F A C U L T Y O F S C I E N C E U N I V E R S I T Y O F C O P E N H A G E N

Graduate Course ProjectDepartment of Mathematical SciencesKang Li

The Baum-Connes Conjecture with Coefficients

Thesis adviser: Ryszard Nest

11. November 2011

Contents i

Contents

1 Introduction 1

2 K-Theory 22.1 Graded C∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 A Spectral Picture of K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Long Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Asymptotic Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Bott Periodicity in the Spectral Picture . . . . . . . . . . . . . . . . . . . . . 162.9 The Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 E-Theory 243.1 The Asymptotic Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 The E-Theory Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 The Equivariant E-Theory Category . . . . . . . . . . . . . . . . . . . . . . . 36

4 The Baum-Connes Conjecture 414.1 Proper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 The Assembly Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 The Green-Julg Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 The Dirac and Dual Dirac Method . . . . . . . . . . . . . . . . . . . . . . . . 484.5 Conjecture for the Free Abelian Group G = Zn . . . . . . . . . . . . . . . . . 504.6 Conjecture for the Groups with the Haagerup Property . . . . . . . . . . . . 524.7 Conjecture for the infinite Property (T) Groups . . . . . . . . . . . . . . . . . 62

Appendix A Self-adjoint Operators 67

Appendix B Clifford Algebras 68

Appendix C The Central Cover 71

References 72

Introduction 1

1 Introduction

The Baum-Connes conjecture (with coefficients) is part of Alain Connes’ noncommutativegeometry programme. It is in some sense the most commutative part of this programme,since it bridges with classical geometry and topology.

Consider a countable group G and a G-C∗-algebra D. The Baum-Connes conjecture withcoefficients identifies the K-theory of the reduced crossed product C∗-algebra C∗λ(G,D) withthe equivariant K-homology group with coefficients Ktop(G,D). The conjecture states thata particular map between these two objects, called the assembly map,

µλ : Ktop(G,D)→K(C∗λ(G,D))

is an isomorphism of abelian groups for every coefficient D.

Part of the importance of this conjecture is due to the fact that it is related to many otherrelevant conjectures in different areas of mathematics [34]. There are following corollaries ofthe Baum-Connes conjecture:

• Novikov conjecture

• Stable Gromov-Lawson-Rosenberg conjecture

• Idempotent conjecture

• Kadison-Kaplansky conjecture

• Mackey analogy

• Construction of the discrete series via Dirac induction (Parthasarathy, Atiyah, Schmidt)

• Homotopy invariance of ρ-invariants (Keswani, Piazza, Schick)

Nevertheless, the conjecture itself allows the computation of the K-theory of C∗λ(G,D) fromthe equivariant K-homology group Ktop(G,D). The conjecture has been proved for somelarge families of groups. In particular, Higson and Kasparov [22] proved the conjecture forgroups having the Haagerup property, that is, groups which admits a metrically proper iso-metric action on some affine Euclidean space. On the other side, a group has Kazhdan’sproperty T if every isometric action of G on an affine Euclidean space has a fixed point.Thus infinite groups with the property T do not have the Haagerup property.

Consequently, the group SL(3,Z) becomes relevant in this context since the conjectureis unknown for SL(3,Z), n ≥ 3 and these groups have property T . On the other hand, theassembly map is known to be injective for SL(n,Z) (in general, for all exact groups, see [18]and [20]). Finally, note that there are counterexamples to the conjecture for groupoids thatcan be constructed from SL(3,Z), and more generally for a discrete group with property Tand such that the assembly map is injective [30].

In this project we start to prove the Baum-Connes conjecture with coefficients for groupshaving the Haagerup property and then discuss the Baum-Connes conjecture for infiniteproperty T groups, in particular for SL(3,Z). Finally, we will explain why it is not possibleto prove the conjecture for certain groups (for example uniform lattices in Sp(n, 1)) byworking purely within E-theory (or for that matter within KK-theory).

K-Theory 2

2 K-Theory

2.1 Graded C∗-Algebras

In this section, we develop the general theory of graded C∗-algebras which will be neededin the sequel. Our main reference of graded C∗-algebras are [7] and [23].

