½  R  ý‚

then the set of all ½o$ for which is a vector subspace of

¼n

ýG  (this is thanks to the Cauchy-Schwarz inequality) and the quotient

Group C*-Algebras and K-theory 199

¼c ¼

k  ý‚´  ýG

has the structure of a Euclidean vector space. Consider now the set of all finitely

K

supported functions on ý which sum to . Let us say that two functions in this set

¼

ýG

are equivalent if their difference belongs to  . The set of equivalence classes is

È È‚ J•



k C¼ý')¿ M

then an affine Euclidean space  over . If is defined by then

¡

HȂ J• Ȃ J• È J• •

9 ¡ L9 ¡

âSC‚ý®)õ¨â T  Œ;—  , as required. If is another such map then

the unique isometry Ž as in the statement of the lemma is given by the formula

Õ

J• È J•

â

Ž ½M ½  

(note that in an affine space one can form linear combinations so long as the coeffi-

cients sum to K ).

Exercise 3.1. Justify the parenthetical assertion at the end of the proof. Prove that if

Ž is an isometry of affine Euclidean spaces then

Õ Õ Õ

7 7 F 7 8F

¯ ¯ ¯ ¯ ¯

6K  Ž M Ž  t

This completes the uniqueness argument above.

¼

I

—AC‚ý ý') Definition 3.3. Let ý be a set. A function is a negative-type kernel if

— has the properties (a), (b) and (c) listed prior to Proposition 3.1.

Thus, according to the proposition, maps into affine Euclidean spaces are classi- fied, up to isometry, by negative-type kernels.

3.2 Isometric Group Actions

h 

Let  be an affine Euclidean space and suppose that a group acts on by isome-

F F

}d,)M}V™ h  tries. If is any point of  then the function maps into , and there is

an associated negative-type function

¡

F F

:9 ¡ 9 ¡

— } } MQT } ™ } ™ 

t

— h

Since h acts by isometries the function is -invariant, in the sense that

L9 ¡ ¢9 ¡ 9 j 9 š9 ¡ 9

— } } M— }¾} } }  } } } $

JF

9

} >o— } and as a result it is determined by the one-variable function — , which is

a negative-type function on h in the sense of the following definition.

¼ —ACAh§) Definition 3.4. Let h be a group. A function is a negative-type function

on h if it has the following three properties:

JF MR

(a) — ,

¡

 } M— } }Š$

200 Nigel Higson and Erik Guentner

¡

U ¤ 7 7

 

!



9 9

g } } $ ¯"$òh ¯3— } }  súR

!

‘ ð ¯

(c) ¯ , for all , all , and all

tStšt

7 7 ¼ 7

!

ï

¢9

¯

!d$ ‘ R ð

such that ¯ .

tStSt

— h

Proposition 3.2. Let h be a set and let be a negative-type function on . There is

F

 $H

an isometric action of h on an affine Euclidean space and a point such that F

the orbit of generates  , and such that

¡

JF F

9 9

— } NQT }¨™  b$dh

for all } .

¡

9

¡ ¡

— } } M˜— } } 

Proof. Let  be the affine space associated to the kernel , as in 

the statement of Proposition 3.1. There is therefore a map from h into , which we

„

V,) } 

shall write as } , whose image generates , and for which

¡

„ „

¡ :9 L9 ¡

}  } — } } >QT

t

d$

Fix Ž and consider now the map . Since

¡ ¡

„ „

9 9 9

¡

¡ ¡

} MQT } — } } M#T Ž4} Ž¶} 

it follows from the uniqueness part of Proposition 3.1 that there is a (unique) isometry

„

} Ž4} ŽÀ$%h

of  mapping to . The map which associates to this isometry is the

F F

„  required action, and  is the required point in .

Remark 3.2. There is also a uniqueness assertion: if ©â is a second affine Euclidean

F

â†$H — } M

space equipped with an isometric h -action, and if is a point such that

¡

8F F

9

â }©™ âã }H$rh h ŽßCÝÀ)ž§â

T , for all , then there is a -equivariant isometry such

JF F

â M that Ž .

Remark 3.3. Proposition 3.2 is of course reminiscent of the GNS construction in ¥V¦ -

algebra theory, which associates to each state of a ¥§¦ -algebra a Hilbert space repre- sentation and a unit vector in the representation space.

Exercise 3.2. Let  be an affine Euclidean space over the Euclidean vector space

h 

k . Suppose that a group acts on by isometries. Show that there is a linear

h k

representation v of by orthogonal transformations on such that

8F F

œ œ

9

}Q™  j}Q™  v } 

F

œ

b$dh $d $lk for all } , all , and all .

Exercise 3.3. According to the previous exercise, if k is viewed as an affine space k

over itself then for every isometric action of h on there is a linear representation

h k

v of by orthogonal transformations on such that

œ œ

}Q™ }Q™LRŽuv }

t

¤ ¤w

œ œ

9

$X R KS }ߙ  }ߙ1Rw]

Show that for every the ‘scaled’ actions ¢ are also  isometric actions of h on . Group C*-Algebras and K-theory 201

3.3 The Haagerup Property h Definition 3.5. Let h be a countable discrete group. An isometric action of on an

affine Euclidean space  is metrically proper if for some (and hence for every) point F

of  ,

JF F

9 ;

y{z}| T }§™ M

è

GJI

t éë

In other words,an action is metrically proper if for every R there are only finitely

JF F

9

é

}Š$

Definition 3.6. A countable discrete group h has the Haagerup property if it admits a metrically proper isometric action on an affine Euclidean space.

In view of Proposition 3.2, the Haagerup property may characterized as follows:

Proposition 3.3. A group h has the Haagerup property if and only if there exists on

¼ —ACAhÞ) h a proper, negative-type function (that is, a negative-type function for

which the inverse image of each bounded set of real numbers is a finite subset of h ).

OP

Groups with the Haagerup property are also called (by Gromov [5]) a-T-menable. This terminology is justified by the following two results. The first is due to Bekka, Cherix and Valette [8].

Theorem 3.1. Every countable amenable group has the Haagerup property.

JF 12

Z >K Z }j

Proof. A function ZQCÝh6)¿« is said to be positive-definite if , if

¡

9 9 9 9

} $dh ý $<« Z }  } ý !

, and if for all ! , and all ,

tštSt tStšt

!

„

Õ

¡

 

¯

ý Z } } xý '¨R

¯



t

ð

¯

ï

¦

~

K Z Observe that if Z is positive-definite then is a negative-type function. Now

one of the many characterizations of amenability is that h is amenable if and only h if there exists a sequence 5¢ZN!AB of finitely supported positive-definite functions on

which converges pointwise to the constant function K . Given such a sequence we can

¦

~

u

‘ K Z\! 

find a subsequence such that the series m converges at every point of

h . The limit is a proper, negative-type function.

The next result is essentially due to Delorme [18].

º h Theorem 3.2. If h is a discrete group with Kazhdan’s property , and if has in

addition the Haagerup property, then h is finite. ¢8£

This normalization is not always incorporated into the definition, but it is convenient here.

¢

i i

d _d 

$ $¥) !

We should also remark that! the next condition is actually implied by the

g!#" $%

¢

i

( (

¢'&



 d condition .

202 Nigel Higson and Erik Guentner

º h Proof. If h has property then every isometric action of on an affine Hilbert space has a fixed point (this is Delorme’s theorem).13 But if an isometric action has

a fixed point it cannot be metrically proper, unless h is finite. Remark 3.4. The reader is referred to [17] for a comprehensive introduction to the

theory of property º groups. We shall also return to the subject in the last lecture. Various classes of discrete groups are known to have the Haagerup property. Here is an incomplete list.

* Amenable groups (see above), * Finitely generated free groups [30], or more generally, groups which act properly on locally finite trees.

* Coxeter groups [9],

‡,+V ‡Ü¦V

*

9 9

K¢ g K¢ Discrete subgroups of g and [56, 55], * Thompson’s groups [20, 51]. For more information about the Haagerup property consult [12].

3.4 The Baum-Connes Conjecture The main objective of this lecture is to discuss the proof of the following theorem:

Theorem 3.3. Let h be a countable discrete group with the Haagerup property.

9

¥Q¦ *  ­ $a * «N

There exists a proper h - -algebra and -theory elements

’ ’

ß ß

9 9

« *© ­  6K($H « «ß

and $H such that .

ß ß Thanks to the theory developed in the last lecture this has the following conse- quence:

Corollary 3.1. Let h be a countable discrete group with the Haagerup property and

¦

h ¥

let Û be a - -algebra. The maximal Baum-Connes assembly map with coeffi- h cients in Û is an isomorphism. Moreover if is exact then the reduced Baum-Connes

assembly map with coefficients in Û is also an isomorphism Remark 3.5. The theorem and its corollary are also true for locally compact groups with the Haagerup property.

Remark 3.6. In fact the final conclusion is known to hold whether or not h is exact, but the proof involves supplementary arguments which we shall not develop here. In any case, perhaps the most striking application of the corollary is to amenable groups, and here of course the full and reduced assembly maps are one and the same

(since the full and reduced crossed product ¥§¦ -algebras are one and the same). In connection with the last remark it is perhaps worth noting that the following problem remains unsolved:

Problem 3.4 Is every countable discrete group with the Haagerup property ¥V¦ -

exact? ¢J¬ In fact the converse is true as well. Group C*-Algebras and K-theory 203

3.5 Proof of the Main Theorem, Part One

Let  be an affine Euclidean space equipped with a metrically proper, isometric

 h ¥©¦

action of a countable group h . In this section we shall build from a proper - -

Q 

algebra - . In the next section we shall construct equivariant -theory elements

’ ’

­ K ­ and , as in Theorem 3.3, and in Section 3.7 we shall prove that .

Notation 3.5 From here on we shall fix an affine Euclidean space  over a Eu-

clidean vector space k . We shall be working extensively with finite-dimensional

 

affine subspaces of  , and we shall denote these by , and so on. We shall

Ç Æ k

denote by k the vector subspace of corresponding to the finite-dimensional affine

Ç

 ¢" k

subspace  . If then we shall denote by the orthogonal complement of

Ç Ç Æ ÆˆÇ

 k k

 in . This is the orthogonal complement of in . Note that

Ç Æ Ç Æ

9

 €k f

Æ Æ3Ç Ç

and that this is a direct sum decomposition in the sense that every point of  has a

F F

Æ

œ 

unique decomposition  .

Æ Æ3Ç Ç The following definition extends to affine spaces a definition we previously made

for linear spaces. The change is only very minor. 

Definition 3.7. Let  be a finite-dimensional affine Euclidean subspace of . Let

Ç

9

 >¥  mjy}zon k Œ

 .

Ç Ç Ç

Here is the counterpart of Proposition 1.13:

¢  

Lemma 3.1. Let  be a nested pair of finite-dimensional subspaces of .

8F

F

Ç Æ

œ œ

¡ ¡

 -|Ž ¹Ž ŽÞ§ Ž Ž  HMŽ The correspondence à , where determines an

isomorphism of graded ¥§¦ -algebras

  M‘  k šÃ-   OP



Æ Æ3Ç Ç t

In Lecture 1 we made extensive use of the Clifford operator ¥ . Recall that this

œ œ

& k

was the function ¥ from the Euclidean vector space into the Clifford

jy{zn k§ algebra m . In the present context of affine spaces the Cliiford operator is not generally available since to define it we have to identify affine spaces with their underlying vector spaces, and we want to avoid doing this, at least for now. But we shall work with Clifford operators associated to various vector spaces which appear as orthogonal complements.

The following is a minor variation on Definition 1.26. k

Definition 3.8. Let k be a finite-dimensional linear subspace of and denote by

Ç Y

¥ the corresponding Clifford operator. Define a -homomorphism

Ç

’

/.10

C]»¨)æ»(Ã-  Ç

by the formula

’

JU

½M½ -QKj'K -¥ 

à Ã

Ç Ç t 204 Nigel Higson and Erik Guentner

Remark 3.7. The definition uses the language of unbounded multipliers. An alterna- ’

tive formulation, using the ‘comultiplication’ å , is that is the composition

Ç

Sí

î

2.

0

9

» - 

Ã

» ì £ -Ž»

/ »Ã /

’ ’

2.10

  Y ½M˜½ ¥ 

where C]»") is the -homomorphism of Definition 1.26.

Ç

- § ¥Q¦

We are now going to construct a ¥Q¦ -algebra as a direct limit of -

0

- Å   

algebras »(Ã associated to finite-dimensional affine subspaces of .

Ç

¢'

Definition 3.9. Let  be a nested pair of finite-dimensional affine subspaces

Ç Æ Y

of  . Define a -homomorphism

’

0

Cl» - Å M)æ» -‰ Å43J

à Ã

Æ Ç

ï

2. 0

0

‘

»QÃ- Å53J »¨Ã- 3 šÃ- Å 

by using the identification  and the formula

’

0 2. 0

0 0

9

~

‘

»(Ã- Å 76ž5Ã-®Æ‰, ) 3 5:šÃ-®Æ"$ »Ã- 3 šÃ- Å  »(Ã- Å53J



’

2.

0

C»")^» - 3  Y

where à is the -homomorphism of Definition 3.8.

Æ3Ç

¢  ¢ å 

Lemma 3.6 Let  be finite-dimensional affine subspaces of . We

’ ’ ’

Ç Æ

å å 

have  .

Æ Æ Ç Ç

ï ï ï

¾ F J¾ ¾ÝF

¡ ¡

“ œ

j ; »

2 2 Ò

Proof. Compute using the generators Ò and of .

0

»¨Ã-‰ Å  

As a result the graded ¥Q¦ -algebras , where ranges over the finite- Ç dimensional affine subspaces of  , form a directed system, as required, and we can

make the following definition:

¥§¦ 

Definition 3.10. Let  be an affine Euclidean space. The -algebra of , denoted

Q ¥Q¦

- , is the direct limit -algebra

0

- §> y{z}| »Ã- Å 

~

)

8:9§;'8 t fin. dim.

affine sbsp.

 - Q h ¥§¦ An action of h by isometries on makes into a - -algebra. To see

this, first define Y -isomorphisms

} C  M)Ø }] 

¦ ¦

Ç Ç

8F x F

¡

} CUmjy{zn ½ )} ½ } x k )Wmjy{zn }¶k 

by } , where here is induced

¦ ¦ ¦ ¦

Ç Ç

}jCe)+ from the linear isometry of k associated to (see Exercise 3.2). Group C*-Algebras and K-theory 205

Lemma 3.7 The following diagram commutes:

9

0

»Ã- Å53

£ <>=

- Å 

»(Ã /

Sí Sí

î î

è è

¹ 1

 0



» - BDÅ43J

Ã

-‰ BFÅ  9

»(Ã /

t

£ ?@<>= ?

OP

The lemma asserts that the maps } are compatible with the maps in the directed

¦ ¦

Q }

system which is used to define - . Consequently, we obtain a map on the

¦É¦

Q h ¥§¦ direct limit. In this way - is made into a - -algebra, as required.

Theorem 3.8. Let  be an affine Euclidean space equipped equipped with a metri-

¥©¦ - Q

cally proper action of a countable discrete group h . Then the -algebra is a ¥Q¦

proper h - -algebra.

  ¥§¦

Proof. Denote by A the grading-degree zero part of the center of the -algebra

0

Ç

» - Å 

à . It is isomorphic to the algebra of continuous functions, vanishing at infin-

’

9N;

I

  R   A

ity, on the locally compact space  . The linking map embeds

Ç Ç Æ Ç

ï

  A § ¥§¦

into A , and so we can form the direct limit , which is a -subalgebra

Æ

Q - §

of - , and is contained in the grading-degree zero part of the center of (in

A Q

fact it is the entire degree zero part of the center). The ¥§¦ -subalgebra has the

Q\™B- Q - Q A Q

property that A is dense in . The Gelfand spectrum of is the

9<;

I

o§ R   

locally compact space ý , where is the metric space completion of

ý 

 and is given the weakest topology for which the projection to is weakly con-

¡ ¡

JF F

14 9

`T 

tinuous and the function ` is continuous, for some (hence any) fixed

F

h k $' . If acts metrically properly on then the induced action on the locally

compact space ý is proper.

