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Doctoral Thesis

Künneth formula for Bredon homology and group C*-algebras

Author(s): Leonardi, Fausta

Publication Date: 2006

Permanent Link: https://doi.org/10.3929/ethz-a-005208995

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Doctoral Thesis ETH No. 16478

Künneth Formula for Bredon Homology and Group

C*-Algebras

A dissertation submitted to the

Swiss Federal Institute of Technology Zürich

(ETH Zürich)

for the degree of

Doctor of Mathematics

presented by

Fausta Leonard!

Dipl. Math. ETH

born March 23, 1977

citizen of Bedretto (TI)

accepted on the recommendation of

Prof. Dr. Guido Mislin, examiner

Prof. Dr. Max-Albert Knus, co-examiner

Prof. Dr. Jacques Thévenaz, co-examiner

Zürich, February 2, 2006

Acknowledgments

It is a great pleasure to thank my supervisor Prof. Guido Mislin for offering me the op¬

portunity to explore this fascinating subject and for his guidance throughout my doctoral

studies. T appreciate his extraordinary competences, his constant and generous support and

his human qualities.I have profited from fruitful discussions with many people at several conferences and sem¬

inars. In particular, a special thank goes to the professors and colleagues of the assistance

group 1 at ETHZ, also for being good office neighbours. Furthermore, I thank Prof. Max-

Albert Knus and Prof. Jacques Thévenaz for accepting to be co-examiners.

A big hug to my friends for the wonderful moments spent playing music, talking, laughing,

feeling.

Lovely thank to my family for always believe in me and encourage my choices.

Live as if you were to die tomorrow. Learn as if you were to live forever.

(Mahatma Gandhi)

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Abstract

The aim of this thesis is to prove a Künneth formula for cquivariant if-homology theory.

Thanks to the Atiyah-Hirzcbruch spectral sequence, in case of rational coefficients the prob¬

lem reduces to finding a Künneth formula for Bredon homology with coefficients in the

complex representation functor. So as a first result we prove a Künneth formula and a

universal coefficient theorem for the Bredon homology with those particular coefficients. Af¬

terwards we prove a Künneth formula for the cquivariant À'-homology, under the assumption

that one of the involved spaces admits a geometric realisation. We give some examples for

which this condition is satisfied, namely in the context of Moore spaces and free group ac¬

tions. We consider the already existing Künneth formula for À'-theory of C*-algebras. For

the particular case of group C*-algebras this allows to compare the validity of a Künneth

formula in the right hand side of the Baum-Connes conjecture - K-thcory of a reduced group

C*-algcbra - with that in the left hand side - equivariant A'-homology of a classifying space -

and "to mix" the conditions of validity of the Künneth formula depending on the context one

is dealing with. Thanks to this last result, we state some good conditions under which the

invariance of the Baum-Connes conjecture with respect to finite direct product is satisfied.

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Riassunto

Scopo di questa tesi è dimostrare una formula di Künneth per la A^-omologia equivariante.

Grazie alla sequenza spettrale di Atiyah-Hirzebruch, nel caso di cocfficienti razionali il prob-

lema si riduce a mostrare una formula di Künneth per l'omologia di Bredon a coefficienti

nel funtore di rappresentazioni complesse. Come primo risultato dimostriamo quindi una

formula di Künneth per 1'omologia di Bredon con questi particolari coefficienti. In seguito.

dimostriamo una formula di Künneth per la A-omologia equivariante assumcndo l'esistenza

di una realizzazione gcometrica per uno dci due spazi coinvolti. Illustriamo succcssivamentc

alcuni casi in cui questa condizione viene soddisfatta, ad esempio nel contesto degli spazi di

Moore e azioni libère di gruppo. Consideriamo la formulazione già nota di una formula di

Künneth per la A'-teoria di C*-algebre. Nel caso di C*-algebre di gruppo, cio perinette di

confrontare il lato destro della congettura di Baum-Connes - ossia A'-teoria di C*-algebrc

ridotte - con il suo lato sinistro - ossia A^-omologia equivariante di uno spazio classificantc -

e di combinare lc condizioni a seconda del contesto di lavoro. Grazie a qucst'ultimo risultato

siamo anche in grado di dare buone condizioni che garantiscono l'invarianza della congettura

di Baum-Connes rispetto a prodotti diretti finiti.

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Contents

Acknowledgments i

Abstract iii

Riassunto v

Introduction 1

1 Bredon (Co-)Homology Groups and Künneth Formula 11

1.1 Basic Definitions 11

1.2 Bredon (Co-)Homology Groups 13

1.3 The Künneth Formula in the Bredon Homology Theory 17

1.4 The Universal Coefficient Theorem in the Bredon Homology Theory 24

2 Ä-theory of C*-Algebras 27

2.1 Basic Definitions on C*-Algebras 27

2.2 K0 of C*-Algebras 31

2.3 Higher A-Groups of C*-Algebras 34

2.4 Relations between A0 and Ky 35

2.5 Kasparov's Equivariant ATA^-Groups 39

3 The Künneth Formula for the A'-Theory of C*-Algebras 45

3.1 Tensor Product of C*-Algebras 45

3.2 Nuclear and of Type 1 C*-Algebras 48

3.3 The Bootstrap Class B 51

3.4 The Künneth Formula for the A'-Theory of C*-Algebras 52

3.5 The Universal Coefficient Theorem for the A'-Theory of C*-Algcbras 55

4 Equivariant A-Homology Theory 59

4.1 Equivariant (Co)-Homology Theories 59

4.2 Equivariant Kasparov ATA-Homology Theory 61

4.3 Categorical Definition of the Equivariant A-Homology Theory 63

vii

4.4 The Künneth Formula 67

4.5 Universal Coefficient Theorem for the G-Equivariant A'-Homology Theory .75

4.5.1 AT-Theory of Moore Spaces 75

4.5.2 Künneth Formula in the G-Equivariant A'-Homology Theory with Co¬

efficients in 'L/nli 77

4.5.3 Universal Q-Coefficient Theorem in the G-Equivariant A"-Homology

Theory 78

4.6 Free Action of Groups and Künneth Formula 79

5 The Baum-Connes Conjecture 81

5.1 The Index Map 81

5.1.1 The Index Map: Approach à la Bauin-Connes-IIigson 81

5.1.2 The Index Map: Approach à la Kasparov 82

5.2 The Baum-Connes Conjecture 82

5.3 Status of the Conjecture and Consequences 82

5.4 The Künneth Formula for the Classifying Space for proper Actions 83

5.5 Stability of the Baum-Connes Conjecture 85

Bibliography 87

Index 90

Curriculum Vitae 91

Introduction

The aim of this thesis is to study the validity of the Künneth formula for various homology

theories, defined on various categories, like the category of topological spaces, C*-algebras,

or G-CW-complcxes.A homology theory h on a (sub-)category C of topological spaces is a collection of co-

variant functors hn,n G Z, to the category Ab of abelian groups (or modules), together with

natural transformations dn (connecting homomorphisms) and satisfying axioms of additivity,

homotopy invariance, exact sequence and excision. In this context, a Künneth type formula

takes the following form:

Definition (Künneth Formula)Let h = (hn)n& be a homology theory with hn : C —» Ab. We say that h satisfies a Künneth

type formula if for all n e Z and all X, Y G Obj(C) there is a natural exact sequence

0^ ]T hm(X)®hq{Y)^hn{XxY)~> Yl Tor(hm(X),h(l{Y))-*Q.m+q=n m+q=n-l

The Baum-Connes conjecture is one of the most fascinating ways to link two different

fields of mathematics. In fact, the conjecture identifies via the so called assembly or index

map two different objects, one of analytic type, and the other of gcornetrical-topological

type, associated to a group. The Baum-Connes conjecture was first set forth in a 1982

preprint [BaCo] of P. Baum and A. Connes, which was published only in 2000, 18 years

later. A current description was given in the paper [BaCoHi] published in 1994 together

with N. Higson. In those years the conjecture has been proved for many classes of groups,

and counterexamples arc still not known. More precisely,

Conjecture (Baum-Connes)Let G be a (discrete) group. Then the index map

tf : Kf(eg) - k(c;g)

is an isomorphism, i = 0,1.

The left hand side of the conjecture, or geometrical-topological side K^(EG), i = 0,1,

involves the G-equivariant AT-homology of the classifying space EG for proper actions. The

2 Introduction

right hand side Aj(G*G), i = 0,1, of analytic type, is the A'-theory of the reduced C*-algebra

C*G. The definition of the assembly map $ goes back to G. Kasparov.

We will be interested in comparing the Künneth formula for the left-hand side of the

Baum-Connes Conjecture, involving equivariant A'-homology, with the right-hand side, in¬

volving a Künneth formula for G*-algcbras.

First we approach the study of the Künneth Formula for Bredon homology theory. As we

will see, this turns out to be related to the Künneth Formula for cquivariant A'-homology.

Bredon homology groups were introduced by Bredon in [Br] in the case of finite groups

and afterwards generalized to the case of infinite groups. For a (discrete) group G and

a family T of subgroups of G, we denote by v?(G) the orbit category, whose objects are

left coset spaces G/H with H G J7, and morphism sets mor(G/H, G/K) are the G-maps

G/H -» G/K, H, K G T. We write G-Mod?, resp. Mod^-G for the category of covariant,

resp. contravariant functors from vj?(G) to the category Ab of abelian groups, and natural

transformations as morphisms. For N G G - Mod^ the Bredon homology groups of G with

coefficients in N are defined as

H?(G;N):=Tori{Z,N),

for i G N. Here Tor,-(-, N) is the z-th derived functor of the categorical tensor product functor

- <g>r N : Mud? -G-* Ab,

and Z denotes the constant cofunctor Z(G/H) = Z, for fief. For a G-CW-complex X

we consider its cellular chain complex (C*(X), d*). For i G Z we get a contravariant functor

C_i(X) from the orbit category vj?(G) to the category Ab of abelian groups by setting

Ci(X)(G/H) = Cl(XH),

where Xn denotes the subspace of iJ-fixed points of X, II G T. The Bredon homology

group of X with coefficients in N G G — Mod^ are defined by

H?(X;N) = IIi(Q,(X)®rN),

for i G N, for T containing the family of isotropy subgroups of the G action on X. For

the study of the Künneth Formula we focus our interest in Bredon homology groups with

particular functors as coefficients. Namely, the functor Rfë which associate to a finite sub¬

group H of G the underlying group of the complex representation ring R^(H). Using several

properties we prove a Künneth Formula for the Bredon homology of groups.

Theorem (Künneth Formula)Let G, H be groups and Tin, TinG, Tina the families of all finite subgroups of G x H,G and

Introduction 3

H respectively. Then for every nçN there is a natural exact sequence:

°^ E H^{G;R^)®H^(H-,Ri)^Ilfr{GxH-,R^H)^iri-\-q—n

- Y, Tor(H^(G-R£),II^(H-R»))^V.m+q—'i> — \

More generally we prove the Formula for the Bredon homology of group-CW-complexes.

Similar results have recently also been independently obtained by Ruben Jose Sanchez-Garcia

(private communication).

Theorem (Künneth Formula)Let G, H be groups, X a G-CW-complex and Y a H-CW-com,plex. Let Tin, TinG,TinH be

the families of all finite subgroups of G x H,G and H respectively. Then for every n G N

there is a, natural exact sequence:

0^ Y, H^(X;R^)®II^(Y;RIé)^H^(XxY;R^H)^m-\-q—H

- J2 Tor(n?nin«(X;R£),H^(Y-,R»))^0.m-\-q=n—l

We consider also Bredon homology groups with extended coefficients in a field k and we

prove a universal coefficient theorem.

Afterwards we study the A'-theory of G*-algcbras. The introduction of algebraic K-

theory in the study of G*-algebras consists in the association of two abelian groups KÜ(A)and Ki(A) to each G*-algebra A, which reflect some properties of A. Our purpose is to

state a general Künneth Formula for the A-theory of G*-algebras referring to the works of

J. Rosenberg and C. Schochct in [RoSc], based on earlier works of L. Brown [Bro], and to

better understand the right hand side of the Baum-Connes Conjecture. We recall that a

G*-algebra A is a Banach algebra (A, || ||) together with an involution * which satisfy the

C*-condition

\\a a\\ = \\a\\ ,

for all a G A. An example is given by the reduced G*-algebra C*G of a group G, which

plays a central role in the study of the Baum-Connes Conjecture. Technical unitalization

problems apart, for a G*-algebra A we define Kq(A) as the Grothendieck group associated

to the semigroup of equivalence classes of idempotents in M^A) and Ki(A) as the group

^{Gl^A)) of connected components of G/00(A). With a categorical point of view, A0 and

A'i are covariant functors from the category of G*-algebras to the category of abelian groups.

Higher A'-groups can also be defined in an analogous way. Like the case of A'i, the definition

of higher AT-groups involves topological properties of the G*-algebras. However, it is very

4 Introduction

interesting to notice that Ko and A^ are strictly related. Denoting SA the suspension of a G*-

algebra A, there are in fact natural isomorphisms Ky(A) = Kq(SA) and Kq(A) = K\_(SA),which carry over to the Bott periodicity

K„{A) <* Kn+2(A),

for all n > 0, making this A'-theory into a Z/2Z-graded A'-theory. At this point we can

extend the definition of A"-groups to negative degrees simply by Bott periodicity. Weak

exactness and long exact sequence arc also properties of the A-theory of G*-algebras.

A generalization of the A'-theory of G*-algebras consists in the definition of an equivariant

AA-theory for pairs of G-G*-algebras. This is due to the work of G. Kasparov published in

the papers [Ka], [Kal] and [Ka2]. For a (discrete) group G, a G-G*-algebra is a G*-algebra

A endowed with an action of G by ^-automorphisms. Hilbert modules are a generalization

of Hilbert spaces, having an inner product with values in a G*-algebra A. Given a pair

of G-G*-algebras (A, B), a Kasparov triple over (A, D) is a triple (£,n,T), where £ is a

Hubert G*-module over B which admits a unitary representation U of G, -w : A —» CB{£) is

a covariant *-homomorphism, and T is a selfadjoint operator in £#(£). Here Cb{£) denotes

the G*-algebra of i?-linear and continuous operator on £. Moreover, we require for T some

generalized Fredholm properties. For a pair (A, B) of G-G*-algebras Kasparov AA'-groups

KKf(A, B) are defined as the set of equivalence classes of Kasparov triples over (A, B),

even in the case i = 0 and odd for i = 1. The equivalence between Kasparov triples is given

by the conjugation via some G-equivariant isometrical isomorphism. The functors A'A^Gare covariant in the second and contravariant in the first variable. The connection with the

earlier defined A-theory of G*-algebras is given by

KK\e\C,B)^Kl{B),

for any G*-algebra B, i G {0,1}.If H and K are Banach spaces, there arc in general many possible norms on the algebraic

tensor product H ® K which are related in a suitable manner to the norms of H and A'

and which are not necessarily complete. In particular, the tensor product of two infinite

dimensional normed spaces is never complete. In the case of G*-algebras, there are many

possible choices for a G*-norm on their algebraic tensor product, each of which carries out to

different G*-completions. The Gelfand-Naimark-Segal construction allows for instance the

construction of the minimal (or spatial) G*-norm. An other example is the maximal C*-

norm, which is simply the supremum taken over all G*-scminorms, and every other G*-norm

on the tensor product will lie in fact between the minimal and the maximal G*-norm. For

C*-algebras A and B we denote by A®1B the G*-complction of the algebraic tensor product

A® B with respect to the C*-norm 7.

In this context, an important role is played by the class of nuclear C*-algebras. A C*-

algebra A is called nuclear if for every C*-algcbra B the algebraic tensor product A® B

admits a unique G*-norm. This class is pretty big, for instance all finite dimensional and

Introduction 5

all commutative G*-algebras are nuclear, while the reduced G*-algcbra C*G of a group G is

nuclear if and only if G is amenable. The property that the tensor product functor - ®mln D

is an exact functor if D is a nuclear G*-algebra plays a central role in the proof of the Künneth

Formula for A'-theory of G*-algcbras. In order to prove the Künneth Formula we use also

the fact that a (separable) G*-algcbra admits a geometric realisation for S2B ®w;m JC(II),

where K(H) denotes the G*-algebra of compact operators on a Hilbert space II. Following

the formulations given in [Bl], we refine the class of nuclear G*-algebras by considering the

smallest class of nuclear G*-algebras containing the commutative G*-algebras and closed

under A'A'-equfvalencc, the so called Bootstrap class B, and we get the following classical

Theorem.

Theorem (Künneth Formula)Let A and B be C*-algebras, with Ae B. Then for every i G {0,1} there is a natural short

exact sequence

0 - (K*(A) ® MB))t - KM ®min B) -+ Tor(K,(A), ^*(#))/-i - 0.

G. Skandalis illustrated in [ChEcOy] that there are G*-algebras which do not satisfy the

Künneth Formula.

The last part of the study of G*-algebras is dedicated to the study of a Universal Coefficient

Theorem. For a G*-algebra A, the A'-theory of A with coefficients in Q, resp. in Z/nZ is

defined by

Ki{A; Q) = Ki{A ®min C), resp. Ki(A; Z/nZ) = K{A ®min Dn),

for i {0, l}where the G*-algebras C and Dn constructed by Cunz are contained in the

Bootstrap class B and have the property that K0(C) = Q and KL(C) = 0, resp. K()(Dn) ^

Z/nZ and KX(DV) = 0. The Universal Coefficient Theorem states the following.

Theorem (Universal Coefficient Theorem)Lei A be a C*-algebra, then for i G {0,1} the following assertions are true:

i) There is an isomorphism of groups Ki(A;Q) = Ki(A) ®z Q-

n) The sequence 0 -> Ki(A)®Zjn% - Ki(A;Z/nZ) - Tor{K-i(A),Z/nZ) -> 0 is an

exact sequence of abelian groups.

We prove the following application of the previous Theorem.

Theorem

Let A,B be C*-algebras and (p : A -* B be a *-homomorphism. The induced morphism

ip« : K*{A) -* K*iB) is an isomorphism if and only if <p(-,Q) : K«(A',Q) ^ A\.(£;Q) and

for dip prime <p{-;Z/pZ) : A'*(/l;Z/pZ) -> K*(B;Z/pZ) are isomorphisms.

6 Introduction

In the following chapter we recall an important Z/2Z-graded equivariant homology the¬

ory, namely for a proper G-CW-complex X we give the definition of the G-equivariant

Kasparov homology groups KKq(X) and ATA'f (X). The cycles were defined by M. Atiyah

in [Atl] and their equivalence relation were studied by G. Kasparov in [Ka]. We note that

for a G-compact proper G-CW-complex X, the definition of those groups agrees with the

more general one in the context of C*-algebras given in the previous chapter, in the sense

that

AA7G(AO = ArAf(Go(X),C),

for i G {0,1}, where C0(X) denotes the G*-algebra of continuous complex-valued functions

on X vanishing at infinity. More generally, for a (not necessarily G-compact) proper G-

CW-complex X, the i - th G-equivariant Kasparov homology group is defined by taking the

colimit on all G-compact proper G-subcomplcxes, in other words

RKf{X) = colimYcX

KK?{Y),

Y G-compact

for i G {0,1}. To generalise the definition to arbitrary (that is, not necessarily proper) G-

CW-complexes, we use the definition of the equivariant A'-homology-theory K^'(-) proposed

by J. Davis and W. Lück in [DaLu], This definition involves concepts of the theory of

spectra. For a (discrete) group G and the orbit category u(G) — uAu(G) of G wc consider

the j>(G)-il-spectrum

IgP = Eff(G/'0 : v(G) -> S2-SP, G/H » K^(G///)

constructed in [DaLu]. The A-homology groups of X are defined as

Af (X) := tt,(X; ®„(G) £§"(<?/?)),

where ieZ. In the case of a proper G-CW-complex X, the definition agrees with the one

for Kasparov ATA'-homology groups, that is

Af(A) - f?Af(X),

for i G {0,1}. For the rest of the paper we forget therefore the A'A' and RK notations and

use the unique notation A'f (•) for all G-CW-complexes.

The connection with Bredon homology is given by the "Atiyah-Hirzebruch type" spectral

sequence. For instance for a proper G-CW-complex X there exists a strongly convergent

spectral sequence of the form

Elq = Hpn-(X;K£(G/7)) * K^(X).

The main point is the study of a Künneth Formula of the following type.

Introduction 7

Conjecture (Künneth Formula)Let G, H be groups. Let X be a G-CW-complex and Y a H-CW-complex. Then there exists

a natural short exact sequence

0 - (Af(X) <g> K?(Y))t - AfxH(X xF)-

- (Tor(Af(X), Af (y))),_! -> 0,

/or a/H G {0,1}.

Note that here we do not require X or Y to be proper G- (resp. if-) CW-complexes.

First we prove some special cases of the Künneth Formula, i.e. with vanishing Tor-term.

The first case requires both CW-pairs (X, A) and (Y, B) to be proper.

PropositionLet G,H be groups. Let (X,A) be a proper G-CW-pair such that for every n G {0,1}the groups K^(X,A) are torsionfree. Then for every proper Il-CW-pair (Y,B) and all

n {0,1}, the abelian groups K%xH((X,A) x (Y, B)) and {KC:{X,A) ® K?(Y,B))„ are

naturally isomorphic.

The generalization to the statement requiring just one CW-pair to be proper is not

difficult. If we want to drop the condition to be proper for both CW-pairs (A, A) and

(Y, B), the statement holds if we require some conditions on the reduced G*-algebra of the

involved groups and we refer to the Künneth Formula for C*-algcbras. The statement is the

following.

PropositionLet G,H be groups. Let (X,A) be a G-CW-pair such that for every n G {0,1} the groups

K^(X,A) are torsionfree. Then for every H-CW-pair (Y, B) such that for any isotropy

subgroup L of the H-action on Y, C*L G B and Kn(C*L) are torsionfree, n G {0,1}, the

abelian groups K%xH ((X, A) x (Y, B)) and ( Af(X, A)®K?(Y, B))n are naturally isomorphic,

n G {0,1}.

The next Theorem is the formulation of a Künneth Formula and consists in a reduction

to the torsionfree case.

Theorem (Künneth Formula)Let G,H be groups, X a proper G-CW-complex and Y a proper H-CW-complex. Suppose

there exists a G-CW-complex Z with K*(Z) torsionfree and a morph-ism Z — X such that

the induced morphismji : Kf{Z) —> Af(X) is surjective, i G {0,1}. The following sequence

is exact

0 - (A-f (X) ® K?(Y))t - AfxH(X xY)^

^(Tor(Af(A-),Af(y)))2_i^0,

for i G {0,1}.

8 Introduction

Like for the torsionfree case wc have a similar statement requiring Y proper and X not

necessarily proper. Again if we want to skip the request of properness to both CW-complexes

X and Y we have to require conditions on the reduced G*-algebra and refer to the Künneth

Formula for G*-algebras. The statement is the following.

Theorem (Künneth Formula)Let G,II be groups, X a G-CW-complex and Y a H-CW-complex. Suppose there exists a

G-CW-complex Z with K^(Z) torsionfree and a morphism Z A X such that the induced

morphism % Kf(Z) — Af(AT) is surjective, i G {0,1}. If for all subgroups L of H which

are isotropy groups for the H-action on Y the reduced C* -algebras C*L G B and the groups

KTI(C*L) are torsionfree, n G {0,1}, the following sequence is exact

0 -> (Af (X) ® K?(Y))i - K^H(X xY)^

^(Tor(Af(X),Af(y)))^i^0,

fori G {0,1}.

Differently than in the case of G*-algebras, where it is known that for any (separable)

G*-algebra B there exist a (separable) G*-algebra F with free A'-theory, a Hilbert space

II and a homomorphism </> : F — SB ®min K(H) such that 0* : K*(F) —> K*(SB) is

surjective, the existence of a geometric realisation for group CW-complexes still remains an

open question. We reserve the last part of the chapter to the study of some interesting

examples of CW-complexes admitting a geometric realisation.

The n-th Moore space is defined by the mapping cone Mn :— 51 Une2 over the map n : S1 —>

S1 given by the n-power map, where e2 is the two dimensional cell. As the A-homology of

Mn is given by„,.,, / Z

,if i = 0

A'<M"> =

J Z/nZ ,if i = 1,

the inclusion S1 <—> Mn induces surjections K^S1) —> K-L(Mn), where AfS1) are torsionfree,

i = 0.1, that is Mn admits the desired geometric realisation. Defining

Kf(X;Z/nZ) - K?+\ie}{X x (M„,*)),

we state the following Universal Coefficient Theorem.

Proposition (Universal Coefficient Theorem for the G-equivariant A-homology Theory)

Let G be a group and X be a proper G-CW-complex. For every i G {0,1} there is a natural

short exact sequence:

0 -> Af{X) ® Z/nZ -* Kf{X;Z/nZ) -> Tor{K?_v{X),Z/nZ) -» 0.

For the characteristic zero case the Universal Coefficient Theorem is stated thanks to the

Formula in the Bredon homology via the "Atiyah-Hirzcbruch type" spectral sequence. The

result is the following.