Definition 2.1. Let A be a C∗-algebra. A grading on A is a ∗-automorphism α of Asatisfying α2 = 1. Equivalently, a grading is a decomposition of A into a direct sum of twoself-adjoint closed linear subspaces, A = A0 ⊕ A1, with the property that AiAj ⊆ Ai+j ,where i, j ∈ Z/2.1 An element of An (for which α(a) = (−1)na) is said to be homogeneousof degree n. The degree of a homogeneous element a is denoted ∂a.

Example 2.2. The trivial grading on A is defined by the ∗-automorphism α = id, or equiv-alently by setting A0 = A and A1 = 0.

Example 2.3. Let H be a graded Hilbert space, that is, a Hilbert space equipped with anorthogonal decomposition H = H0⊕H1. The C∗-algebras K(H) and B(H) are graded. Wedeclare that the diagonal matrices to be even and the off-diagonal ones to be odd.

Example 2.4. Let S = C0(R) be the C∗-algebra of continuous, complex-valued functions onR which vanish at infinity, and define a grading on S by the decomposition

C0(R) = even functions ⊕ odd functions

or equivalently by the ∗-automorphism f(x) 7→ f(−x).

Definition 2.5. A graded C∗-algebra A is inner-graded if there exists a self-adjoint unitaryε in the multiplier algebra M(A) of A which implements the grading automorphism α onA:

α(a) = εaε

for all a ∈ A.

Example 2.6. The trivial grading on a C∗-algebra A is inner with ε = 1. The grading onK(H) and B(H) are inner with

ε =

(1 00 −1

)∈ B(H) =M(K(H)).

However the grading on S is not inner, since a self-adjoint unitary in M(C0(R)) = Cb(R)should be ±1.

Definition 2.7. Let A and B be graded C∗-algebras. Let AB be the algebraic tensorproduct of the linear spaces underlying A and B with a grading and a new product andinvolution given by

(a1b1)(a2b2) = (−1)∂b1∂a2a1a2b1b2(ab)∗ = (−1)∂a∂ba∗b∗

∂(ab) = ∂a+ ∂b, (mod 2).

for all homogeneous elements a, a1, a2 ∈ A and b, b1, b2 ∈ B.

Remark 2.8. (AB)0 = A0B0⊕A1B1, (AB)1 = A0B1⊕A1B0. Alternately, if αand β are the grading automorphisms on A and B, then αβ is the grading automorphismon AB.

Proposition 2.9. The algebraic graded tensor product is associative and commutative:

1Note that A0 is a C∗-subalgebra of A and A1 is of course not a subalgebra.

K-Theory 3

A(BC) ∼= (AB)CAB ∼= BA

Proof. Note first the fact that if a and b are both homogeneous, then so is the product aband ∂(ab) = ∂a+ ∂b. It follows easily from the fact that the isomorphisms are given by

a(bc) 7→ (ab)cab 7→ (−1)∂a∂bba

Definition 2.10. The graded commutator of homogeneous elements a and b in a gradedC∗-algebra is given by the formula

[a, b] = ab− (−1)∂a∂bba

This is extended by linearity to all elements.

We notice that it’s not true in general that [a, b] = −[b, a]: if a has degree 1 then [a, a] =2a2. The next proposition gives some standard identities valid for graded commutators.

Proposition 2.11. Let A be a graded C∗-algebra, and a, b, c homogeneous elements of A.Thena) [a, b] + (−1)∂a∂b[b, a] = 0b) [a, bc] = [a, b]c+ (−1)∂a∂bb[a, c]c) (−1)∂a∂c[[a, b], c] + (−1)∂a∂b[[b, c], a] + (−1)∂b∂c[[c, a], b] = 0.

It follows from the universal property of ordinary tensor products we have the followingpropositions:

Proposition 2.12. Given two graded ∗-homomorphisms2 of graded C∗-algebras ϕ : A→Cand ψ : B→D, then ϕψ : AB→CD is also a graded ∗-homomorphism.

Proposition 2.13. If C is a graded C∗-algebra and if ϕ : A→C and ψ : B→C are graded∗-homomorphisms whose images graded-commute (i.e. all graded commutators [ϕ(a), ψ(b)]are zero), then there is a unique graded ∗-homomorphism from AB into C which mapsab to ϕ(a)ψ(b).