Remark 3.8. The above elegant argument is due to G. Skandalis.

3.6 Proof of the Main Theorem, Part Two

In this section we shall assume that  is a countably infinite-dimensional affine Eu-

clidean space on which h acts by isometries (this simplifies one or two points of our

presentation). For later purposes it will be important to work with actions which are h not necessarily proper. Note however that if h has the Haagerup property then will act properly and isometrically on some countably infinite-dimensional affine space

 .

’

9 9

«M - - Q $H « Qx

We are going to construct classes ­u$< and . We

’

F

ß ß

shall begin with the construction of , and for this purpose we fix a point $" .

Ð

¢

D C

Observe that C is an affine space over the Hilbert space ; by identifying as an orbit of

D C D we can transfer the weak topology of the Hilbert space to . 206 Nigel Higson and Erik Guentner

This point is, by itself, an affine subspace of  , and there is therefore an inclusion

Y -homomorphism

’

C]»¨)E- Åc

t

’

FL

0

» - Å  $a

The image of lies in all those subalgebras à for which , and

’

0

Ç

» - Å  Y

considered as a map into à the -homomorphism is given by the formula

’

9

Ce½d,)ê½ ¥ 

Ç

ï

w 8F F FL

~

Ce )Šmjy{zn k  ¥ M $lk

where ¥ is defined by .

Ç Ç Ç Ç Ç

ï ï

’

QANADCF- Ån

Lemma 3.2. If C]» is the asymptotic morphism defined by

’ ’

9

[ [

½M ½ 

’

¾ ¾

¡

>½ `  h

where ½:[ , then is -equivariant.

F F

Proof. We must show that if and are two points in a finite-dimensional affine ½r$b»

space  , then for every ,

Ç

Ô Ô

¡ ¡

9

~

y{z}| ½ ` ¥  ½ ` ¥  R

[HGxI

Ç Ç

ï ï

JF F F

~

¨ ¥

where ¥ is as above and similarly . It suffices to compute the

¾ ¾ÛŸ

±

Ç ¡ Ç

ï ï

> 

limit for the functions ½ . For these one has

Ô Ô Ô Ô

¡ ¡ ¡ ¡ JFL F

9

~ ~

½ ` ¥  ½ ` ¥  –` ¥ ¥ '` T 

Ç Ç Ç Ç

t

ï ï ï ï

The proof is complete.

’

9

« - §x 

Definition 3.11. The element $X is the -theory class of the equiv-

’ ’ ’

ß

[ [

½  QANADCG- Ån CݽH,)

ariant asymptotic morphism C]» defined by .  The definition of ­ is a bit more involved. It will be the -theory class of an

asymptotic morphism

9

­§CH- §PANADC©¸ ¶ QŒ

Q and our first task is to associate a Hilbert space ¶ to the infinite-dimensional

affine Euclidean space  . We begin by broadening Definition 1.27 to the context of

affine spaces. 

Definition 3.12. Let  be a finite-dimensional affine subspace of , with associ-

Ç 

ated linear subspace k . The Hilbert space of is the space of square integrable

Ç Ç

jy{zn k  

m -valued functions on :

Ç Ç

¡

9

mjy{zn ¶  Mi  k Œ

Ç Ç Ç

t

jy}zon k 

This is a graded Hilbert space, with grading inherited from that of m . Ç The following is the Hilbert space counterpart of Lemma 3.1.

Group C*-Algebras and K-theory 207

° 

Lemma 3.3. Let  be a nested pair of finite-dimensional subspaces of the

Ç Æ

k  

affine space  and let be the orthogonal complement of in . The corre-

F 8F

Æ3Ç Ç Æ

œ œ

¡ ¡

Žl§ Ž -|Ž Ž  jBŽ ¹Ž 

spondence à , where determines an isomorphism

¶  M‘ ¶ k  -߶   OP Ã

of graded Hilbert spaces  .

Æ ÆˆÇ Ç

Following the same path that we took in the last section, the next step is to as-

 

semble the spaces ¶ into a directed system.

Ç k

Definition 3.13. If w is a finite-dimensional Euclidean vector space then the basic

Iò$d¶ w6

vector ½ is defined by

F

¡ ¡ —NM¡—

I

™:JBK L

ª

2

½ >v

0 I 0

ö ø

2

t

F

¡ —NM¡— ¡

I

™:JBK L

ª

½ $bw v

I

2 0

Thus maps to the multiple 0 of the identity element

ö ø

2

jy{zn w6

in m .

Ô Ô

¡

I

™'JOK L

v ½PI 6K

Remark 3.9. The constant 0 is chosen so that .

ö ø

$d¶ k  ¶  

Using the basic vectors ½ we can organize the Hilbert spaces

Æ3Ç Ç

into a directed system as folloÆ3Ç ws.

¢£

Definition 3.14. If  then define an isometry of graded Hilbert spaces

Ç Æ

C¶  >)æ¶  

k by

Æ3Ç Ç Æ

~

‘

¶   ¶  Q6<½H, )¿½ -½&$ ¶Ã k šÃ-Ž¶  



Æ Ç Æ3Ç Æ3Ç Ç

t

å

¢6 ¢6 

Lemma 3.9 Let  be finite-dimensional affine subspaces of . Then

Ç Æ

å å

#k k OP

k .

Ç Æ Æ3Ç We therefore obtain a directed system, as required, and we can make the follow-

ing definition:

¶ Q Definition 3.15. Let  be an affine Euclidean space. The graded Hilbert space

is the direct limit

9

¶ QM y{z}| ¶  

~

)

Ç

9

8 8 ö fin. dim. affine sbsp.

in the category of Hilbert spaces and graded isometric inclusions.

 ¶ Q

If h acts isometrically on then is equipped with a unitary representation

- Q h of h , just as is equipped with a -action.

We are now almost ready to begin the definition of the asymptotic morphism

- QPA ADCV¸ ¶ Qx ­QC . What we are going to do is construct a family of asymptotic

morphisms,

0

9

­\ljC »(Ã- Å PANADC©¸ ¶ ÅnŒ

208 Nigel Higson and Erik Guentner

 ¢'

one for each finite-dimensional subspace of  , and then prove that if then

Ç Æ

the diagram 9

0

¸ ¶ Qx

» - Å 

à _ _ _/

9

ð

£R<

 

- Å53 ¸ ¶ Qx

»Ã _ _ _/

<

is asymptotically commutative. Once we have done that we shall obtain a asymptotic

0

»(Ã- Å  y}z{|

morphism defined on the direct limit ~ , as required.

To give the basic ideas we shall consider) first a simpler ‘toy model’, as follows. 

Suppose for a moment that  is itself a finite-dimensional space. Fix a point in ;

 k

call it RX$ ; use it to identify with its underlying linear space ; and use this

F F

¡

,)Ü` `J' K

identification to define scaling maps on  , for , with the common

ŽH$š   RV$H Že[>$š  

fiexed point RV$H . If and if then define by the usual

JF F

¡ Ç Ç Ç

e[ M€Ž `  formula Ž .

Lemma 3.4. Let  be an affine subspace of a finite-dimensional affine Euclidean

Ç

Û  * ¥ bÛ

space  . Denote by the Dirac operator for and denote by

Ç Ç

ÇOS ÇOS ÇOS UT

the Clifford-plus-Dirac operator for  . The formula

Ç

F [

* -§Kj'KÃà -Û  K]Ã-  ­\Ç Cݽ\Ã-UŽb,)¿½

[

Ç

ÇOS

defines an asymptotic morphism

0

Ç

­ C »(Ã- Å PANADC©¸ ¶ ÅnŒ t

Proof. The operator * is essentially self-adjoint and has compact resolvent (see

S

Ç

0

[ [

½M ¯ C]»¨)^¸ ¶ Å4T x ¯

Section 1.13). So we can define Y -homomorphisms by

[

* 

½ . Moreover we saw in Section 1.12 that the formula

S

Ç

D [ [

Û x ­ CݽMÃ-UŽb,)¿½

Ç

0 0

­QC»QÃ- Å xANADC&¸ ¶ Å Œ ­ defines an asymptotic morphism . The formula for Ç

in the statement of the lemma is nothing but the formula for the composition

í

î

í

î

0 0

ì

- Å  »¨Ã-Ž»(Ã- Å WV ¸ ¶ ¡T xšÃ-¸ ¶  x

»Ã / _ _ _/

Ç

t

Ç ­

So Ç is an asymptotic morphism, as required.

¢¨

Lemma 3.5. Let  be a nested pair of affine subspaces of a finite-dimensional

Ç Æ

Û Û  

affine Euclidean space  . Denote by and the Dirac operators for and ,

Æ Ç Æ

and denote by Ç

0

3

­ C» - Å °ASAFCV¸ ¶ Ånx ­ C» - Å538°ANADC©¸ ¶ ÅnŒ

Ã Ã Ç and

the asymptotic morphisms of Lemma 3.4. The diagram

Group C*-Algebras and K-theory 209

9

0

¸ ¶ Qx

- Å 

»(Ã _ _ _/

9

ð

£R<

 

- Å53 ¸ ¶ Qx

»Ã _ _ _/

<

is asymptotically commutative.

¾ F ¾

¡

“ œ

© V 2

Proof. We shall do a computation using the generators Ò and

¾ÝF

¡

»    2

Ò of . Denote by the orthogonal complement of in , so that

Æ3Ç Ç Æ

T

   f

Æ3Ç Ç Æ

and

T

¶ Q>‘ ¶   -Ž¶   -߶  

à Ã



Æ3Ç Ç

t Æ

To do the computation we need to note that under the isomorphism of Hilbert spaces

¶  ‘ ¶  šÃ-Ž¶   Û Û -QK‚®K]Ãà -Û

 the Dirac operator corresponds to

Æ3Ç Ç Æ Æ3Ç Ç

(to beÆ precise, the self-adjoint closures of these essentially self-adjoint operators cor-

* -§K;˜KÃà -*

respond to one another). Similarly * corresponds to under the

Æ3Ç

S S

Ç Æ

‘

XT  ¶ XT šÃ-Ž¶   ¶

isomorphism  . Hence by making these identifications

Æ3Ç Ç

of Hilbert spaces we get Æ

¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡

¦ § ¦S§ ¦ §

~ ~ ~

Ø ` Û M Ø ` Û  - Ø ` Û 

Ã

Æ Æ3Ç Ç

and

¡ ¡ ¡ ¡ ¡ ¡

¡

¡

¡

¦ § ¦S§ ¦S§

~ ~ ~

Ø ` * M Ø ` * šÃ- Ø ` * 

t

Æ3Ç

S S

Ç Æ

0

“

­ -‰ŽÃ <$ »Ã- Å  Ç

Now, applying [ to the element we get

¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡

¦ § ¦S§ ¦ §

~ ~ ~



Ø ` *  - Ø ` *  - Ø ` Û x b

à Ã

ÆˆÇ Ç

S

Æ

’

“

¸ ¶ XT xLÃ-¸ ¶  xLÃ-¸ ¶  Œ ­  -UŽÃ Æ

in , while applying [ to we get

Æ3Ç Ç Æ3Ç

Æ

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¦ § ¦ § ¦ § ¦ §

~ ~ ~ ~



Ø ` *  - Ø ` Û  Ø ` ¥  - Ø ` Û x y

à Ã

t

Æ3Ç Æ3Ç Ç

Æ@S

¡ ¡

¦S§

~

Ø ` *

But we saw in Section 1.13 that the two families of operators  and

¡ ¡ ¡ ¡

¡ ¡

Æ3Ç

¦ § ¦ §

;

~ ~

Ø `‚) ` Û  Ø ` ¥

 are asymptotic to one another, as . It follows

’

Æ3Ç Æ3Ç

“ “ œ

-‰ŽÃ A ­ ­ -|ŽÃ ÝŒ -|ŽÃ

Ç Æ [ that [ is asymptotic to , as required. The calculation for is similar. Æ3Ç

Turning to the infinite-dimensional case, it is clear that the major problem is

to construct a suitable operator * . We begin by assembling some preliminary

S Ç

facts. Suppose that we fix for a moment a finite-dimensional affine subspace  of

Ç

T

 

 . Denote by its orthogonal complement in . This is an infinite-dimensional Ç

subspace of k , but in particular it is a Euclidean space in its own right, and we can

UT 

form the direct limit Hilbert space ¶ as in Definition 3.15. Ç

210 Nigel Higson and Erik Guentner

 UT

Lemma 3.6. Let  be a finite-dimensional affine subspace of and let be its

Ç Ç

orthogonal complement in  . The isomorphisms

¶  M‘ ¶ k LÃ-Ž¶    ¢¨ 



Æ Æ3Ç Ç Ç Æ

of Lemma 3.3 combine to provide an isomorphism

T

¶ QM‘ ¶   -Ž¶   OP

Ã



Ç

t Ç

Definition 3.16. Let  be a finite-dimensional subspace of an affine Euclidean

Ç



  

space  . The Schwartz space of , denoted is

Ç Ç



M®5 mjy}zon k   B

 Schwartz-class -valued functions on

Ç Ç Ç

t

 

 

The Schwartz space Q is the algebraic direct limit of the Schwartz spaces :

Ç

 

9

QM y{z}|  

~

)

Ç

9

8 8 ö fin. dim.

affine sbsp.

 

C  M)  

using the inclusions k .

Æ3Ç Ç Æ



XT  ¶ ¡T 

We now want to define a suitable operator * on with domain . A

S

Ç

Ç Ç

"UT

very interesting possibility is as follows. If kY¢ is a finite-dimensional subspace

Ç



* ü¥ 6Û w6

­ ­ then the operator ­ acts on every Schwartz space , where

kY¢`w : just use the formula

! !

Õ Õ

¾ F F

E½

ª ª ª

9

¾ ¯ ¯ ¯

* ­j½ > Œ ½ Œn

Ã

 ¯

F F ¾ ¾

:9 9 ¢9 9

k !

from Lecture 1, where ! is an orthonormal basis for and are

tštSt tštSt w

the dual coordinates on k , extended to coordinates on by orthogonal projection.

 6

The actions on the Schwartz spaces w are compatible with the inclusions used to

 

w6 XT Gy{z}|

define the direct limit ~ , and we obtain an unbounded, essentially

)

Ç



XT  XT 

self-adjoint operator on ¶ with domain . Let us now make the following Ç

key observation: Ç UT Lemma 3.7. Suppose that  is decomposed as an algebraic direct sum of pairwise

orthogonal, finite-dimensionalÇ subspaces,

¡

T

 €k yk zk '™š™S™

t

Ç



¡T 

If ½<$ then the sum

Ç

¡ 9

* ½d* ½©f* ½V * ½V'™š™S™

S

Ç

   

a¥ Û k

where * is the Clifford-Dirac operator on , has only finitely many



GT 

nonzero terms. The operator defined by the sum is essentially self-adjoint on 

Ç FT

and is independent of the direct sum decomposition of  used in its construction. Ç Group C*-Algebras and K-theory 211

Proof. Observe that

 

T

k ™S™š™Lyk   My{z}|

!

~

)

t

Ç

!

 

¡T  ½ k À™S™S™ BkÝ!e

Therefore if ½ç$ then belongs to some . Its image in

Ç



¨™S™š™Ó kÝ!lk‰m€ k under the linking map in the directed system is a function of the

form

F¡ — — F€¡ — —

Y

Y

u

œ œ œ œ

2

½€m ™S™š™ !lknmlM ™L½ ™S™š™¢ !  Î ™ ™ Î 0

constant 0

0

2 2

tštSt t

Î

Î

F

— — ¡

Y

u

*!lk‰m *!lknm:½" R z'aK

2 Î

Since 0 is in the kernel of we see that for all . 2 This proves theÎ first part of the lemma. Essential self-adjointness follows from the

existence of an eigenbasis for * , which in turn follows immediately from the BS

existence of eigenbases in the finite-dimensionalÇ case (see Corollary 1.1). The fact XT that * is independent of the choice of direct sum decomposition follows from the

formulaÇ

 

9

T

* ½H* ½ ½r$ w6‡¢  

I if

S

Ç

Ç

Z

* ¨* *

I I I

which in turn follows from the formula I in finite dimensions.