Introduction 9

Theorem (Universal Coefficient Theorem)Let G, H be groups, X a proper G-CW-complex and Y a proper H-CW-complex. Then there

is a natural isomorphism

[K?(X;Q) ®Q K'JiY-Mi * K?xU(X x Y;Q),

fori G {0,1}.

The aim of the next example is to study a Künneth Formula for CW-complexes on which

a group action is free. For a finite CW-complex W, the /-th suspension SlW admits the

following geometric realisation, studied by P.E. Conner and L. Smith in [CoSm].

Lemma

Let W be a finite CW-complex. Then there exists a CW-complex V and a morphism V —>

S'W,l G N>i, such that K*(V) is torsionfree and the induced morphism j-L : KX(V) —>

Ki(SlW) is surjective, i G {0,1}.

Using the tricky fact that for a ü-CW-complcx Y on which H acts freely holds true the

isomorphism

Af(Y) = K.ÇY/H),

wc prove the following Künneth Formula for the free action.

Proposition (Künneth Formula for free Actions)Let G, H be groups, X a proper G-CW-complex and Y a proper H-CW-complex, on which

II acts freely. For i G {0,1}, there is a natural short exact sequence

0 -> (Af(X) ® K?{Y))t - K?*U(X xY)^

^(Tor(A-f(X),Af(y))U^0.

The last Chapter is devoted to the study of the Baum-Connes Conjecture. In particular

we want to give an overview of how the assembly map is defined and see the progress made

in the proof of the Conjecture itself and the consequent validity of some related conjectures.

The main feature of this thesis is to compare the validity of the Künneth Formula in the left

hand side of the Baum-Connes Conjecture with that in the right hand side and "to mix"

the conditions depending on the context one is dealing with. For instance we approach the

Künneth Formula given in Chapter 3 by stating a Künneth Formula for the classifying space

for proper actions via the assembly map.

Theorem (Künneth Formula)Let G, H be groups. Suppose that C*G G B and that G, H and G x H satisfy the Baum-

Connes Conjecture. Then the following sequence is exact for all i G {0,1}.'

0 -> (Af (AG) ® Af (EH))t -> KTH(EG x EH) ->

- (Tor(Af (EG), Af (Fi/)));_i - 0.

10 Introduction

Analogously, wc can state a Künneth Formula for the group G*-algebras via the assem¬

bly isomorphism by assuming conditions on the classifying space for proper actions. The

following is an example.

Theorem (Künneth Formula)Let G, H be groups such that G, H and GxH satisfy the Baum-Connes Conjecture. Suppose

that there exist a G-space Z with torsionfree K-theory and a morphism Z —» EG such that

the induced morphism, jl : Af(Z) —> Af(AG) is surjective, fori G {0,1}. Then the following

sequence is exact:

o - (a;(g;g) ® #.(<?;#))< - Afc;g ®miv c;h) -+ ror(Afg;g), k^c;h))^ -+ o,

for alii G {0,1}.

It is already known that the Baum-Connes Conjecture is compatible with direct limit of

groups, whereas invariance of the Conjecture with respect to finite direct product or taking

subgroups of finite index still remain open questions. Work of H. Oyono-Oyono, Chabcrt

and Echteroff shows that a certain stronger version of the Baum-Connes Conjecture (the

Baum-Connes Conjecture with Coefficients in a G*-algebra) is stable with respect to free or

amalgamated products and direct and some semidirect products. In this thesis we give some

good conditions under which the invariance of the Baum-Connes Conjecture with respect to

finite direct product is satisfied. The approach is to take groups whose classifying spaces for

proper actions satisfy the Künneth Formula.

Theorem

Let G,H be groups satisfying the Baum-Connes Conjecture. Suppose that C*G G B and

Ki(C*G) is torsionfree for i G {0,1}. Then GxH also satisfies the Baum-Connes Conjec¬

ture.

We illustrate the consequence for some examples.

• If G is a finite group and H a group which satisfies the Baum-Connes Conjecture, then

also the product G x H satisfies the Baum-Connes Conjecture.

• If G is a finitely generated abelian group and H a group which satisfies the Baum-

Connes Conjecture, then also the product GxH satisfies the Baum-Connes Conjecture.

• If G is an elementary amenable group with At(G*G) torsionfree for i G {0,1} and H

a group which satisfies the Baum-Connes Conjecture, then also the product GxH

satisfies the Baum-Connes Conjecture.

Chapter 1

Bredon (Co-)Homology Groups and

Künneth Formula

1.1 Basic Definitions

Most topological spaces we consider in our work will be either locally compact or CW-

complexes.

Definition 1.1.1 (G-space). Let G be a (discrete) group with identity e. A G-space is a

topological space X equipped with a (continuous) left G-action

G x X -> X, (g,x) ^ gx

satisfying ex — x and g(hx) — (gh)x, for all g, h G G.

Example 1.1.2 (cartesian product). The cartesian product X x Y of two G-spaces is a

G-space via the diagonal G-action (g, (x,y)) i—> (gx,gy).

Definition 1.1.3 (G-pair). Let G be a group. A pair of G-spaces, or G-pair, is a pair (X, A)

of G-spaces such that A C X.

Definition 1.1.4 (product of pairs). Let G, H be groups. Let (X, A) be a G-pair and (Y, B)

a H-pair. The product of (X, A) with (Y, B) is defined as the G x H-pair (X, A) x (Y, B) :=

[X xY,X xBliAxY).

The following definition generalizes the concept of a CW-complex to G-spaces.

Definition 1.1.5 (G-CW-complex). A G-CW-complex consists of a G-space X together with

a filtration X° C X1 C X2 C • • C X by G-subspaces such thai the following axioms hold:

(1) Each Xn is closed in X.

12 Bredon (Co-)Homology Groups and Künneth Formula

(2)[jnmX^X.

(3) X° is a discrete subspace of X.

(4) For each n > 1 there is a discrete G-space Afl together with G-maps f : /Sn_1 x A.„ —

Xn~l and f : Bn x An —> Xn such that the following diagram is a push-out diagram:

S""1 x An —^ X*1-1

Ba x An -^-^ Xn,

where Sn~l and Bn denote respectively the standard unit sphere and unit ball in Eu¬

clidean n-space.

(5) A subspace Y of X is closed if and only ifYd Xn is closed for each n > 0.

Definition 1.1.6 (G-CW-pair). Let G be a group. A G-CW-pair is a pair (X, A) ofG-CW-

complexes such that A C X.

Definition 1.1.7 (induced space). Let G and H be groups, X a H-CW-complex and ip :

H —» G a group homomorphism. We denote by indv(X) the space G x X modulo the

equivalence relation given by (g, x) ~ (g(p(h), h^x), for all g G G, h G H,x G X.

Notation

We denote by [g,x] the equivalence class of (g,x). The space indip(X) is a G-CW-complex

for the action on the "first factor", i.e. 7• [g, x] := [yg, x] for 7, g G G, x G X.

For a if-CW-pair (X,A), we denote (ind^X), ind^A)) simply by indv(X,A).When <p is an inclusion, we also write ind^(X) and G xH X for indip(X) and similarly for

a pair.

Definition 1.1.8 (proper G-space). A G-space X is called proper if there are finite subgroups

Hi < G and open Ht-invariant subspaces Xl C X such that the natural G-maps G xIit Xi —>

X are G-homeomorphisms onto their images, and

X^Gx^Xl1

Notation

For an abelian category C, we denote by CX the category of chain complexes over C.

Definition 1.1.9 (chain homotopy equivalence). Let (C, d), (C, &) be chain complexes in

an abelian category C. Two chain morphisms f,g : C —> C' in the category GJZ are called

chain homotopic if for every n there exists a morphism s,n : Cn —> C'n+l such that

(^n+lSn 1 Sn_iOn = Jn— gn-

1.2 Bredon (Co-)Homology Groups 13

The complexes (C, d) and (C, 0') are called chain homotopy equivalent if there are chain

morphisms f : C — C' and g : C' —> C such that f o g is chain homotopic to idc and go f

is chain homotopic to idc, and will be denoted by (C, d) ~ (C, d'), or simply by C ~ C'.

Notation

Let C\,C2 be categories and F, G : C\ — C2 functors. If v : F —> G is a natural transformation

between F and G, we denote by vc the morphism from F(c) to G(c), for every object c G C]_.

Remark 1.1.10. We will be considering the case where C is the abelian category Horn(C\_, C2)

of additive functors from Ci to C2 for two abelian categories Ci,C2- In this case, a chain

complex (F,d) in the category Hom(Ci,C2) is a sequence of functors Fi : d -> C2 together

with a sequence of natural transformations di : F_ —> I)-i which are required to satisfy the

condition

<9i_i^ = 0,

for alii. Two chain complexes (F,d),(G,d') G C*Ii"ora(Ci,C2) are called (naturally) chain

homotopy equivalent if for every i there exist natural transformations Vi : Ft —> G,;,'0i : G'i —»

Fi, st : Fi —> Fi+i and s[ : G,: —> Gi+\ such that for every c G C-,

i&i+iWi)* + «-M&ùc. - (idG.X - (i/, o A)c

and

{di+i)c(Si)c + (Si-i)c(di)c = (idFi)c - (i>i o Vi)c.

Two naturally chain homotopy equivalent chain complexes (F,d),(G,D') will be denoted by

(F, d) ~ (G, &), or simply by F ~G.

1.2 Bredon (Co-)Homology Groups

In this section we define Bredon (co-)homology groups first of groups and afterwards of G-

CW-complexes.

Let G be a group. A family of subgroups of G is a set T of subgroups of G that is

non-empty and stable under taking conjugates in G. Typical notations for families are the

trivial family Tr — {{e}}, the family of finite subgroups Tin, the family of cyclic subgroups

TC, the family of proper subgroups Vr of a non-trivial group and the family of all subgroups

All. Let 5bca family of subgroups of G. We denote by Vp{G) the orbit category, whose

objects are left coset spaces G/K with K G T, and morphism sets mor(G/K,G/L) are the

G-maps G/K — G/L, K, L G T. Let A, L G T and ip : G/K —> G/L be a morphism in

vf{G). Then p> is completely determined by the image (p(K) = gL, where g is an element

of G which satisfies t/-1 Kg C L.

Wc write G — Mod^, resp. Mod? — G for the category of covariant, resp. contravariant

functors from Vj?(G) to the category Ab of abelian groups, and natural transformations as

14 Bredon (Co-)Homology Groups and Künneth Formula

morphisms. In the sequel we will call contravariant functors for short cofunctors. We note

that every abelian group A defines a constant (co-)functor A : v?{G) —> Ab by setting

A(G/II) — A for every II e T and its value on morphisms is the identity homomorphism

of A. The category Mod? — G is abelian (similar remarks for the category G — Mod?).

In particular, a sequence M —> N —> L is exact in Mod? - G, if and only if the sequence

M(G/H) -> N(G/II) -> L(G/H) is exact for every H G T. We want to see now that

every cofunctor M G Mod? - G admits a projective resolution F*(M) -» M. An object

P G Mod? — G is called projective if the functor

mor(P, -) : Mod? - G -> Ab

is exact. The following construction gives rise to projective objects. For K G T we consider

the cofunctor

Pk : v?(G) -* Ab

defined by PK(G/H) = Z\mor{G/H,G/K)\ and on morphisms in the obvious way. We

recall that Z[mor(G/H,G/K)] is the free abelian group with basis mor{G/H,G/K). Let

M G Mod? - G. If / : PK -+Misa morphism, then the evaluation evK(f) is defined by

evaluating f{G/K) at the identity 1 G rnor(G/K,G/K). Moreover, we can also show that

this evaluation map

evK mor(PK,M) -> M{G/K)

is bijective. The proof of this result is in [MiVa]. As a result, mor(PK, -) maps a short

exact sequence 0 —> M -» N — L —> 0 in the category Moiijr - G to a short exact

sequence 0 -> M{G/K) -> N(G/K) -^ L(G/K) -> 0 of abelian groups, showing that PK

is indeed projective. Next, we want to prove that for every M G Mod? - G there is an

epimorphism P -> M with P projective. For K & T, every x G M(G/K) gives rise to a

unique 0(ar) : PK{G/K) -* M{G/K) mapping 1 G mor[G/K, G/K) to x. We can therefore

define a map

<MG/A) : 0 PK(G/K) - M(G/A)

xeM(G/K)

by putting the component of <&K(G/K) corresponding to the index x in M(G/K) to be the

map <fi(x), so that <&K{G/K) is onto. It is not hard to show that we can extend $K(G/K)

to a unique morphism <&K : TJ PK — M. Repeating this construction for every A' G J*7 we

define a morphism $ with components <£>/<-:

*:II( II Pk)^M,A'e^ x<=M((7/A')

which is an epimorphism of a projective object onto M. Every M G Mod?-G admits there¬

fore a projective resolution, which is unique up to chain homotopy. Wc write P*(M) -» M for

a projective resolution of M. For each N G Mod? - G wc have therefore a cochain complex

1.2 Bredon (Co-)Homology Groups 15

of abelian groups mor(P*(M),N) and wc can define derived functors Extl(M,N),i > 0,

which are contravariant in the first variable and covariant in the second, by putting

Ext%M,N) = Hl(mar(P*{M),N)), i > 0.

Bredon cohomology groups of G are defined as follows.

Definition 1.2.1 (Bredon cohomology group of G). Let G be a group, T be a family of

subgroups of G and M G Mod? - G. The Bredon cohomology groups of G with coefficients

in M are defined as

Hr{G- M) = Exé(Z, M),i>0.

If T consists only of the trivial group, by identifying Mod? — G with the category of

(left) G-modules, Bredon cohomology reduces to the usual cohomology. If T contains G,

Z - PG is projective and thus F>(G; M) = 0 for all i > 0, and H%{G; M) = M(G/G). A

short exact sequence 0—> M ^ N — L —> 0 in Mod? - G gives rise to a long exact sequence

of Bredon cohomology groups

//>(G; M) -> #>(G; N) - #>(G; £)->•

induced from the short exact sequence of cochain complexes

0 -> mor(P*(Z), M) -> mor(P*(Z), N) - mor(P*(Z), L) - 0.

In the sequel, for a group G, a G-space X and a subgroup H of G wc denote by XH the

subspace of H-fixed points of X.

Definition 1.2.2 (isotropy group). Let G be a group operating on a set X. For an element

x G X, the isotropy group of x in G is defined as G,x = {g G G\gx = x}. We denote the

family of isotropy groups of X by T(X) — {G,4\x G X).

Definition 1.2.3 (cellular chain complex). Let G be a group, T a family of subgroups

and X a G-CW-complex. The cellular chain complex (C*(X),dA consists of G-modules

Ci{X) = Z[Aj], where Ai is the G-set occurring in the definition of the i-th-skeleton of X,

and di : Ci(X) — Ci-\{X) is the natural boundary homomorphism, for every i G N.

Let G be a group, T a family of subgroups of G and X a G-CW-complex. For subgroups

H, K G T, a morphism *p : G/H —* G/K is determined by the image p(H) = gK, for some

g G G such that g~lHg C K. We obtain a well defined morphism ip* : XK — XH given by

tp*(x) = gx. Indeed, ip* is well defined since Hgx = gx for all x G XK as g~lHg C A', and

(p* is independent on the choice of g. In view of that the following definition makes sense.

Definition 1.2.4. Let G be a group, T a family of subgroups ofG. Let X be a G-CW-complex

and (C*(X),d*) be the cellular chain complex of X. For every i G Z, the contrnvariant

functor Ç_i(X) from, the orbit category u?(G) to the category Ab of abelian groups is defined

by associating to any coset G/H G v?(G) the abelian group Ci{XH). Together with the

obvious natural transformations dt : C^X) —» Gi_1(X), the pair (Q*(X), dA defines a chain

complex in the category Mod? — G.

16 Bredon (Co-)Homology Groups and Künneth Formula

Let G be a group, X a G-CW-complcx, and T a family of subgroups containing the

family T(X) of isotropy groups of X. For II G T, wc have that

a(xH) = z[Af\,

and, as Aj is a disjoint union of orbits G/Ka with A'„ G T(X), we see that Af is the disjoint

union of sets of the form (G/Ka)H = rnor(G/H, G/K„), what shows that

C\(XH) = Z[]lmor(G/H,G/Ka)} * ® PKa(G/H).a n

It follows that the contravariant functor C_M(X) is a sum of contravariant fuctors of the form

Pkbi with Ka G T(X). In particular, Q*(X) is projective for all *, wheter it is considered

as an object in Mod? — G or Mod?(x) — G. Note also that there is a natural augmentation

map Ç_q(X) — Z, defined on an object G/K by mapping each basis element of Gq(Xk) to

1 G Z, and the zero map if XK is empty. The following definition states the generalization

of Bredon cohomology groups to G-CW-complexes.

Definition 1.2.5 (Bredon cohomology group of X). Let G be a group, X a G-CW-complex

and T a family of subgroups of G containing the family T(X) of isotropy groups of X. Let

M G Mod? — G. The Bredon cohomology groups ofX with coefficients in M are defined by

#>(X; M) = Hl(mor(QSX), M)),i> 0.

Good references for the basic properties of Bredon cohomology groups arc [Br] for the

case of finite G, and [MiVa] for general G. Note that the definition of Bredon cohomology

groups is independent of the choice of the family T, as long as it contains the family of

isotropy subgroups.Next we want to describe the concept of Bredon homology groups. Again, we do it first

for groups and afterwards for G-CW-complexes.

Let G be a group, J7 be a family of subgroups of G and N G G — Mod?. The i — th left

derived functor of the categorical tensor product functor

- ®? N : Mod? - G —> Ab

is denoted by Tor%(-, N), i G N. In other words, the groups Tori(M, N) can be computed

as the homology groups of the chain complex P*(M) ®? N, where P*(M) is a projective

resolution of M. By definition, M ®? N is the abelian group

Y M(G/K) ®z N(G/K)/ ~,

A"e^

where the equivalence relation is generated by (f>*m®n ~ m(g><f>*n, with (f) G rnor(G/K, G/L),

0* G mor(M(G/L),M(G/K)), 0* G mor(N(G/K), N(G/L)), m G M(G/L),n G N(G/K),

K, LeT.

Bredon homology groups are defined as follows.

1.3 The Künneth Formula in the Bredon Homology Theory 17

Definition 1.2.6 (Bredon homology group of G). Let G be a group, T be a family of

subgroups of G and N G G - Mod?. The Bredon homology groups of G with coefficients in

N are defined byHf(G;N)=Tori(Z,N),i>Q.

For instance, Il£(G; N) = Z®? N.

Definition 1.2.7 (Bredon homology group of X). Let G be a group, X a G-CW-complex

and T a family of subgroups of G containing the family T(X) of isotropy groups of X. Let

N G G — Mod?. The Bredon homology groups of X with coefficients in N are defined by

üf(X; N) = Hi(C*(X) ®? N), i > 0.

As in the case of Bredon cohomology groups, the definition yields groups independent of

the family T, as long as T(X) C T. Also for Bredon homology, a good reference is [MiVa].

1.3 The Künneth Formula in the Bredon Homology

Theory

The aim of this section is to prove a Künneth formula for the Bredon Homology groups first

of groups and then of group-CW-complexes with coefficients in a particular functor.

Let G be a group and T a family of finite subgroups of G. We denote by R<c(H) the

underlying abelian group of the complex representation ring of a finite group H G T- Let

H, K G T. To any morphism ip : G/H — G/K in the orbit category v?(G), we can associate

an equivalence class of group homomorphisms from H to A modulo the conjugation by

elements in A in the following way. The morphism ip is determined by the image (p(H) = gK,

for some g G G such that g~lHg C A'. Up to conjugation by elements of A' we get a unique

injective group homomorphism H —> K,h i-> g~lhg. Indeed, if <p(H) = gYK = g2K for

gi G G such that g^Hgt C K,i — 1,2, then g2 = gik,k G K. Since in the group AcA'

the conjugation by elements of A corresponds to the identity, we get a well defined induced

morphism (p* : Peu —* PcK of abelian groups. Therefore, the following definition makes

sense.

Definition 1.3.1 (Pc)- Let G be a group and T a family offinite subgroups of G. We write

Re for the covariant functor G/H h- RC(H) from the orbit category v?(G) to the category

Ab of abelian groups.

Remark 1.3.2. If necessary, in order to avoid confusions, we specify the ground group G

writing P£' instead of Pc.

Lemma 1.3.3. Let H\_,H2 be finite groups. There is a natural group isomorphism

Rc(HL x H2) S PC(PX) ® RC(H2).

18 Bredon (Co-)Homology Groups and Künneth Formula

Proof. For a finite group H, the irreducible C-representations form a basis of the free abelian

group Pc(P). Further, by [Se] (Thm. A3.10.zz)), any irreducible representation of Pi x H2 is

isomorphic to a representation p\®p2, with pr is an irreducible representation of Ht, i = 1,2.

It follows then easily that the map

Ac(Pi) ® Pc(#2) - Pc(Pi x H2)

[pi] ® [p-i] ^ [pi ® p-i]

is a natural isomorphism. D

In the next Lemma let Gf denote the singular chain complex functor.

Lemma 1.3.4 (Eilenberg-Zilber Theorem). For any two topological spaces X,Y, there exists

a natural chain map C*(X x7)-» Cl(X) ® C*(Y) which is a chain homotopy equivalence.

Proof. [Ma, Thm. Vl.4.3.1.] D

The previous Lemma implies that if X is a G-CW-complex and Y is an P-CW-complcx,

there is a natural chain homotopy equivalence of cellular chain complexes of Z[G x H]-

modulcs

0 : C*(X x Y) - C*(X) ® C,(Y).

In particular, because (X x Y)KxL = XK x YL, for a subgroup KxLcGxHwe obtain

a chain homotopy equivalence

<W : a {(x x y)KxL) - c*(xK) ® a(rL),

which is compatible with the action of the normalize!' group Nqxh(K x L) of K x L in

GxH.

Corollary 1.3.5. Let G,H be two groups, Tg,Th be families of subgroups of G, resp. H,

X and Y be G-, resp. H-CW-complexes. Consider the cofunctors V,i/(/;2 from the category

v?G(G) x u?H(II) to CAb defined by

MG/K x H/L) = C„ ((X x Y)KxL)

and

MG/K x H/L) = C,(XK) ® C*(YL).

Then ipi, ip2 considered as chain complexes in the product category Mod?G — Gx Mod?H — H

are chain homotopy equivalent.

Definition 1.3.6 (KG). Let G be a group. A universal space for proper actions ofG, denoted

by EG, is a proper G-CW-complex such that ifX is any proper G-CW-complex, then there

exists a G-map f : X —» EG and any two G-maps from X to KG are G-homotopic.

1.3 The Künneth Formula in the Bredon Homology Theory 19

Remark 1.3.7. The universal space KG exists and is unique up to G-homotopy equivalence.

This space is uniquely characterized by the following property. Let K be a subgroup of G,

then

(EG)K ~{*} , if \K\ <co

(KG)K =0, if \K\ = oo.

A construction of the universal space EG and a proof of those facts are given in ([MiVa],

Thms. 2.2 and 2.4).

Example 1.3.8. a) If G is a finite group, then every action is proper, and EG = {pt}.

b) If G is torsionfree, we can take EG = EG, the usual universal G-space with orbit space

the Eilenberg Mac Lane space BG = K(G, 1).

c) If G is a crystallographic group acting on Rn, then KG = Wl, [Vaj.

d) If G acts properly on a tree X, then EG — X. In general, if G acts properly and

isometrically on a metric space X admitting a notion of barycenter and the existence

and uniqueness of geodesies, then KG — X ([Va], Ex. 4.1.8,).

Similar statements arc in general true for a universal G-space ET for an arbitrary non¬

empty family T of subgroups of G, which is closed under conjugation and passing to sub¬

groups. In this case, one obtains a G-CW-complex ET such that (ET)H is contractible for

K G T and empty, if K is a subgroup of G not belonging to T. In this context, ETin = KG,

where Tin denotes the family of all finite subgroups of the (discrete) group G. More about

in [Lu].

Notation

Let G,H be groups. For a subgroup K of G x H, we denote by ttqK the projection of A in

G and by tthK the projection of K in H.

Lemma 1.3.9. Let G, H be groups and K C G x II a subgroup. Then

(KG x EH)K = (EGYaK x (EIIfHK.

Proof. Let (x,y) G (EG x EH)K and k = (kg,kh) G A C n0K x irHK. Then (x,y) =

k(x,y) = (kgx,khy), which shows that (x,y) G (EGYaK x (KHfnK. Let now x G EG, y G

EH, kg G nGK,kh G tthK with kgx = x,khy = y. Then (kg,kh) G 71qK x irnK and

(kg,kb)(x,y) = (x,y), which shows that (x,y) G (EG x EH)1TgKx7ThK. Therefore (x,y) G

(KG x KH)K, since A C ircK x tthA. Ü

Proposition 1.3.10. Let G, H be groups. Then the product space EG x EH is a model for

the universal space of G x II, in other words:

EG x KH = K(G x H).