Corollary 2.14. Let H and K be graded Hilbert spaces with grading automorphisms α andβ and H⊗K be the graded Hilbert space tensor product with grading automorphism α⊗β,then there is a unique injective graded ∗-homomorphism from B(H)B(K) into B(H⊗K)which takes the homogeneous elementary tensor ST to the operator

v ⊗ w 7→ Sv ⊗ (−1)∂v∂TTw.

Proof. Note first that B(H), B(H)B(K) and B(H⊗K) are graded C∗-algebras with grad-ing automorphisms Adα, AdαAdβ and Adα⊗β , then we define two ∗-homomorphisms

ϕ : B(H)→B(H⊗K) given by S 7→ (v⊗w 7→ Sv⊗w) and ψ : B(K)→B(H⊗K) givenby T 7→ (v⊗w 7→ (−1)∂v∂T v⊗Tw). Since ∂v = ∂α(v) and ∂(Adβ(T )) = ∂T , we see thatϕ and ψ are graded. Their images are graded-commute since an even operator T mapseven/odd vector v to even/odd vector Tv and an odd operator T maps even/odd vector v toodd/even vector Tv. In particular ∂S + ∂(Sv) = ∂v. The unique graded ∗-homomorphismfrom B(H)B(K) into B(H⊗K) follows from the previous proposition. Now we prove theinjectivity: if 0 =

∑ni Si⊗Ti ∈ B(H⊗K), we may assume that the operators Si ⊆ B(H)

are linearly independent. For all vectors v, w ∈ H and ξ, η ∈ K we have

〈(n∑i

Si⊗Ti)v⊗ξ, w⊗η〉 = 〈(n∑i

(−1)∂v∂Ti〈Tiξ, η〉Si)v, w〉.

It follows the 0 = Ti ∈ B(K) for all i, and the proof is complete.

2A ∗-homomorphism ϕ : A→B of graded C∗-algebras is graded if ϕ(An) ⊆ Bn for n = 0, 1. Equivalentlya ∗-homomorphism that respects the gradings.

K-Theory 4

Definition 2.15. Let A and B be graded C∗-algebras and let AB be their algebraic gradedtensor product. The maximal graded tensor product, which we will denote by A⊗maxB, isthe completion of AB in the norm

||∑

aibi|| = sup ||∑

ϕ(ai)ψ(bi)||,

where the supremum is taken over graded-commuting pairs of graded ∗-homomorphisms,mapping A and B into a common third graded C∗-algebra C.

We are going to show that A⊗maxB is the universal enveloping C∗-algebra of AB, butbefore we prove it we need a lemma:

Lemma 2.16. Any Γ-C∗-algebra A can be faithfully represented on a Γ-Hilbert space H ⊗l2(Γ), where Γ is a discrete group. In particular, any graded C∗-algebra A can be faithfullyrepresented on H ⊕H with the standard odd grading i.e. transposing the two copies of H.

Proof. We begin with a faithful representation A ⊆ B(H). Define a new representation ofA on H ⊗ l2(Γ) by

π(a)(v ⊗ δg) = (αg−1(a)(v))⊗ δg,

where δgg∈Γ is the canonical orthonormal basis of l2(Γ) and α is the action of Γ onA. Under the identification H ⊗ l2(Γ) ∼=

⊕g∈ΓH we have simply taken the direct sum

representation

π(a) =⊕g∈Γ

α−1g (a) ∈ B(

⊕g∈Γ

H).

Hence it’s a faithful representation of A on H ⊗ l2(Γ). Since the left regular representationof Γ spatially implements the action α, so this representation is Γ-equivariant:

(1⊗ λs)π(a)(1⊗ λ∗s)(v ⊗ δg) = π(αs(a))(v ⊗ δg).

Proposition 2.17. The graded ∗-representations of AB on a graded Hilbert space H are innatural one-one correspondence with the graded-commuting pairs of graded ∗-homomorphismsof A and B into the same graded C∗-algebra C.

Proof. One direction follows from Proposition 2.13 and previous lemma. Now let π :AB→B(H) be a nondegenerate graded ∗-representation and α, β, γ are the gradings forA,B,H. For x =

∑i aibi ∈ AB and v ∈ H define πA : A→B(H) given by

πA(a)(π(x)v) = π(∑i

aaibi)v

and πB : B→B(H) given by

πB(b)(π(x)v) = π(∑i

(−1)∂b∂aiaibbi)v.