0 0

2 2

Unfortunately the operator * above does not have compact resolvent. Indeed

S

Ç

¡ ¡ ¡ ¡

9

¡

* * f* f* ™S™š™

S

Ç ¡

from which it follows that the eigenvalues for * are the sums

S

Ç

¡ 9

ýb˜ý "ý fý ™S™š™

¡

 



* ý

where ý is an eigenvalue for and where almost all are zero. It therefore K

follows from Proposition 1.16 that while the eigenvalue R occurs with multiplicity , ¡

each positive integer is an eigenvalue of * of infinite multiplicity.

ÇOS

Because * fails to have compact resolvent we cannot immediately follow

S

Ç ­ Lemma 3.4 to obtain our asymptotic morphisms Ç . Instead we first have to ‘per-

turb’ the operators * in a certain way.

ÇOS

¡

¢6 ¢6 ¢

Notation 3.10 We are now going to fix an increasing sequence 

 

™S™S™ of finite-dimensional affine subspaces of whose union is . We shall denote by

¡

e! ! ß! k ˜ k the orthogonal complement of in (and write ), so that there is

an algebraic orthogonal direct sum decomposition

¡

€k zk yk zkÏËj'™š™S™ t

Later on we shall want to arrange matters so that this decomposition is compatible  with the action of h on , but for now any decomposition will do.

Having chosen a direct sum decomposition as above, letus make the following definitions:

212 Nigel Higson and Erik Guentner  Definition 3.17. Let  be a finite-dimensional affine subspace of . An algebraic

orthogonal direct sum decompositionÇ

¡

T

 #w zw yw ™S™š™ Ç

is standard if it is of the form

9

T

 #k zke!¨zke!lk ™S™S™

Ç

Ç

k gQ' K k

for some finite-dimensional linear space and some , where the spaces ! Ç

are the members of the fixed decomposition of  given above.  Definition 3.18. Let  be a finite-dimensional affine subspace of . An algebraic

orthogonal direct sum decompositionÇ

¡

T

 €ý zý zý '™š™S™ Ç into finite-dimensional linear subspaces is acceptable if there is a standard decom-

position

¡

T

 #w zw yw ™S™š™ Ç

such that

w '™š™S™¢zwD! ¢QýN!('™š™S™:zýM! ¢`w ™S™š™¢zwr!€k

for all sufficiently large g . *

We are now going to define perturbed operators [ which depend on a choice

S

Ç

ï

9<; 

of acceptable decomposition, as well as on a parameter `j$u{K . 

Definition 3.19. Let  be a finite-dimensional affine subspace of and let

Ç

¡

T

 €ý zý zý '™š™S™

Ç UT

be an acceptable decomposition of  as an orthogonal direct sum of finite-

Ç

`'ÀK *

dimensional linear subspaces. For each define an unbounded operator [

ÇOS



ï

XT  ¡T 

on ¶ , with domain , by the formula

Ç Ç

¡ ¡

* ` * u` * u` * ™S™š™

[

S

Ç

ï



¡

²

K\D` * ¥;!uÛ©! ¥;! ÛV!

where ` , where , and where and are the Clifford

Æ ;! and Dirac operators on the finite-dimensional spaces ý .

It follows from Lemma 3.7 that the infinite sum actually defines an operator



XT  *

with domain . The perturbed operators [ have the key compact resolvent

S

Ç Ç property that we need: ï

Group C*-Algebras and K-theory 213 

Lemma 3.8. Let  be a finite-dimensional affine subspace of and let

Ç

¡

T

 €ý zý zý '™š™S™

Ç T be an acceptable decomposition of  as an orthogonal direct sum of finite-

dimensional linear subspaces. The operatorÇ

¡ ¡

* ` * u` * u` * ™S™š™

[

S

Ç ï is essentially self-adjoint and has compact resolvent.

Proof. The proof of self-adjointness follows the same argument as the proof in



* XT 

Lemma 3.7: one shows that there is an orthonormal eigenbasis for [ in .

S

Ç Ç

As for compactness of the resolvent, the formula ï

¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡

* '` *  ` * u` * 

[

S

Ç

ï

¡ *

implies that the eigenvalues of [ are the sums

S

Ç

ï

¡ ¡ ¡

9

¡

¡

ý u` ý u` ý ™S™š™ ýb'`

¡

 



* ý

where ý is an eigenvalue for and where almost all are zero. Since the lowest

 

; ² ;

K ` ) ) `

positive eigenvalue for * is , and since as (for fixed ), it follows

¡

é é *

that for any there are only finitely many eigenvalues for [ of size or less.

S

Ç

ï *

This proves that [ has compact resolvent, as required.

S

Ç

ï

0

­ C» - Å ‡ANAeC ¸ ¶ ÅnŒ Ã

We can now define the asymptotic morphisms Ç that

 ŽD,) ŽE[

we need. Fix a point R in and use it to define scaling automorphisms on

  RV$H

each  for which .

Ç Ç

 R $

Proposition 3.4. Let  be a finite-dimensional affine subspace of for which

Ç

 *

and let [ be the operator associated to some acceptable decomposition of

Ç

S

Ç

ï T

 . The formula

Ç



­ Cݽ -UŽb,)¿½:[ * -§KjK -Û  K - b 

Ç Ã Ã Ã Ã

[

[

Ç

S

Ç

ï

0 0

0

­ CE»¨Ã- Å A AFC®¸ ¶ Å4T xšÃ-¸ ¶ Å Œ defines an asymptotic morphism Ç , and

hence, thanks to the isomorphism of Lemma 3.6, an asymptotic morphism

0

Ç

­ C » - Å PANADC©¸ ¶ ÅnŒ

à t

Proof. This is proved in exactly the same way as was Lemma 3.4. *

It should be pointed out that operator [ does depend on the choice of ac-

ÇOS

ï ­

cepable decomposition, and so our definition of Ç appears to depend on quite a ;

bit of extraneous data. But the situation improves in the limit as `Ž) . The basic calculation here is as follows:

214 Nigel Higson and Erik Guentner 

Lemma 3.9. Let  be a finite-dimensional affine subspace of and denote by

Ç

*Ž[bæ* *Qâ æ*§â

[ [

and [ be the operators associated to two acceptable de-

S

Ç

ÇOS

ï

ï

T ½&$ »

compositions of  . Then for every ,

Ç

Ô Ô

~

â

[

y}z}| ½ *  ½ *  R

[

[HGJI t

Proof. We shall prove the following special case: we shall show that if the summands

ý ýßâ !

in the acceptable decompositions are and ! , and if

â â

ý '™š™S™Lý ¢Qý ™S™S™:zý ¢:ý ™S™š™Lý

! !lk

!

Ô Ô

~

g y{z}|V[HGJI ½ *[Í ½ *§â  aR for all , then [ . (For the general case, which is not

really any harder, see [32] and [34].)

U

! ý>! ýŽâ +A!

Denote by the orthogonal complement of in ! , and by the orthogonal

¡

ýŽâ ýM! + Lý

complement of ! in (set ). There is then a direct sum decomposition

U U

9

T

 Q+  y+  ™S™š™

Ç

* *§â [

with respect to which the operators and [ can be written as infinite sums

ÇOS

S

Ç

ï

ï

¡

§[ \[  

[

* ` * u` *  ` * u` * ™S™š™

0 0

and

§[ 5[  

â

* '` *  ` *  ` *  ` * ™S™š™

[

t

0 0

 

¡

~

¡

` –`

Since ` it follows that

¡ ¡

9

~

\[ 

â

[

* * ` * u` * '™š™S™

[ 0

and therefore that

¡ ¡ ¡ ¡ ¡

¡ ¡

~

\[ 

â

[

* *  ` *  ` * ™S™S™

[

t 0

In contrast,

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

9

¡

[ [

   

* '` *  ` * u` *  ` * '™š™S™

[

¡

Ô Ô Ô Ô

¡

0 0





~

XT  ` *[ *§â x½ so` *Ž[1½ ½ $ 'çK

and since it follows that [ for every .

¾ ¾ÛŸ

±

Ç

¡

> 

This implies that if ½ then

Ô Ô Ô Ô Ô Ô

¡ ¡ ¡ ¡

Ÿ Ÿ

Ÿ

± ± ±

~ ~ ~

â â â â

[ [ [ [ [

 sf`  s * *  * * *  *  ½ *  ½ *   *

[ [ [ [ t An approximation argument involving the Stone-Weierstrass theorem (which we have seen before) now finishes the proof. For later purposes we note the following simple strengthening of Lemma 3.9. It

is proved by following exactly the same argument.

¤

9N; 

Lemma 3.10. With the hypotheses of the previous lemma, is $W{K

Ô Ô

Ĥ ¥¤

9

~

â

[

y{z}| ½ *  ½ *  R

[

[HGxI

¤ P uniformly in . O

Group C*-Algebras and K-theory 215 ­ It follows from Lemma 3.9 that our definition of the asymptotic morphism Ç is independent, up to asymptotic equivalence, of the choice of acceptable decomposi-

15 T

tion of  (compare the proof of Lemma 3.4). Ç

Proposition 3.5. The diagram

9

0

¸ ¶ Qx

» - Å 

à _ _ _/

9

ð

<

£

 

¸ ¶ Qx

- Å53

»Ã _ _ _/

< is asymptotically commutative.

Proof. Using the computations we made in Section 1.13, as we did in the proof of

’

­  Lemma 3.5, we see that the composition Æ is asymptotic to the asymptotic

morphism Æ3Ç

9

â



½ -UŽb,)¿½¢[ * -QKjK -Û  K - b 

à à à Ã

[

Ç ­

where, if Æ is computed using the acceptable decomposition

¡ 9

T

 #ý ý ý ™S™S™

Æ *§â

then [ is the operator of Definition 3.19 associated to the decomposition

¡

T

   ý nzý zý '™š™S™

Æ3Ç

t

Ç

’

UT ­  ­ Ç

But this is an acceptable decomposition for , and so Æ is asymptotic to , Æ3Ç

as required. Ç ­ It follows that the asymptotic morphisms Ç combine to form a single asymptotic

morphism

O

H- Q ¸ ¶ QŒ

­§C _ _ _/

t

9

$˜ Q «N

Our definition of the class ­ is therefore almost complete. It re- ß

mains only to discuss the equivariance of ­ . 

Suppose that the countable group h acts isometrically on . Using the point 

R6$a that we chose prior to the proof of Proposition 3.4, indentify with its 

underlying Euclidean vector space k , and thereby define a family of actions on ,

¤

9

R KT

parametrized by $W by

F ¤w F

œ œ

9

}Q™  }Q™LR† v }  RŽ  

¢

F F

b™ }d™ (see Exercise 3.3). Thus the action } is the original action, while has a

global fixed point (namely RV$H ).

¢Ä6

It should be added however that ]'^ does depend on the choice of initial direct sum decom- position, as in 3.10. 216 Nigel Higson and Erik Guentner

Lemma 3.11. There exists a direct sum decompostion

¡

#k zk zk "f™š™S™

‹

Qk ¨™S™š™Ó kÝ! }Š$

for which ‹

¤

l9 9

ë

 }Q™Sß!¢"Ž!lk $X R KT OP

g for all t

Proposition 3.6. If the direct sum decomposition

¡

#k zk zk "f™š™S™

H- Q‰ANADCb¸ ¶ §x is chosen as in Lemma 3.11 then the asymptotic morphism ­§C

is equivariant in the sense that

Ô Ô

¾ 7

9

~

­n[ Œ R y{z}| ­n[ }Q™  }Q™

[

¸

[HGxI

0

7

`- Q }b$dh

for all $ and all .

0

- Å 

Proof. Examining the definitions, we see that on »QÃ the asymptotic morphism

7

7

¡

[

} ™ Œ ­ ,) } ™ ­

[ Ç

¸ is given by exactly the same formula used to define ,

0 FT except for the choice of acceptable direct sum decomposition of  . But we al- ready noted that different choices of acceptable direct sum decompositionÇ give rise

to asymptotically equivalent asymptotic morphisms, so the proposition is proved.

9

- Q «M

Definition 3.20. Denote by ­u$< the class of the asymptotic morphism

ß

- QPA ADCV¸ ¶ Qx ­QC .

3.7 Proof of the Main Theorem, Part Three

’

9

l ÀK§$& « «ß Here we show that ­ . The proof is almost exactly the same as

the proof of Proposition 2.21 in theß last lecture.  Lemma 3.12. Suppose that the action of h on the affine Euclidean space has a

fixed point. Then the composition

- §

£ «

« / / « in equivariant  -theory is the identity morphism on .

Proof. The proof has three parts. First, recall that in the definition of the asymptotic ’

morphism we began by fixing a point of  . It is clear from the proof of Lemma 3.2 that different choices of point give rise to asymptotically equivalent asymptotic mor-

phisms, so we might as well choose a point which is fixed for the action of h on

’ ’

[ [

Y ½D ½ 

 . But having done so each -homomorphism in the asymptotic ’

morphism is individually h -equivariant. It follows that the equivariant asymptotic

Group C*-Algebras and K-theory 217

’

» A AFCa- Ån

morphism C is equivariantly homotopy equivalent to the equivariant

’

Ce»ç)b- Ån

Y -homomorphism . Using this fact, it follows that we may compute

’

 

the composition ­ in equivariant -theory by computing the composition of the

’ Y

asymptotic morphism ­ with the -homomorphism . But the results in Section 1.13

C]»¨)^¸ ¶ Ånx

show that this composition is asymptotic to ¯ , where

9

[ [ [

¯ ½M½ * 

and *[ is the operator of Definition 3.19 associated to any acceptable decomposition

½H,)¿½ *[1

of  . This in turn is homotopic to the asymptotic morphism . Finally this

9

« «‚

is homotopic to the asymptotic morphism defining K($H by the homotopy

ß

Ĥ ¤

9<;

½ *[1 $u K 

Â

½H,)

¤

;'9

½ Rw1# 

*[ *[

where # is the projection onto the kernel of (note that all the have the same h

K -dimensional, -fixed kernel).

’

9

 $< « «ß

Theorem 3.11. The composition ­ is the identity.

ß

¤

9

R KT -®¢ Q ¥¨¦ - Q

Proof. Let $W and denote by the -algebra , but with the scaled

9

} ŽAŠ,) } ™ ¢PŽ -®¢ Q h ¥Q¦

h -action . The algebras form a continuous field of - -

ô

Q - h ¥¨¦

algebras over the unit interval. Denote by Ú the - -algebra of continuous

ï ¥©¦

sections of this field. In a similar way, form the continuous field of h - -algebras

ô

¶ QŒ ¸|¢ ¶ §x ¸ h ¥¨¦

and denote by Ú the - -algebra of continuous sections.

ï

H- Q ¸ ¶ §x The asymptotic morphism ­§C _ _ _/ induces an asymptotic mor-

phism

„

9

ô ô

Q ­QCc- ¶ §x ¸ Ú

_ _ _/ Ú

ï ï

’

- Q and similarly the asymptotic morphism C]» _ _ _/ determines an asymp-

totic morphism

„

’

ô

Q - Ú

C» _ _ _/

t ï

From the diagram of equivariant  -theory morphisms

¨

¨

9 9

ô

§ -

Ú

¥V R KS £

« / /

ï

R

ð

 ð 

Ê Ê

  

- Q

¢

« «

/ /

£

¤

9

‹ $u R KT

where ¢ denotes the element induced from evaluation at , we see that if the

¤

9

R KT

bottom composition is the identity for some $o then it is the identity for all

¤ ¤

9

R KT oR

$" . But by Lemma 3.12 the composition is the identity when since the

F F

9

} )Ø}™

action N, has a fixed point. It follows that the composition is the identity ¤ when ®K , which is what we wanted to prove. 218 Nigel Higson and Erik Guentner

3.8 Generalization to Fields

We conclude this lecture by quickly sketching a simple extension of the main theo- rem to a situation involving fields of affine spaces over a compact parameter space. This generalization will be used in the next lecture to prove injectivity results about

the Baum-Connes assembly map.

dfe ý

Definition 3.21. Let ý be a set. Denote by the set of negative-type kernels

¼

I

¼ý ý) dfe ý

—AC . Equip with the topology of pointwise convergence, so that

J• • • • 8• •

¢9 ¡ ¢9 ¡ :9 ¡

— )¿— dfe ý — M)¿— $*ý in if and only if  for all .