20 Bredon (Co-)Homology Groups and Künneth Formula

Proof. Let if be a finite subgroup of G x P. By Rem. 1.3.7, since K is finite, the space

(EGY°K x (KHY"K is contractible. Therefore, by Lemma 1.3.9, (EG x KH)K is also

contractible. Let now K be an infinite subgroup of GxH. Since K C 7r<-;A x tvhK, also

ttgK or 7THK are infinite. Thus, by Lemma 1.3.9 and Rem. 1.3.7, P(G x H)K = 0, so also

(KG x EH)K is empty. The proposition follows from the characterisation of the universal

space illustrated in 1.3.7.

Theorem 1.3.11. Let G,H be groups and Tin, Tine, TinH the families of all finite sub¬

groups respectively of G x H,G and H. There is a natural chain homotopy equivalence in,

C*Ab

QM(G x H)) ®?m RclxH ~ (CMG) ®^nC Re) ®z {QMH) ®nnH Re)

Proof. From the category u?in(G x H) to C*Ab we define the following cofunctors

0i : (G x II)/K i- C,(E(GxH)K)02 : (G x H)/K .-> C,(K(G x HY°KHK)

and functors

V>i : (G x H)/K h+ PC(A)

rj,2:(GxH)/K ^ Rci-naK x nHK).

We check that the map ip2 is actually a functor. Indeed, let A', L G Tin and <p : (GxH)/K —

(G x H)/L be a morphism in the orbit category v?in(G x II). The morphism ip is determined

by the image p(K) = (g, h)L, for some (g,h) eGx H satisfying (g~\ h~1)K(g, h) C L and

therefore, (g'1, /T^ttgAT x 7THK)(g, h) C ttgL x tïhL. Thus ^ induces a well defined G x H-

morphism

Tp : (G x H)/(nGK x tthK) -f (G x H)/(tzgL x tthL)

defined by ^(ttgA' x tthA) = #7TGL x /wr^L. The morphism ^(<p) can be therefore defined

as Rc(~!p), which shows the functionality of ip2.

Let K G .Fin. By applying the (co)-functors above, the morphism

/ : (G x H)/K -+ (G x P)/(ttgA x tt^A)

induces a chain map via restriction

/* : C*(E(G x HYaKx7I"K) -+ C*(E(G x Hf)

and a homomorphism

/. : Rc{K) - ^(ttgA x tt^A)

via induction. We get natural transformations induced by /* and /*, respectively:

A : 4>2 - 0i, M :'0i -» ^2-

1.3 The Künneth Formula in the Bredon Homology Theory 21

By Lemma 1.3.9 and Prop. 1.3.10, /* is a chain homotopy equivalence, therefore A is a

natural equivalence and wc may assume /* — id and A — id. We remark that C„(E(G x

II)) ®?in Pc'x is equal to 0i ®?in ipi in our new notation. We show now that the chain

complexes 0i ®?in rpi and 4>2 ®?m tp2 arc chain homotopy equivalent. Wc consider the chain

map

id ® fl : 0! ®?ul V»i -+ 02 ®?in '02

x ® y i—> x ® fiy.

Since x®y ~ x®py, we can view x®/.ix as an element of 0i®^n'0i or of <t>2<8>?inip2- Therefore,

it is plain that id ® p, is an isomorphism on each chain group. Moreover, it is compatible

with boundary maps by naturality and therefore, it is a chain homotopy equivalence.

We consider now the following (co)-functors from the category v?inGxJrmH(G x II) to

C*Ab:_

(h:(GxH)/(UxV) ^ C*(K(G x H)UxV)

^i : (G x H)/(U x V) ^ RC(U x V).

Sinccevery subgroup K G Tin can be included in 7rGA'x7T//A, the chain complexes 4>2®?jnih

and 0! ®?tnGx?inH ^i arc equal. Moreover, we compute the tensor product as follows:

Jl ®Fina*FmB Ti - Z<pxv)&ncxrina C*(K(G x H)"*v) ® R(U x V)/ ~

w^i.3.4 ^^G,^) ® C*(EHV) ® Pc(P) ® «c(V)/ ~=

E^^atEC")®«^)/-®®£ve*»fl a(PPK) ® RC(V)/ ~=

(ac^c) ®^«Q «c) ® (£*(£#) %«ff «c) ,

which is exactly the right part of the assertion of our theorem. D

Definition 1.3.12 (flat chain complex). Let R be a ring with 1. A chain, complex of R-

modules is called flat if its constituent R-modules are flat.

Lemma 1.3.13. Let (C,d), (D,&) be chain complexes over the p.i.d. R, and suppose that

one of the two complexes is flat. Then there is a natural short exact sequence

0^ J2 IPn(C)®RHq(D)^Hn(C®HD)n

£ Tor«(Hm(C),Hq(D))-^Q.

m+q—.n

m-\-q=ri—l

Proof See [Thm. K2.1, [HiSt]]. D

Remark 1.3.14. Let R be a commutative ring with 1. Let A, B be R-modules. If A (or B)

is projective (or fiat), then Tor^(A, B) — 0, for all n>l.

22 Bredon (Co-)Homology Groups and Künneth Formula

Proof. See [Prop. 7/7.8.1, [HiSt]]. D

Lemma 1.3.15. Let G be a group and TinG the family of all finite subgroups of G. The

chain complex C^KG) defines a projective resolution of'Z in the category Mod?ina - G.

Proof. The cellular chain complex C^EG) consists of G-modules such that for every i G Z,

Ci(EG) = Z[Aj]. The space A.< is the G-set occurring in the definition of the i-skcleton of

E_G and it can be written as a disjoint union of orbits G/Ka, for some Ka G TinG. For all

i G Z, C_t(KG) are projective, as they are isomorphic to a direct sum ®a PKa of projective

cofunctors PKa G Mod?inc - G defined above, Ka G TinG. The sequence C^EG) -» Z is

exact in the category Mod?inG — G. In fact, evaluated on each orbit G/H, H G TinG, we

get exact sequences of abelian groups C*(EGU) -» Z, as for finite subgroups H G Tina,

KGH ^ {*}.

By definition, the group Pfm°(G;Pg) is Tor;(Z, Pg), i G N. By the previous Lemma

this is the i — t/i-homology group of the chain complex G^PG) ®?inc Re- The following

corollary is therefore a direct consequence of the previous Lemma.

Corollary 1.3.16. Let G be a group and TinG the family of all finite subgroups of G.

Then for every functor N G G-Mod?inc the groups Hfinc(G; N) and Pf,VlG(PG; N) are

naturally isomorphic.

We state now a Künneth Formula for the Bredon homology of groups.

Theorem 1.3.17 (Künneth Formula). Let G, H be groups and Tin, TincTinn the families

of all finite subgroups respectively ofGx H, G and H. Then for every n G N there is a natural

exact sequence:

°^ E H^(G-R^)®H^(H-R^)^H^(GxII-,R^H)^m-\-q=n

-, J2 Tor(H^(G-RGc),H^«(H;R£))-,Q.

Proof. By Cor. 1.3.16, the groups Hfina(G; Pg) and Il(ina(EG; Pg) arc naturally isomor¬

phic, i G N. By Thm. 1.3.11,

Hi(QM{G x II)) ®?m R£xH) ê* H% {(CJEG) ®?inG Pg) ®z {QMH) ®*n„ #c ))

Wc show that the complex of Z-modules C_^(EG) ®?inG Pg IS projective (and therefore

fiat). For every L G TinG, the abelian groups Ci(KGL) are isomorphic to the direct sum

©a-PK«(G/P) of the projective cofunctors PKa evaluated on G/L, Ka G Tina- Let A G

TinG. We consider the tensor product

PK®TinGRc= E Z[mor(G/I,G/A)]®Pg(L)/~.

1.3 The Künneth Formula in the Bredon Homology Theory 23

Every clement x ® y G Z[mor(G/L, G/K)] ® Pg(L) is equivalent to the element id ® x*y G

Z[mor(G/K,G/K)] ® Pg(A~), where z« is the induced morphism from Pg(L) to Pg(A').Therefore

Pk ®Fina Rl = Z[mor(G/A, G/K)} ® P.g(A")/ - .

Let NG(K) be the normaliser of A in G, then the set mor(G/K,G/K) is naturally iso¬

morphic to the Weil group WG(K) = NG(K)/K. The Weil group WG(K) acts on Pg(A')

by conjugation, since NG(K) acts on Rq(K) by conjugation and A' C NG(K) acts trivially.

Moreover, the equivalence relation on Z[WG(K)]®Pg(A') is generated by w®x ~ l®)W*(x),

where w G WG(K) and w* : Pg(A') - Pg(A") is defined by the iyG(A')-action. We get

Pk ®^nG Pg ^ Z[WG(K)] ®Wg(k) Pg(A) - Pg(A),

which is a free abelian group. Therefore, since C^(E_G) is a sum of projective functors Pxa,

(2*(EG) ®?inG Pg is a projective complex of Z-modules. By the Künneth-formula of Lemma

1.3.13, for every n G N there exists therefore a natural exact sequence

0 -+ Yl ^m(Ç*(PG')®^mGPg)®if(3(G,(PP)®>-m/7PcJ) - Hn (QM(G x II)) ®?i.r, Pgx/y)m+q—n

- Yl T^l(Hm(CMG) ®MnG Pg), Hq(C*(EH) ®Trn,H Re)) - 0,

m+g=n—1

which shows our theorem. D

More generally, we have the following.

Theorem 1.3.18 (Künneth Formula). Let G,H be groups, X a G-CW-complex and, Y a

H-CW-complex. Let Tin, Tina, TinH be the families of all finite subgroups respectively of

G x H, G and II. Then for every n G N there is a natural exact sequence:

0^ Yl IlfnnG(X-,Ri)®Hpn-(Y;R^)^H^(XxY-,R^1)~,m+q—n

-> ]T ror(p^(^;^c)^f'lH(^^c))-o.rn+q—n—l

Proof. Let A G Tin be a finite subgroup of G x H. The same argument of Lemma 1.3.9

shows that

(X xY)K =X*cK xY*"K.

Analogously to the proof of Thm. 1.3.11, the chain complex of abelian groups G* (X x Y) ®?in

RqXH is naturally chain homotopy equivalent to (C_*(X) ®?inG Pg) ® (d*(Y) ®?mH Pc)-By similar arguments as before we can show that ^(X) ®?ina Pg and C_*(Y) ®?rnH R$ are

complexes of projective Z-modules. The assertion follows therefore by 1.3.13. D

24 Bredon (Co-)Homology Groups and Künneth Formula

Remark 1.3.19. The Künneth Formula does not require the spaces X and Y to be proper.

Moreover, for any G-CW-complex X Bredon homology ofX is the same as Bredon homology

of the (proper) G-CW-complex X x KG, in symbols, Ilfina(X; N) s* Hf"lG(X x KG; N),

for any coefficient functor N G G — Mod?inc.,i G N. To see this, for every finite subgroup

H G Tina we have that (X x EG)11 = XH x (EG)11 ~ XH, which implies thai C,(X) is

chain, homotopy equivalent to Ç_*(X x EG) as complexes in Mod?ina — G, which shows the

assertion.

1.4 The Universal Coefficient Theorem in the Bredon

Homology Theory

In this section we want to consider Bredon Homology groups with extended coefficients in a

field k.

We recall that the characteristic of a field k is the unique natural number n such that

nZ is the kernel of the unique ring homomorphism from Z to k which sends 1 to 1&. We

denote the characteristic of k with the standard notation char(k). In this chapter we will

consider fields of characteristic zero, like for example Q, R, etc., which are in particular also

Z-modules. For our interest this section will be useful in particular for the case k — Q. We

remark that many assertions of the chapter hold true more generally for all characteristics.

Definition 1.4.1 (Rc,k)- Let k be a field of char(k) = 0. Let G be a group and, Tina the

family of all finite subgroups of G. For H G Tina, we define R^}k(H) := R-c(H) ®i k and

we denote by Rc,k the functor from the orbit category u?i„G to the category k — Mod of

k-modules, which associates to G/H G v?ina the k-module Pc,fc(P).

If necessary, in order to avoid confusions, we specify the ground group G writing Pgfcinstead of Pc,fc. We recall that for II G Tina we denote by PH the projective cofunctor

from the orbit category u?ina(G) to Ab defined by PH(G/L) :— Z[mor(G/L,G/II)}, for all

L G TinG.

Lemma 1.4.2. Let k be a field of char(k) 0, G be a group and Tina the family of all

finite subgroups of G. For all, H G Tina, there is an isomorphism of groups

PlI ®FinG Rc,k = Pc,fc(#)-

Proof. Let II G Tina- By definition,

Ph ®nna Rc,k - Y Amor(G/L, G/II)] ®z Rc,k(L)l -,

where every element x ® y G Z[mor(G/L,G/H)} ®z Rc,k{L) is equivalent to the clement

id ® x*y G Z[mor(G/H,G/H)) ®% Rc,k(H). Here x* denotes the induced morphism from

1.4 The Universal Coefficient Theorem in the Bredon Homology Theory 25

Rc,k(L) to Rc,k(H)i which is of the form Rq(x) ® idk, with Rc(x) the induced morphism

from RC(L) to RC(H). Therefore

Ph ®?ina Pc,fc = Z[mor(G/H, G/H)] ®z Rc,k(H)/ ~ .

The set mor(G/H,G/H) is naturally isomorphic to the Weil group WG(H) = NG(H)/H,

where NG(H) is the normalize! group of H in G. The Weil group WG(H) acts by conju¬

gation on Rc>k(H), since it acts by conjugation on Rc(H) and trivially on k. Moreover,

the equivalence relation on Z[WG(H)} ® Rc}k(H) is generated by w ® x ~ 1 ® w*(x), where

w G WG(H) and w* : PC)fc(P) -* Pc,fc(P) is defined by the WG(P)-action. We get therefore

Ph ®n»G Rc,k = Z{Wa(H)] ®Wa(H) Rc,k(H) = Rc,k(H).

D

By the previous Lemma it follows that the group Ph ®?ma Rc,k inherits in a natural way

the structure of a k-module, which is additionally projective in the category of A'-modules

(since k is a field) and therefore also flat. Moreover, with the same argument as in the Proof

of Thm. 1.3.17, C_^(KG) ®?inG Rc,k is a projective and flat complex of fc-modules.

Remark 1.4.3. The previous Lemma holds for more general cases. Namely, for a group G,

T a family of subgroups of G and N G G — Mod? the following groups are isomorphic:

PH®?N^N(G/H),

for every H G T.

Lemma 1.4.4. Let k be a field of char(k) = 0, G be a group and TinG the family of all

finite subgroups of G. Then for every n G N there is a group isomorphism,

Iirna(G; Rc,k) = Pfno(G; Pc) ®z k.

Proof. The chain complex C^(KG) ®?inG Rc,k is natural chain homotopy equivalent to

(C_*(EG) ®?inG Re) ®z k and the assertion follows since k is a field of characteristic 0. D

By the previous Lemma we can view the abelian groups H^"lG(G; Pc,fc) with the structure

of Avmodulcs, which are free (since A: is a field), hence projective and flat in the category of

fc-modules, for every n G N.

Theorem 1.4.5 (Universal Coefficient Theorem). Let k be a field of char(k) = 0. Let

G, H be groups and TinG, Tinn1 Tin be the families of all finite subgroups of G, II, GxH

respectively. Then there is a natural isomorphism, of k-modules

[H^(G; Pg,) ®fc H?n"(H; ßgfc)]„ = H^^G x H; R°xH),

for every n G N.

26 Bredon (Co-)Homology Groups and Künneth Formula

Proof. We consider the flat complexes of A>modules C*(EG) ®?ina Rc,k anc^ G-*(EM) ®?inH

R^k. By the same reason as in Lemma 1.3.13, for every n G N wc get the following natural

short exact sequence:

0^ Yl Hm(CMG)®rmGRc,k)®kHq(C^(EH)®?innRlk)^m+q=n

- Hn((£MG) ®TinG Rc,k) ®k (QSKH) ®?inH Pg,)) -

- Yl T<(Hm(C_MG) ®?ina Pg>), Hq{C*(EH) ®?jnH Pg,)) - 0.

rri+q=n— 1

The term in the middle, by the same argument used in proof of Thin. 1.3.17, can be expressed

as follows:

Hn((CMG) ®?inG Ri,k) ®k (GMH) ®^nH Rlk)) s Hn((QM(G x H)) ®?in J$f).

By definition, Pfnc(G; PgJ = Pn(G,(PG) ®?iliG Pgi/fc)), n G N. Therefore our theorem is

proved, since by Lemma 1.4.4 Hpn°(G; flgk) = #f*"G(G; Rc) ®z k is a flat A--module which

annuls the Tork{(-, -)-group. D

More generally, the same statements can be proved for G-CW-complcxes.

Lemma 1.4.6. Let k be afield, of char(k) = 0, G be a group and X a G-CW-complex. Let,

Tina be the family of all finite subgroups of G. Then for every n G N,

H?n°(X; Rc,k) ^ H?nG(X; Rc) ®i k.

Proof. The proof is analogous to that of Lemma 1.4.4, since the chain complex C„(X) ®?inG

Rc,k is naturally isomorphic to (C„(X) ®?ina Rc) ®i k and the functor - ®z k is exact. D

Theorem 1.4.7 (Universal Coefficient Theorem). Let k be a field of char(k) = 0. Let

G,H be groups, X a G-CW-complex and Y a H-CW-complex. Let TinG,Tinu,Tm be the

families of all finite subgroups of G, II, G x H respectively. For n G N there is a natural

isomorphism,

[Hrma(X; Pg,fc) ®k Hfin"(Y; Rlk)]n - Pfn(X x Y; Pgf<).

Proof. Analogously to Thm. 1.3.17, the assertion follows with the same proof as of Thm.

1.4.5 and by noting that the chain complex G,(X x Y) ®?in Pgjj" of Avmodulos is naturally

chain homotopy equivalent to (G^(X) ®?ina Ltçk) ®k (C^(Y) ®?inH R^k)- ^

Chapter 2

if-theory of C*-Algebras

2.1 Basic Definitions on C*-Algebras

We introduce some basic definitions and notions about C"-algebras.

Definition 2.1.1 (Banach algebra). A Banach space A (overC) is a, complete norm,ed space

(A, || ||) (over C). If furthermore A is an algebra, (over C) and, its multiplication satisfies

the inequalityINI<INI-||2/II

for each, x, y G A, then A is called a Banach algebra.

Definition 2.1.2 (G*-algebra). Let A be a Banach algebra.

a) A map x i—» x* is called an involution if it satisfies

i) (x + y)* = x* + y*

ii) (Ax)* = Ax*

Hi) x** ~ x

iv) (xy)* = y*x*,

for every x, y G A, AG C.

b) If A admits an involution *, we say that an element a G A is positive, denoted by

o > 0, if it has the form bb* for some b G A.

c) If A admits an involution * with

\\x*x\\ = \\x\\2.yxeA,

then (A,*) is called a C*-algebra.

28 A'-theory of G*-Algebras

d) An algebra-homomorphism </> between C*-algebras A and B is called a *-homom,orphism

if

0(s') = (4>(z)Y,

for all x G A. A bijective *-homomorphism is called ^-isomorphism.

Example 2.1.3. a) For a locally compact topological space X, we denote by C0(X) the

algebra of continuous functions f from X to C vanishing at infinity, which means

that for every e > 0 we can find a compact subset Kt of X such that \f(x)\ < t

for all x G X\Kf. Pointwise addition and multiplication, involution given by f*(x) =

f(x),x G X, f G Cq(X) and sup-norm, endow C0(X) with the structure of a C*-algebra.

This is an abelian C*-algebra, with, unit if and only if X is compact, and in this case

we have that Cq(X) = C(X) is the algebra of continuous functions from X to C.

b) Forn G N, the algebra Mn(C) ofnx n-matrices with complex entries together with the

operator norm and involution M* := MT, M G M„(C), is a C*-algebra. We can also

generalize this example by considering the C*-algebra, M„(A) of n x n-matrices with

entries in a C*-algebra A.

c) Let H be a Hilbert space. The algebra B(H) of bounded operators on H with norm, given

by the operator norm and *-operation given by the usual adjunction is a C*-algebra.

This C*-algebra is non abelian as soon as the dimension of H is bigger than one. Any

closed subalgebra of B(H) which is invariant under involution is a C*-algebra. An

example is the closed subalgebra K(H) of compact operators on II. Conversely, using

Gelfand-Naimark-Segal-construction we will see that any C*-algebra may be viewed as

a closed subalgebra (invariant under involution) of some B(H) (Example 2.1.10).

Rings and G*-algcbras do not necessarily have a multiplicative unit, as the previous

examples show. We can add a unit to a ring or a G*-algcbra by the following construction.

Definition-Proposition 2.1.4 (ring with unit). Let A be a ring. Then the set

Ai :={(a,A)|aG.4,AGZ}

with operations

(a, A) + (b, ß) = (a + b,\ + /i),

(a,\)-(b,u) — (a,b + pa + \b,\p),

for all a, b G A, A, p G Z, is a ring with unit (0,1).

Definition-Proposition 2.1.5 (C-algcbra with unit). Let A be an algebra over C. Then

the set

Aijc :-{(a,A)|üG A^gC}

2.1 Basic Definitions on G*-Algebras 29

with operations

(a,X) + (b,p) - (a + b,X + p),

(a,X)-(b,p) — (a,b + pa + Xb, A//),

and scalar multiplicationp(a,X) = (pa,p,X),

for all a, b G A, X, p G C, is a C-algebra with unit (0,1).

Definition-Proposition 2.1.6 (Banach algebra and G*-algebra with unit). Let, now (A, \\

||) be a Banach algebra non necessarily unital. Then the algebra AjiC with norm

||(a,A)||=8up{||a6 + A6|| | b G A, \\b\\ = 1},

for all a G A, X G C, is a Banach algebra with the unit (0,1). Moreover, if A is a C* -algebra

with involution *, then the Banach algebra -Aj,c with involution

(a,XY = (a*,Ä)

for all a G A, X G C, is a C* -algebra with unit (0,1).

An important example of G*-algebra is the reduced G*-algebra of a group. Let G be a

group and consider its associated complex group algebra CG, which is the C-vector space

with basis G. The algebra CG can be also viewed as the space of complex valued functions

on G with finite support, with product given by the convolution, namely, if / = ^2gec fy9

and I = YlheG ^h are elements of CG, then

/ * 1= ^2 hhgh.9,heG

We denote by l2G the Hilbert space of square summable complex valued functions on G.

The left regular representation AG on l2G is defined by

\G(g)1>(h) = iP(g~lh),

for g, h G G, 0 G l2G, and it can be extended to CG by

geG

where / = £ffeG fQg e CG, so that

In the next definition wc denote by || ||op the operator norm on the space of bounded

operators B(l2G) on l2G.

30 i\-theory of G*-Algebras

Definition 2.1.7 (C*G). Let G be a group. The reduced C*-algebra ofG is the norm closure

of XG(CG) in B(l2G) and is denoted by C*G. In other words,

C*G = XaJCG)Mop.

Example 2.1.8. a) The reduced, C*-algebra C*G of a finite group G is its group algebra

CG.

b) IfG^ is an abelian (discrete) group, then there is an isomorphism, of'C*G with the algebra

C(G) of continuous functions on G, where G := Hom,(G,Sl) is the Pontryagin dual

of G, S1 = {z G <£\\z\ = 1}. This is given by the Fourier transformation

*

:CG^ C(G), f ~ / : {X -> Y, fis)x(9)hgeG

as illustrated in [Va], Ex. 1.7.

b) For G = Z, CZ is the algebra of Laurent polynomials C[£,£-1]. The previous Fourier

map sends a Laurent polynomial to the associated, trigonometric polynomial, and C*Z =

C(S').

In the theory of G*-algebras it is very useful to regard every G*-algebra as a G*-subalgebra

of the G*-algebra B(H) of all bounded operators on some Hilbert space H (via the so called

GiVS-construction).

Definition 2.1.9 (representation). A representation of a G*-algebra A is a pair (H,<p),

where H is a Hilbert space and p : A — B(H) is a *-hom,omorphism. A representation

(H, <p) is called faithful if ip is injective.

Example 2.1.10 (GNS and universal representation). Let A be a C*-algebra and r be a

positive linear functional on A. It is not hard to show that the set NT — {a G A\r(a*a) = 0}

is a closed left ideal of A and that the map

(A/NT)2 -> C, (o + iVr, b + NT)^ r(b*a)

is a well-defined inner product on A/NT (Thm,. 3.3.7 [Mu]). We denote by HT the Hilbert

completion of A/Nr. For every a & A we define an operator <p(a) G B(A/NT) by

(p(a)(b + Nr) =ab + NT,

for all b G A. The operator (p(a) is norm-decreasing. Indeed, for all b G A, we have that

\\<p(a)(b + NT)\\2 - r(b*a*ab) < ||a||2r(&*fe) = ||a||2||6 + 7Vr||2 (the latter inequality is again

given by by Thm,. 3.3.7 [Mu]), and therefore, ||<p(a)|| < ||a||. So, the operator ip(a) has a

unique extension to a bounded operator <pT(a) on HT, and the map

<pT : A —> B(Hr), a h-»- p>T(a)

2.2 Kq of G*-Algebras 31

is a *-homomorphism. The representation (HT,tpT) of A is the Gelfand-Naimark-Segal rep¬

resentation (or GNS representation) associated to r. We define the universal representation

of A to be the Hilbert direct sum, of all representations of the form, (HT,pT) of A, where r

ranges over all positive linear functionals on A.