Since παβ = Adγπ, it follows that πAα = AdγπA and πBβ = AdγπB . It’s routine tocheck that we get a pair of ∗-homomorphisms with graded commuting ranges such thatπA × πB = π.

Corollary 2.18. A⊗maxB is the universal enveloping C∗-algebra of AB i.e.

||∑

aibi|| = sup ||π(∑

aibi)||,

where the supremum is taken over all (ungraded) ∗-representations π : AB→B(H).

K-Theory 5

Proof. If A is a graded C∗-algebra with the grading α and π : A→B(H) is a (ungraded)∗-representation, then we can define a graded ∗-representation π : A→B(H ⊕ H) givenby π(a) = π(a) ⊕ π(α(a)), where H ⊕ H has the standard odd grading. It’s clear that||π(a)|| ≥ ||π(a)||. The proof is complete by previous proposition.

Remark 2.19. From this Corollary we see that ⊗max is the largest possible C∗-norm onAB, hence functorial: if ϕ : A→C and ψ : B→D are graded ∗-homomorphisms thenthere is a unique graded ∗-homomorphism ϕ⊗maxψ : A⊗maxB→C⊗maxD mapping a⊗b toϕ(a)⊗ψ(b) for all a ∈ A and b ∈ B.

Definition 2.20. Let A and B be graded C∗-algebras and faithfully represent A and B ongraded Hilbert spaces H and K. The minimal graded tensor product A⊗B of A and B isthe completion of AB in the faithful representation AB→B(H⊗K).

Remark 2.21. Since Mn(C)A ∼= Mn(C)A for any C∗-algebra A, we see that the minimalnorm is independent of the choices of faithful graded representations. And as the maximaltensor product we can show that ⊗ is also functorial.

Proposition 2.22. If A is inner-graded, then A⊗B ∼= A ⊗ B and A⊗maxB ∼= A ⊗max B.If A and B are both inner-graded with grading operators εA and εB respectively, then A⊗Band A⊗maxB are inner-graded with grading operators εA⊗εB and εA⊗maxεB respectively.

Proof. It’s straightforward to check that ab 7→ aε∂bA b is an isomorphism of AB ontoAB. This isomorphism induces isomorphisms on the maximal and minimal graded tensorproduct of A and B. Moreover, εA⊗εB ∈ M(A)⊗M(B) ⊆ M(A⊗B) and εA⊗maxεB ∈M(A)⊗maxM(B) ⊆M(A⊗maxB).

Corollary 2.23. If A is inner graded by ε and K(H) has the standard even grading, thenA⊗K(H) ∼= M2(A⊗K(H)) with standard even grading given by η = diag(ε⊗ 1,−ε⊗ 1).

Remark 2.24. If A or B is nuclear, then using same ideas for ordinary tensor product it’snot difficult to show that the quotient map from A⊗maxB to A⊗B is an isomorphism.

2.2 Amplification

The graded C∗-algebra S = C0(R) will play a special role for us. Using it we shall enrich oramplify the category of graded C∗-algebras and ∗-homomorphisms.

To do this we introduce a ∗-homomorphism ∆ : S→S⊗S as follows: Denote by SR thequotient of S consisting of continuous functions on the interval [−R,R] (the quotient mapis the operation of restriction of functions) and denote by XR ∈ SR the function x 7→ x.If f ∈ S, then we can apply the continuous functional calculus to the self-adjoint elementXR⊗1 + 1⊗XR ∈ SR⊗SR to obtain an element f(XR⊗1 + 1⊗XR) ∈ SR⊗SR. It followsthat S→SR⊗SR given by f 7→ f(XR⊗1 + 1⊗XR) is a graded 3 ∗-homomorphism for everyR > 0, thus we get a graded ∗-homomorphism S→

∏R SR⊗SR.

Lemma 2.25. There is a unique graded ∗-homomorphism ∆ : S→S⊗S whose compositionwith the quotient map S⊗S→SR⊗SR is the ∗-homomorphism f 7→ f(XR⊗1 + 1⊗XR) for

every R > 0. In fact ∆(u) = u⊗u and ∆(v) = u⊗v + v⊗u, where u(x) = e−x2

and

v(x) = xe−x2

are the self-adjoint homogeneous elements in S, which generate S.