Suppose now that U is a compact Hausdorff space and that we are given a con-

¾ U ¾ U

)þ— dfe ý $

tinuous map , from into . For each we can construct a

Ò

 k

Euclidean vector space k and an affine space over following the prescrip-

Ò Ò Ò

tion laid out in the proof of Proposition 3.1 (thus for example k is a quotient of the

Ò

R 

space of finitely supported functions on ý which sum to , and is a quotient of

Ò K the space of finitely supported functions on ý which sum to ). We obtain in this way some sort of ‘continuous field’ of affine Euclidean spaces over U (we shall not

need to make this notion precise).

-   ¥§¦

The ¥¨¦ -algebras may be put together to form a continuous field of - U

algebras over . (See [19]Ò for a proper discussion of continuous fields.) To do so we

¾

) ½ $g-  

must specify which sections , are to be deemed continuous, and Ò

for this purpose we begin by deemingÒ to be continuous certain families of isometries

¼ !

from into the affine spaces  .

Ò

¦ U ¼

!

'h>C )ó¤h

Definition 3.22. Let be an open subset of and let Ž be a family

¼ ¦

!

“

fh $

of isometries of into the affine Euclidean spaces  ( ) defined above. We ¢òý

shall say that the family is continuous if there is a finite subset  and if there

 ¦ ¼

“

² 9 9

½ 6K g $ MC¼ý')

are functions h (where , and ) such that

tStšt

ï

 

½  ½ h

(a) Each function h sums to one, and is supported in (and therefore each

ï ï ¤h

determines a point of  ).

• J• ¦



“

²

$%ý ½  $

(b) For each and each , the value h is a continuous function of .

F ¼



!

ï

Ž  h

(c) The isometry h maps the standard basis element of to the point of

 ½

determined by h .

ï

¾

) ½

Definition 3.23. Let us say that a function , which assigns to each point

¾ U

Ò

¥Q¦ -  

$ an element of the -algebra is a continuous section if for every

¾ U ¦ ¾

Ò

ë

‹ R

$ and every there is an open set containing , a continuous family of

¼ J¼ji

!

ch>C )+¥h ½<$d»¨Ã- 

isometries Ž as above, and an element such that

Ô Ô

¦

“

j

t

~

Žch ½ ½Rh ‹ $

¦ ¦

t

ï

J¼

i

ch »§Ã- 

Here by Ž we are using an abbreviated notation for the inclusion of

¦É¦

¼ ¼

! !

ï

¥h Žch>C )íŽchÝ J¢ ¥h into - induced from the isometry by forming the

composition

î

ö

Bk

J¼ ¼

i i

1

- Q

=

 @ » - » - ÆNl‰ Ã

à / / t Group C*-Algebras and K-theory 219

Lemma 3.13. With the above definition of continuous section the collection of ¥©¦ -



m-   B Ù ¥©¦

algebras 5 is given the structure of a continuous field of -algebras

U

Ò Ò P

over the space . O

U

9

- ¥Q¦ Definition 3.24. Denote by Q the -algebra of continuous sections of the

above continuous field.

}X™ ý h d¤e ý

If a group h acts on the set then acts on by the formula

8• • • •

¡ ¡

:9 ¡ ¢9 ¡

—T § — } }

 . In what follows we shall be solely interested in the h

case where ý and the action is by left translation. U

Definition 3.25. Let be a compact space equipped with an action of a countable

¾ U

,)_—

discrete group h by homeomorphisms. An equivariant map from into

‡

Ò

h¨ '–R ò¢¨h

d¤e is proper-valued if for every there is a finite set such that

¡

‡

:9 ¡ ¡

— } } °s  } } $<

Ò t

The following is a generalization of Theorem 3.8.

U

)nd¤e h¨ h

Proposition 3.7. If —AC is a -equivariant and proper-valued map then

U

9

h ¥Q¦ - OP the - -algebra Q is proper.

By carrying out the constructions of the previous sections fiberwise we obtain

the following result (which is basically due to Tu [18]). U

Theorem 3.12. Let h be a countable discrete group and let be a compact metriz- U

able h -space. Assume that there exists a proper-valued, equivariant map from into

JU

9

¦

h¨ - § h ¥ 

d¤e . Then is a proper - -algebra and there are -theory classes

’

U U JU U

9 9 9 9

­ $H - Q ¥ x ¥  - Qx

and $d

ß ß

’

U JU

9

  ¥  ¥ x OP

for which the composition ­ is the identity in . ß By trivially adapting the simple argument used to prove Theorem 2.20 we obtain

the following important consequence of the above: U

Corollary 3.2. Let h be a countable discrete group and let be a compact metriz-

U

d¤e h¨

able h -space. If there exists a proper-valued, equivariant map from into

¥¨¦ Û

then for every h - -algebra the Baum-Connes assembly map

4

¤ U ¤" U

[Hõ

ó

9 9

¦

C h Û Œ>) ¥ h Û Œx

is an isomorphism. If h is exact then the same is true for the assembly map into

¤f U

9

ì

¥¨¦ h Û xx P . O 220 Nigel Higson and Erik Guentner 4 Injectivity Arguments

The purpose of this lecture is to prove that in various cases the Baum-Connes assem-

bly map

4

¤" ¤

[Hõ

9 9

¦

h ÛbM) ¥ h Ûbx is injective. A great deal more is known about the injectivity of the assembly map than its surjectivity. In a number of cases, injectivity is implied by a geometric prop-

erty of h , whereas surjectivity seems to require the understanding of more subtle

issues in harmonic analysis.

9

h Ûb

In all but the last section we shall work with the full crossed product ¥©¦ ,

9

ì

h Ûb h but all the results have counterparts for ¥§¦ . If is exact then arguments below applied in the reduced case; otherwise different arguments are needed.

4.1 Geometry of Groups ‡

Let h be a discrete group which is generated by a finite set . The word-length of

‡

ã

}  $oh

an element } is the minimal length of a string of elements from and

‡

¡ h whose product is } . The (left-invariant) distance function on associated to the

length function ã is defined by

¡

ã

¢9 ¡ ¡

T } } M } }  t The word-length metric depends on the choice of generating set ‡ . Nevertheless,

the ‘large-scale’ geometric structure of h endowed with a word-length metric is

independent of the generating set: the metrics associated to two finite generating sets ‡

and º are related by inequalities

K

¡ :9 ¡ ¢9 ¡ 9

~

™DTpo } }  ¥)s`T } } Ps¨¥˜™STqo } } †"¥

Ï

¥

‡

¡

ë

R º } }

where the constant ¥ depends on and but not of course on and .

T  ý

Definition 4.1. Let ý be a set and let and be two distance functions on . They

‡

é%ë ë R

are coarsely equivalent if for every there exists a constant R such that

‡ J• • 8• •

¢9 ¡ :9 ¡

t t

é

T    

and

8• • J• • ‡

9 9

¡ t ¡ t

é

   T 

Thus any two word-length metrics on a finitely generated group are coarsely equivalent. When we speak of ‘geometric’ properties of a finitely generated group we

shall mean (in this lecture) properties shared by all metrics on h which are coarsely equivalent to a word-length metric. This geometry may often be visualized using Theorem 4.1 below.

Group C*-Algebras and K-theory 221 U

Definition 4.2. A curve in a metric space is a continuous map from a closed in-

U 7 U

9

¯C†

terval into . The length of a curve —ÉE) is the quantity

!

Õ

» r

¦

9

¡

¯ ¯

y  ´ ¯†M ÖŒ× Ø T ¯ `  ¯ ` Œ

[s s s

¦t¦t¦

ð ð

t

[ [ [

ð

¯

Ç Æ

0

Î

U ¾ ¾ ¾ ¾ U

:9 ¡ :9 ¡

T 

A metric space is a length space if for all $ , is the infimum of

¾ ¾ ¡ the lengths of curves joining and .

Theorem 4.1. Let h be a finitely generated discrete group acting properly and co-

¾ U compactly by isometries on a length space U . Let be a point of which is fixed

by no nontrivial element of h . Then the distance function

¾ ¾

¢9 ¡ 9 ¡

 } } >QT } ™ } ™ 

h OP on h is coarsely equivalent to the word-length metric on . See [48, 64] for the original version of this theorem and [10] for an up to date

treatment. In the context of the above theorem we shall say that the space h is coarsely equivalent to the space U (see [57, 58] for a development of the notion of coarse equivalence between metric spaces, of which our notion of coarse equiva-

lence between two metrics on a single space is a special case).

Example 4.1. If h is the fundamental group of a closed Riemannian manifold

u then h is coarsely equivalent to the universal covering space . Example 4.2. Any finitely generated group is coarsely equivalent to its Cayley graph. For example free groups are coarsely equivalent to trees.

4.2 Hyperbolic Groups

Gromov’s hyperbolic groups provide a good example of how geometric hypotheses

¤ ¦ on groups lead to theorems in ¥ -algebra -theory. In this section we shall sketch very briefly the rudiments of the theory of hyperbolic groups. Later on in the lecture

we shall prove the injectivity of the Baum-Connes assembly map

4

¤f ¤

[Hõ

ó

9 9

¦

h Ûb>) ¥ h Ûb C for hyperbolic groups.16 The first injectivity result in this direction is due to Connes and Moscovici [15] who essentially proved rational injectivity of the assembly map

in the case Ûa« . Our arguments here will however be quite different.

U U

Definition 4.3. Let be a metric space. A geodesic segment in is a curve

7 U

9

C‰ —ÉA)

¯ such that

Ĥ ¤

9 = =

~

T ¯ ¯  `xx> `

7 ¤

sf`°s'—

for all s .

¢H]

We shall see that every hyperbolic group is exact; hence the reduced assembly map v:w is injective too.

222 Nigel Higson and Erik Guentner

¾ ¾

¡ ¯

Observe that if ¯ is a geodesic segment from to then the length of is

¾ ¾

9

¡

 precisely T .

Definition 4.4. A geodesic metric space is a metric space in which each two points

are joined by a geodesic segment.

U U Definition 4.5. A geodesic triangle in a metric space is a triple of points of , to-

gether with three geodesic segments in U connecting the points pairwise. A geodesic

Û ''R triangle is Û -slim for some if each point on each edge lies within a distance

Û of some point on one of the other two edges. é

Example 4.3. Geodesic triangles in trees are R -slim. An equilateral triangle of side

Ë

é y

in Euclidean space is x -slim. U

Definition 4.6. A geodesic metric space is Û -hyperbolic if every geodesic triangle

U

Û Û Û '–R å in is -slim and hyperbolic if it is -hyperbolic for some .

Thus trees are hyperbolic metric spaces but Euclidean spaces of dimension µ or more are not. Definition 4.6 is attributed by Gromov to Rips [24]. It is equivalent to a wide variety of other conditions, for which we refer to the original work of Gromov [24] or one of a number of later expositions, for example [22, 10]. (The reader is also referred to these sources for further information on everything else in this section.)

Definition 4.7. A finitely generated discrete group h is word-hyperbolic, or just hy- perbolic, if its Cayley graph is a hyperbolic metric space.

This definition leaves open the possibility that the Cayley graph of h constructed with respect to one finite set of generators is hyperbolic while that constructed with respect to another is not. But the following theorem asserts that this is impossible:

Theorem 4.2. If a finitely generated group h is hyperbolic for one finite generating P set then it is hyperbolic for any other. O

Examples 4.3 Every tree is a R -hyperbolic space and a finitely generated free group

å h

is R -hyperbolic. The Poincare´ disk is a hyperbolic metric space. If is a proper h and cocompact group of isometries of å then is a hyperbolic group. In particular,

the fundamental group of a Riemann surface of genus µ or more is hyperbolic.

¤ ¤

9



'¨R z{ØÝÖ h 

If h is a finitely generated group and if then the Rips complex

? 9 9

4

oKL } } 

is the simplicial complex with vertex set h , for which a -tuple

¤



±

tStšt

? 9 ²

}¯ } qs

is a -simplex if and only if T , for all and . In the case of hyperbolic ¦

groups the Rips complex provides a simple model for the universal space h :

¤{z R

Theorem 4.4. [7, 47] Let h be a hyperbolic group. If then the Rips complex

¤

9



h OP z{ØÝÖ h  is a universal proper -space.

In the following sections we shall need one additional construction, as follows:

Group C*-Algebras and K-theory 223 U

Definition 4.8. A geodesic ray in a hyperbolic space is a continuous function

U

|

9<;

Ce R M)

|

9}

R  such that the restriction of to every closed interval  is a geodesic segment. Two

geodesic rays in U are equivalent if

| |

; 9 ¡

t

`xx `x y}z{|–Öx×eØ T

t

[HGJI

The Gromov boundary of a hyperbolic metric space U is the set of equivalence

U h classes of geodesic rays in . The Gromov boundary Eh of a hyperbolic group is the Gromov boundary of its Cayley graph.

The Gromov boundary Eh does not depend on the choice of generating set. It

is equipped in the obvious way with an action of h . It may also be equipped with a

compact metrizable topology, on which h acts by homeomorphisms. Moreover the

~<Eh

disjoint union h„çh may be equipped with a compact metrizable topology

h h

in such a way that h acts by homeomorphisms, that is an open dense subset of ,

¾

;

! $Hh $dEh }!V) h

and that a sequence of points } converges to a point iff in ¾

and there is a geodesic ray | representing such that

|

; 9

t

ÖŒ× Ø T } 

!

t

!

€~ŠEh h h From our point of view, a key feature of h6h is that the action of on is amenable. We shall discuss this notion in Section 4.5.

4.3 Injectivity Theorems

In this section we shall formulate several results which assert the injectivity of the Baum-Connes map ó under various hypotheses.

Our first injectivity result is a theorem which is essentially due to Kasparov (an ¤r¤ improved version of it, which invokes his é -theory, underlies his approach to the Novikov conjecture).

Theorem 4.5. Let h be a countable discrete group. Suppose there exists a proper

’

9 9

¥¨¦ * ­„$6 * «N $6 « *§

h - -algebra and elements and such that for

’

ß ß

9

h ¯d­Û $H « «ß

every finite subgroup Á of the composition restricts to the

ß

9

±

« «ß h ¥¨¦ Û identity in  . Then for every - -algebra the Baum-Connes assembly

map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h ÛbM) ¥ h Ûbx is injective.

Proof. We begin by considering the same diagram we introduced in the proof of Theorem 2.20:

224 Nigel Higson and Erik Guentner

¤ ¤f

4

[Hõ

9 9

h « - Ûb ¥¨¦ h « -©ÛbŒ

à Ã

ú /

 

£ £

¤ ¤"

4  

[Hõ

9 9

R

h *dÃ-Ûb ¥¨¦ h *dÃ-ÛbŒ ú

ð /

 

¤ ¤f

4  

[Hõ

9 9

h « - Ûb ¥¨¦ h « -VÛbŒ Ã

/ Ã

t

ú

* -Û h ¥¨¦ The middle assembly map is an isomorphism since à is a proper - -algebra.