Theorem 2.1.11 (Gelfand-Naimark). Let A be a C*-algebra. Then it admits a faithful

representation. Specifically, its universal representation is faithful.

Proof. Thm. 3.4.1. [Mu]. D

Example 2.1.12. Let G be a group and A = C;G. For f = ^geGfgg, let r(f) - fe. Then

HT — l2(G) and ipT : C*,G —> B(l'2(G)) is just the left regular representation.

2.2 K0 of C*-Algebras

To define the K0 group of a ring (and successively of a G*-algebra) we need the following

algebraic construction.

Definition 2.2.1 (semigroup). A semigroup (S,+) consists of a set S together with an

associative operation + .A semigroup (S,+) is abelian when the operation is commutative.

Let (S, +) be an abelian semigroup and consider the product S x S with the equivalence

relation (x, y) ~ (u, u) if and only if there exists an element r G S such that

x + v + r — y + u + r,

for elements x,y,u,v G S. Let Gr(S) := S x S/ ~ and denote by [(x, y)\ the equivalence

class of (x, y) G S xS. The quotient Gr(S) with componentwise addition is an abelian group

with the distinguished neutral element [(x, x)], and [(x/, x)] the inverse of [(x,y)],x,y G S.

We call Gr(S) the Grothendieck group of S and the semigroup morphism ß : S —> Gr(S)

given by fi,(x) := [(x + x, x)] the Grothendieck morphism of S. Wc note that the image of /i

generates Gr(S) as a group, since [(x, y)] = pt,(x) — fi(y).

Proposition 2.2.2 (Universal Property). Let (S,+) be an abelian semigroup with corre¬

sponding Grothendieck group (Gr(S),+). Then for any group G and semigroup morphism

ip : S —> G there exists a unique homomorphism 4> : Gr(S) — G satisfying cf> o /.i = p.

Proof. The morphism defined by (f>([(x,y)]) := p>(x) — p(y) is well defined and satisfies

0 o n — p. Uniqueness follows since the image of /i generates Gr(S). [cfr. [Ro], Thm.

1.1.3.]a

32 A'-theory of G*-Algebras

Let A be a ring with unit. For n G N we consider the ring Mn (A) of n x n-matrices with en¬

tries in A, M^A) :^ Un>iMn(A), the group Gln(A) of invertible n x n-matrices, Gl^A) :=

Un>].Gln(A) and the set Pn(A) of idempotcnt n x n-matrices, P^(A) :— Un>iPn(A). Two

idempotents p G Pn(A),q G Pm(A),n, m G N, are called equivalent, denoted by p ~ q, if

there exist k G N, fc > n,m, and w G Glk(A) such that

P © Ofc-n = 1X(Q © Ofc-m)«-1,

where 0n is the zero matrix in M„(A) and © denotes the direct sum defined as

© : Mn(A) x Mm(A) - Mn+m(,4)

(Aß) - A®B:=(£ iyRemark 2.2.3. Let A be a unital ring. The set P0O(^4)/ ~ wit/i t/ie tiireci sum © is art

abelian semigroup.

Definition 2.2.4 (A'0 of a unital ring). Let A be a ring with unit. The K0-group of A,

denoted by K0(A), is defined as the Grothendieck group associated to (P0O(J4)/ —,©).

Remark 2.2.5. Let (S, +) be an abelian semigroup. We may alternatively define Gr(S) to be

the free abelian group on generators [x],x G S, divided out by the relations that ifx+y = z in

S then [x] + [y] = [z] in Gr(S). We note that [(x,y)) in the previous construction corresponds

to [x] — [y] in this second construction. Thus each element of Ko(A) can be written as

a difference of two equivalence classes of idempotents \p] — [q], for some p G Pn(A),q G

Pm(A),n,m,eN.

Remark 2.2.6. We may view A"0 as a covariant functor from the category of unital rings

to the category of abelian groups Ab. Indeed, a homornorphism p : A —> B of unital rings

extends to a homornorphism p : Mri(A) — Mn(B), which maps Pn(A) to Pn(B) for each

n G N and defines a group homornorphism, p* : K0(A) —> K0(B).

Remark 2.2.7. A (right) A-module M is called projective of finite type if there exist an

A-module N and n > 1 such that

M®N9éAn

(as A-modules), or equivalently if there exists an idempotent p G Pn(A) such that

M^p(An).

Wc may therefore view Kq(A) as the Grothendieck group of the semigroup Va of isomorphism

classes of projective modules of finite type over A, namely

K0(A) =VAx VA/ ~,

2.2 Po of G*-Algebras 33

where (M0, Mi) ~ (iV0, N{) if there exists n G N such that

M0 © iVi © A71 3* Mi © N0 © An,

with M0, ML, N0, ^ eVA-

Example 2.2.8, a) If A = C, a projective module of finite type is a finite dimensional

vector space, therefore,

K0(C) = Gr(Vc, +) s Gr(N, +) = (Z, +).

b) If G is a finite group and A — CG, a projective module of finite type over CG is a

complex finite dimensional representation of the group G, thus K0(CG) — R<c(G), the

additive group of the complex representation ring.

Let now A be a (not necessarily unital) C-algcbra. We want to give the definition of the;

Kq group of A. Let AIiC be the unital C-algebra associated to A defined in Def. 2.1.5. Let

<Pa : Ai}c — C be the projection whose kernel is A and pA* K0(Altc) —> Ao(C) = Z be the

induced morphism.

Definition 2.2.9 (Ka of a C-algebra). Let A be a (not necessarily unital) C-algebra. The

K'Q-group of A, denoted, by K0(A), is defined as the kernel of the induced morphism pA*-

Remark 2.2.10. A morphism p> : A — B of arbitrary C-algebras extends to a unital ring

homornorphism pi,c AItC - P/,c, <fi,c(a, A) := (p(a),X), which, sends Prl(ATj) to P„(P/,c)

for each n G N and induces therefore a homornorphism y>/,c* : Kq(At,c) —> A^(P/,c)- Since

<Pb o <Pi,c = P>a, the map (pi,c* induces a homornorphism p* : K'0(A) —> K'n(B), making K'0

a covariant functor from the category of (not necessarily unital) rings to the category Ab.

Remark 2.2.11. IfA is a unital C-algebra, the unital ring homornorphism AjtC —> A, (a, X) i—>

a + X-1 induces a homornorphism Kq(Aj^) —» Kq(A), which maps Ker(Kü(Aj^c) —* A0(C))

isomorphically to K0(A), which means by definition that K'0(A) = KÜ(A). This shows the

equivalence of definitions 2.2-4 and 2.2.9. From now on we will use the notation K0(A)

also for the K'Q-group of a not necessarily unital C-algebra A. Therefore, K0 is a covariant

functor from the category of C-algebras to the category of abelian groups.

Definition 2.2.12 (KQ of G*-algcbras). Let A be a G* -algebra. The KQ-group of A is defined

as the Ko-group of the underlying C-algebra A.

Remark 2.2.13. Ä"o is a covariant functor from the category of C*-algebras to the category

of abelian groups. In opposition to the case of higher K-groups, the definition of the KQ

group does not involve topological properties of the C*-algebras.

34 P-theory of G*-Algebras

2.3 Higher K-Groups of C*-Algebras

Definition 2.3.1 (A~i of unital Banach algebras). Let A be a unital Banach algebra. The

K\-group of A is defined, as the group of connected components o/G700(J4) viewed with the

union topology. In symbols,

K1(A) = ir0{Gloo(A)).

Proposition 2.3.2. For a unital Banach algebra A, K\(A) is an abelian group.

( cost — siirt \Proof. We first notice that the arc defined by t i-> I '.

,

'

, J connects the identity\sint cost J

2 x 2-matrix to I ] in G/2(C). This fact extended to the n x n-matrices allows us

by left (resp. right) multiplication to switch any two rows (resp. columns) of a matrix in

Mn(A). We denote by ~ the equivalence relation "being in the same connected component".

Then for S,T G Gln(A) we have that (' t) ~

\0 s)' S° that ^with 7 the identity

n x n-matrix):

ST 0\

0 I

Remark 2.3.3. A continuous, unital homornorphism p : A —> B between unital Banach

algebras A, B induces a homornorphism, p* : KX(A) —* K\(B). We can therefore view K\ as

a covariant functor from, the category of unital Banach algebras to the category Ab of abelian

groups.

Example 2.3.4. a) The groups Gln(C) are connected for each n G N. Indeed, to connect,

M G Gln(C) with the identity n x n-matrix I, we consider the complex affine line

(1 - z)I + zM, z G C

through M and I. It meets the set of singular matrices in at most n points (since

det((l — z)I + zM) is essentially the characteristic polynomial of M). So we conclude

our proof using connectedness of C minus n points. Therefore, A'i(C) = 0.

b) Let G be a finite group, then CG is isomorphic to a direct sum Mni (C) © • • • © M,ik (C)

of matrix algebras over C, ni, ,nk G N, and, for n G N, Gln(CG) = Gln.m(C)) x

x Glank(C), in particular it is connected. Therefore, A~i(CG) = 0.

2.4 Relations between Kq and Pi 35

Remark 2.3.5 (K[ of Banach algebras). Again we first assumed that A had a unit. Let

now A be an arbitrary Banach algebra. The K[-group K[(A) is defined by the kernel of

the morphism pa*' P~i(-4/,c) —* ^i(C), which is induced by the projection on the second

argument </?/,c : Ar,c —> C. Since by the previous example A"i(C) is trivial, we have K[(A) =

Ki_(Ai,c). Again we show that the definitions of i<i(-) and K[(-) are equivalent in case of

unital Banach algebras, which allows us again to use the notation Pi(-) also for the group

K[(-) for any Banach algebra. Let A be a unital Banach algebra. Then we have a C-

algebra isomorphism Aj^c = AxC, which in view of following Ex. 2.4-12 gives Ki(Atic) —

KX(A) x iCi(C), and since Pi(C) is trivial, Ki(AIjC) =* K\(A). Therefore, K[(A) ^ K^(A)

and we are done. Finally, we can consider also K\ as a covariant functor from the category

of Banach algebras to Ab.

Definition 2.3.6 (Ki of G*-algebras). Let A be a C* -algebra. The Ki-group of A is defined

as the K\-group of the underlying Banach algebra A.

Next we generalize and state the definition of higher P-groups.

Definition 2.3.7 (Kn of unital Banach algebras). Let A be a unital Banach algebra and

n G N, n > 1. The Kn-group of A is defined as the (n - \)-th homotopy group of Gl^A)

viewed with the union topology. In symbols,

Kn(A) = 7rn_1(G/0O(^)) = colimk^.^GlkiA)).

In general, for not necessarily unital Banach algebras, higher P~-groups are defined by

K'n(A) := Ker(Kn(AIiC) -+ Kn(C)). As An(C) is 0 if n is even and Z otherwise, we get that

if A is unital both definitions agree, that is K'V(A) = Kn(A).

Definition 2.3.8 (Kn of G*-algebras). Let A be a C*-algebra and n G N,n > 1. The

Kn-group of A is defined as the Kn-group of the underlying Banach algebra A.

Remark 2.3.9. For every n G N,n > i, Kn is a covariant functor from, the category of

G* -algebras to the category of abelian groups.

2.4 Relations between Ko and K\

The aim of this section is to give first a relation between K0 and Kx and successively to

prove a relation of periodicity between K0 and K± with higher P-groups.

Proposition 2.4.1 (weak exactness). Let A be a Banach algebra and I C A a closed ideal

with the obvious inherited structure of Banach algebra. The short exact sequence 0 —> / —>

A -^ A/1 —> 0 induces exact sequences

K*(I)±K*(A)^K*(A/I),

for *G {0,1}.

36 A"-theory of G*-Algebras

Proof. [Va], Prop. 3.3.1. Q

Proposition 2.4.2 (long exact sequence). Let A be a Banach algebra and I C A a closed

ideal with the obvious inherited structure of Banach algebra. There exists a connecting ho¬

momorphism, 6 such that the following sequence is exact:

KX(I) % KX(A) ^ K,(A/n ^ K0(I) * Kq(A) ^ Kq(A/I).

Proof. [Va], Prop. 3.3.2. D

To a given Banach algebra we define the following (non unital) Banach algebras.

Definition-Proposition 2.4.3 (cone and suspension). The cone of a Banach algebra A is

defined as the Banach algebra,

CA = {f&C{%l],A)\f(0) = 0},

while the suspension of A is the Banach algebra

SA = {f eC([0,l],A) \ f(0) = f(l) = 0},

with operations defined pointwise and norm given by the supremum norm. If A is a C*-

algebra, cone and suspension of A are also C*-algebras with involution defined pointwise.

Remark 2.4.4. A Banach algebra is said to be contrantible if the identity m,ap id : A —> A is

homotopic to the zero map. Notice that C is not contractible as a Banach algebra. The cone

CA is contractible for any Banach algebra A ([Mu], Thm. 7.5.2,), and if A is contractible,

so is its suspension, SA ([Mu], Ch. 7.5). By the contractibility of CA we can also show that

KX(CA) = Kq(CA) = 0 ([Va], Ex. 3.3.4;.'

Proposition 2.4.5. Let A be a Banach algebra. There is a natural isomorphism

dA : K,(A) - Kq(SA).

Proof. Wc consider the natural extension

Q~>SA^CAA-*0,

where cv(f) — /(I) is a surjective homomorphism from GA onto A with kernel SA (in

general this sequence is called mapping cone sequence). We get the long exact sequence

K^SA) - KX{CA) -> K\(A) % Kq(SA) - K0(CA) -* KQ(A).

Since K\(CA) — Kq(CA) = 0, the connecting morphism Oa is the desired isomorphism.

[Prop. 3.3.5, [Va]]D

2.4 Relations between KQ and P\ 37

Proposition 2.4.6. Let A be a Banach algebra. There is a natural isomorphism,

ßA:K0(A)^K,(SA).

Proof. This isomorphism is given by the so called Bott map Pa : Kq(A) —> K]_(SA). Con¬

struction and more details about this map arc given in [Va], Prop. 3.3.G. D

Theorem 2.4.7 (Bott periodicity). Let A be a Banach algebra. For each n > 0, there is a

natural isomorphism

Kn(A) <* Kn+2(A).

Proof. Composing the Bott map ß of Prop. 2.4.6 with the isomorphism d of Prop. 2.4.5,

we obtain that

Kq(A)0-A K^SA)3^ Kq(S2A).

Analogously, we get

Px(A) H Kq(SA) ßSt KX(S2A).

This means that

K{(A) s K(S2A), i &{(),!}. (2.1)

For n > 1, with f2 denoting the loopspace, 7Yn(GL^(A)) — rKn-1(i\GL00(A)) = nn^i(GL^(SA)),

so

Kn+l(A) = Kn(SA), n>l. (2.2)

We show now our assertion by induction on n. For n — 0, using the isomorphism /?a of Prop.

2.4.6 and by (2.2) we getßA (2.2)

KQ(A)^K1(SA) =* K2(A).

For n = 1,(2.1) (2.2) (2.2)

Ky(A) ^ üfi(SM) * K2(SA) =* K3(A).

By induction step, for n > 1 we get

Kn(A) (2é Kn-i(SA) = Kn+y(SA) ^ Kn+2(A).

D

Remark 2.4.8 (negative A'-groups). Fori < 0 and A a Banach algebra, one puts Ki(A) ~

Kq(A) if i is even and Ki(A) = K\(A) if i is odd.

An important corollary of Bott peridiocity is the construction of six-term cyclic exact

sequences:

38 P-theory of G*-Algebras

Corollary 2.4.9 (six-term cyclic exact sequence). Let A be a Banach algebra and I C A

be a closed ideal with the obvious inherited structure of Banach algebra. The short exact

sequence 0 —» / —> A — A/1 —> 0 gives a natural six-term cyclic exact sequence

Kq(I) -ÜU Kq(A) -=5- K0(A/I)

Ö p

Kx(A/n ^~ Ki{A) <^— Ky(I).

Proof. The morphism 5 is the connecting homornorphism of the long exact sequence de¬

scribed in Prop. 2.4.2. The morphism ß is obtained in the following way. We compose the

Bott map ßA/T for A/1 with the connecting homornorphism 6sa of the long exact sequence

applied to the short exact sequence 0 —> SI —> SA —> SA/Si —» 0 and we obtain

K0(A/I) ß^ Ki(S(A/I)) = K,(SA/SI) ^ KQ(SI).

Note that we used the fact that S(A/1) — SA/SI. By composing this Osa ° ßA/i with

the inverse of the isomorphism dj : P"i(P) —> Kq(SI) of Prop. 2.4.5 we get \l. [[Va], Cor.

3.3.8]D

Remark 2.4.10. in view of the definition of K-groups for C*-algebras, all claims and re¬

marks of this section are true for C*-algebras, too.

Example 2.4.11. As an application of Bott periodicity we can compute all K-theory groups

of the C*-algebra C and of the complex group algebra CG of a finite group G. Namely, in

view of Examples 2.3.4 a) a'nd 2.3.4 b),

#.(C) = { I ' ^luS eVen'

/

\ 0, otherwise,

and

K-(CG) =

{ ^(G) ' iji is evtn>

' ^ 0,

otherwise.

Example 2.4.12. Let A be the product Ay x A2 of two Banach algebras Ay and, A2. We get

the short exact sequence

0 -* Ay A A = Ax x A2 A A/Ax =- A2 - 0,

which splits. Therefore, we get the following six-term cyclic exact sequence

KoiAx) -^-> K0(A) -2^- K0(A2)

s /*

K,(A2) <-=^- K,(Ä) ^— KiiAJ,

2.5 Kasparov's Equivariant ATAT-Groups 39

and, because of the splitting, 5 — 0 and /j, — 0. This allows to compute the K-groups of A

using those of Ay and A2, namely,

K,(A) S K*(AX) x K+(Ai),

for* = 0,1-

Definition 2.4.13 (mapping cone). Let A, B be C*-algebras and 0 : A —> B a *-homornorphism,.

The mapping cone C^ of tft is defined as

C* = {(a,f) £ A®CB \ f(l) = <P(a)},

where CB denotes the cone of B.

In other words C$ is the pull-back of A and CB along </> and the evaluation map at 1.

The injection i(f) = (0, f) and the projection ir(a, f) = a give a short exact sequence

0 -> SB -^ C^ ^ A -> 0,

called the mapping cone sequence. By Cor. 2.4.9 we can associate a 6-term exact sequence,

which combined with the suspension isomorphisms of Prop. 2.4.5 and 2.4.6 becomes a

sequence of typeKi(B) * P0(G(/1) > Kq(A)

KX(A) +— tfi(C*) *— Kq(B).

In view of those remarks the next proposition holds.

Proposition 2.4.14. Let A,B be C*-algebras and 0 : A —» B a *-homornorphism. The

induced morphism 0* : AT*(A) —> K*(B) is an isomorphism, if and only if the K-theory of

the mapping cone C^ vanishes.

2.5 Kasparov's Equivariant KK-Groups

In what follows, all G*-algcbras arc usually assumed to be separable, even if this is not stated

explicitly.

Definition 2.5.1 (G — G*-algebra). Let G be a (discrete) group. AG— C*-algebra is a

C*-algebra endowed, with an action of G by *-automorphisms.

Example 2.5.2. Let X be a locally compact, proper G-space. Then the group G acts by

*-automorphisms on the C*-algebra C0(X), which turns Cq(X) into a G — C*-algebra.

40 A'-theory of G*-Algebras

Definition 2.5.3 (Hilbert G*-module). For a C*-algebra A, a Hilbert C*-module over A is

a right A-module £ with an A-valued scalar product, namely a map

< -,- >A- £ x £ -* A

which satisfies for all, x, x' G £, a G A

<x,y>A = <y,x>*A<x + x',y>A = < x, y >a + < x', y >A

<x,ya>A = <x,y>Aci

< x, x >a *> 0, with equality if and only if x = 0.

Moreover, £ is required to be complete with respect, to the norm,

\\x\\£ = || < x,x >A Ha-

For a Hilbert G*-module £ over A, the ring

£a(£) '— {T : £ —> £ v4-linear and continuous |3 T* : £ —> £ such that

for every -0, v G £ : < Tip, v >a=< V^ T*v >a}

together with involution * and operator norm is a G*-algebra.

Example 2.5.4. a) In case A = C, a Hilbert C*-m,odule over A is just an ordinary

Hilbert space over C.

b) Any C*-algebra A is itself a Hilbert C*-module over A with A-valued scalar product

given by < x, y >a'.— x*y, x,y G A.

c) Let A be a C* -algebra. Forn G N, £ := An is a Hilbert C*-module over A with A-valued

scalar product defined by < x,y >a'-= YH-i^Vh xiV £ A",x = (x,\,--- ,xn),y =

(yii '- '

i Vn)- The associated operator C*-algebra is CaIA71) = Mn(A).

d) Let l2(N) be the space of square summable (complex) sequences and A be a, C*-algebra.

Then the algebraic tensor product space Ha := Z2(N) ® A with an A-valued scalar

product defined by < f,g >a'.= Y^irieNfn9n, f,0 £ TLa, is a Hilbert C*-module over A.

e) Let G be a group and A a G — C*-algebra. We denote by l2(G,A) the space of

functions f : G —> A such that YigeC f(9)*f(y) converges in A. Then l2(G,A)

is a Hilbert C*-module over A with A-valued scalar product given by < f,h >a'-=

E.yeG figYKa), f,he l2(G, A) (Cauchy-Schwarz Inequality).

Definition 2.5.5 (compact and finite rank in the sense of Hilbert G*-modules). An operator

T G Ca(£) is compact in the sense of C* -modules if it is a norm, limit of finite rank operators.

It is finite rank in the sense of C*-modules if it is a finite linear combination of operators of

the form Q^y, where £, v G £ and, for x G £

ßtAx) = t < l^x >a

2.5 Kasparov's Equivariant KAT-Groups 41

Note that if T is of finite rank, its image is a finitely generated A-modulc.

Definition 2.5.6 (Kasparov triple). Let A,B be G — G*-algebras. A Kasparov triple over

(A,B) is a triple (£,-k,T), where £ is a Hilbert C*-module over B which admits a uni¬

tary representation U of G (unitary in the sense that < U(g)£,,U(g)v >b= g- < £, ^ >b,

Vg G G,^,v G £), it : A —> Cß(£) is a covariant * -homornorphism (covariant in the sense

that U(g)'ïï(a)U(g~1) = 7r((/a), Vry G G, a G A), and T is a selfadjoint operator in CB(£)-

Moreover, we require 7r(a)(:T2 - 1), [n(a), T\, [U(g), J"] to be compact in the sense of C*-

modules, yg G G, a G A.

Definition 2.5.7 (equivalence between Kasparov triples). Two Kasparov triples

(£i,7Ti, JFi) and (£2,'K2,J-2) on (A, B) are called equivalent, if there exists a G-equivariant

isometrical isomorphism u : £± — £2 such that

Va G A : 7T2(a) = «^(a)«-1, J^ = u^vT1.

In the case of graded Hilbert C*-modules the map u présentes the graduation (that is, u is of

degree 0).

Definition 2.5.8 (even and odd Kasparov triples). A Kasparov triple (£,7r,.P) over (A, B)

is even (odd otherwise) if the Hilbert C*-module £ is Z/2Z-graded, and U,-k preserve the

graduation whereas T reverses it. This means that £ = £q CD £\, and in that decomposition

n (Uq 0\ /tt0 0\_ [0 V*\

Definition 2.5.9 (Kasparov A'AT-groups). Let A, B be G - C*-algebras. Then KKf(A, B)

is the set of equivalence classes of Kasparov triples over (A,B), even, in the case i — 0 and

odd, for i = 1.

The sets KKf(A, B) arc abelian groups in natural way ([Va], Prop. 4.2.7).

Notation

In the case where G is trivial, we just drop it and we write A'A,''(^4, B) =- KI<i(A, B),i =

0,1.

Remark 2.5.10. Fori = 0,1, the functors KKf are covariant in the second and contravari¬

ant in the first variable. Indeed, let A, B, C be C* -algebras and a — [(£, tt, J7)] G KKf'(A, B)

be an equivalence class of Kasparov triples. A * -homornorphism 9 : G —» A defines

9*a = [(£, wo 9, F)] g PPf(G, B).

On the other hand, consider the Hilbert C*-module over C given by £ ®# G and a *-

homornorphism 9 : B —* C, then

B,_a = [(£ ®B C, n ® 1, T ® 1)] G KKf(A, C).

42 A'-theory of G*-Algebras

Example 2.5.11. For any C*-algebra B, taking for G the trivial group,

KKi(C,B)^Ki(B),i = 0,l.

(Ex. 5.7.(2) [Va]). In section 4-8 we will also see that for a G-compact proper G-space X,

KKf(X) * KKf(Cq(X), C), % = 0,1.

(example 5.7.(1), [Va]).

We introduce now the reduced crossed product. For a group G and a G — G*-algcbra

A, we denote by CC(G, A) the algebra of finitely supported functions / : G —> A. The

multiplication is given by the twisted convolution, which for /, h G CC(G, A),f — YlqeG /(flO#

and h = X) eG h(g)g is defined by

f*ah= ]T f(9Ùa92(h(92))gig2,

where a : G — Aut(A), g \-> ag. For every g G G, we have (f*ah)(g) = Y^.,eG f(s)aÂHs~) 9))-

The algebra GC(G, A) is a *-algebra with involution given by

r(9) = og(f(g-1)T,

for all / G CC(G, A), g G G. Similarly we can define the space l2(G, A) of functions f : G —+ A

with Eg£c7/(#)*/(#) convergent. The norm given by ||/|| := || £(/eG/(</)*/(<?)IU turns

£2(G, ^4) into a Banach space, / G l2(G, A). The left regular representation \g,a of GC(G, A)

on l2(G, A) is given by

(\Uf)h)(g)^Y,a^{f^))K^l9),seG

for / G Cc(G,A),h G /2(G,A),<? G G, so that GC(G,,4) acts on l2(G, A) by bounded opera¬

tors.

Definition 2.5.12 (reduced crossed product). Let G be a group and A be a G — C*-algebra.

The reduced crossed product A x>r G is the operator norm, closure of AG a(Cc(G,A)) in

B(l2(G,A)).

Example 2.5.13. The reduced crossed product C xr G is the reduced C*-algebra C*G of G.

One can also see that reduced crossed products give rise to a generalized formulation of

the Baum-Connes Conjecture, namely the Baum-Connes Conjecture with Coefficients. More

about in [ChEcOy].

2.5 Kasparov's Equivariant A'A'-Groups 43

Remark 2.5.14. It is interesting to see that the action ofCc(G,A) on l2(G,A) comes from

the combination of the shift action A ofG on l2(G, A) given by (X(g)h)(s) — h(g~ls), with the

action n of A on l2(G, A) given by (7r(a)h)(s) = as-i(a)h(s), for a G A, h G l2(G, A), g, s G

G.

Theorem 2.5.15 (Kasparov). Let A,B,C be (separable) G — C*-algebras. Then there is a

bi-additive pairing, for i,j, i + j G Z2, called Kasparov product,

KKf(A,B)xKKf(B,C) -> KK^(A,C)(x,y) ^ x®bV

which is associative and functorial in all possible senses. Moreover, for any separable G—C'*-

algebra D there is a homornorphism of extension of scalars

rD:KKf(A,B) - KKf(A®minD,B®rtli/nD),

where the minimal tensor product ®Tn,in wdl be defined in the next Chapter 3, and a descent

homom,orphism

jG : KKf(A, B) - KKi(A xv G, BxrG).

Both are functorial in all possible senses.

Remark 2.5.16. For an explicit description of the descent homom,orphism, jo, see Rem.

5.9.(d) [Va,]. Later we will use the descent, homornorphism in, a particular context to define

the Ba,um-Connes assembly map.

An important notion is the i<P'-equivalence of G*-algcbras.

Definition 2.5.17. Let A, B be C*-algebras. A class a G KK*(A, B) is a KK-equivalence if

there exists ß G KK^B, A) with ct®Bß = id KK*(A, A) and ß®Aa = id& KK*(B, B).

The C*-algebras A and B are KK-equivalent if there exists a KK-equivalence in KK*(A, B).

44 A-theory of G*-Algebras

Chapter 3

The Künneth Formula for the

if-Theory of C*-Algebras

3.1 Tensor Product of C*-Algebras

Let II, K be (complex) vector spaces. We denote by H ® K the (complex) algebraic tensor

product of H and K, which is the (complex) linear span of elements x ®y,x G II, y G K. If

H, K, L are vector spaces and <p : H x K — L a bilinear map, then there is a unique linear

map ip' : H ® K —> L such that <p'(x ® '</) = <p(x,y), for every x e H,y K (Universal

Property of the tensor product). If A and B are algebras, by the Universal Property there

exists a unique multiplication on A ® B such that (a ® 6) (a' ® 6') = aa' ® bb', which endows

A® B with an algebra structure. If A, B are ^-algebras, the following proposition ensures

the existence of an involution on A® B, which makes it into a *-algcbra.

Proposition 3.1.1 (Tensor product of *-algcbras). Let A, B be *-algebras. Then there exists

a unique involution on A® B such, that

(a® 6)*-a*® 6*,

for all a G A,b G B.

Proof. We have to show that it is well defined. For this, we show that ELi ai ® ^ = 0

implies ELi a* ® tf, = ®- Choose linearly independent C\,--- , cm G B having the same

linear span as bi, ,bn and write bt = EJli ^jcj- Since the c,-'s are linearly independent,

E,: j Xijai ® c:i = ° implies^* A^a* = 0, for every j = 1, • •

,m. Therefore, E;Li a* ® b* =

Eij < ® A«^ = £,(E; A^-aJ) ® cj = E, 0 ® a* = 0. D

An immediate consequence is the following.

Theorem 3.1.2 (Universal Property). Let A,B,C be *-algebras, ip : A x B — C a, *-

homornorphism, then there exists a unique * -homornorphism ip' : A ® P —> G suc/i t/iot

<//(a ® &) = <£>(«, &), for alla G A 6 G P.

46 The Künneth Formula for the A'-Theory of G*-Algebras

Corollary 3.1.3. Let A, B, C be *-algebras and ip : A —> C,ip : B —> C be *-hom,omorphisms

such that every element ofip(A) commutes with each element of'tp(B). Then there is a unique

*-homornorphism it : A® B — G such that ir(a®b) = <p(a)ip(b), for all a G A,b G B.

Example 3.1.4. For all n > 1 and ^-algebra A, A ® Mn(C) is ^-isomorphic to Mn(A).

If P, K are Banach spaces, there are in general many possible norms on H ® K which

are related in a suitable manner to the norms of P and K and which are not necessarily

complete. In particular, the tensor product of two infinite dimensional normed spaces is

never complete. When the spaces are Hilbert spaces, however, matters are simple. In fact,

if H, K are Hilbert spaces, there is a unique inner product < -, > on H ® K such that

< x ® y, x' ® y' >h®k=< x, x' >h< V, y' >k, for all x, x' G P, y, y' G A. With this inner

product we shall regard H ® K as a pre-IIilbert space and we denote by H®K its Hilbert

space completion. In the case of G*-algebras, the next proposition assure the existence of

a G*-norm on their tensor product. It is easy to prove that if A,A',B,B' arc *-algebras

and ip : A —> A', 'ip : B —'> B' *-homomorphisms, then </? ® ip : A ® B —» A' ® B' is a

*-homomorphism.

Proposition 3.1.5. Let (H, ip), (K, ip) be Gelfand-Naim,ark representation of the C*-algebras

A, B, respectively. There exists a unique * -homornorphism it : A® B — B(H®K), such that

ir(a®b) = ip(a) ®ip(b),

for all a G A,b G B. Moreover, if <p,ip are infective, so is it.

Proof. [Mu], Thm. 6.3.3.. D

Let A, B be G*-algebras with faithful Gelfand-Naimark representation (H,ip),(K,'p),

respectively, and consider the unique ^-homornorphism it : A®B — B(H®K) of the previous

proposition. Then the map

\\-\\min:A®B^R+,c^\\7T(c)\\

is a G*-norm on the *-algebra A ® B (note that it is injectivc!). The G*-norm || • ||m;„ is

called the minimal (or spatial) G*-norm.

Remark 3.1.6. One can prove that the spatial C*-norm is in fact the minimal C*-norm on

the *-algebra A®B ([Mu], Ch. 6.4J.

Remark 3.1.7. The C*-norm 11 |\min is a cross norm. In other words, \\a®b\\min = ||a||• 116||,

for all a G A,b G B, as \\a ® b\\min - ||7r(a ® &)|| - \\<p(a)\\ \\ip(b)\\ = \\a\\ \\b\\. The

last equation follows from the fact that injectivc *-homom,orphisms between C*-algebras are

necessarily isometric ([Mu], Thm. 3.I.5.J.

3.1 Tensor Product of G*-Algebras 47

Definition 3.1.8 (minimal tensor product). Let A, B be C*-algebras. The C*-completion

of A® B with respect to the minimal C*-norm \\ \\mm is called the minimal tensor product

C*-algebra and is denoted, by A ®min B.

In general, for G*-algebras A, B there may be more than one G*-norm on A ® B. We

denote by A ®7 B the G*-completion of A ® B with respect to a G*-norm 7 on A ® B.

Lemma 3.1.9. Let A, B be C*-algebras and 7 a C* -seminorm on A® B. Then

7(a®b)<\\a\\\\b\\,

for alla G A,b G B.

Proof. [Mu], Cor. 6.3.6.

For two G*-algcbras A, B, the following construction gives rise to an other G*-norm on

A ® B. Let P be the set of all G*-norms on A ® B. The map

|| • \\max •A ® B -* E+, \\c\\max := jsup7er7(c)

is a well defined G*-norm on A®B. To see that, let c ~ Ej aj®bj, with a3 G A, b3 G B. Then

by Lemma 3.1.9, for any 7 G P we have 7(c) < E, "f(aj ® M ^ Ej IKIIIIM- Therefore,

11 c||max < 00. || • \\riiax is in fact a G*-seminorm, because it is submultiplicative and inherits

the G*-property ||c*c||TOaa: = llc-llmox ^rom ^ne seminorms 7 G T of which it is a supremum.

As the minimal norm is a G*-norm wc have || ||maj. > || • ||Amn, so ||c||max only vanishes

when c — 0. Thus || • ||„ttix. is a G*-norm that by construction majorizes any G*-norm. The

G*-norm || • \\max is called the maximal G*-norm. We denote the G*-completion of A ® B

with respect to the maximal G*-norm by A®max B, the maximal tensor product G*-algebra.

Remark 3.1.10. By Lemma 3.1.9 every C*-norm is a cross norm. In, fact, let a be a

C*-norm, on A® B, then

IH|||6|| = ||o®6||mt-ri <rt(a®6)< ||a®6|U.„< ||a||||6||,

for all a®b G A® B.

There is an interesting relation between the G*-complctions with respect to ordered C*-

norms.

Proposition 3.1.11. Let A,B be C*-algebras and a,ß be C*-norms on, A® B, such that

a < ß. The A®a B is a quotient of A ®ß B. In particular, for all C*-norm 7 on A® B,

A ®mir< P is a quotient of A®^ B and A ®7 B is a quotient of A ®max B.

Proof. The inclusion map id : A ® B — A ®a B extends by continuity to a morphism

id : A®ßB -> A®aB when a < ß. As id(A®B) = ,4®P is dense in A®aB, and id(A®,3B)

is closed, it follows that id is surjective and A ®a B is isomorphic to A ®ß B/Ker(id) ([We],

Prop. T.6.24.).

48 The Künneth Formula for the P-Theory of G*-Algebras

Example 3.1.12. Let X be a locally compact Hausdorff space and A a C*-algebra. Then

the embedding

<p:Co(X)®A^CQ(X,A),

defined asip(f®a)(x) = f(x)a, f G C0(X),a G A,x G X is dense. In particular, CQ(X)®minA = CQ(X, A) ([We], Lemma T.6.K5).

Remark 3.1.13. For arbitrary groups G,H the reduced C*-algebra C*(G x H) of G x H

is ^-isomorphic to C*(G) ®min G*(H). We see that in the following way. We recall that by

definition the reduced C* -algebra C*(G x H) is the closure of the group algebra C(G x H) in

B(l2(G x H)). It is easy to see that group algebras satisfy the fact that C(G x H) £* CG®CII.

We have the following inclusions:

c(g xh)^cg®chc c;(G) ® c;(ii) c c;(G) ®min c;(H).

Here ® denotes the algebraic tensor product. Notice that C*(G)®minC*(H) is complete. As

l2(G x H) = l2(G)®l2(H), ® denoting the Hilbert space tensor product, we can therefore see

that the C*-algebra C*(G) ®min C*{H) is also complete in B(t2(G x H)), which show our

assertion.

3.2 Nuclear and of Type I C*-Algebras

Definition 3.2.1 (nuclear). A C*-algebra A is called nuclear, if for every C*-algebra B, the

algebraic tensor product A® B has a unique C*-norm.

Example 3.2.2. a) For each n>\, Mn(C) is nuclear. In fact, for every C*-algebra B,

Mn(C) ® B is ^-isomorphic to Mn(B), which has in fact, a unique CT-norm,.

b) Every finite-dimensional C* -algebra is nuclear. In fact, if A is a finite-dimensional

C* - algebra, it is *-isomorphic to Mni (C) © • • © Mnh (C), for some integers m, , nk.

If B is an arbitrary C-algebra, then A®B is ^-isomorphic to (Mm(C) ® B) © • • ©

(Mnk(C) ® B), which admits a unique C*-norm.

c) Commutative C*-algebras are nuclear ([We] T.6.20.).

d) Let S be a non-empty, upwards-directed set of C*-subalgebras of a C*-algebra A. If

ÖBesB is dense in A and all elements of S are nuclear, then A is also nuclear ([Mu],

Thm. 6.3.10.;.

e) LetlC(l2(N)) be the C*-algebra of compact operators on I2(N). Then Mn(C) C Mn+1(C)

C /C(/2(N)) and UnMn(C) c K(l2(N)) is dense. Therefore, by d), IC(l2(n)) is nuclear.

f) More generally, we can prove analogously to e) that the space of compact operators

fC(H) on a Hilbert space H is a nuclear C*-algebra ([Mu], Ex. 6.3.2.;.

3.2 Nuclear and of Type I C*-Algebras 49

It is very interesting task to determine for which classes of groups the corresponding

reduced group G*-algebras arc nuclear. First we recall the definition of amenable group.

The amenability property has a large number of equivalent formulations, here we give the

most convenient for our context. We recall that the maximal group G*-algebra C^axG of a

group G is defined by the universal property for which any *-homomorphism from CG to the

G*-algebra B(H) of bounded operators on a Hilbert space II factors through the inclusion

CG - C*maxG.

Definition 3.2.3 (amenable). A group G is called amenable if the natural *-rnorphism

C*WJ.G —» C*(G) is an isomorphism.

Proposition 3.2.4. The reduced C*-algebra C*(G) is nuclear if and only if the group G is

amenable.

Proof. The assertion follows directly from our ad hoc definition of amenability. If we are

using other definitions of amenability involving probability measures or Fölner sequences,

the fact that the reduced G*-algebra C*(G) is nuclear if G is amenable is proved in ([HiRo],Ex. 3.3.4). A proof of the converse is given in ([HiRo], Rem. 3.3.5.). O

The class of amenable groups is quite large. Finite, abelian, solvable groups are all

examples of amenable groups. Moreover, subgroups of amenable groups are amenable and

a (finite) direct product of amenable groups is amenable. On the other hand, there are

non-amenable groups, and consequently non-nuclear G*-algebras. The reduced G*-algebra

of a non-abelian free group is for example not nuclear.

Remark 3.2.5. In general, the space of bounded operators B(H) on a, Hilbert space H is

not nuclear ([We], T.N.).

The next two propositions give two important properties of nuclear G*-algebras. The

first one states in particular that the tensor functor — ®min T> is an exact functor if D is

a nuclear G*-algebra. The second one says that the class of nuclear G*-algebras is closed

under extensions.

Proposition 3.2.6 (short exact sequences). Let A, B, J, D be C*-algebras and

be a short exact sequence of C*-algebras. If B ® D has a unique C-norm, (this is the case if

B or D is nuclear), then

-> J <SW D — A ®mm Ü -* B ®min D ~> 0

is a short exact sequence of C*-algebras.

50 The Künneth Formula for the A'-Theory of C*-Algebras

Proof. [Mu] Thm. 6.5.2. G

Proposition 3.2.7 (extension of nuclear by nuclear). Let A, B, J be C-algebras and

be a short exact sequence of C*-algebras. If J and B are nuclear, so is A.

Proof. [Mu] Thm. 6.5.3.

Remark 3.2.8. The class of nuclear C-algebras is also closed with respect to quotients,

inductive limits, crossed products by amenable groups and tensor products. It is not known

whether the class of nuclear C*-algebras is the smallest class of C*-algebras closed with

respect of those operations (see [Bl], p. 155;. It is not true in general, that a C-subalgebra

of a nuclear C*-algebra is always nuclear ([Bl], p. 155;.

Remark 3.2.9. Let G be an amenable group and X a locally compact proper G-space. Then

the reduced crossed product C0(X) xr G is nuclear.

Definition 3.2.10 (type I). A separable C-algebra A is called of type I, if every non-zero

C* -algebra quotient, of A admits a non-trivial closed ideal L, such that for every irreducible

representation it of L and every element x G L, tt(x) is compact.

Example 3.2.11. Commutative and, finite dimensional C*-algebras are of type I.

Proposition 3.2.12. Let M be a closed ideal, in a C-algebra A. Then A is of type I if and,

only if both M and A/M are of type I.

Proof. [Mu] Thm. 5.6.2.

Remark 3.2.13. A consequence of the last proposition is that a C*-algebras A is of type I

if and only if its unitalisation Aj is of type I.

Theorem 3.2.14. The class of nuclear C-algebras contains all C* -algebras of type I.

Proof. [Bl] Thm. 15.8.2.

Remark 3.2.15. In the literature, type I algebras are often called postlimmal or CCR alge¬

bras.

3.3 The Bootstrap Class B 51

3.3 The Bootstrap Class B

A very important class of nuclear G*-algebras is the Bootstrap class B, which is defined as

follows.

Definition 3.3.1 (Bootstrap class B). The Bootstrap class B is the smallest class of (sepa¬

rable) nuclear C^-algebras with the following properties:

(B\) B contains C.

(B2) B is closed under inductive limits.

(B'i) I/O—>J—»A—>B—>-0isa short exact sequence of C* -algebras, and two of the terms

are in B, so is the third.

(BA) B is closed under KK-equivalence.

Example 3.3.2. i) All type I algebras and inductive limits of type I algebras are in the

Bootstrap class B. In particular, commutative and finite dimensional algebras are in

B.

ii) The nuclear C*-algebra JC(H) of compact operators on a Hilbert space H is in the

Bootstrap class B.

We recall that the class of elementary amenable groups is the smallest class of groups

which contains all finite groups, all abelian groups and is closed with respect to taking

subgroups, extensions, factor groups and union of directed systems of such groups. This

class is strictly larger than the class of solvable groups, but properly contained in the class

of all amenable groups, in particular the corresponding reduced G*-algebras of elementary

amenable groups are nuclear. More is explained in the next example.

Example 3.3.3. By the inductive construction of the Bootstrap class B we can see that the

reduced C*-algebras of elementary amenable groups are in the Bootstrap class B.

Remark 3.3.4. The Bootstrap class B is closed under tensor products. We can see this

again by its inductive construction. If a C*-algebra B lies in B, it is built up from, C by the

operations (Bl) - (BA). Then for A G B also a tensor product A® B can be built up from

A ® C by the same operations ([Bl], Rem. 22.3.5;.

Remark 3.3.5. We can see that the Bootstrap class B is the smallest class of nuclear C*-

algebras containing the commutative C-algebras and closed under KK-équivalence, cfr. [Bl],

23.10.3.

52 The Künneth Formula for the P-Theory of C*-Algebras

3.4 The Künneth Formula for the i^-Theory of C*-

Algebras

For a (separable) G*-algcbras A and P there is a Z/2Z-gradcd pairing

a(A, B) : KP(A) ® Kq(B) - Kp+q(A ®mm B),

p, q G {0,1}, which is natural in A and B. This map is defined using the Kasparov product;

we can find the details in [Bl], 23.1. When it is clear we will denote a(A,B) just by a.

For the formulation of the Künneth Formula we refer to Chapter 23 of [Bl], based on the

standard arguments given in the work of J. Rosenberg and C. Schochet [RoSc],

Notation (graded tensor product and torsion)Let A, B be G*-algebras. We use the following usual notation:

(K(A)^K(m-J (Ko(A)®Kq(B))®(K](A)®K,(B)), ifi = 0

1 *l } ® A ))% -\ (K0(A) ® K,(B)) © (K,(A) ® Kq(B)), if i = 1

'

The same for the torsion groups:

Tnr(K(A) tC(P,)) _ { ToT(Kq(A),Kq(B)) ®Tor(K,(A),K,(B)), */ * = 0

1 *l h A )h~\ Tor(KQ(A),Ki(B))(BTor(Kl(A),K0(B)), if i = l

'

Theorem 3.4.1 (special Künneth Formula). Let A and, B be C-algebras, with A G B and

B with torsionfree K-theory. Then the map

a : (K*{A) ® K*(B))t -> K;(A ®mm B)

is a natural isomorphism, i G {0,1}.

Proof. Let B be a G*-algcbra with torsionfree P-theory, and let A be the class of all C*-

algcbras A for which a(A, B) is an isomorphism. We want to show that A contains the

Bootstrap class B. The class A contains obviously C. If A — lim^An, then A ®rrun B =

lim^(An®nullB), so if each An is in A, and A also since A'-theory commutes with G*-algebra

inductive limits, and this shows that A is closed under inductive limits. Wc consider a short

exact sequence of G*-algebras 0—>./—>.A—»A/J—>0 and we tensorize the corresponding

six-term P-theory exact sequence with K*(B). If two of A, J, A/J are in A, we apply the

Five Lemma 4.4.3 to conclude that also the remaining algebra is in A. Then, A is closed

with respect to PP-equivalence. Therefore, A contains B. D

Remark 3.4.2. The special Künneth Formula 3.4-1 holds true even if we require A to be in,

the Bootstrap class B and A (instead of B) to have torsionfree K-theory.

3.4 The Künneth Formula for the P-Theory of G*-Algebras 53

Example 3.4.3. The C-algebra K(H) of compact operators on a Hilbert space H is in the

Bootstrap class B and has K-groups given by

Therefore for any C*-algebra A there is a natural isomorphism

Ki(A®rrvinK(H))^Ki(A),

for i G {0,1}.

In order to prove a general Künneth Formula for A'-Theory of G*-algcbras we need the

following lemma. Wc recall that wc denote by SB the suspension of B, by CB the cone on

B and by K(H) the space of compact operators on a Hilbert space P.

Lemma 3.4.4. Let B be a (separable) C-algebra. Then there exist a (separable) C*-algebra

F with free K-theory, a Hilbert space H and a homornorphism <p : F -» SB ®min K,(H) such

that <p* : AT*(P) — K*(SB) is surjective.

Proof. Prop. 23.5.1., [Bl].

Remark 3.4.5. The previous Lemm,a allows to construct a so called geometric realisation of

a given (separable) C-algebra B. Let C be a C*-algebra such that A*_i(G) = K*(B), e.g.

C = SB. Let F be as in previous lemma, and <p : P -> SC ®mtn K.(H) with (p* surjective.

Then Ki(SC ®min K(H)) =* K(SC) ^ Kl+i(C) = K(B). So we get a geometric realisation

ofK.(B).

In order to prove the next Theorem, we need also the following property. Let A and B

be G*-algebras, then S(A ®min B) = A ®min SB. We see this in the following way. Let

g G SB, then for an arbitrary chosen a G A, the map

/ : [0,1] -» A ® B, t^a® g(t)

defines an element in S(A ® B), as /(0) = a ® g(0) - 0 = a ® g (I) = /(l). As A ® B C

A ®miu B, the map / can be viewed also as an clement in S(A ®miu B). Conversely, let

/ G S(A ®mm B) and consider an arbitrary linear functional a : A -» C. Consider the

induced linear map

a ® idB : A ®min B — C ®min B.

Then (a ® idB)(f) defines an element in SB, as (a ® idB)(f)(0) = (a ® idB)(f(0)) = 0 -

(a®idB)(f(l)) ^ (a®idD)(f)(l).

Theorem 3.4.6 (Künneth Formula). Let A and B be C-algebras, with A G B. Then for

every i G {0,1} there is a natural short exact sequence

0 - (K*(A) ® K*(B))i - Ki(A ®m,„ B) - Tor(K*(A), K^B))^ -* 0.

54 The Künneth Formula for the P-Theory of C-Algebras

Proof. By Lemma 3.4.4, let P be a G*-algebra with free P-theory and p : P —> B a

homornorphism such that 0* is surjective (if necessarily replace B by SB ®min JC(H)). As

in Rem. 3.4.5, we consider the mapping cone sequence

0 - SB -> G^A P - 0.

Since A is nuclear, By Prop. 3.2.6 the following sequence is also exact:

0 -> A ®min SB -* A ®min Cfï>A ®min F -> 0,

where \x — id ®min v. The associated A'-theory exact sequence is

Ki(A ®min Q) -^ Ki(A ®min F) -> K(A ®min B) -

-> Pi_i(A ®min C0) ^ Äj_i(4 ®„„-„ P) -> • • •

.