We want to show that the top assembly map is injective, and for this it suffices to

’

¤ ¤

4 4

[Hõ [Hõ

9 9

h Û -*§ h ÛbX) C show that the top left-hand vertical map à is

injective. For this we shall show that the composition¦

¤ ¤ ¤

4 4 4

[Hõ [Hõ [Hõ

9 9 9



h «ŽÃ-©Ûb h *dÃ-Ûb h «ßÃ-VÛb 

£ / /

¤

4

[Hõ

is an isomorphism. In view of the definition of it suffices to show that if ý is a h

h -compact proper -space then the map

’

9 9

~

ý Ûb )+ ý Ûb Ce ¯ ­ 

¦ ¦ ¦

ß ß

is an isomorphism. The proof of this is an induction argument on the number g

¦ ý

of h -invariant open sets needed to cover , each of which admits a map to a

gr%K ý

proper homogeneous space hQ´lÁ . If , so that itself admits such a map, then

I

±

'h w w Á ý , where is a compact space equipped with an action of . There is

then a commuting diagram of restriction isomorphisms

9 9

I I

 h ±yw Ûb  h ±yw Ûb F

V /

ß ß

R R

÷0ÿ ÷0ÿ

ð ð

þ þ

 

9 9 9

± ±

w Ûb w Ûb 

 /

 V

(see Proposition 2.8), and the bottom map is an isomorphism (in fact the identity)

¦

9

ë

±

« «ß ¢:ý HK  g K h

since ¯ in . If then choose a -invariant open set which

¦

ï

Bý

admits a map to a proper homogeneous space, and for which the space ý

~ Kh may be covered by g -invariant open sets, each admitting a map to a proper

homogeneous space. By induction we may assume that the map ¯ is an isomorphism

¦

for ý . Applying the five lemma to the diagram

©¦

:9 9 9

™S™š™ ™S™š™

ý Ûb  ý Û   Ûb

/  / / /

ß ß ß

R R

ð ð

VF VF VF

©¦

  

9 9 9

™S™š™ ™S™š™

Ûb ý Ûb  ý Û  

/  / / /

ß ß ß ý

we conclude that ¯ is an isomorphism for too. ¦ Group C*-Algebras and K-theory 225

Remark 4.1. The proof actually shows that the assembly map is split injective.

The second result is taken from [10] and is as follows.

U U Á

Theorem 4.6. Let be a compact, metrizable h -space and assume that is -

h Û

equivariantly contractible, for every finite subgroup Á of . Let be a separable ¥¨¦

h - -algebra. If the Baum-Connes assembly map

4

U ¤" U ¤

[Hõ

ó

9 9

¦

h Û Œ>) ¥ h Û Œx C

is an isomorphism then the Baum-Connes assembly map

4

¤" ¤

[Hõ

ó

9 9

¦

h Ûb>) ¥ h Û x C

is split injective.

U

Z

Û  Proof. The inclusion of Û into as constant functions gives rise to a commu-

tative diagram

¤" U

¤ U

4

[Hõ

9

9

¦

¥ h Û Œx

h Û x

ú /

O O

 

 

¤f

¤

4

[Hõ

9

9

¥¨¦ h ÛbŒ Ûb

h /

t ú

We shall prove the theorem by showing that the left vertical map is an isomorphism.

¤

4

[Hõ h In view of the definition of it suffices to show that if ý is any -compact proper

h -space then the map

JU

Z

9 9

Ce ¥ ý ÛbM)+ ¥ ýŽ Û x

¦

ß ß is an isomorphism. By a Mayer-Vietoris argument like the one we used in the proof

of Theorem 4.6 it actually suffices to consider the case where ý admits a map to a w

proper homogeneous space hQ´:Á . In this case there is a compact space equipped

I

±

ýh w with an action of Á such that . Consider now the following commuting

diagram of restriction isomorphisms:



] w U

9 9

I I

 ¥ h ±yw6 Ûb  ¥ h ±bw6 Û x

 /

ß ß

R R

÷0ÿ ÷0ÿ

ð ð

þ þ

U

 

9 9

± ±

¥ w6 Û  ¥ w6 Û x  

 /

t 

The bottom horizontal map is an isomorphism (since Z is a homotopy equivalence of ¥¨¦ Á - -algebras) and therefore the top horizontal map is an isomorphism too. The last injectivity result is an analytic version of a result of Carlsson-Pedersen [11]. We will not discuss the proof, but refer the reader to the original paper of Higson [10, Thm. 1.2 & 5.2]. We include it only because it applies more or less directly to the case of hyperbolic groups.

226 Nigel Higson and Erik Guentner

U

h h

Definition 4.9. Let h be a discrete group, let be a -compact, proper -space,

U U h and let be a metrizable compactification of h to which the action of on

extends to an action by homeomorphisms. The extended action is small at infinity if

¤ U

for every compact set ¢ ,

r ½

¤

¦ ¦

9

y}z{| ¿zÌ$l| } M'R

è GJI

where the diameters are computed using a metric on U . h

Theorem 4.7. Let h be a countable discrete group. Suppose there is a -compact

¦ ¦ h

model for h having a metrizable compactification satisfying

¦ ¦

h h

(a) the h action on extends continuously to ,

¦ h

(b) the action of h on is small, and

¦

Á Á h

(c) h is -equivariantly contractible, for every finite subgroup of .

¥Q¦ Û

Then for every separable h - -algebra the Baum-Connes assembly map

4

¤ ¤"

[Hõ

ó

9 9

¦

C h Ûb>) ¥ h Û x P is injective. O

4.4 Uniform Embeddings in Hilbert Space

We are now going to apply the second theorem of the previous section to prove

injectivity of the Baum-Connes assembly map for a quite broad class of groups.

U U +

Definition 4.10. Let and + be metric spaces. A uniform embedding of into

U

)&+

is a function ½C with the following two properties:

‡

é

'–R ¨R

(a) For every ' there exists some such that

J¾ ¾ J¾ ¾ ‡

l9 ¡ 9 ¡

é

T °s  T ½  ½ x°s

t

‡

é

–R '¨R

(b) For every ' there exists such that

J¾ ¾ J¾ ¾ ‡

l9 ¡ 9 ¡

é

T °'  T ½  ½ x°'

t

U +

Example 4.4. If ½ is a bi-Lipschitz homeomorphism from onto its image in then U

½ is a uniform embedding. But note however that if the metric space is bounded U then any function from to + is a uniform embedding (in particular, uniform em-

beddings need not be one-to-one). Á

Exercise 4.1. Let h a finitely generated group and let be a finitely generated sub-

h Á

group of h . If and are equipped with their word-length metrics then the inclu-

¢–h sion Á is a uniform embedding. Group C*-Algebras and K-theory 227

Remark 4.2. In the context of groups, any function satisfying condition (a) of Def- inition 4.10 is in fact a Lipschitz function. But condition (b) is more delicate. For example it is easy to find examples for which the optimal inequality in (b) is some-

thing like

¾ F J¾ ¾ ‡ ¾

o

9 9

¡ ¡

°'  T ½  ½ x°' T

If a finitely generated group h acts metrically-properly on an affine Euclidean

F F

$H }V,)}Q™ h  space  , and if , then the map is a uniform embedding of into . We are going to prove the following result, which partially extends the main theorem of the last lecture: This theorem is due to Tu [18] and Yu [68] (in both cases, in a somewhat dis-

guised form). h

Theorem 4.8. Let h be a countable discrete group. If is uniformly embeddable in

¥Q¦ Û

a Euclidean space then for every h - -algebra the Baum-Connes assembly map

4

¤" ¤

[Hõ

ó

9 9

¦

h Ûb>) ¥ h Û x C is (split) injective. The first step of the proof is to convert a uniform embedding, which is something

purely metric in nature, into something more h -equivariant. For this purpose let us

recall the following object from general topology:

U U

Definition 4.11. Let be a discrete set. The Stone-Cech compactification of is

’

U

9 KlB

the set of all nonzero, finitely additive, 5¢R -valued probability measures on the

’ U algebra of all subsets of U . We equip with the topology of pointwise convergence

(with respect to which it is a compact Hausdorff space).

’

U U

ó

9 KlB

Thus a point of is a function from the subsets of into 5LR which is ad- ó

ditive on finite disjoint unions and which is not identically zero. A net converges

U

ó ó ó

Q Y¢

to if and only if converges to Q , for every .

¾ U

Example 4.5. If is a point of then the measure ó , defined by the formula

Ò

¾

K $H

if

ó

Q>aÂ

¾

Ò

$H´

R if

’ ’

U U

is a point of U . In this way is embedded into as a dense open subset.

ó K Remark 4.3. The fact that the measures assume only the values R and will matter

little in what follows, and we could equally well work with arbitrary, finitely additive

U ó

measures for which MK .

U ó

If — is a bounded complex function on , and if is a finitely additive measure

on U , then we may form the integral

‚

J¾ ¾

ó

— 4T  

228 Nigel Higson and Erik Guentner — as follows. First, if — assumes only finitely many values (in other words if is a

simple function) then define

‚

Õ

J¾ ¾ ¾ ¾

ó ó

— 4T j ýŠ™ 5 C]— Mý†B



t

ì — Second, if — is a general bounded function, write as a uniform limit of simple func-

tions and define the integral of — to be the limit of the integrals of the approximants.

¾ J¾

ó ó



,)„ƒ — ¶T 

Exercise 4.2. If — is a bounded function then the map is a con-

’ U

tinuous function from into « .

9 KlB Remark 4.4. The virtue of 5¢R -valued measures is that this integration process makes sense in very great generality — it is possible to integrate any function from U into any compact space.

Suppose now that h is a finitely generated discrete group. The compact space

’ h

h is equipped with a continuous action of by the formula

ó ó

}©™  Q> |} 

t

) 

Let ½ŽC‰h be a uniform embedding into an affine Euclidean space and let

¼

I h)

—ACeh be the associated negative type kernel:

¡

¢9 ¡ 9 ¡

— } } M#T ½ }  ½ } x

t

9

¡

}¾}  ©,)¿— }¾}

According to part (a) of Definition 4.10 the function } is bounded, for

’

ó

9

¡

} $Dh — $ h every } . As a result, we may define negative type kernels , for ,

by integration: ú

‚

ó

¢9 ¡ L9 ¡

} } > — } } }¾} 1T }  —

t

ú

ß

¡ ¡

9 9

¡ ¡

触

} }  } ( — } } }

Observe that — , so that our integration construction

’

ú ú h

defines an equivariant map from h into the negative type kernels on .

‡

9

¡

é é

} {' òR 'òR T }

Lemma 4.1. For every ' there exists so that if then

’

‡

ó

¢9 ¡

} } °' $ h

— , for every . ú

Proof. This is a consequence of part (b) of Definition 4.10.

é 

Since h is finitely generated, for every there is a finite set so that if

¡

ó

¡ ¢9 ¡

t

é

} $6 } }  } ,) — T then . The map is therefore proper-valued in the sense of Definition 3.25, and we have proved the ú following result:

Proposition 4.1. If a finitely generated group h may be uniformly embedded into an

affine Euclidean space then there is an equivariant, proper-valued continuous map

’

dfe h¨ h OP from h into the space of negative-type kernels on . Group C*-Algebras and K-theory 229

It will now be clear that to prove Theorem 4.8 we mean to apply Theorems 3.12

’ h

and 4.6. To do so we must replace h by a compact -space which is smaller (sec- ’ ond countable) and more connected (in fact contractible) than h . This is done as

follows.

U h

Lemma 4.2. Let h be a countable group, let be a compact -space and let

U U

) d¤e h¨ h

—AC be a continuous and -equivariant map from into the nega-

h + h

tive type kernels on h . There is a metrizable compact -space and a -map from

U —

to + through which the map factors.

U ¥©¦

Proof. Take + to be the Gelfand dual of the separable -algebra of functions on

¾

¢9 ¡ 9 L9 ¡

è

)¿— } }  } } } $Hh

generated by the functions , , for all .

Ò

+ h

Lemma 4.3. Let h be a countable group, let be a compact metrizable -space

‚+Þ)†d¤e h¨ h

and let —AC be a proper-valued, -equivariant continuous map. There

ý Á

is a metrizable compact h -space which is -equivariantly contractible, for every

h h ý

finite subgroup Á of , and a proper-valued, -equivariant continuous map from

h¨

into dfe . +

Proof. Let ý be the compact space of Borel probability measures on (we give

¦ ¥ +© ý the weak topology it inherits as a subset of the dual of ; note that we are

speaking now of countably additive measures defined on the Borel \ -algebra). If

ó

*ý — $`dfe h¨

$ then define by integration:

‚

ú

ó

9 9

¡ ¡

} T 8  } M —‡ } — }



t

ú

ó )¿— The map , has the required properties. Proof (Proof of ú Theorem 4.8). Proposition 4.1 and the lemmas above show that the hypotheses of Theorem 3.12 and Corollary 3.2 are met. Theorem 4.6 then implies injectivity of the assembly map, as required.

4.5 Amenable Actions In this section we shall discuss a means of constructing uniform embeddings of

groups into affine Hilbert spaces.

Pˆ

½

Ø h¨

Definition 4.12. Let h be a discrete group. Denote by the set of functions

Pˆ

½

9

è

½ŽC†hú)  R KS Ø h¨ ½ } r K ‘

such that Ù . Equip with the topology of

ß

} W) ½ }  ) ½ ½

pointwise convergence, so that ½ if and only if for every

ˆ

½

$oh Ø h¨ h

} . Equip with an action of by homeomorphisms via the formula

¡

Q™¢½ ŽAM˜½ } ŽA

} . h Definition 4.13. Let h be a countable discrete group. An action of by homeo-

morphisms on a compact Hausdorff space U is amenable if there is a sequence of

ˆ

½

U

½l! C }b$Hh h¨

continuous maps )^Ø such that for every

Ô Ô

¾ J¾

~

y}z{| ÖŒ× Ø ½:! }©™  }Q™ ½:! Œ R

GJI

! 

t

Ù

Ò

Ô Ô

= =

è

}  h  ‘

Here, if is a function on , then we define Ù . ß 230 Nigel Higson and Erik Guentner

We are going to prove the following result:

Proposition 4.9 If a finitely generated group h acts amenably on a compact space U

then h is uniformly embeddable in a Hilbert space.

Remark 4.5. The method below can easily be modified to show that if a countable

group h (which is not necessarily finitely generated) acts amenably on some compact U space U then there is an equivariant, proper-valued map from into the negative-

type kernels on h . The methods of the previous section then show that the Baum-

Connes assembly map is injective for h .

Examples 4.10 Every hyperbolic group acts amenably on its Gromov boundary. If

Á h

h is a discrete subgroup of a connected Lie group then acts amenably some h

compact homogeneous space Á ´:# . If is a discrete group of finite asymptotic di-

’ h mension then h acts amenably on the Stone-Cech compactification . See [33] for

a discussion of all these cases (along with references to proofs).

I

Z§Cý ýò) «

Definition 4.14. Let ý be a set. A function is a positive-definite

• • • • • 8• • 8• J•

9 9 9 9

¡ ¡ ¡

$ j  Z >%K Z Z kernel on the set ý if for all , if , for all

ý , and if

m

Õ

J• •S 

9

¯ ¯

ý Z xý '¨R



ð

¯

ï

• •

9 9 9 9

ý $H« ý $*ý m

for all positive integers , all and all m .

tStšt tStšt

J• •

9 WÜK Remark 4.6. The normalization Z is not always made, but it is useful

here. As is the case with positive-definite functions on groups (which we discussed

J• • J• •

¢9 ¡ ¡9

d Z  in Lecture 4), the condition Z is actually implied by the last condition.

Comparing definitions, the following is immediate:

¦



ý Z

Lemma 4.4. If Z is a positive-definite kernel on a set if denotes its real part,

¦

~‰

Z ý OP then K is a negative type kernel on .

Proof (Proof of Proposition 4.9). Suppose that h acts amenably on a compact space

ˆ

½

U U

)_Ø h¨

, and let ½l!;C be a sequence of functions as in Definition 4.13. g

After making suitable approximations to the ½! we may assume that for each there

ˆ

½

¾ U ¾

 ¢Àh $ ½ ©$"Ø h¨

is a finite set such that for every the function ! is

‹ŠÉ¡

J¾ J¾ ¾ U

9

 Ž }>˜½  } $ !

supported in . Now let ! . Then fix a point and define

I

Z C h h®)¿«

functions ! by

Õ

¾ ¾

9 9 9

¡ ¡ ¡

Z } } > Ž } } ¹Ž } }  } }

! ! !

t

è

Ù

ß

‹

I

h Š¢ah

These are positive definite kernels on h . For every finite subset and

ë

R $Œ_ every ‹ there is some such that

‹ Group C*-Algebras and K-theory 231

¡

= 9 =

¡ ¡ t

ë ~

g } } $<  Z } }  K ‹

and !

t

_ E¢"h

In addition, for every gW$ there exists a finite subset such that

¡

¡ ¢9 ¡

} } $<´  Zc! } } >R

t



¦

~Ž

K Z\!Rš

It follows that for a suitable subsequence the series ‘ is pointwise

¦

I

~a

h h K  convergent everywhere on . But each function ZM! is a negative type kernel, and therefore so is the sum. The map into affine Euclidean space which is associated to the sum is a uniform embedding.