Here we use the previously showed property that S(A®minB) = A®minSB, hence, by Prop.

2.4.5 and Prop. 2.4.6 we get K,^(A ®min SB) =* I<i-i(S(A ®mUl B)) = K(A ®min B), for

every i G {0,1}. From the previous long exact sequence we obtain the following short exact

sequence:

0 -> Coker(m) -» K(A ®min B) -> Kerfa^) -> 0.

On the other hand, since the resolution of B is free, the following sequence is exact:

0 -> Tan^{KM),K.{B)) -> (P,(A)®P,(G^)), ^ (A^(J4)®P;(P))i - (P;(A)®P*(P))7- - 0,

where ^ = id ® z/j. By Lemma 3.4.4, since F and C^ have torsion-free A'-theory, the group

K*(A) ® A'* (P), resp. P* (A) ® P* (G^,), is isomorphic to K* (A ®rnin F), resp. AT, (4®min C^).This identification replaces ipi by /i,/. Thus Coker(/j,i) = (A'*(A) ® K*(B))j, and A'er(/^.„i) =

Torv^AT,^), P»(P)), for all z{0,1}. D

Remark 3.4.7. There are C-algebras which do not, satisfy the Künneth Formula. This fact

was illustrated by G. Skandalis by observing that any C-algebra which satisfy the Künneth

Formula is automatically K-exact, i.e. the functor B i— K*(A ®„än B) is half-exact. To see

this we refer to [HiLaSk] and [ChEcOy]. By [HiLaSk], any exact, sequence 7 : J —> B — B/J

admits a double mapping cone C(-y) with Ä*(G(/y)) — 0. This means equivalently that

the sequence P*(7) : AT*(J) —> K*(B) —> K*(B/J) is exact. Now, if A is not K-exact,

there exists a short exact sequence 7:0—> J ^ B —> B/J —> 0 such that K*(A ®n

J) —> K*(A ®min B) —> K*(A ®min B/J) fails to be exact. So we have K*(C(-y)) = 0 and,

K*(A ®min G(7)) = K*(C(A ®mia l)) ¥" 0, which means that A cannot satisfy the Künneth

Formula. An example of a non-separable C-algebra A which is not K-exact is given by

A = \\*L1Mn(C) ([ChEcOy], Rem. 4.3;. It is still an open question whether all, nuclear

C-algebras satisfy the Künneth Formula.

3.5 The Universal Coefficient Theorem for the PT-Theory of G*-Algebras 55

3.5 The Universal Coefficient Theorem for the

if-Theory of C*-Algebras

In this section we discuss a Universal Coefficient Theorem for the A'-Theory of G*-algebras

and a related interesting application.

Proposition 3.5.1. There exists a C-algebra C G £ such that Kq(C) = Q and P'i(G) = 0.

For every n G N there exist C-algebras Dn G B such that, KQ(Dn) = Z/nZ and I<i(Dn) = 0.

The C-algebras C and Dn are unique up to KK-equivalence.

Proof See [Bl], Ex. 23.15.6 - 7. The uniqueness follows by Cor. 8.3 [RoSc]. D

By the previous proposition, the next definition makes sense.

Definition 3.5.2 (A'-Theory of G*-algebras with coefficients). Let A be a C-algebra. We

define the K-Theory ofA with coefficients in Q by K,(A; Q) := K,(A®miriC)- For every n G

N, we define the K-Theory ofA with coefficients in Z/nZ by K*(A; Z/nZ) := K*(A<8>mmDn).

In the literature the A'-Theory of G*-algebras with coefficients in Z/nZ is also called

m,od — n A'-theory.

Theorem 3.5.3 (Universal Coefficient Theorem). Let A be a C*-algebra, then fori G {0,1}

the following assertions hold true:

i) There is an isomorphism of groups Ki(A; Q) = Kt(A) ®% Q.

it) The sequence 0 -> Ki(A)®Z/nZ -> KL(A;Z/nZ) - Tor(K1.1(A),ZjnZ) -> 0 is an

exact, sequence of abelian groups.

Proof. i) By definition, let G G B such that K»(A;Q) = K*(A ®mtn C). Recall that

the AT-theory of G is given by KU(C) = Q and Ki(C) = 0, in particular torsionfree.

Therefore by the special Künneth Formula 3.4.1 we have

KQ(A ®min C) * [K*(A) ® K*{C)]o - Kq(A) ® A'o(G) © K,(A) ® KL(C) = K0(A) ® Q,

and analogously

K:(A®minC) * [K,(A)®K\(C)}1 = K,(A) ® K0(C) @KQ{A) ® Ky{C) = K\(A)®Q,

which shows the assertion.

ii) Let now Dn G B such that Kt(A;Z/nZ) - K*(A ®mm D). By the Künneth Formula

3.4.6, for i G {0,1} we have the following short exact sequence:

0 - (K*(A) ® K*(Dn))L -+ Ki(A ®mm Dn) - Tor(K*(A), P*(Dn));_i -+ 0.

56 The Künneth Formula for the P-Theory of G*-Algebras

Therefore, recalling that the Af-theory of Dn is K0(Dn) = Z/nZ and K1(Dn) ^ 0, in

case i — 0 we have

0 - K0(A) ® Z/nZ - K0(A ®min Dn) -> To^K^A), Z/nZ) -+ 0,

and for i — 1

0 -> Kl(A) ® Z/nZ -> Pi (A ®min Dn) - Tor(A'0 (A), Z/nZ) - 0,

hence the assertion follows.

D

Lemma 3.5.4. Let M be an abelian group with the property that M ®-£ Q = 0 a?id ./'or a//

prime p Torf(M, Z/pZ) = 0, so M = 0.

Proof. Consider the short exact sequence 0 - Z -> Q -> Q/Z -» 0. By tensoring with M

we get the exact sequence

Tor(M, Q/Z) -> M ®z Z -» M ®z Q - M ®z Q/Z,

where by assumption M®ZQ - 0. We write Q/Z = ®pZ/p°°Z, where Z/p°°Z = wlim^nZ/pnZ.

By assumption Tor(M,Z/pZ) = 0, therefore also Tor(M,Z/pnZ) = 0 and since taking col-

imit commutes with the Tor(M, -)-functor, Tor(Af, Q/Z) = 0. Therefore M £ Af ®z Z =

0.

In order to prove the next theorem, we recall Prop. 2.4.14, which asserts that for a

-homornorphism (p : A —> B between two G*-algcbras A, B, the induced morphism <p* :

K*(A) — A^(P) is an isomorphism if and only if K*(CV) = 0, for * = 0,1.

Theorem 3.5.5. Let A,B be C -algebras and, <p : A —> B be a *-homornorphism. The in¬

duced, morphism ip* : K*(A) —> AT*(P) is an isomorphism if and only ifp(-;Q) K*(A;Q) ->

K*(B;Q) and ip(-;Z/pZ) : K*(A;Z/pZ) - K*(B;Z/pZ) are isomorphisms, for all p prime.

Proof. If K+{A) ^ K*(B), using Thm. 3.5.3 i) wc have that Ki(A;Q) ^ K{(A) ®z Q ^

I<i(B) ®z Q = Ki(B; Q), which shows that ip(-\ Q) is an isomorphism. By Thm. 3.5.3 ii) we

get a square commutative diagram

0 Ki(A)®Z/pZ > K(A;Z/pZ) > Tor(Ki.1(A),Z/pZ) > 0

a* v(-;/VpZ) -

0 > Ki(B)®Z/pZ Ki(B;Z/pZ) Tor(K^(B),Z/pZ) —+ 0.

By assumption the two external vertical arrows are isomorphisms. By the 5-lemma so is

ip(-;Z/pZ) and the assertion follows. Let us now show the other direction of our theorem.

Let C,DP&B such that K,(A; Q) = K»{A®min C) and K*(A\ Z/pZ) = A'*(A ®min Dp). For

3.5 The Universal Coefficient Theorem for the P-Theory of G*-Algebras 57

the morphism ip : A —> B we denote by ipc A®minC —> B®nanC and by <pDp ' A®mmDv —>

B®nunDp the induced morphisms. By assumption, the induced morphisms K*(tpc) — ¥>(; Q)and K*(<pv) — <p(-;Z/pZ) are isomorphisms for all p's. This is equivaleiitly with the fact

that the A'-theory of the corresponding mapping cones vanish, in symbols A'*(GV(7) ~ 0 and

k*(cv>dp) = °- We note that ^'Vc = C^^nCand CVDv = Cv®min Dp. Using the definition

of the K-theory with coefficients we get therefore that

T/im.3.5.3

0 = MCvo) = K>(CV ®min C) = K*(CV; Q) * üf.(C„) ® Q,

and

0 = K*(CVDp) = Ä,(C„ ®mm Dp) = K+iCv, Z/pZ),

for all p's. Therefore, by Thm. 3.5.3 we get a short exact sequence of the type

0 -> #,(0,,) ® Z/pZ - KL(C^ Z/pZ) = 0 -^ Tor(AVi(CV), Z/pZ) -+ 0,

from which follows that Tar{Ki-x{Cv),Z/pZ) = 0, for all i G {0,1}. This shows that all

the assumptions of Lemma 3.5.4 arc satisfied for the abelian groups AT*(GV). Therefore

K*(Ctp) = 0, and the assertion is proved. D

58 The Künneth Formula for the P-Theory of G*-Algebras

Chapter 4

Equivariant Ä-Homology Theory

4.1 Equivariant (Co)-Homology Theories

In this section we give the abstract definition and state the main properties of equivariant

(co)-homology theories.

Definition 4.1.1 (G-equivariant homology theory). Let G be a group. A G-equivariant

homology theory hG(-) is a collection of covariant functors hG(-), n G Z, from the category

of G-CW-pairs to the category Ab of abelian groups, together with natural transformations

0G : hG(X,A) -> hG_x(A) := h^_,(A,Hl) for any G-CW-pair (X,A), such that the following

conditions are satisfied:

• (G-homotopy invariance) If f,g : X —> Y are two G-homotopic maps between G-CW-

complexes X and Y, then h°f = hGg for every n G Z.

• (exact sequences) For any G-CW-pair (X, A), the sequence

• • • -> hGn(A) S t%{X) ^ hG(X, A) S h'UA) ->

is exact. In this diagram,, i is the inclusion map i : A^ X, and j is the map of pairs

(X, 0) — (X, A) induced by the identity map on X.

• (excision) If (X, A) is a G-CW-pair, B a G-CW-complex and f : A -> B is a cellular

G-map, then there is an isomorphism

hG(i,f):hG(X,A)^hG(XUfB,B),

where i is the inclusion.

60 Equivariant P-Homology Theory

• (additivity) If {(Xa, Aa)} is a collection of G-CW-pairs, then for every n G Z there is

an isomorphism

(&hGia :Q)hG(Xa,Aa) -> hG([[Xa,l[Aa),a cv

where ia : Xa ^-> ]J Xa are the inclusion maps.

Definition 4.1.2 (equivariant homology theory). LetQrps be the class of groups. An equiv¬

ariant homology theory h~(-) is a collection {ha(-)}Gegrps of G-equivariant homology theo¬

ries equipped with an induction structure, that is, for a group homornorphism (p : H —> G and

an H-CW-pair (X, A) such that Ker(ip) acts freely on X, there is a natural isomorphism

indv : hj[(X,A) -> hG(ind^X, A))

satisfying the following properties:

• (compatibility with boundaries) dG o ind^ = indv o d" ;

• (functoriality) If ip : G —> V is a second group homornorphism, such that Ker(yj o ip)

acts freely on X, then

ind^0lf> = hln(hQ) o ind^ o indv,

where h0 : inditindv(X, A) ^ indlpolfi(X,A),(k,g,x) i-> (ktp(g),x) is the canonical

r-homornorphism;

• (compatibility with conjugation) If cQ denotes the conjugation of G by the element

g G G, then indCg = hc(h) holds true, where h : (X,A) => indCij(X,A),x i-* (e,0_1x)

is the canonical G-homeomorphism.

Let TinQrps denote the class of finite groups. A collection h~(~) = {hc(~-)}ceFinÇrpS with

the same properties is called an equivariant homology theory for finite groups.

Remark 4.1.3. We define equivariant homology theory of arbitrary G-spaces in a similar-

way. We just gave the definition for G-CW-complexes, since in the future our interests will

focus in particular on these G-spaces. In a more general setting one has to be sure to use a

good notation of 'pair' (X, A).

Similarly, a G-equivariant cohomology theory hG(~) is a collection of contravariant func¬

tors hG(-),n G Z, from the category of G-CW-pairs to the category Ab of abelian groups,

together with natural transformations dG : hG(X,A) -> hG+1(A) := h%+l(A,V\), for any

G-CW-pair (X,A), which satisfies the analogous G-homotopy, exactness and excision ax¬

ioms. Moreover, the following additivity axiom should hold. If {(Xa,Aa)} is a collection of

G-CW-pairs, then for every n G Z there is an isomorphism

h'6(]l^,l[Aa) ^Y[hUXa,Aa),a tt

4.2 Equivariant Kasparov ATA"-Homology Theory 61

where ia : Xa ^-> X are the inclusion maps.

A collection {hG(—)}ceprp« of G-cquivariant cohomology theories equipped with an in¬

duction structure satisfying the axioms corresponding to Dcf. 4.1.2 is called an equivariant

cohomology theory /i" (—).

4.2 Equivariant Kasparov ifif-Homology Theory

In this section we define of an important Z/2Z-graded cquivariant homology theory, namely,

the G-equivariant Kasparov K-homology groups KK^X) and KKG(X) of a proper G-

space X. This theory is a special case of the Kasparov P"A'-Theory defined in Chapter 2.

Good references for basic definitions and properties of Kasparov P-homology theory are [Va]and [Ka].

Definition 4.2.1 (G-compact). A proper G-space X is called G-compact if G\X is compact.

We notice that, since G is assumed to be discrete, a proper G-space X is locally compact

if G\X is compact.

Notation

We denote by C0(X) the algebra of continuous functions on X, vanishing at infinity and by

CC,(X) the algebra of compactly supported continuous functions on X.

Hereafter we define Kasparov AT-homology groups for G-compact, proper G-spaces. Anal¬

ogously to Chapter 2 we begin with the definition of Kasparov triples, equivalence of Kas¬

parov triples and afterward Kasparov A'-homology groups. We will also precisely consider

the analogy with Chapter 2.

Definition 4.2.2 (Kasparov triple). Let X be a G-compact, proper G-space. A Kasparov

triple (also called abstract elliptic operator) on X is a triple (H,tt,F) consisting of an Hilbert

space H endowed with a unitary representation U of G, a G-covariant representation tt of

Cq(X) on H (covariant in the sense that ir(fog-x) = UgTT(f)U^, for all f G CQ(X), g G G)

and a bounded self adjoint operator F on H, which is G-invariant (that is, FUy = UgF, for

all g G G) and such that [F, ir(f)] and ir(f)(F2 - 1) are compact operators on H for any

fzC0(X).

It is very important to remark that Dcf. 4.2.2 agrees exactly with Def. 2.5.6 of Kasparov

triples over the pair of G - G*-algebras (C0(X),C).

Definition 4.2.3 (G-equivariant, properly supported). A Kasparov triple (H,tt,F) on X

is called G-equivariant if the operator F is G-equivariant, i.e. if [g,F] = 0 for any g G

G. It is called properly supported if for any f G CC(X) there exists h G CC(X) such that

(7T(h) - l)FTT(f) = 0.

'

62 Equivariant P-Homology Theory

Remark 4.2.4. By Lemma 2.2.2 and Prop. 2.2.1 of[VaKu], from now on we shall assume

that all Kasparov triples are G-equivariant and properly supported.

Definition 4.2.5 (even and odd Kasparov triples). A Kasparov triple (H,ir,F) on X is

even (odd otherwise) if the Hilbert space H is Z/2Z-graded and U, it preserve the graduation

whereas F reverse it. This means that H = Hq®'Hi, and, in that decomposition

Definition 4.2.6 (equivalence between Kasparov triples). i) A Kasparov triple a = (H, it, F)

is called degenerate if [P, 7r(/)] = 0 and tt(J)(F2 - 1) = 0 for every f G Cq(X).

ii) Two Kasparov triples a0 — (H0,ttq,Fq) and a\ = (TCi,tti,Fi) are called homotopic if

Uq = Ui,ttq = iti and there exists a norm continuous path (Ft)je[o,i]> connecting F0 to

Fi, such that the triples at = (7it,7Tt,Ft) are Kasparov triples (of the same parity).

Hi) Two Kasparov triples aa and a\ are called equivalent and denoted by a0~ oi if there

exist two degenerate cycles ßo,ßi such that, up to unitary equivalence, a0 © ßo is ho¬

motopic to tti © ßi.

Wc can verify that this definition of equivalence of Kasparov triples agrees with that

given in the more general context of G*-algebras given in Chapter 2.

Definition 4.2.7 (G-equivariant Kasparov P-homology group of proper G-compact spaces).

Let X be a G-compact, proper G-space. We denote by KKG(X) the set of equivalence classes

of even Kasparov triples and by KKG(X) the set of equivalence classes of odd Kasparov

triples. In case G is the trivial group we write only KK0(X) and KKi(X).

Example 4.2.8. For a G-compact proper G-space X, as a comparison of Def, 2.5.6 with

Def. 4.2.2 we get that KKG(X) = KKG(C0(X), C).

More generally, for proper G-spaces, hence in particular for proper G-CW-complexes, we

define G-cquivariant Kasparov PA~-homology groups as follows.

Definition 4.2.9 (G-equivariant Kasparov PP-homology group). Let X be a, proper G-

space. For i G {0,1}, we define the i-th G-equivariant Kasparov K-homology group by

RKG(X) := colimy x

KKG(Y).

Y G-compact

Definition above represents a so called compactly supported P"G'-homology Theory.

4.3 Categorical Definition of the Equivariant A'-Homology Theory 63

Remark 4.2.10. A result of Kasparov (see [Kal]) shows that for proper G-spaces the defini¬

tion of RKG(-) is invariant under proper G-homotopies. That is if f,l : X — Y are proper

G-homotopic maps between proper G-spaces X, Y, then

U = U:RKG(X)-*RKf(Y),

for i G {0,1}. One could give a weaker definition of homotopy, by dropping the requirement

of the representations ttq and tti to be equal. Instead, one requires to admit a continuous

path t i—> ni of representations joining ttq with tti .In [Kal] Kasparov proves that the groups

constructed by means of this weaker definition are isomorphic to the one we previously de¬

fined.

Proposition 4.2.11. Fori G {0,1}, RKG : X h+ RKG(X) is a functor from the category

of proper G-spaces to the category of abelian groups. More, the collection {RKa(—)}aegrpa

is a Z/2Z-graded equivariant homology theory on the category of proper G-CW-pairs, for

i£{0,l}.

Proof. See Prop. 4.2.7, [Va]. D

4.3 Categorical Definition of the Equivariant

if-Homology Theory

The aim of this chapter is to define equivariant P"-homology-theory KG(-) proposed by J.

Davis and W. Liick in [DaLu], whose arguments are arbitrary (that is, not necessarily proper)

G-CW-complcxes, and in case of proper G-CW-complcxcs the definition agrees with the

Kasparov KK-homology theory, that is KG(X) = RKf(X) for all proper G-CW-complex

X.

Let us first introduce some basic material on spectra. A pointed space is a compactly

generated space with a distinguished base point. We recall that the smash product X AY

of two pointed spaces X, Y is the quotient of the product space X x Y (with its compactly

generated topology) under the identification (x',yo) ^ (xo,y)i lor all x G X, y G Y and

basepoints x0 G X, y0 G Y. We denote by Sn the n-sphere, and wc recall that SnASl = Sn+}.

Definition 4.3.1 (spectrum). A spectrum S is a collection ofpointed spaces {Si}iez, together

with pointed m,aps o,\ : St AS1 —-> St+\, called structure maps. A morphism f : S —> S' between

two spectra S,S' is a sequence of pointed maps ft : St -+ S[ which are compatible with the

structure maps a\, a\, i.e. /t+1 o aL — a[ o (f\ A id$i), i G Z.

Spectra and morphisms between spectra define a category denoted by SV-

Example 4.3.2. a) The sphere spectrum Sph has SL — S7 and a-t = IdSt+i (up to home-

ornorphism), ieN.

64 Equivariant A'-Homology Theory

b) Let S = {.%}iez be a spectrum and Y be a pointed space. The smash product of Y and

S is the spectrum, Y A S — {Y A Si}iez with obvious structure maps.

In the next definition we use the standard notation [Sk+t,St], to indicate homotopy

classes of base point preserving maps Sk+t —» Sj. Wc write susp to indicate the morphism

which associates to a space its reduced suspension and wc identify susp(Sn) with Sn A S1.

Definition 4.3.3 (homotopy groups). Let S = {St}iez be a spectrum. The k-homotopy

group of S is defined by TTk(S) = colimiTTk+i(Si), where the direct limit is given by using the

structure maps o~i:

TTk+i(SA = [Sk+\ St],s^' [Sk+i+\ St A S% (^1* [Sk+i+1,6'f+1]. = TTk+l+i(S,l+i),

where k,i G Z.

Example 4.3.4. The homotopy groups of Sph are the stable homotopy groups of the zero

sphere 7rfc(Sph) — 7r£l(S°). In general, for a pointed space Y, irk(Y A Sph) = ^(Y), k G Z.

Let us define the category il — SV of ü-spectra.

Definition 4.3.5 (il — SV). A spectrum S is called a il-spectrum if for each structure map,

its adjoint Sj —* flSi+i '= rn,ap,(Sl, Si+i) ^s a weak homotopy equivalence of spaces. A weak

homotopy equivalence of spectra is a map f : S —> S' of spectra inducing an isomorphism on

all homotopy groups. The category il — SV is the corresponding full subcategory of SV.

Example 4.3.6. The most famous examples of fl-spectra are the Eilenberg-Mac Lane spec¬

trum H and the Bott spectrum BU. The Eilenberg-Mac Lane spectrum is given by H = {Pt|,

with Hi = P(Z, i) the Eilenberg-Mac Lane CW-complex characterized by 7r,;(P(Z, i)) = Z

andiTj(K(Z,i)) = 0 ifi ^ j, i,j G N. Structure maps are ui : K(Z,i)AS1 —> K(Z,i + l) are

the adjoint of the natural equivalence K(Z, i) —* Q,K(Z, i + 1) — map,(S1, K(Z, i + 1)). The

Bott spectrum is given by BU — {BUi} with BU;L -Zx BU for i even, and BV-, — U for i

odd, BU denoting the classifying space of the unitary group U. The adjoint of the structure

maps ^ : BUi A S1 —> BUi+i corresponds to Z x BU ^ VtU respectively 11 ~ Q,(Z x BU).

Bott periodicity is given by the property that i12U ^ i1(Z x BU) ~ ilBU c± U, and

1T2(Z x BU) ~ W ~ Z x BU.

For a category C, a C-space, a pointed-C-space, a C-spcctrum, respectively a C-O-spectrum

is a functor from C to the category of spaces, of pointed spaces, to the category SV of spectra

and respectively to the category Ü — SV of Sl-spectra.

Definition 4.3.7 (tensor product of (pointed-) C-spaces). Let C be a sm,all category. Let X

be a contravariant, and Y be a covariant (pointed,-) C-space. The tensor product of X and Y

over C is defined to be the (pointed) space

X®c.Y= ]l X(c)xY(c)/~,c.£Obj(C)

4.3 Categorical Definition of the Equivariant P-Homology Theory 65

where (X(tp)(x), y) ~ (x, Y(<p>)(y)), for all morphism ip : c —> dmC and points x G X(d), y G

Y(c).

A C-spectrum S can also be thought of as a sequence {Sl}iez of pointed C-spaces and the

structure maps as maps of pointed C-spaces. With this interpretation, in view of the previous

definition, it is obvious what the tensor product spectrum X ®c S of a contravariant pointed

C-space X and a covariant C-spcctrum S over a category C means. The same definition in

case of C-O-spcctra.

Let us now define equivariant homology theory. The main reference to this approach is

[DaLu].

Let G be a (discrete) group and u(G) = vMl(G) the orbit category of G, whose objects

are orbits G/H, where H runs over all subgroups of G, and with G-maps as morphisms. We

consider the z/(G)-f2-spectrum

l£p = K^(G/7) : u(G) -> il-SV, G/H » K%?{G/H)

constructed in [DaLu]. It is important to recall its fundamental property, that for all sub¬

groups H of G and i G Z there is a canonical isomorphism between TTt(K!Gp(G/H)) and

Kt(C*H), where C*H is the reduced G*-algcbra of P.

We define A'-homology groups of G-CW-complexes as follows. For a G-CW-complex X wc

consider the contravariant functor X\ from v(G) to the category of pointed CW-complexes,

taking G/H to X+ (the "+" means disjoint union with a base point).

Definition 4.3.8 (AT-homology groups of G-CW-complexes). Let G be a, group and X be a

G-CW-complex. The K-homology groups ofX are defined as

KG(X) := tt,(XI ®„(C) K£p(G/?)),

where i G Z.

For a G-CW-pair (X, A) we define KG(X, A) merely as the reduced group KG(XJA), i G

Z. Here, for Y with a G-invariant base point {y0}, KG(Y) - Ker(KG(Y) -» KG({y0})),so that KG(Y) £* KG(Y) © KG({yo})- If A = 0, we set X/A := X+, so that KG(X,0) -

KG(X+) = KG(X). This theory is a G-equivariant homology theory on G-CW-pairs, which

satisfies Bott periodicity, that is, for a G-CW-pair (X, A) there is a canonical and natural

isomorphism

B{X,A):K(:(X,A)^KG]_2(X,A),for all i G Z. We also note that for a subgroup P of G, (G/H)\®u{g)&{GIT) identifies with

the ^-spectrum K]GP(G/H), so that Kf(G/H) =* K,(C;H) in a canonical way. In particular,

taking P - G, we have G/G - {pt} and KG({pt}) £ Pt(Cf*G). It is important to note that

66 Equivariant A'-Homology Theory

for a proper G-CW-complex X, there is a natural isomorphism between A'-homology groups

and Kasparov KA'-homology groups of X, that is,

Af(X) s RKf(X)

for all i G {0,1}. For details we refer to [Ka2] and [MiVa]. In view of this fact we will

not distinguish anymore between Kasparov ATA'-homology theory and Davis and Lück A'-

homology theory as long as wc are dealing with proper G-CW-complexes. We forget therefore

the PP" and RK notations and use the unique notation Pf(•) for all G-CW-complexes.

The connection with Bredon homology is given by the following "Atiyah-Hirzebruch type"

spectral sequence:

Theorem 4.3.9. [Atiyah-Hirzebruch] Let G be a group and X a proper G-CW-complex.

Then there exists a strongly convergent spectral sequence of the form

El = B?^(X;KG(Gm) = KG+q(X),

for p,q G Z.

Good references on basic material about spectral sequences are [CaEi] and [Sp], while

a compete description of the Atiyah-Hirzebruch spectral sequence is given in [MiVa], Note

that if P is a finite subgroup of G, KG(G/II) ^ Rc(H) in case of q even and KG(G/H) = 0,

otherwise. This will be used in the following way, combining the spectral sequence of Thm.

4.3.9.

Corollary 4.3.10. Let G be a group and f : X -> Y a G-rnap of proper G-CW-complexes

inducing an isomorphism

Hpna(X; Rc) ^ HpnG{Y; Rc)

of Bredon homology. Then f induces an isomorphism

KG(X) ^ KG(Y).

Remark 4.3.11. An analogous result holds true when we work with coefficients in a general

field k.

Remark 4.3.12. Every spectrum, S gives rise to a homology theory on CW-complexes, and

conversely every homology theory h on CW-complexes is representable by a spectrum in the

sense that

hi(X) = 7Ti(X+AS),

for all i G Z, and analogously in case of cohomology theories. The Bott spectrum B seen in

Example 4.3.6 represents the usual K-homology theory K\(-) of CW-complexes,

Ki(X)=7T,;(X+AB\J),

i G Z. In case G is trivial, I<\a](X) = K,(X) for all CW-complexes X, i G {0,1}. We can

find more details in [MiVa].

4.4 The Künneth Formula 67

In the sequel we state some more properties of cquivariant A'-homology theory.

Example 4.3.13. Let G be a finite group. Then

KG({pt}) = Kf(G/G) = K(C;G) = Pi(CG) = | Rc(G) , ifi is even,

0,

otherwise.

Equivariant P-homology groups possess an important property if the action of the group

on the space is free.

Definition 4.3.14 (free action). A left action of a group G on a set X is called free if, for

every x G X, the only element in G that stabilizes x is the identity in G, that is, gx — x

implies g ~ e. A G-space is called free if the action of G on X is free.

Proposition 4.3.15. Let X be a free G-CW-complex. Then there exists a canonical iso¬

morphism

KG(X) * Kt(G\X),

fori G {0,1}, where the CW complex G\X is endowed with the quotient topology.

Proof. Prop. 4.2.9, [Va]. For more details see [Ka2]. D

4.4 The Künneth Formula

In this section we want to state a Künneth Formula for the G-equivariant P-homology

theory. Let us first introduce the following usual notation.

Notation (graded tensor product and torsion)Let G, P be groups and X, Y be G- (resp. H-) CW-complexcs. We use the following usual

notation:

_

/ (KG(X) ® KF(Y)) ©(A'f

(X) ® K?(Y)),if

i - 0(KG(Y\<^KH(V\\ -J ^o \^> «^o V )) W^vi ^;>»Ai \x ))i iJ l~v

tA„(AJ®A. (Y)h~<^ (KG{X]®KH{Y))(B{KG{X)®KH{Y)^ ifi=il

The same for the torsion groups:

Tor(KG(X) K»(Y\\ ^

Î Tor(KG(X), K»(Y)) ®Tor(KG(X), K^Y)), if i = 0

i0nA*lAj,A*^'-\ Tor(KG(X),K«(Y))®Tor(KG(X),K«(Y)), if i = 1

'

In view of what follows, it is natural to conjecture that a Künneth Formula of the following

type holds true always.

Conjecture 4.4.1 (Künneth Formula). Let G, H be groups. Let X be a G-CW-complex and

Y a H-CW-complex. Then there exists a natural short exact sequence

0 - (KG(X) ® K?(Y))i -> KGxH(X xY)^

->(Tor(KG(X),K?(Y)))^^0,

for alii G {0,1}.

68 Equivariant P-Homology Theory

Note that wc do not require X or Y to be proper G- (resp. P-) CW-complexes. In order

to prove some Künneth Formulas we state the following Lemmata.

Lemma 4.4.2. Let G, H be groups and (X, A) a G-CW-pair, such that for every n {0,1}

the groups KG(X,A) are torsionfree. Then the following functors

h% : (Y,B) -> h%(Y,B) := KG*H((X,A) x (Y, B))

and

fc? : {Y, B) ~ k»(Y, B) := (KG(X, A) ® K?(Y, B))n

from the category of H-CW-pairs to the category of abelian groups define equivariant homol¬

ogy theories.

Proof. The first assertion follows because KGxH define itself an equivariant homology theory.

To prove that k/J define also an equivariant homology theory, we use the fact that P'f define

an equivariant homology theory; the axiom of exact sequences holds true because KG(X, A)

are torsionfree.D

Lemma 4.4.3. Consider the following diagram of abelian groups and homomorphisms.

Al _JL_ A2 ^L^ A2 ^L^ Al _Ji_> A5

h h U-

h

D Jl D12 D 33 n J4 D

Assume that each row is exact, that each square is commutative, that f\ is an epimorphism,

f'2 and f\ are isomorphisms, and /5 is a monomorphism. Then /3 is also an isomorphism.

Proof. A simple diagram chase reveals this fact ("5-Lcmma"). D

The next Lemma states a very useful property of homology theories of G-CW-complexes,

Lemma 4.4.4. Let G be a group and X a G-CW-complex. Let h,k be homology theories

on the category ofG-CW-pairs and r : h —» k be a natural transformation of G-equivariant

homology theories. Suppose that r(G/K) is an isomorphism for all subgroups K of G which

are isotropy groups for the G-action on X. Then r(X) is also an isomorphism.

Proof. Let X — UneN Xn be the given G-CW-complex. By definition, for every n G H we

have the following push-out diagram:

Uh.cgS^xG/IL. —+ X-1

O^^xGR X\

4.4 The Künneth Formula 69

where the Pa's run over the isotropy groups of the G-action on X. By the properties of

push-out diagrams, the relative homology theories of the columns are the same:

K(Xn,Xn~') = K( U (Bn,Sn-r) x G/Ha),UaCG

and

K(Xn,Xn~l) = h( ]J {Bn,Sn~1) x G/IQ.Ha<zG

By the Suspension isomorphism (see [MiVa]), we have

K((Bn,Sn~l) x G/Ha) Sä h^i((Bn~\Sn~2) x G/Ha) ^i---

- ^ h^n+i((B\ S°) x G/IIa) * h*_n(G/Hn),

and analogously

K((Bn,Sn~v) x G/IIa) ^ K-n(G/Ha).

Therefore, by additivity of the homology theory and since r(G/Ila) is an isomorphism for

every a, the groups K(Xn, Xn~l) and K(Xn,X7'^) are isomorphic. For every n G N, we

have the following commutative diagrams with exact rows:

- h*+i(Xn,Xn-l^ h^X11-1)

k*+i(Xn,Xn )

r{Xn-1)

- K(Xn-V\

-> h*(Xn)

T(Xn)

+ K(Xn)

K(X\Xn-l^

k*(Xn,Xn-l\

Since X° is discrete, it can be written as X° = \\G/Ka. As t(G/K) is an isomorphism

for every isotropy group AT C G, it follows from the additivity of h and k that t(X°) is an

isomorphism, too. We assume that t(Xt'~1) is an isomorphism. Then by 5-Lemma 4.4.3,

r(Xn) is an isomorphism, too. By induction, r(Xn) are isomorphisms for each n G N. The

assertion of our lemma follows since r(X) = colim r(Xn). D

Proposition 4.4.5 (special Künneth Formula). Let G,H be groups. Let (X, A) be a proper

G-CW-pair such that for every n G {0,1} the groups KG(X,A) are torsionfree. Then for

every proper H-CW-pair (Y, B) and all n G {0,1}, the abelian groups KGxH((X, A) x ("V, B))

and (KG(X, A) ® K^(Y, B))n are naturally isomorphic.

Proof. Consider the equivariant homology theories (restricted on proper P-CW-pairs)

h» : (Y,B) - h*(Y,B) := KGxll((X,A) x (Y, B))

and

A£ : fV, B) » kJr[(Y, B) := (Kc(X, A) ® K?{Y, B))n

70 Equivariant P-Homology Theory

defined in Lemma 4.4.2. Considering (X, A) as a fixed proper G-CW pair, the product

ti : (KG(X, A) ® Af(Y, B))n -, KGxH((X, A) x (Y, B))

induces a natural transformation r : k^ —> h^ of homology theories on proper P-CW-pairs.

By definition, r is a natural isomorphism if t(Y, B) are isomorphisms, for every proper P-

CW-pair (Y, B). In view of Lemma 4.4.4, this is exactly the case if r(H/L) are isomorphisms

for every finite subgroup L of H. Let A be a finite subgroup of II. Then

Pf(H/L) - Kï({pl}) s ifc(L).

The first isomorphism follows from the induction property of the equivariant A'-homology

theory. Since P is finite wc have the second equivalence. By the same induction argument

we have:

KGxH((X,A)x(H/L)) = KGxn(IndG^((X,A)x{pt})) s KGxL((X,A)x{pt}) - KGxL(X,A),

where L acts trivially on (X, A). Resuming, we get

r(P/P) : KG(X, A) ® PC(P) -> KGxL(X, A).

We can now think of r(H/L) as a natural transformation between G-homology theories on

proper G-CW-pairs. By the same argument of before applied on the G-CW-pair (X, A), the

map r(H/L) is an isomorphism if it is an isomorphism on every orbit-space G/N, N C G

finite. So, let N be a finite subgroup of G. Then by the same arguments of before, the

left-hand side of r(H/L) can be viewed as

KG(G/N) ® PC(P) * K?({pt}) ® Rc(L) = Rc(N) ® PC(P).

As well, the right-hand side can be written as

KG*L(G/N) = KG*L(IndGlLL{pt}) - K^L({pt}) ^ RC(N x L) -

^PC(7V)®PC(/,).

The last equivalence is Lemma 1.3.3. Therefore, t(H/L) is an isomorphism for every orbit-

space H/L and the proof is concluded. ETJ

In the next proposition we prove a more general statement. Namely, previous proposition

holds also without requiring the G-CW-pair (X, A) to be proper.

Proposition 4.4.6 (special Künneth Formula). Let G,H be groups. Let (X,A) be a G-

CW-pair such that for every n G {0,1} the groups KG(X, A) are torsionfree. Then for every

proper H-CW-pair (Y, B) and all n G {0,1|, the abelian groups KGxH((X, A) x (Y, B)) and

(KG(X, A) ® K^(Y,B))n are naturally isomorphic.

4.4 The Künneth Formula 71

Proof. Using the same notation as in the previous proof 4.4.5, wc just have to pay more

attention when we show that the map r(H/L) is an isomorphism. In this case we show that

it is an isomorphism on every orbit G/N, where N is any (not necessarily finite) isotropy

subgroup of G. Let A?" be a subgroup of G. By the same arguments, the left-hand side of

t(II/L) can be viewed as

KG(G/N) ® RC(L) £ K?({pt}) ® RC(L) * K*(C*TN) ® PC(P).

The right-hand side can be written as

KGxL(G/N) = KGxL(IndGxxLL{pt}) Ç* K?xL({pt}) ^ K*(C*r(N x L)) * K,(C;N ®min CL)

^k,(c;n)®rjc(l).

The last equivalence is Thm. 3.4.1, since Pc(P) is torsionfree and CL G B.

The same generalisation is possible if we do not require to all P-CW-pairs (Y, B) to be

proper. In this case we need conditions on the PT-theory of some reduced G*-algebras.

Proposition 4.4.7 (special Künneth Formula). Let G, H be groups. Let (X, A) be a proper

G-CW-pair such that for every n G {0,1} the groups KG(X,A) are torsionfree. Then for

every H-CW-pair (Y,B) such that K^(C*L) are torsionfree for all isotropy subgroups L of

the H-action on Y, the abelian groups KGxH((X, A) x (Y, B)) and (Af (X, A)®K?(Y, B))n

are naturally isomorphic, for all n G {0,1}.

Proof. Using the same notation and arguments as in the proof of Prop. 4.4.5, but considering

all subgroups L of H (and not just finite), the map t(H/L) becomes

t(H/L) : KG(X, A) ® K*(C*rL) -* KGxL(X, A).

We note that r(H/L) is in fact a natural transformation between G-homology theories on

proper G-CW-pairs, since we required Kt.(C*L) to be torsionfree for all isotropy subgroups L

of the P-action on Y. To show that it is an isomorphism, we show that it is an isomorphism

on every orbit-space G/N, where N is in this case a finite subgroup of G. The left-hand side

becomes

kg(g/n) ® p,(g;p) s* Pc(ao <g> p*(g;p),

and the right-hand side

KGxL(G/N) = KGxL(IndGlLL{pt}) =* K?xL({pt}) - K*(C;(N x Lj) * P*(CA ®m,in C;.L)

^rc(n)®k,(c;l).

The last equivalence is again Thm. 3.4.1, since Rc(N) is torsionfree and CN GO.

72 Equivariant A'-Homology Theory

If we want to drop the condition to be proper for both CW-pairs (X, A) and (Y, B), the

statement holds true if wc require that for any isotropy group L of the P-action on Y, the

abelian groups K„(C*L) are torsionfree and the reduced G*-algebra C*L lies in the Bootstrap

class B (or the same requiring K*(C*L) to be torsionfree, for all isotropy subgroup L of the

/Paction on Y, and the reduced G*-algebra G*N to lie in B, for all isotropy subgroup N

of the G-action on X), as we have seen in Chapter 3 dedicated to the Künneth Formula of

G*-algebras. The statement is given in the next proposition.

Proposition 4.4.8 (special Künneth Formula). Let G, II be groups. Let (X, A) be a G-CW-

pair such that the groups KG(X, A) are torsionfree for all n G {0,1}. Then for every H-CW-

pair (Y, B) such that for any isotropy subgroup L of the H-action on Y the reduced C*-algebra

C*L lies in B and has torsionfree K-theory, the abelian groups KGxH((X,A) x (Y, B)) and

(KG(X, A) ® K^(Y, B))n are naturally isomorphic, for all n G {0,1}.

Proof. Using the same notation and arguments as in the proof of Prop. 4.4.5 we get the map

r(P/P) : KG(X,A)®KM(C;.L) -+ KGxL(X,A),

for all (not necessarily finite) subgroups L of II. We note that t(I1/L) is in fact a natural

transformation between G-homology theories on G-CW-pairs, since we required A'*(G*P)

to be torsionfree for all isotropy subgroups L of the P-action on Y. To show that it is an

isomorphism, we show that it is an isomorphism on every orbit-space G/N, for all isotropy

groups N of the G-action on X. The left-hand side becomes

kg(g/n) ® a;(g;l) ^ a;(g;ao ® k*(c;l),

and the right-hand side

KGxL(G/N) = KGxL(IndGxxLL{pt}) « K?xL({pt}) * K*(C*r(NxL)) = K*(C;N®nunC;.L).

The last equivalence is given by 3.1.13. In view of Thm. 3.4.1, both sides are isomorphic if

Gr*P G B and all groups K*(C*L) arc torsionfree, which hold true in our case. D

Note that Prop. 4.4.5, 4.4.6, 4.4.7 and 4.4.8 are special cases of Conjecture 4.4.1, saying

that (Af(X)®Af(Y))n ^ KGxH(XxY) as here the Tor-term vanishes. The next Theorem

consists in a reduction to the torsionfree case.

Theorem 4.4.9 (Künneth Formula). Let G,H be groups, X a proper G-CW-complex and

Y a proper H-CW-complex. Suppose there exist a G-CW-complex Z with KG(Z) torsionfree

and a morphism, Z ±> X such that the induced morphism, jl : KG(Z) -* Af(AT) is surjective,

i G {0,1}. The following sequence is exact

0 - (Af (X) ® K?(Y)\ - KGxH(X xY)^

^(Tor(KG(X),KIJ(Y)))i_i^0,

fori G {0,1}.

4.4 The Künneth Formula 73

Proof. Let Z —> X be a G-CW-complex with KG(Z) torsionfree and induced morphism

ji : KG(Z) -» KG(X), i G {0,1}. Wc may assume that j is an inclusion, by replacing X by

the mapping cylinder of /. Note that Z is proper and (X, Z) is a proper pair. We have the

following short exact sequence:

0 -> KG+l(X, Z) - Af(Z) -> Af(X) - 0,

where the groups Afj (AT, Z) and KG(Z) are torsionfree. Consider the abelian group K^(Y).

By applying the functor — ®ZK^(Y) to the previous sequence, we obtain the exact sequence

0 -> Tor(KG(X), K?(Y)) - Af+1(A, Z) ® K?(Y) - (4.1)

£ Pf(Z) ® Kf(y) - Af(X) <g> Af(Y) - 0.

Wc denote by 0 the homornorphism from KG+1(X,Z) ® K?(Y) to Kf(Z) ® K?(Y). The

exactness means that the cokerncl of 4> is Af(AT) <g> Kf(Y) and the kernel of 0 is the torsion

group Tor(KG(X),K[i(Y)). Wc consider now the following long exact sequence:

KG+xH((X, Z) x Y) H1 KGxH(Z xY)^ KGxII(X xY)^

-> KGxH((X, Z)xY)h KG:xH(Z x Y) - •

,

which yields a short exact sequence

0 -> coker(<j>i+i) -> AfxH (AT xY)^ fcer^) -» 0.

Consider the following commutative diagram:

(KG(X,Z)®K?(Y))i+i (KG(Z)®K?(Y))t

K^xH((X,Z)xY) -*»U KGxH(ZxY),

whose vertical morphisms are isomorphism in view of Prop. 4.4.5, since the groups Af_x (AT, Z)

and KG(Z) are torsionfree and the space arguments are proper. By the remarks about the

sequence (4.1), it follows that

coker(<i>i+i) ^ (AfPO ® K?(Y))i.

Similarly,(KG(X,Z)®K?(Y))t —» (Af(Z)®Af (y)).t_!

KGxH((AT, Z) x Y) -£-> KG_\H (ZxY),

shows

kerifa) - Tor(KG(X), Äf(T))i-i-D

74 Equivariant P-Homology Theory

We have the following more general statement, where wc do not require to one space to

be proper.

Theorem 4.4.10 (Künneth Formula). Let G,H be groups, X a G-CW-complex and Y a

proper H-CW-complex. Suppose there exist a G-CW-complex Z with KG(Z) torsionfree and

a morphism Z -^ X such that the induced morphism ji : KG(Z) —> Af(X) is surjective,

i G {0,1}. Then the following sequence is exact

0 - (KG(X) ® Kil(Y))i - KGxn(X x Y) ->

^(Tor(Af(X),A'f(y)))2_1^0,

for i G {0,1}.

Proof. The proof uses the same argument of the previous proof of Thm. 4.4.9. The condi¬

tions of Prop. 4.4.6 are here in fact satisfied, since the groups Kf+1(X, Z) and KG(Z) are

torsionfree and y is a proper P-CW-complex.

Alternatively, wc can skip the request of properness for both CW-complexes X and Y,

by requiring conditions on the reduced G*-algcbras.

Theorem 4.4.11 (Künneth Formula). Let G,H be groups, X a G-CW-complex and Y

a H-CW-complex. Suppose there exist a G-CW-com,plex Z with KG(Z) torsionfree and, a

morphism Z -4 X such that the induced morphism ji : Kf(Z) -> Af (AT) is surjective,

i G {0,1}. Iffor all subgroups L of H, which are isotropy groups for the H-action on Y, the

reduced C*-algebras C*L lies in B and the groups Kn(C*L) are torsionfree, ri G {0,1}, the

following sequence is exact

0 - (Af(*) ® K?(Y))i -> KGxH(X x Y) ->

-(Tor(Af(X),Af(y)))i_1-0,

fori G {0,1}.

Proof. The proof uses the same argument of the previous proof of Thm. 4.4.9. In our case,

in view of Prop. 4.4.8 we have isomorphisms Af (A, Z) ® Af(y) = KGxH((X, Z) x Y) and

KG(Z) ® Pf (V) S* KGxH(Z x y), since KG(Z) and KG(X,Z) are torsionfree and for all

isotropy subgroups L of II, C*L G B and K„(C*L) are torsionfree.

Example 4.4.12. Let X = Y = {pt}, on which the groups G and II act trivially. We have

Kf({pt}) = Ki(C;G) and Kj*({pt}) - Ki{C*H). If the reduced, C*-algebra C*G ofG lies

in the Bootstrap class B (or the sam.e if C*H G B), by Thm,. S4.6 we have the following

Künneth Form,ula for the reduced C*-algebras of G and II :

0 -> [k*(c;g) <g> K*(c;ii)]i - k(g;.g ®min c;ii) - tot(k*(c;g), p,(g;p)),_i ^ o.

Since by Rem. 3.1.13 C;G®minC;H = C*(G x H), the term in the middle is AfGT*G®mi.„

C*H) = I<i(C*(G x H)) = KGxH({pt}), and the sequence represents a Künneth Formula

forX = Y=\jpt}.

4.5 Universal Coefficient Theorem for the G-Equivariant P'-Homology Theory75

Example 4.4.13. Let G,H be groups, X = {pt} a G-CW-complex and Y a proper H-

CW-complex. We require P*(G*G) to be torsionfree. Then for all n G {0,1} we have the

following isomorphism

KGxH({pt} x Y) - [K*(C;G) ® Af(Y)]n.

The next example relies on techniques developed in Section 4.6.

Example 4.4.14. Let G be a group acting trivially on a G-CW-complex X. Then we have

the following short exact sequence:

0 -, [K>(X) ® K*(C*rG)\i - KG(X) -, Tor(K*(X), P^^G)),-! - 0.

This is a consequence of Prop. 4-6.2.

Later, after a chapter dedicated to the Baum-Connes conjecture, wc will also see what

happens to our Künneth Formulae for the proper spaces EG, EH, G, H being groups for

which the Baum-Connes Conjecture holds true.

4.5 Universal Coefficient Theorem for the

G-Equivariant if-Homology Theory

4.5.1 if-Theory of Moore Spaces

Definition 4.5.1 (mapping cone Mf). Let A,X be topological spaces and f : A -, X a

continuous map. We denote by CA := A x I/(A x {1}) the cone over A. The mapping cone

Mf of f is defined as

Mf := X U/ CA := A Û CA/ ~,

where the cone CA is attached to X along A x {0} via the identification (a, 0) ~ f(a),

(o,0) eCA.

Remark 4.5.2. For A -^ X -, X U/ CA a sequence of CW-complexes and any homology

theory h, we have connecting homomorphisms S and a, long homological exact sequence

• -^ hi(A) - hi(X) -, ht(X U/ CA, *) -^ • •

.

Let S1 be the 1-dimensional sphere. For a positive integer n we consider the map n :

5-1 _ 51 given by the n-th power map. The cone CS1 over S1 can be identified with the

two dimensional cell e2.

Definition 4.5.3 (n-th Moore space). Let S1 be the 1-dimensional sphere and, n G N. The

mapping cone Mn := S1 Un e2 over the n-th power map n : S1 -, S1 is called the n-th Moore

space.

76 Equivariant A'-Homology Theory

Remark 4.5.4. The second Moore space is the real projective 2-dimensional space MP2 =

S'ö2e2.

Remark 4.5.5. For a natural number n, the homology groups of the n-th Moore space are

given by

{Z, if i = Q

Z/nZ , if i = l

0, ieZ\{0,l}.