Remark 4.7. In fact it is possible to characterize the amenability of a group action in terms of positive definite kernels. See [2] for a clear and rapid presentation of the facts relevant here, and [3] for a comprehensive account of amenability. The exis-

tence of a sequence of positive definite kernels on h which have the two properties

displayed in the proof of the lemma is equivalent to the amenability of the action of

’ h h on its Stone-Cech compactification . See [2] again, and see also Section 5.6 for more on this topic. Remark 4.8. The theory of amenable actions is very closely connected to the theory

of exact groups. To see why, suppose that h admits an amenable action on some compact space U . Then using the theory of positive-definite kernels it may be shown

that

U U

9 9

ì

¦ ¦

¥ h >¥ h 

and moreover that the crossed product ¥§¦ -algebra is nuclear. This means that the for

¦ Û

any ¥ -algebra ,

U

9 9

› ›

¦ ¦

Ûa¥ h Ûbn- ¯ Û ¥ h ‰-

!

ÇÓÒ t

See [2] for a discussion of these results. It follows of course that the crossed product

ì

¥©¦ h¨

¥¨¦ -algebra is exact, in the sense of Definition 2.10. But then it follows that ,

U U

9 9

ì

h M˜¥¨¦ h 

which is a subalgebra of ¥Q¦ , is exact too. Therefore, by Propo- h sition 2.6 the group h is exact. To summarize: if acts amenably on some compact

space then h is exact. In fact the converse to this is true too: see Section 5.6.

4.6 Poincare´ Duality

We conclude this lecture with a few remarks concerning a ‘dual’ formulation of the Baum-Connes conjecture for certain groups. With an application to Lecture 6 in mind

we shall formulate the following theorem in the context of reduced crossed products.

Theorem 4.11. Let h be a countable exact group and let be a separable proper

9

«M ¥¨¦ ­6$¨

h - -algebra. Suppose that there is a class with the property that

ß

= 9

± ±

h ­ $D «N for every finite subgroup Á of the restricted class is invertible.

Then the Baum-Connes assembly map

4

¤ ¤f

[Hõ

ó

ì

9 9

ì

¦

C h Ûb>) ¥ h Ûbx

232 Nigel Higson and Erik Guentner

¥§¦ Û

is an isomorphism for a given separable h - -algebra if and only if the map

¤f ¤"

9 9

ì ì

¦ ¦

­ C ¥ h ©Ã-Û xM) ¥ h Ûbx ¦

induced from ­ is an isomorphism.

ó ó

Remark 4.9. In the proof we shall identify with ­ , so that will be for example ¦

injective if and only if ­ is injective. As usual, analogous statements may be proved ¦ for reduced crossed products, either in the same way if h is exact, or with some additional arguments otherwise.

Proof. Consider the diagram

¤f

¤

4

[Hõ

9

9

ì

¥¨¦ h ©Ã-Ûb

h -Ûb

à 

ú /

 

¤f

¤

4  

[Hõ

9 9

9

ì

¥¨¦ h ÛbŒ Ûb

h /

ú 

in which the horizontal arrows are the Baum-Connes assembly maps and the vertical 

arrows are induced by composition with ­ in -theory and by composition with

ß  the element descended from ­ in nonequivariant -theory. The diagram commutes. By Theorem 2.20 the top horizontal map is an isomorphism. Furthermore, an argu- ment like the ones used in Section 4.3 shows that the left hand vertical map is an isomorphism. Therefore the bottom horizontal map is an isomorphism if and only if the right vertical map is an isomorphism, as required.

The theorem in effect reformulates the Baum-Connes conjecture entirely in the

framework of ¤ -theory (hence the term ‘Poincare´ duality’, since we have replaced

¤ ¤ ¤

4

[Hõ the -homological functor of h with -theory). It has an important applica- tion to groups which act isometrically on Riemannian manifolds. We shall not go into details, but here is a rapid summary of the relevant facts. The Clifford algebra

constructions we developed in Lecture 1 may be generalized to complete Rieman-

 6 ¥§¦

nian manifolds . We denote by the -algebra of sections of the bundle

jy{zn º 6

of Clifford algebras m associated to the tangent spaces of . There is a Ò

Dirac operator on (an unbounded self-adjoint operator acting on the Hilbert space

¡

of i -sections of the Clifford algebra bundle on ), and it defines a class

9

­ $H  6 «\

in almost exactly the same way that we defined ­ for linear spaces. Moreover if a

group h acts isometrically on then the Dirac operator defines an equivariant class

9

­ $H  6 «N

ß

t h Now if happens to be a universal proper -space then the hypotheses of Theo- rem 4.11 are met: Group C*-Algebras and K-theory 233

Proposition 4.12 Let be a complete Riemannian manifold and suppose that a

countable group h acts on by isometries. Assume further that is a universal



proper h -space. The Dirac operator on defines an equivariant -theory class

9 9

 ÛVn$H  6 «N

ß

Á ¢oh which, restricting from h to and finite subgroup , determines invertible ele-

ments

= 9

± ±

 ÛV $H  6 «N OP t

The proposition applies for example when h is a lattice in a semisimple group

(take to be the associated symmetric space), and in this case (which is perhaps the most important case of the Baum-Connes conjecture yet to be resolved) the con- jecture reduces to a statement which can be formulated purely within ¤ -theory.

5 Counterexamples

In this lecture we shall present a miscellany of examples and counterexamples. To- gether they show that the Baum-Connes conjecture is the weakest conjecture of its type which one can reasonably formulate. They also point to shortcomings in the machinery we have developed in these lectures. The counterexamples involve Kazh-

dan’s property º and expander graphs.

5.1 Property T º Definition 5.1. A discrete group h has property if the trivial representation is an

isolated point in the unitary dual of h .

See the monograph [17] for an extensive discussion of property º . We shall use

the following equivalent formulations of property º :

Theorem 5.1. Let h be a discrete group. The following are equivalent: º (a) h has property .

(b) Every isometric action of h on an affine Euclidean space has a fixed point.

? h¨

(c) There is a central projection $˜¥§¦ with the property that in any unitary

? ¶

representation of h , on a Hilbert space the operator acts as the orthogonal

¶ OP

projection onto the h -fixed vectors in .

? h¨

The projection $<¥Q¦ will be called the Kazhdan projection for the property h

º group . ?

Remark 5.1. If h is finite then the Kazhdan projection is the sum

Õ

K

?

 }

= =

h

è

Ù ß

234 Nigel Higson and Erik Guentner

hN˜¥Q¦ h¨ h

in the group algebra «G (in the formula we are regarding as a unitary

?

h¨ h

subgroup of ¥Q¦ ). If is infinite then is a very mysterious object. For example if

7

? ?

è

 ™

we (mistakenly) regard as an infinite formal series of group elements, ‘

? ?

} ™ 

} , then from the easily proved relation we conclude that all the scalars

7

? è

are equal, while from the fact that acts as K in the trivial representation we

7

? è conclude that the scalars sum to K . Thus we arrive at a formula for like the

one displayed above, where the sum is infinitely large and the normalizing constant

= = h K¢´ is infinitely small.

It is quite difficult to exhibit infinite property º groups, but they do exist. For

example Kazhdan proved that lattices in semisimple groups of real rank µ or more

have property º . It is also known that there are many hyperbolic groups with property

º . º

Lemma 5.1. If h is an infinite property group then the quotient mapping from

¤

ì

¦ ¦

h¨ ¥ h¨

¥ onto does not induce an isomorphism in -theory.

¤"

? h¨Œ

Proof. The central projection generates a cyclic direct summand of ¥©¦

¤f

ì h¨x

which is mapped to zero in ¥Q¦ . º

It follows immediately that if h is an infinite property group then the Baum-

¤f ¤"

ì

h¨Œ ¥¨¦ h¨Œ Connes assembly maps into ¥Q¦ and cannot both be isomor-

phisms. We shall not go into the matter in detail here but in fact it is the assembly

¤f h¨Œ

map into ¥¨¦ which is the problem. This can be seen quite easily in some º examples. For instance it is not hard to show that if h has property then associated

to each irreducible, finite-dimensional and unitary representation of h is a distinct

h¨

central projection in ¥Q¦ (the Kazhdan projection is the one associated to the h trivial representation). So if a property º group has infinitely many irreducible,

finite-dimensional and unitary representations (this will happen if h is an infinite

¤" h¨Œ

linear group) then ¥Q¦ will contain a free abelian subgroup of infinite rank,

¤

4

[Hõ whereas h¨ will very often be finitely generated. Unfortunately the main method we have applied to prove cases of the Baum-

Connes conjecture treats the full and reduced ¥§¦ -algebra more or less equally. Hence

property º causes the method to fail:

º h Proposition 5.1. If h is an exact, infinite property group then does not satisfy the hypotheses of Theorem 2.20.

Proof. If h did satisfy the hypotheses then by Theorem 2.20 the quotient mapping

¤f ¤f

ì

h¨Œ ¥¨¦ h¨x ¥¨¦ to would be an isomorphism.

5.2 Property T and Descent

Proposition 5.1 indicates that our basic strategy for proving the Baum-Connes con-

jecture for a group h , which involves proving an identity in equivariant, bivariant ¤

-theory, will not work for infinite property º groups (at least if these groups are exact). Group C*-Algebras and K-theory 235

However one can ask whether it is possible to prove the conjecture for a given ¤

group h by proving an identity in bivariant -theory for crossed product algebras.

¦

We noted in Lecture 5 that if h is a complete manifold then the Baum-Connes

conjecture for h is equivalent to the assertion that the map

¤ ¤

9 9

ì ì

¦ ¦

­ C ¥ h  6xŒ>) ¥ h¨x

¦ ¦ ¦

9

 6 «N induced from the Dirac operator class ­$  , is an isomorphism. One

might hope that in fact the descended class ß

9 9

ì ì

¦ ¦

­ $H ¥ h  6x ¥ h¨Œ is an isomorphism. This is not (always) the case, as the following theorem of Skan-

dalis [60] shows:

ì

º ¥V¦ h¨

Theorem 5.2. Let h be an infinite, hyperbolic property group. Then is not ¥Q¦

equivalent in  -theory to any nuclear -algebra.

›

¯

VÃ- ! Û 

Recall from the last lecture that a ¥Q¦ -algebra is nuclear if

9

ì

›

- Û Û ¥Q¦ ¥¨¦ h  6x ©Ã , for all . Since the -algebra is easily proved to be nu-

clear weÇÓÒ obtain the following result:

º h Corollary 5.1. Let h be an infinite, hyperbolic, property group and assume that

acts on a complete Riemannian manifold by isometries. The Dirac operator class

9 9

ì ì

¦ ¦

h¨Œ h  6x ¥ ­ $H ¥ P is not invertible. O

Remark 5.2. The corollary applies to discrete, cocompact subgroups of the Lie

‡

? 9

K¢ groups g ( is quaternionic hyperbolic space). See [17]. Despite this, it fol-

lows from the work of Lafforgue [44] that in this case ­ as above does induce an ¤ isomorphism on -theory. This shows that  -theory is not a perfect weapon with which to attack the Baum-Connes conjecture.17

To prove Theorem 5.2 we shall use the following result. Eh

Theorem 5.3. Let h be a hyperbolic group and let be its Gromov boundary.

Ž~ Ah There is a compact, metrizable topology on the disjoint union h6h with the

following properties: h

(a) The set h is an open, discrete subset of . h (b) The left action of h on itself extends continuously to an amenable action of on

h . h

(c) The right action of h on itself extends continuously to an action on which is

OP

trivial on h .

à ¢

Exactly the same remarks apply here to ê ê -theory. 236 Nigel Higson and Erik Guentner

Remark 5.3. Item (c) is essentially the assertion that the natural action on the Gromov compactification is small at infinity, in the sense of Definition 4.9.

We shall also require a few simple representation-theoretic ideas.

Definition 5.2. Let h be a discrete group. The left regular, right regular and adjoint

¡

ã h¨

representations of h on are defined by the formulas

¡

è

• •

ý  ŽÝ> } ŽÝ

è

• •

ü  ŽÝ> Ž¶}

¡

è

• •

­  ŽÝ> } Ž4} 

¡ ¡

ã ã

9

I

•

ŽD$Xh $ h¨ h h h¨ for all } and . The biregular representation of on is

defined by the formula

¡

¡

è è

• •

­  ŽÝ> } Ž4} 

0

2

¡

ã

9 9

¡

•

} Žd$

for all } and . ü

The left and right regular representations determine representations ý and of

¡

ã

ì

h¨ ¹ h¨x ¥¨¦ in . Since these representations commute with one another, together

they determine a ¥Q¦ -algebra representation

¡

’

ã

9

~

ì ì

›

¦ ¦

¥ h¨ )+¹ h¨Œ Ce¥ h¨n-

ÇÓÒ

I

ì ì

›

hY¢¨¥©¦ h¨‰- ¥¨¦ h¨

which is of course the biregular representation on h .

ÇÓÒ /

Definition 5.3. Denote by the kernel of the quotient homomorphism from

ì ì ì ì

› ›

¯

h¨ h¨n- ¥¨¦ h¨c- h¨ ¥¨¦ !¥¨¦

¥¨¦ onto , so that there is an exact sequence

ÇÓÒ

/

ì ì ì ì

› ›

¥¨¦ h¨ ¥¨¦ h¨‰- ¥¨¦ h¨c- ¯ ¥¨¦ h¨

! R

R / / / /

ÇÓÒ

t

’

/

ì

›

h¨‚- ¥§¦

Lemma 5.2. The ¥Q¦ -algebra representation maps the ideal of

¡ ¡

ÇÓÒ

ã ã

ì

¥¨¦ h¨ h¨Œ ¹ h¨Œ

into the ideal ¸ of .

¡ ¡

ã ã

h¨ h¨Œ Proof. Denote by ‘ the Calkin algebra for — the quotient of the

bounded operators by the ideal of compact operators. We are going to construct a Y -

¡

ã

ì ì

›

¥Q¦ h¨S- ¯ ¥¨¦ h¨ ‘ h¨x homomorphism from ! into which makes the following

diagram commute:

¡

ã

ì ì

›

¦ ¯ ¦

h¨‰- h¨ ¥ !(¥ h¨Œ / ‘

O O

¡

ã

ì ì

›

¥¨¦ h¨n- ¥¨¦ h¨

¹ h¨x

ì “’

î /

ÇÓÒ t

Here the vertical arrows are the quotient mappings. Commutativity of the diagram will prove the lemma.

Group C*-Algebras and K-theory 237

¡

ã

¥ Eh¨ ‘ h¨x

We begin by constructing a Y -homomorphism from into , as

Ah¨ ½ h

follows. If ½%$¥ then extend to a continuous function on , restrict the

¡

ã

¢ h h¨

extension to the open set h , and then let the restriction act on by point-

¥ Eh¨ wise multiplication. Two different extensions of ½„$ will determine two

pointwise multiplication operators which differ by a compact operator. Hence our

¡

ã

ZQC†¥ Eh¨ )”‘ h¨x

procedure defines a Y -homomorphism , as required. Now

¥ Eh¨ h let h act on via the (nontrivial) left action of (see Theorem 5.3) and define

a Y -homomorphism

¡

ã

9

¦

Z§CÝ¥ h Eh¨M)†‘ h¨x

by the formula

Õ Õ

è è

Z ½ ™N} M Z ½ Œý }

è è

Ù Ù

ß ß

¡

ã

h¨ }  (we are using ý to denote both the unitary operator on and its image in

the Calkin algebra). Next, thanks to part (c) of Theorem 5.3 the right regular repre-

¡

ã

¥ Eh¨3¢•‘ h¨x sentation commutes with the algebra ZG . We therefore obtain a

Y -homomorphism

¡

ã

9

~

ì

›

¦ ¦

¥ h Eh¨‰- ¥ h¨ )–‘ h¨Œ

ÇÓÒ

t

9

Eh ¥§¦ ¥¨¦ h Eh¨ But since the action of h on is amenable the -algebra is nu-

clear, so that the maximal tensor product above is the same as the minimal one.