Remark 4.5.6. For a natural number n, the relative K-homology groups of the n-th Moore

space Mn are given by'

0, if i = 0

and the K-homology groups are

'

Z, if i = 0

Proof. For a natural number n, the sequence S1 -, S1 -, Mn induces the following long

exact sequence:

• • •4 Ki(S')^ Ktf1) - Ki(Mn, {pt}) 4 Kq(S') K-^] MS1) -, Ko(Mv, {pt})

Since AfP1) = Z, i = 0,1, we get

... ^ Z1^ Z ^ Ki(Mn,{pt}) ^ ZK-^} Z ^ I<Q(Mn,{pt}) ^ .

Since S1 is connected, K0(n) is the identity map, which shows that Kq(Mu1 {pt}) = 0

and the connecting homornorphism Ô is the zero map. The morphism Kx(n) is induced by

powering by n, therefore it is injective and Ki(M„, {pt}) = Z/nZ. Wc consider now the long

exact sequence

Ki({pt}) -, Ki(Mn) -, Ki(Mn, {pt}) ± Kq({pI}) -, K0(Mn) -, KÜ(M„, {pt}) - • •

By the previous computations and since AT0({p*|) = Z and Af{p£}) = 0, we get

0 - Ki(Mn) -> Z/nZ -^Z-^ K0(Mn) ->()->--•,

with 8 — 0 and therefore our remark. CTTJ

Remark 4.5.6 states that Moore spaces admit geometric realisations with torsionfree A'-

homology groups.

Lemma 4.5.7 (geometric realisation). Forn G N, let Mn denote the n-th Moore space. The

inclusion SL -> Mn induces surjections AfS1) -, Kt(Mn), where K^S1) are torsionfree,

2 = 0,1.

Proof This follows directly from Rem. 4.5.6 and since Kt(Sl) = Z,i = 0,l. D

4.5 Universal Coefficient Theorem for the G-Equivariant A'-Homology Theory77

4.5.2 Künneth Formula in the G-Equivariant K-Homology Theory

with Coefficients in Z/nZ

Definition 4.5.8 (G-equivariant A'-homology Theory with coefficients in Z/nZ). Let G be

a group and X be a proper G-CW-complex. For n G N let Mn denote the n-th Moore space.

For i G {0,1}, the i-th G-equivariant K-homology group of X with coefficients in Z/nZ is

defined byKG(X;Z/nZ) := K^{e}(X x (Mnj*)).

Notation

In the case of G - {e} we write K(X;Z/nZ) instead of K}e](X;Z/nZ).

Example 4.5.9. Let X = {pt}. By Rem,. 4.5.6 we get

is ft +i n? i n>\ \ Z/nZ , if i — 0

K({pt};Z/nZ)=

^{l if. = l

The following Künneth Formula is a relative version of Thm 4.4.9.

Lemma 4.5.10 (Künneth Formula in the one-point-relative G-equivariant A'-homology The¬

ory). Let G, H be groups. Let X be a G-CW-complex and Y a proper H-CW-complex with

YH / 0 and for which there exists a H-CW-complex Z such that K"(Z) are torsionfree

and the induced morphisms Kf(Z) -, Kf{Y) are surjective, i = 0,1. Then the following

sequence is exact:

0 -, (Af(AT) ® Af(y {pt})). -, KGxH(Xx(Y, {pt})) -, ToTi_i(KG(X),Af(y {pt})) -> 0,

for i = 0,1.

Proof. Let {pt} C YH. Note that P has to be a finite group, otherwise Y — 0 since Y

is a proper P'-CW-complcx. Since {pt} is fixed with respect to the action of H, there

exists an P-map p : Y -, {pt} which is a retraction of the inclusion j in the sequence

p

{pt} ,

> Y —> (y {pt}). Therefore for the long exact sequence

• • • - K?({pt})^ K?{Y) - KtH(Y, {pt}) -, K^({pt})

wc have Af(y) = K?(Y,{pt}) (D K?{{pt}), i = 0,1. Let Z be as required, then Af(Z)

projects surjectively also to Af ({pt}), as the following commutative diagram shows:

Af(y)^Af({pt})

By Thm 4.4.9, Y and {pt} satisfy the Künneth Formula. Therefore our proof is done, since

Af(Y) = K»(Y, {pt}) e Af({pt}). a

78 Equivariant P'-Homology Theory

Proposition 4.5.11 (Universal Coefficient Theorem for the G-equivariant AT-homology The¬

ory). Let G be a group and X be a proper G-CW-complex. For every i G {0,1} there is a

natural short exact sequence:

0 -, Af (AT) ® Z/nZ -, KG(X;Z/nZ) -> Tor(KG_l(X),Z/nZ) -, 0.

Proof. Note that by definition KG(X;Z/nZ) = K^^(X x (Mn, *)). By Lemma 4.5.7, the

conditions of Lemma 4.5.2 arc satisfied for (Y,*) = (Mn,*) and II — {e}. We therefore

apply Lemma 4.5.2 to Y = Mn. By Rem. 4.5.6 wc get for the case i = 1

0 -, Af(A) ® Z/nZ -, KG(X; Z/nZ) -, Tor(KG(X), Z/nZ) -, 0,

and analogously for i = 0 we have

0 -, KG(X) ® Z/nZ -> P0G(AT; Z/nZ) -, Tor(KG(X), Z/nZ) -, 0.

The reader should notice that X docs not need to be a proper G-CW-complex in Prop.

4.5.2. For instance, the case X = {pt} yields

0 -, Ki(C;G) ® Z/nZ -, Af ({pi}; Z/nZ) -+ Tor(Afx(G;G), Z/nZ) -, 0.

From this point of view it would be natural to define K*(C*G; Z/nZ) to be Af ({p£}; Z/nZ).

We do not address here the question whether this new definition of PfG*G; Z/nZ) is really

naturally isomorphic to the one introduced in 3.5.2. But both definitions fit into natural

coefficient sequences.

4.5.3 Universal Q-Coefficient Theorem in the G-Equivariant K-

Homology Theory

As a consequence of the Universal Coefficient Theorem 1.4.7 for the Bredon homology and

of Cor. 4.3.10 following from the Atiyah-Hirzebruch spectral sequence we have:

Theorem 4.5.12 (Universal Coefficient Theorem). Let G, H be groups, X a proper G-CW-

complex and, Y a proper II-CW-complex. Then there is a natural isomorphism,

[Af(X;Q) ®Q K?(Y;Q)}i = KGxH(X x Y;Q),

for ie {0,1}.

Proof. By Thm. 1.4.7 we see that for i G N there is a natural isomorphism

[Pf"fX; PgQ) ®Q HrnH(Y; i2gQ)]< = Hfin{X x Y; RGxQH).

By Cor. 4.3.10 with Q-coefficients (i.e., by tensoring everything by Q), we get the desired

natural isomorphim

[KG(X; Q) ®q Pf(Y; Q)]; ^ KGxH(X x Y; Q),

for i G {0,1}.D

4.6 Free Action of Groups and Künneth Formula 79

4.6 Free Action of Groups and Künneth Formula

The aim of this example is to find a Künneth Formula for CW-complexes on which a group

acts freely. Wc first show that the Künneth Formula holds true when one group is trivial,

then we use a peculiarity of free actions, namely that equivariant A'-homology theory of a

space on which a group acts freely can be expressed by A'-theory of a quotient.

We recall that we use the notations S'A, resp. SlX, to indicate the suspension, resp.

l-th. suspension of a topological space Af G N>-i. We also recalljthat the reduced homology

groups of a pointed space X with base point x0 are defined as hn(X) = ker(p* : hn(X) -,

K({xo})),n £ N; wherep : X -, {x0} is the constant map. Then /ifAT) = hn(X)®hn({x0}).The suspension axiom of a reduced homology theory asserts that;Jin(X) = hn+i(SX). For a

(not necessarily pointed) space X and X+ = X U {x0} we have hn(X+) = hn(X).

Lemma 4.6.1. Let W be a finite CW-eomplex. Then there exists a CW-complex V and

a morphism V -4 SlW,l G N>i, such that ATfV) is torsionfree and the induced, morphism,

ji : AfV) -, Ki(SlW) is surjective, i G {0,1}.

Proof. [CoSm], Frop. 2.4.

Proposition 4.6.2 (Künneth Formula for the trivial group). Let G be a group, X a proper

G-CW-complex and Y a CW-complex. Fori G {0,1}, there is a natural short exact sequence

0 - (Pf (A-) ® K.(Y))i -, Pfx{ft}(AT x y) -,

-,(Tor(KG(X),K(Y)))i.i^0.

Proof. We consider Y+ = Y\J {y0}. By Lemma 4.6.1 let V a CW-complex such that AfV)

is torsionfree and the morphism jt : AfV) -> Ki(S2i(Y+)) is surjective, i G {0,1}. By

Thm. 4.4.9, the proper G-CW-complex AT and the CW-complex S2l(Y+) satisfy the Künneth

formula. By the properties of the reduced A'-homology theory and by Bott periodicity wc

have the following equivalences:

K*(S2l(Y+)) = K*(S2l(Y+)) ®z=- A\(Y+) e z ^ K,(Y) e z.

Therefore, by the 5-lemma, the proper G-CW-complex X and the CW-complex Y also satisfy

the Künneth Formula. ^

Lemma 4.6.3. Let II be a group and Y a II-CW-complex, on, which H acts freely. Then

we can express the H-equivariant K-homology theory of Y in the following way:

K?{Y) = PfY/H),

where Y/H is equipped with the quotient topology.

80 Equivariant P-Homology Theory

Proof. Prop. 4.2.9, [Va]. For more details see [Ka2].

Proposition 4.6.4 (Künneth Formula for the free action). Let G, H be groups, X a proper

G-CW-complex and Y a proper H-CW-complex, on which II acts freely. For i G {0,1},

there is a natural short exact sequence

0 - (KG(X) ® K?{Y))i - KGxH(X x Y) -,

-^(Tor(K?(X),K?(Y))),^i^0.

Proof. Since P acts freely on Y, by Lemma 4.6.3, Af (Y) = K»(Y/H). The proposition

follows therefore by applying Prop. 4.6.2 on the CW-complex Y/H. O

Chapter 5

The Baum-Connes Conjecture

5.1 The Index Map

Let G be a (discrete) group and X a proper G-compact space. The aim of this chapter is

to define a group homornorphism p,f : KG(X) -, AfG,!G), for i = 0,1, called index or

assembly map. This morphism is very interesting, as it connects a topological object, the

P-homology of X, to an analytical object, the P-theory of C*G. A good reference for the

next arguments is [BaCoHi].We recall that if (H,it,F) G Af (AT) is a Kasparov triple, the representation it is called

essential if ix(Cq(X))'H is dense in H. We recall also that we may assume that the repre¬

sentation 7T is essential (sec Ex. 4.2.16, [Va]) and the operator F properly supported (see

Lemma 6.1.2, [Va]).

5.1.1 The Index Map: Approach à la Baum-Connes-Higson

Let (H,tt,F) G Af(AT) with it essential, F properly supported and let U be a unitary

representation of G on H. Let H := it(Cc(X))H; it is a dense subspace of H with the

property that F(H) C P. The action defined by

^g:=U(g-')^,

ij) G H, g G G, gives a right CG-module structure to II and the map

< tpi,iß-2 > G?) :=< '0i, U(g)ip-2 >H

defines a G*G-valued scalar product in H, ipx,ip2 G P, f? G G ([Va]). With respect to

this CG-valued scalar product we complete II into a Hilbert G*-module £ over C*G. The

operator F extends continuously to a bounded operator T on £ (Lemma 6.1.3 [Va]), and

(£, it, T) defines an element in ATAfC, C;G) = Ki(C*rG). We set therefore

/f(H,7r,P):=(£,7r,^),

which extends continuously to the direct limit.

82 The Baum-Connes Conjecture

5.1.2 The Index Map: Approach à la Kasparov

Let X be a proper locally compact G-space. Let h G CfAT), h>0, J2gœ K9^x) = 1> Vx G

X and e G GC(G x AT) given by

e(g, x) = y/h(x)h(g~lx),

for (g,x) G G x X. The map e is an idempotent in CC(G x A"), using the product law in the

crossed product G0(AT) xr G (see [Va]). We get therefore an element [e] G K0(Cq(X) xr G)

which does not depend on the choice of the function h, since the set of such functions

is convex. On the other hand, by Kasparov's Thm. 2.5.15 we have Kasparov's descent

homornorphism

jG : KG(X) = KKG(C0(X), C) - KIU(CQ(X) xr G, C*rG).

For an element x G Af(AT) we define its image under the index map as the Kasparov product

tiG(x) = [e] ®Cü(x)xC 3g(x) e AfG;G).

5.2 The Baum-Connes Conjecture

The Baum-Connes Conjecture was first set forth in a 1982 preprint [BaCo] of P. Baum and

A. Conncs, which was published only in 2000, 18 years later. The current description was

given in the paper [BaCoHi] published in 1994 together with N. Higson. The Conjecture

draws a link between two different fields of mathematics and identifies via the assembly

map two different objects, one of analytic type and the other of geometric-topological type,

both associated to a group. It is also very important as it gives rise to many other famous

Conjectures in topology, geometry and functional analysis. We will use the validity of the

Conjecture in some cases to prove several Künneth Formulas.

Conjecture 5.2.1 (Baum-Connes). Let G be a group. Then the index map

pG : Af(PG) -, Ki(C;G)

is an isomorphism, i = 0,1.

5.3 Status of the Conjecture and Consequences

We dedicate this section to an enumeration of the most important groups known to satisfy

the Baum-Connes Conjecture 5.2.1 and we state some important consequences.

Example 5.3.1. a) The Baum-Connes Conjecture holds true for finite groups. In fact,

in this case EG ~ {pt}. Therefore, Af(PG) = Af({pt}) = AfGf*G).

5.4 The Künneth Formula for the Classifying Space for proper Actions 83

b) The Baum-Connes Conjecture is satisfied by one-relator groups (proved in 1999 by G

Béguin, H. Bettaieb and A. Valette in [BeBeVaj).

c) The Baum-Connes Conjecture holds true for a-T-menable1 groups (N. Higson, G.

Kasparov, 2001, [HiKa]). Since (countable) amenable groups are a-T-menable (see

[MiVa]), Baum-Connes holds true also for (countable) amenable groups.

d) V. Lafforgue ([La]) proved the validity of the Baum-Connes Conjecture for groups which

admit property (RD) (this notion is discussed in Chapter 8 of [Va]) and, which admit

a proper, cocompact, isometric action on a, strongly bolic metric space (a notion intro¬

duced, by G Kasparov and G. Skandalis in [KaSk]).

We refer to Appendix 7 of [MiVa] for an overview of the importance of the Baum-Connes

Conjecture related to few Conjectures in the field of algebra and topology like the (strong)

Idempotent Conjecture, the (strong) Trace Conjecture, the (strong) Novikov Conjecture,

Gromov-Lawson-Rosenberg Conjecture, (strong) Zero-in-the-spectrum Conjecture, Zero Di¬

visor Conjecture, (strong) Atiyah Conjecture, Embedding Conjecture, (weak, strong) Bass

Trace Conjecture and the Projective Class Group Conjecture.

5.4 The Künneth Formula for the Classifying Space for

proper Actions

We recall that the equivariant P-homology theory of the classifying space for proper actions

is possible to compute (up to torsion) via an equivariant version of the classical Chern

character. This result was announced by Baum-Connes in 1989 and definitly proved by

Matthey in [Mat].The formulation of a special Künneth Formula for the classifying space for proper actions is

just a recall of Prop. 4.4.6 given in Chapter 4, as the next proposition argues.

Proposition 5.4.1 (special Künneth Formula). Let, G,H be groups and suppose that the

groups KG(EG) is torsionfree, for alii G {0,1}. Then,

(KG(EG) ® KlJ(E_H))i 9ä KGxH(EG x EH),

for alii G {0,1}.

Proof. The conditions of the special Künneth Formula given in Prop. 4.4.6 are satisfied,

since KG(EG) is torsionfree and EH is a proper P-space. D

lLet X be a metric space. An isometric action of a group G on X is called metrically proper if, whenever

a sequence (.gn)neN in G ultimately leaves finite subsets of G, then the sequence (gr^')n<?_H ultimately leaves

bounded subsets of X, for every x G X. A group is called a-T-menable if it admits a metrically proper,

isometric action on some affine Hilbert space.

84 The Baum-Connes Conjecture

Example 5.4.2. For a finite group G and any group H, the conditions of Thm. 5.4.1 are

satisfied, as KG(E_G) = KG({pt}) are torsionfree, for all i G {0,1}, and therefore the special

Künneth, Formula holds in this case.

In a similar way we can formulate the Künneth Formula of Thm. 4.4.9 for the classifying

space for proper actions by requiring the existence of a geometric realisation.

Theorem 5.4.3 (Künneth Formula). Let, G, H be groups. Suppose that there exist a G-space

Z with torsionfree K-theory and a morphism Z -^ EG such that the induced morphism

j, : KG(Z) -, Kf(E_G) is surjective. Then the following sequence is exact:

0 -» (Af(PG) ® K?(EH))i - KGxH(EG x EH) -

- (Tor(KG(EG), Ki1 (M))),^ - 0,

for alii G {0,1}.

Proof. Our assertion is a special case of Thm. 4.4.9. D

In the sequel we approach the Künneth Formula for G*-algebras discussed in Chapter

3 in order to state a Künneth Formula for the classifying space for proper actions via the

assembly map.

Theorem 5.4.4 (Künneth Formula). Let G,H be groups. Suppose that C*G G B and that

G, H and GxH satisfy the Baum-Connes Conjecture. Then the following sequence is exact

for alii G {0,1}:

0 - (Af(PG) ® K?(EH))i -, KGxiI(E(G x H)) ->

- (Tor(KG(EG),K?(EH)))^ -, 0.

Proof. Since C*G G B, by Thm. 3.4.6 the following sequence is exact for all i G {0,1}:

0 - (k*(c;g) ® K*(c*rH))i -, a",(g;g ®min c;h) -, Tot(p*(g;g), k*(c;h))^i -+ o.

Recall that by Rem. 3.1.13, C;G ®mm C;H £ C^(G x P), and by Prop. 1.3.10, E(G x

H) = EG x EH. The assertion follows by applying the assembly map in every term of the

sequence.D

Analogously, we can state a Künneth Formula for group G*-algebras via the assembly

map by assuming conditions on the classifying space for proper actions.

Theorem 5.4.5 (Künneth Formula). Let G, P be groups such that G,H and GxH satisfy

the Baum-Connes Conjecture. Suppose that there exist a G-space Z with torsionfree K-

theory and a morphism Z -^ EG such that the induced morphism, j% : KG(Z) —> KG(EG) is

surjective, for i G {0,1}. Then the following sequence is exact:

0 -, (k*(c*g) ® iu(c;h)\ -, k(c;g ®mm c;h) -, tot{k*{c;g), p*(c;p));-i - o,

for alii G {0,1}.

5.5 Stability of the Baum-Connes Conjecture 85

Proof. As G, P and GxH satisfy the Baum-Connes Conjecture, we can apply the assembly

isomorphism to the Künneth Formula of Thm. 5.4.3 and get our assertion. D

Remark 5.4.6. We recall that Ki(C*rG) ^ KG({pt}). Therefore, if G satisfies the Baum-

Connes Conjecture, we have KG(EG) = Kf({pt}).

In view of the Baum-Connes Conjecture and Conjecture 4.4.1, we state our following

Conjecture.

Conjecture 5.4.7. Let G, H be groups. Then the following sequence is exact:

o -> (k*(c;g) ® K.{c;H))i - k^g ®rnin c;h) - tot{k*{c;g), k^h))^ -, o,

for all i G {0,1}.

5.5 Stability of the Baum-Connes Conjecture

It is known that the Baum-Connes Conjecture is invariant with respect to direct limit of

groups (see [MiVa]), whereas invariance of the Conjecture with respect to finite direct product

or with respect to the operation of taking subgroups of finite index still remains an open

question. Work of Oyono-Oyono, Chabert and Echteroff show that a certain stronger version

of the Baum-Connes Conjecture (the Baum-Connes Conjecture with Coefficients in a certain

given G*-algebra) is stable with respect to free or amalgamated product (as a consequence

of Thm. 2.4.1 cited in [Va] and stated by Oyono-Oyono in [Oy]), and under direct and

semidirect products (sec Thm. 2.4.2 of [Va], stated in the original paper [Oyl]). In the

sequel we give good conditions for which the invariance of the Baum-Connes Conjecture

with respect to finite direct product is satisfied. The approach is to take groups whose

classifying spaces for proper actions satisfy the Künneth Formula.

Theorem 5.5.1. Let G,H be groups satisfying the Baum-Connes Conjecture. Suppose that

C*G G B and AT;(G*G) is torsionfree for i G {0,1}. Then GxH also satisfies the Baum-

Connes Conjecture.

Proof. We have the following sequence of isomorphisms:

,-,„

Prop.l.3.10 „ ,,

Prap.bA.l

KGxH(E(G x H)) = KGxH(EGxEH) = (KG(E_G) ® K?(EH)){ =

Conj.5.2.1 T/wn.3.4.1

= (K*{c;G)®K*(c;H))i <= p,(G;G®m,;nG;p) =

RemS.1.13

p,(g;(g x p)).r^j

Therefore also the assembly map pi0xn is an isomorphism, i G {0,1}. D

The following examples are an illustration of the previous theorem.

86 The Baum-Connes Conjecture

Example 5.5.2. a) Let G be a finite group and H a group which satisfies the Baum-

Connes Conjecture. Then by 5.5.1 also the product GxH satisfies the Baum-Connes

Conjecture, as C*G = CG G B, A"; (CG) is torsionfree and, G satisfies itself the Baum-

Connes Conjecture.

b) Let G be a finitely generated abelian group and, H a group which satisfies the Baum-

Connes Conjecture. We verify the conditions of Thm. 5.5.1. First, finitely generated

abelian groups satisfy the Baum-Connes Conjecture. We write G = Zn x P, where F is

a, finite group. Then C*G = C*Zn ®min CF lies in the Bootstrap class B, as this class

is closed under tensor products. The K-theory ofC*G is torsionfree, as by Thm,. 3.4-1

holds that Ki(C*TG) = K(C:.Zn®minCF) = [AT*(G;Z")®A^(CP)]7- a,n,d,K*(C;Zn) and,

iC(CP) are both torsionfree. Therefore the product GxH satisfies the Baum-Connes

Conjecture.

c) Let G be an, elementary amenable group (in particular, C*G G B and G satisfies the

Baum-Connes Conjecture) with Ki(C*G) torsionfree for i G {0,1}, and let H be a

group satisfying Baum-Connes Conjecture. Then by Thm,. 5.5.1, GxH satisfies

Baum-Connes, too.

d) There are also examples of invariance with respect to direct product which do not need

the material discussed before. For example, if G, II are amenable groups, then also

GxH is amenable and in particular satisfi.es the Baum-Connes Conjecture.

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York, 1994.

Index

G*-algebra, 27

G*-algebra, nuclear, 47

G*-algebra, of type I, 49

G-equivariant A-homology Theory with co¬

efficients, 75

G-equivariant Kasparov P-homology groups

60

G - G*-algcbra, 39

A'-homology groups, 63

PP-equivalence, 43

A'o of G*-algebras, 33

Pi of G*-algebras, 35

Kn of G*-algebras, 35

amenable group, 48

Bootstrap class B, 50

Bott periodicity, 37

Bredon cohomology, 15

Bredon homology, 17

cellular chain complex, 15

cone of a Banach algebra, 36

equivariant homology theory, 58

GNS construction, 31

Hilbert G*-module, 40

index map, 79

Kasparov ATAT-groups, 41

Kasparov product, 43

Kasparov triple, 41

Moore space Mn, 73

negative P-groups of G*-algebras, 37

proper, 12

reduced C-algcbra C*G of G. 30

reduced crossed product, 42

representation functor Pc, 17

representation functor with coefficients Pc,*-,

24

spectral sequence, Atiyah-Hirzebruch type, 64

spectrum, 61

suspension of a Banach algebra, 36

tensor product, maximal, 46

tensor product, minimal, 46

Universal space of proper actions EG, 18

mapping cone, 39

Curriculum Vitae

I was born in Airolo, Switzerland, on March 23th, 1977. I attended Primary School (Scuola

Elementare) in Airolo, Secundary School (Scuola Media) in Ambri, and High School (Liceo

Cantonale) in Bellinzona, obtaining a mathematics/science diploma (Maturità Tipo C) in

1996. In September 2001 I obtained the degree of Holder of the Diploma in Mathematics

of the Swiss Federal Institute of Technology in Zürich (Dipl. Math. ElTfZ). Successively,

while working as a teaching assistant in the Group for Algebra and Topology at the ETHZ,

I completed my studies for a Ph.D. Thesis under the supervision of Professor Guido Mislin

in February 2006. During all this time I cultivated my favourite hobby, to play the violin.

Fausta Leonardi

Departement Mathematik

Eidgenössische Technische Hochschule

ETII-Zentrum

8092 Zürich

Switzerland

fausta. leonardi@math. ethz. ch