9 9

ì

h Eh¨ h Eh¨ ¥¨¦ Moreover amenability also implies that ¥§¦ agrees with . See Re-

mark 4.8. It follows that the Y -homomorphism displayed above is the same thing as

a Y -homomorphism

¡

ã

9

~

ì ì

›

¦ ¦

¯

h Ah¨c- h¨ )–‘ h¨Œ ¥ !¥ t

The lemma now follows by restricting this Y -homomorphism to the subalgebra

9

ì ì ì ì

› ›

¥¨¦ h¨n- ¯ ¥¨¦ h¨ ¥¨¦ h Eh¨‰- ¯ ¥¨¦ h¨ !

! of .

¤ ¤f 1/

Lemma 5.3. The -theory group  is nonzero.

ì ì

›

h¨ h¨A- ¥¨¦ h¨>)¿¥¨¦ Y }©,)Ø}ß-%}

Proof. Let å CÝ¥Q¦ be the -homomorphism .

/

ÇÓÒ

? ?

h¨ 2d„å  2&$ Let $–¥¨¦ be the Kazhdan projection and let . Then . To see

this, observe that the composition

¡ ¡

ã ã

ì ì

ì ì

›

›

¥¨¦ h¨ ¥¨¦ h¨š-

¥¨¦ h¨

¯

¥¨¦ h¨š- !†¥¨¦ h¨¢V¹ h¨š- h¨x

ì / / ÇÓÒ corresponds to the tensor product of two copies of the regular representation, and ob-

serve also that this representation has no nonzero h -fixed vectors. Hence, the image

ì ì

›

¯

¥§¦ h¨\- h¨

of the Kazhdan projection in !Q¥¨¦ is zero. We shall now prove that

¤f 0/

p

 2L R

in  . Note first that the representation

¡

’

ã

~

ì ì

›

¦ ¦

Ce¥ h¨‰- ¥ h¨ )^¹ h¨Œ ÇÓÒ

maps 2 to a nonzero projection operator. Indeed the composition

238 Nigel Higson and Erik Guentner

¡

ã

ì ì

›

¥¨¦ h¨‰- ¥¨¦ h¨

¥¨¦ h¨

¹ h¨Œ £

ì / /

ÇÓÒ

¥Q¦ h¨ h

is the representation ­ of associated to the adjoint representation of , which

’ 2

does have nonzero h -fixed vectors, and maps to the orthogonal projection onto ’

these fixed vectors. But by Lemma 5.2 the representation maps / into the compact

¡

¤

ã h¨x

operators, and every nonzero projection in ¸ determines a nonzero -theory

¡

¤" 0/ ¤f

ã

¸ h¨xŒ  2L

class. Hence the map from  to takes to a nonzero element,

¤" 0/

2Lc$ 

and therefore the class  is itself nonzero. ¦

Proof (Proof of Theorem 5.2). Let us suppose that there is a separable nuclear ¥ -

9

ì ì

 Z $H ¥§¦ h¨  ¥¨¦ h¨ algebra and an invertible -theory element . Since is an

exact ¥Q¦ -algebra there are invertible elements

9

ì ì ì

› › ›

¦ ¦ ¦

Z<- K$< ¥ h¨c- ¥ h¨ ¨- ¥ h¨x

ÇÓÒ ÇTÒ ÇTÒ

and

9

ì ì ì

› › ›

¦ ¦ ¦

ZH- ¯ K$< ¥ h¨n- ¯ ¥ h¨ "- ¯ ¥ h¨x

! ! ! t

We therefore arrive at the following commuting diagram in the  -theory category:

9‹˜

î1—

ì ì ì

› ›

¥¨¦ h¨n- ¥¨¦ h¨ f- ¥¨¦ h¨x

R

÷

ð /

ÇÓÒ ÇÓÒ

R

ð

 

ì ì ì

› ›

¥¨¦ h¨‰- ¯ ¥¨¦ h¨ "- ¯ ¥¨¦ h¨x

! !

“—

î /

’

÷ Î

But since is nuclear the right hand vertical map is an isomorphism (even at the

level of ¥¨¦ -algebras). It follows that the left hand vertical map is an isomorphism in ¤

the  -theory category too. As a result, the -theory map

¤"

¤"

~

ì ì ì ì

› ›

¦ ¦ ¦ ¦

h¨x h¨c- ¯ ¥ h¨Œ ) ¥ ¥ h¨‰- ¥

! ÇÓÒ is an isomorphism of abelian groups. But thanks to the ¤ -theory long exact sequence this contradicts Lemma 5.3.

5.3 Bivariant Theories

In the previous section we showed that it is not possible to prove the Baum-Connes

‡

? 9 KL

conjecture for certain groups (for example uniform lattices in g ) by working ¤D¤ purely within  -theory (or for that matter within -theory). In this section we shall prove a theorem, also due to Skandalis [61], which points to another sort of weakness of bivariant ¤ -theory. Recall that the bivariant theory we constructed —

namely  -theory — has long exact sequences in both variables but that we could not equip it with a minimal tensor product operation (since the operation of minimal tensor product does not in general preserve exact sequences). Kasparov’s ¤D¤ -theory has minimal tensor products but the long exact sequences are only constructed under

some hypothesis or other related to ¥Q¦ -algebra nuclearity. One might ask whether or not there is an ‘ideal’ theory which has both desirable properties. The answer is no:

Group C*-Algebras and K-theory 239 ¤

Theorem 5.4. There is no bivariant -theory functor on separable ¥V¦ -algebras which has both a minimal tensor product operation and long exact sequences in both variables.

Remark 5.4. By the term ‘bivariant ¤ -theory functor’ we mean a bifunctor which, ¤r¤ like  -theory and -theory, is equipped with an associative product allowing us to

create from it an additive category. The homotopy category of separable ¥V¦ -algebras should map to this category, and the ordinary one-variable ¤ -theory functor should factor through it. To prove Theorem 5.4 we shall need one additional computation from represen- tation theory.

Lemma 5.4. [41] Let h be a residually finite discrete group. The biregular represen-

¡

’

ã

I

h¨ h

tation of h on extends to a representation of the minimal tensor product

›

¥¨¦ h¨n- ¯ ¥¨¦ h¨

! .

5Lh B h

Proof. Let ! be a decreasing family of finite index normal subgroups of

¾

™\h h $

for which the intersection ! is the trivial one-element subgroup of . If

¾

Ä Ä

«; h « h( «G hQ´lh  !

then denote by ! the corresponding ‘quotient’ element of

’

I

! hQ´lh¨! hQ´lh(! hQ´:h(!

«; and denote by the biregular representation of on

¡

ã

›

¯

hQ´lh(!  -

. Thanks to the functoriality of ! it is certainly the case that

Ô Ô Ô Ô

¾ ¾

Š Š

— —

î î

'¨Öx× Ø !

³ ³ ³ ³

t

’ ß ß ’ ß ß ß ß

   

!

ö ø ö ø ö ø ö ø

Î Î Î Î

In addition

Ô Ô ÔOš Ô

’

¾ ¾

Š Š Š



 '

! ! !

³ ³

ß ß ß ß ß ß

’

 

2

ö ø ö öœ› ö ø øø

Î Î Î Î

hQ´:h(!e (observe that since ¥Q¦ is finite-dimensional the minimal tensor product here

is equal to the maximal one). Now, it is easily checked that

ÔOš Ô Ôš Ô

’ ’

¾ J¾

Š

 Öx×eØ ! !e '

t

ß ß ß

!

2 2

öœ› ö øø öœ› ö øø Î

Putting together all the inequalities we conclude that

ԝš Ô Ô Ô

’

¾ ¾

9

î1—

 '

³ ³

ß ß ß

’

 

2

öž› ö øJø ö ø ö ø Î

as required.

ì

¦ ¦

h¨ v ¥ ¥ h¨ Lemma 5.5. Let q be the kernel of the quotient map from onto , so

that there is a short exact sequence

Õ

ì

¦

¦ ¥ h¨

h¨ ¥

R R

/ q / / /

t

9

*©

If there is a bivariant theory  which has long exact sequences in both vari-

¥ v q ¢'¥ Õ

ables, and if Õ is the mapping cone of , then the inclusion determines an

9

 q ¥  invertible element of Õ . 240 Nigel Higson and Erik Guentner

Proof. Consider the commuting diagram

Õ

ì

¦

¦ ¥ h¨

¥ h¨

R R

/ q / / / ð

  

9

ì

R

¥¨¦ h¨

¥°Õ ý£Õ

R / / / /

7 J7

9

ì

ý Ú5 ½$¥¨¦ h¨M¥¨¦ h¨T R KS&C†v Q ½ RwAB

where Õ . The inclusion of

¥¨¦ h¨ ý into Õ (as constant functions) is a homotopy equivalence, and therefore by

applying  -theory to the diagram and then the five lemma we see that the inclusion

¢–¥°Õ

q induces isomorphisms

R R

ð ð

9 9 9 9

qw  ¥°Õ   ¥°Õ *§  q *§

 / and / *

for every and . It follows that the inclusion determines an invertible element of

9

q ¥°Õe  as required.

Proof (Proof of Theorem 5.4). If the bivariant ‘  -theory’ has a minimal tensor prod-

uct then it follows from the lemma above that the inclusion

› ›

¦ ¦

¯ ¯

q- !¥ h¨P¢¨¥°Õ¨- !(¥ h¨ ¤ determines an invertible element in  -theory and therefore an isomorphism on - theory groups. We shall prove the theorem by showing that the map on ¤ -theory induced from the above inclusion fails to be surjective.

Consider the short exact sequence

î

Õ

9

ì

›

›

¦ ¦ R

h¨ ¥ h¨‰- ¯ ¥

¯ !

¥¨¦ h¨n- !¥¨¦ h¨ R

/ i / / / vD-ÀK

where the ideal i is by definition the kernel of the quotient mapping . The

v<-®K ¥ -¥¨¦ h¨

mapping cone of is (canonically isomorphic to) Õ , and therefore

¤

›

i)¢¥ - ¯ ¥¨¦ h¨ ! the inclusion Õ induces an isomorphism in -theory. Observe

now that we have a sequence of inclusions

› ›

¦ ¦

¯ ¯

q- !¥ h¨P¢"i`¢'¥°Õ- !(¥ h¨ t We wish to prove that the overall inclusion fails to be surjective in ¤ -theory, and since the second inclusion is an isomorphism in ¤ -theory it suffices to prove that the first inclusion fails to be surjective. From here the proof is more or less the same as

the proof of Lemma 5.3, and we shall be very brief. There is a diagonal map

›

¦ ¦ ¦

å CA¥ h¨>)¿¥ h¨c- ¯ ¥ h¨

!

›

¦ ¯ ¦

˜$ ¥ h¨ß- !b¥ h¨ å and we denote by 2 the image under of the Kazhdan

projection. It is an element of the ideal i . According to Lemma 5.4 the biregular

¡

ã

I

›

¯

h h¨ ¥¨¦ h¨\- !§¥¨¦ h¨ representation of h on extends to . From the proof of Lemma 5.2 we obtain a commuting diagram

Group C*-Algebras and K-theory 241

¡

ã

ì ì

›

¯

¥¨¦ h¨‰- !(¥¨¦ h¨ h¨Œ / ‘

O O

¡

ã

9

›

¥¨¦ h¨‰- ¯ ¥¨¦ h¨

! h¨x

/ ¹

£

’ i which shows that the ¥Q¦ -algebra representation maps the ideal into the compact

operators. Consider now the sequence of maps

ö

¡

ã

›

q- ¯ ¥¨¦ h¨

!

¸ h¨Œ £

/ i / t

The composition is zero. But the projection ? is mapped to a nonzero element in

¡

¤ ¤

ã

h¨x  2L

¸ , and the -theory class of is mapped to a nonzero element in the -

¡

¤"

ã

h¨Œ  2Lß$ ij

theory of ¸ . This shows that the class is not the image of any

¤

›

q- ¯ ¥¨¦ h¨ -theory class for ! , and this completes the proof of the theorem.

5.4 Expander Graphs

The purpose of this section and the next is to present a counterexample to the Baum-

Connes conjecture with coefficients, contingent on some assertions of Gromov. K

Definition 5.4. Let Ÿ be a finite graph (a finite, -dimensional simplicial complex)

¡

ã

Ú–k Ÿ Ÿ åfC kQ¨)

and let k be the set of vertices of . The Laplace operator

¡

ã §

k is the linear operator defined by the quadratic form

¡

Õ

Ç œ œ

9 = =

~

â

Æ8½ åV½  ½  ½ 

t

ð

©

Î Î

ï

ö ø

The sum is over all (unordered) pairs of adjacent vertices, or in other words over the

Ÿ ý å

edges of Ÿ . We shall denote by the first nonzero eigenvalue of . å If the graph Ÿ is connected then the kernel of consists precisely of the constant

functions on k . In this case

¡

ã

¡

Ô Ô

K

½&$ kQ

Ç

9

“ ½ s ƉåV½ ½

œ (1)

½ >R ¡

t

‘

Ÿ ý

Ù

­

Ò

9

ë

‹€ ‹ R Ÿ

Definition 5.5. Let be a positive integer and let . A finite graph is a -

expander if it is connected, if no vertex of Ÿ is incident to more than edges, and if

Ÿ°'f‹ ý .

See [46] for an extensive discussion of the theory of expander graphs. The following observation of Gromov shows that expander graphs give rise to examples of metric spaces which cannot be uniformly embedded in affine Euclidean spaces.

242 Nigel Higson and Erik Guentner

I

ë

ð

‹ R 5§Ÿ B !

Proposition 5.2. Let be a positive integer, let , and let ! be a se-

9 = = ;

‹€ y}z{| GxI k Ÿ   k !

quence of -expander graphs for which ! . Let be the

k  k Ÿ  k !

disjoint union of the sets ! and suppose that is equipped with a dis- k tance function which restricts to the path-distance function on each ! . Then the

metric space k may not be embedded in an affine Euclidean space.  Proof. Suppose that ½ is a uniform embedding into an affine Euclidean space . We

may assume that  is complete and separable, and we may then identify it isometri-

¡

ã

\ ½ kA! ½l!

cally with _ . By restricting to each , and by adjusting each by a transla-

¡

ã

N ½w! tion in _ (that is, by adding suitable constant vector-valued functions to each )

we can arrange that each ½l! is orthogonal to every constant function in the Hilbert

¡

¾

ã



½ >R kA! _N ‘

space of functions from to (we just have to arrange that Ù ).

¡

ã Ò

_N Now the Laplace operator can be defined on -valued functions justÎ as it was on

scalar functions, and the expander property (1) carries over to the vector-Laplacian

¡

ã N

(compute using coordinates in _ ). However

¡

Õ

Ç œ œ

9 = =

~

â

ƈåV½ ½  ½  ½ 

! ! ! !

ð

©

Î Î

ï

ø ö

Õ

s K

ð

©

Î Î

ï

ö ø

= =

ke! s

t µ

It therefore follows from the expander property that

¡ ¡

Õ

Ô Ô Ô Ô

K

Ç œ

9 = =

Æ8åV½:! ½l! s ke! ½l!   ½l! s

t

‹ µ:‹

Ù

­

Î

Î

¡

Ô Ô

m

œ œ

g $ kE! ½l!  s

Thus for all , and for at least half of the points , we have  . This

contradicts the definition of uniform embedding since among this half there must be

œ œ œ œ

9 ;

â â

GJI

! y}z}| ! T ! M ! points and ! with .

In a recent paper [26], M. Gromov has announced the existence of finitely gen- erated groups which do not uniformly embed into Hilbert space. Complete details of the construction have not yet appeared, but the idea is to construct within the Cayley graph of a group a sequence of images of expander graphs. Let us make this a little more precise, as follows.

Definition 5.6. Let us say that a finitely generated discrete group h is a Gromov

9

ë

‹ R ‹l

group if for some positive integer and some there is a sequence of -

‰! ZN!GCÏk Ÿ†! M)êh

expander graphs Ÿ and a sequence of maps such that :

œ œ é

(a) There is a constant , such that if and â are adjacent vertices in some graph

œ œ

9

é

†! T Z\!  Z\! âãxPs

Ÿ then .

èvô

¸ Ú

§

0

×p§ Ö¤£ £

GJI€¢

C}b$dh 'R y{z}| ! | $

(b) ­ .

÷

Î

£ £

ö¦¥ ø Î

Group C*-Algebras and K-theory 243

= = ;

y}z{| GJI k Ÿ   ! Remark 5.5. The second condition implies that ! . It appears that Gromov’s ideas prove that Gromov groups, as above, exist. In any case, we shall explore below some of the properties of Gromov groups. We conclude this section with a simple extension of Proposition 5.2, the proof of which is left to

the reader. h

Proposition 5.5 If h is a Gromov group then cannot be uniformly embedded in P an affine Euclidean space. O

5.5 The Baum-Connes Conjecture with Coefficients

We shall prove that, contingent on the existence of a Gromov group as in the last sec- Û

tion, there exists a separable, commutative ¥§¦ -algebra , and an action of a count- Û

able group h on , for which the Baum-Connes map

4

¤ ¤f

[Hõ

ó

ì

9 9

ì

¦

C h Ûb>) ¥ h Ûbx

fails to be an isomorphism.

/

¦

h ¥

Lemma 5.6. Let h be a countable group and let be an ideal in a - -algebra

ó

ì h

. If the Baum-Connes assembly map is an isomorphism for , with coefficients

¤ /

´

all the separable ¥Q¦ -subalgebras and , then the -theory sequence

¤f / ¤f ¤" /

9 9 9

ì ì ì

h x ¥¨¦ h Œ ¥¨¦ h ´ Œ ¥¨¦ / / is exact in the middle. Proof. Since exactness of the sequence is preserved by direct limits it suffices to

consider the case in which itself is separable. The proof then follows from a chase

around the diagram of assembly maps

¤ 4 ¤ 4] /

[Hõ [Hõ

9 9

 h ´ 

h /

/ ¤f ¤" /

¤f  

9 9 9

h x ¥¨¦ h Œ ¥¨¦ h ´ Œ

¥¨¦ / /

/ ¤f ¤" /

¤f   

9 9 9

ì ì ì

h x ¥¨¦ h Œ ¥¨¦ h ´ Œ ¥¨¦ / /

and the fact that the middle row is exact in the middle.

/

We shall prove that if h is a Gromov group then for a suitable and the

conclusion of the lemma fails. ¥Q¦

Definition 5.7. Let be the -algebra of bounded complex-valued functions on

|

I I

_ h 5Lg\B

h for which the restriction to each subset is a -function. Denote by

/

|

I

h _ the ideal in consisting of -functions on .

244 Nigel Higson and Erik Guentner

E /  E

ã

I

| | |

9 9

˜‘ _ h¨Œ ‘ _ h¨x 

Thus  and .

I

h h _

Now let h act on be the right translation action of on .

¡

ã

I

h _N

Lemma 5.7. The (right regular) covariant representation of on deter-

9

ì

h 

mines a faithful representation of the reduced crossed product algebra ¥V¦ as

¡

ã

h¨ P operators on . O

From here on we shall assume that h is a Gromov group. For simplicity we shall

ÏkÝ!Š)+h now assume that the maps ZN!GC which appear in Definition 5.6 are injective.

For the general case see [11].

kA! kÝ! Z\! g

Let k be the disjoint union of the . Let us map via to the th copy of

¡

ã

I I

h _ k h _ k§

h in , and thereby embed into . We can now identify with a

¡

ã

I

_N

closed subspace of h .

¡ ¡

ã ã

I I

h _NŽ) h _N

Definition 5.8. Denote by åfC the direct sum of the Laplace

¡ ¡

ã ã

Ý!eU¢ k§

operators on each k with the identity on the orthogonal complement

¡ ¡

ã ã

I

§P¢ h _\

of k .

¡

ã

9

I

~

ì

q ¥§¦ h P¢¨¹ h _\Œ Lemma 5.8. The operator å belongs to (it is in fact

in the algebraic crossed product).

k Þk Ÿ  !

Proof. First, some notation. Let us continue to identify the vertex set ! ,

9

Z h  } } â G$f Ÿ  } !

via ! , with a subset of . We shall write if the group elements

}â k Ÿ } !

and correspond to vertices in ! which are adjacent in the graph . Finally if

Ÿ } K !

corresponds to a vertex of ! we shall write for its valence, minus .

¡

ã

I

è

h _N ½ The Hilbert space has canonical basis elements ! and in this basis

the formula for å is

>

Õ

~ ~

è è è

,)Š } x½ ½ å qw]CE½ }b$lk

! ! ! !

 if

8

è è ô

>Ã

Ú Ù

Á

ï

ö¦¥ ø

Î

~

è

qw]CE½ !V,)¿R }W´$ kÝ!

å if .

~ q

We can therefore write å as a finite sum

Õ

F 7 7

9

~



å q§ ™  ŽV™

ð

q¨

7 è

where the coefficient functions $H are defined by

! }  }b$lkÝ!

7 if

9

} g‰M Â

}W´$lkÝ!

R if

F

p 

and, for Ž ,

¡

9

~

}w†$< Ÿ   }¶Ž K

!

7 if

9

 Â

} g‰M

¡

9

R }w†$< Ÿ   }¶Ž if ! . (The sum is finite thanks to the first item in Definition 5.6.)

Group C*-Algebras and K-theory 245

9

Ÿ ‹€ R Since the graphs ! are -expanders the point is isolated in the spectrum

of å , and therefore we can make the following definition:

ZM!;Cœke!W) h

Definition 5.9. Let h be a Gromov group and assume that the maps

9

ì

¦

h  å

are injective. Denote by ç$<¥ orthogonal projection onto the kernel of .

¡

ã

I

h _

The operator  is the orthogonal projection onto the -functions on which

Ý! k

are constant on each k and zero on the complement of .

¤f

9

ì

¥Q¦ h x

Lemma 5.9. The class of  in is not in the image of the map

/ ¤" ¤f

9 9

ì ì

x>) ¥¨¦ h Œ h

¥¨¦ .

|

I

h 5Lg\B¢

Proof. Let !V , which is a quotient of , and denote by

9 9

ì ì

¦ ¦

vA!dC¥ h j)¿¥ h Ž!e

the quotient mapping. We get maps

¤" ¤f

9 9

ì ì

¦ ¦

vÝ! C ¥ h x>) ¥ h Ž!ÝxM³

¦

t

v § v  @HžK g !

Since ! is a rank one projection, we find , for all . Therefore

¤ ¤" ¤f /

? 9 9

ì ì

h x ¥Q¦ h x

the -theory class of in ¥Q¦ does not come from (which

 ÙB©Š³ v !

maps to the direct sum ! under ).

/

9

ì

¥Q¦ h ´  Lemma 5.10. The image of  in is zero.

To prove the lemma we shall need some means of determining when elements in

9

ì

h Ûb

reduced crossed product algebras ¥§¦ are zero. For this purpose, recall that the

9

ì

h Ûb ¥¨¦ Û

¥¨¦ -algebra is faithfully represented as operators on the Hilbert -module

¡

ã

9 Ûb

h .

¡

ã

9

è

h Ûb

Exercise 5.1. If # denotes the orthogonal projection onto the functions in

9

ì

è

h Û  S}AB º$X¥Q¦ # ºŽ#

supported on 5 , and if , then is an operator from functions

F

è

B 5S}AB # ºŽ#

supported on 5 to functions supported on . If all the elements are equal ºR

to R then .

è è

ºŽ# º $HÛ Exercise 5.2. The operator # can be identified with an element via the

formula

¡

JF

ã

9 j 9

è è

• • •

 # ºŽ#5  } >º ™ $ h Ûb

t

è è

º º6 T ™F} T $uÛ

If is a finite sum ‘ in the algebraic crossed product (where )

È

è è

 T Z§CEÛ¿)ÙÛbâ h Y

then º . If is a -equivariant -homomorphism and if is the

Ȃ

è è

Z º 

induced map on crossed products then º .

9

ì

h 

By checking on finite sums we see that if an operator º6$D¥©¦ has matrix

¡

ã

9

è è è

º ! ! º $ _M

coefficients for the canonical basis of h then the functions

ï

associated to º are defined by

9

è è

 

º Ž g‰M–º ! !

t ï

246 Nigel Higson and Erik Guentner

¡ ¡

ã ã

I I

h _N) h _N

Proof (Proof of Lemma 5.10). The projection XC is com-

¡

ã

 k  ! prised of the sequence projections ! onto the constant functions in . The

matrix coefficients of  are therefore described by the formula

>

Õ

K

è è

Â

½ uCe½ ,) }b$lk

! !

! if

= =

ke!

Ã

>

è

Ù

­

Á

Î

è

}W´$lke! !Š,)+R

uCe½ if .

è

$<  As a result, the functions  associated to the projection , as in the exercises,

are given by the formula

K

9

Â

Ž4} Žd$lke!

if

Ã

= =

9

k

è !

 Ž g‰M

Á

Ž4}D´$lk Ž ´$ k R !

if ! or

t

/ /

è

$ }$6h Ž!o$6 ´

This shows that  , for all . It follows that the elements

/

9

ì

¥§¦ h ´  R 

associated to the image of  in are , and so the projection is itself

/

9

ì

¥¨¦ h ´  R in .

The two lemmas show that the ¤ -theory sequence

¤f / ¤f ¤" /

9 9 9

ì ì ì

¦ ¦ ¦

h x ¥ h Œ ¥ h ´ Œ ¥ / /

fails to be exact in the middle. Hence:

h ¥©¦ Theorem 5.6. Let h be a Gromov group. There is a separable, commutative - -

algebra Û for which the Baum-Connes assembly map

4

¤ ¤f

[Hõ

ó

ì

9 9

ì

¦

h Ûbx C h Ûb>) ¥ P fails to be an isomorphism. O

5.6 Inexact Groups

The following result (see [28, 29, 15]) shows that Gromov groups fail to be exact. h Theorem 5.7. If a finitely generated discrete group h is exact then embeds uni- formly in a Hilbert space.

To prove the theorem we shall use a difficult characterization of separable exact

¥¨¦ -algebras, due to Kirchberg [42] (see also [66] for an exposition). It involves the

following notion:

È

* ¥§¦ C€ )¿*

Definition 5.10. Let and be unital -algebras. A unital linear map

È

¥¨¦ $`_ Cw >) m

of -algebras is completely positive if for all the linear map m

È

*© m defined by applying entrywise to a matrix of elements of is positive (meaning it maps positive matrices to positive matrices).

Group C*-Algebras and K-theory 247

Y

Theorem 5.8. A separable ¥Q¦ -algebra is exact if and only if every injective -

¹ ¶d homomorphism ž) can be approximated in the point norm topology by

a sequence of unital completely positive maps, each of which factors, via unital, P completely positive maps, through a matrix algebra. O

Kirchberg’s theorem has the following consequence:

Corollary 5.2. If h is a countable exact group then there exists a sequence of com-

¡

È

ã

ì

h¨>)+¹ h¨Œ

pletely positive maps !GCÝ¥¨¦ which converge pointwise in norm to

¡

ã

ì

h¨ ¹ h¨Œ

the natural inclusion of ¥Q¦ into and which have the property that for

È

`_ }V,) ! }  every gW$ the operator valued function is supported on a finite subset

of h .

Proof. By Theorem 5.8 there exists a sequence of unital completely positive maps

¡

ã

ì

h¨ ¹ h¨Œ which converge pointwise in norm to the natural inclusion of ¥©¦ into , and which individually factor through matrix algebras. Let us write these maps as

compositions

¡

ã

ì

¦

¥ h¨”ª

«M «

m h¨Œ

/ / ¹ (2)

Î Î

t

Î

ì

¦

WCE¥ h¨ )û fm «N

Now, a linear map ¬ is completely positive if and only if the

ì

¥¨¦ h¨Œ>)+« ðMCe

linear map m defined by the formula

m

Õ

  

K

¯ ¯ ¯

@M  ð  ½ ¬ ½



ð

¯

ï §ˆð

is a state. Moreover the correspondence ¬ is a bijection between completely

9 9

Ž Ž

positive maps and states. In addition, if m are finitely supported functions

¡ ¡

tStšt

ã ã

h¨No™š™S™€ h¨ on h which determine a unit vector in the -fold direct sum ,

then the vector state

m

Õ

  

Ç

9

ð  ½:¯ @M Æ¥Že¯ ý ½l¯ µŽ



ð

¯

ï

ì

¥¨¦ h¨Œ ¬ on m corresponds to a completely positive map which is finitely sup-

ported, as a function on h , as in the statement of the lemma. But the convex hull

of the vector states associated to a faithful representation of a ¥©¦ -algebra is always

weak ¦ -dense in the set of all states (this is a version of the Hahn-Banach theorem).

ì

¦

¥ h¨ «N It follows that the set of those completely positive maps from into m

which are finitely supported as functions on h is dense, in the topology of point-

ì h¨

wise norm-convergence, in the set of all completely positive maps from ¥V¦ into

«M ¬!

m . By approximating the maps in the compositions (2) we obtain com-

¡

ã

ì

h¨ ¹ h¨x pletely positive maps from ¥§¦ into with the required properties.

Proof (Proof of Theorem 5.7). According to Corollary 5.2 there exists a sequence of

¡

È

ã

ì

h¨>)^¹ h¨x

unital completely positive maps !GCÝ¥¨¦ which converge pointwise

¡

ã

ì

h¨ ¹ h¨Œ in norm to the natural inclusion of ¥§¦ into and which are individually

finitely supported as functions on h . Define a sequence of functions

248 Nigel Higson and Erik Guentner

I

Z Ceh h®)¿« !

by

¡ ¡ ¡

È

Ç

¡ ¢9 ¡ 9

¡

! } } S }  Z\! } } M)ƌ } 

t

Z h

The functions ! are positive-definite kernels on the set , in the sense of Defini-

 

9

ýݯ8Z }€¯ } xý 'R ! tion 4.14. (To prove the inequality ‘ write the sum as a matrix

product

°ž± °ž±

¡ ¡

È È

! } } Œý ! } }mlxýAm ý ý } tštSt

. . . .

} }m

ž`­®¯ ­®¯ ² . .. . ² .

tStSt . . .

¡ ¡

È È

›

}

m

ý } } xý ý } } Œý

m ! m ! m m

m m tštSt

and apply the definition of complete positivity.) The functions Z>! converge pointwise

‹

ë

M¢„h ‹ R

to K , and moreover for every finite subset and every there is some

$`_

such that ‹

¡

¡ = :9 ¡ =

t

ë ~

} } $<  Z\! } }  K ‹

g and

t

°

gW$_ 

In addition, for every there exists a finite subset h such that

¡

9

¡ ¡

} >R } $<´  Z } }

!

t



~

K Z š !

It follows that for a suitable subsequence the series ‘ is pointwise con-

I

~

h h K Z  vergent everywhere on . But each function ! is a negative type kernel, and therefore so is the sum. The map into affine Euclidean space which is associated to the sum is a uniform embedding.

Remark 5.6. This proof is obviously very similar to that of Proposition 4.9. In fact,

according to Remark 4.7 the above argument shows that if a countable group h is

’ h exact then acts amenably on its Stone-Cech compactification h [28, 29, 15]. As

a result: if a countable group h is exact then the Baum-Connes assembly map

4

¤ ¤f

[Hõ

ó

ì

9 9

ì

¦

h Ûbx C h Ûb>) ¥

is injective, for every Û .

References

1. John Frank Adams. Infinite loop spaces. Princeton University Press, Princeton, N.J., 1978.

2. C. Anantharaman-Delaroche. Amenability and exactness for dynamical systems and their ë a -algebras. Preprint, 2000. 3. C. Anantharaman-Delaroche and J. Renault. Amenable groupoids. L’Enseignement Mathematique,´ Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. Group C*-Algebras and K-theory 249

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