Proof. Note first that the intersection of the kernels of the maps S⊗S→SR⊗SR is zero. Thisfollows from the fact that if f1, . . . , fn ⊆ S are linearly independent, there exists R > 0such that f1|[−R,R], . . . , fn|[−R,R] is still linearly independent. So the family of graded ∗-homomorphisms S⊗S→SR⊗SR for R > 0 induces a faithful graded ∗-homomorphism from

3We want this ∗-homomorphism respects the gradings, but by continuity we only need to check the casewhen f are polynomials.

K-Theory 6

S⊗S to∏R SR⊗SR. I want to construct a graded ∗-homomorphism such that the following

diagram commutes:

S

//∏R SR⊗SR

S⊗S //

-

;;

SR⊗SRHowever, this is a easy consequence of the fact Im(S→

∏R SR⊗SR) ⊆ Im(S⊗S →

∏R SR⊗SR).

Since u and v generate the C∗-algebra S, we only need to check this for u and v:

(u(XR⊗1 + 1⊗XR))R = (e−(XR⊗1+1⊗XR)2)R

= (e−(X2R⊗1+1⊗X2

R))R

= (e−X2R⊗1 · e−1⊗X2

R)R

= (e−X2R⊗e−X

2R)R.

i.e. the image of u under S→∏R SR⊗SR equals the image of u⊗u under S⊗S →

∏R SR⊗SR.

Similarily, we can show that v ∈ S and u⊗v+v⊗u ∈ S⊗S have same image. This completesthe proof.

The ∗-homomorphisms η4 and ∆ provide S with a sort of coalgebra structure, i.e. thefollowing diagrams5

S

∆

∆ // S⊗S

1⊗∆

S⊗S∆⊗1

// S⊗S⊗S,

S

=

||∆

=

""S S

S⊗Sη⊗1

aa

1⊗η

==

commute. The coalgebra structure amplifies the category of graded C∗-algebras:

Definition 2.26. Let A be a graded C∗-algebra. The amplification of A is the graded tensorproduct SA = S⊗A. The amplified category of graded C∗-algebras is the category whoseobjects are the graded C∗-algebras and for which the morphisms from A to B are the graded∗-homomorphisms from SA to B. Composition of morphisms ϕ : A→B and ψ : B→C inthe amplified category is given by the following composition of graded ∗-homomorphisms:

SA∆⊗1−→ S2A

S(ϕ)−→ SBψ−→ C.

Remark 2.27. The coalgebra structure implies that the composition law is associative andthat the ∗-homomorphisms SA→A obtained by taking the graded tensor product of η : S→Cwith the identity map on A serve as identity morphisms for this composition.

Remark 2.28. Most features of the category of graded C∗-algebras pass to the amplifiedcategory. One example is the max/min-tensor product operation: given amplified mor-phisms ϕ1 : A1→B1 and ϕ2 : A2→B2 there is a tensor product morphism from A1⊗A2 toB1⊗B2, in other words a graded ∗-homomorphism from S(A1⊗A2) into B1⊗B2 defined bythe composition of graded ∗-homomorphisms

S(A1⊗A2) = S⊗A1⊗A2∆⊗1⊗1−→ S2⊗A1⊗A2

∼= SA1⊗SA2ϕ1⊗ϕ2−→ B1⊗B2.

It turns out that the tensor product is associative and commutative functor.

4η : S→C defined by η(f) = f(0).5It’s easily verified by considering the elements u and v in S.

K-Theory 7

2.3 Stabilization

Stabilization means replacing a C∗-algbera A with A⊗K(H), its tensor product with theC∗-algebra of compact operators. It is a central ideal in K-theory, for instance when A is atrivially graded C∗-algebra with unit, then the semigroup of isomorphism classes of finitelygenerated projective A-modules can be identified with the semigroup of homotopy classes ofprojections in A⊗K(H).

Let us now return to the graded situation. Fix a graded Hilbert space H whose evenand odd grading-degree parts are both countably infinite-dimensional, there are graded ∗-homomorphisms

κ : C→K(H), β : K(H)⊗K(H)→K(H)

defined by mapping λ ∈ C to λe, where e is the projection onto a one-dimensional, grading-degree zero subspace of H, and by identifying H⊗H with H by a grading-preserving unitaryisomorphism. These play a role similar to the maps η and ∆ introduced in the previoussection. Unfortunately, there is no canonical choice of the projection e or the isomorphismH⊗H ∼= H, and for this reason we cannot “stabilize” the category of C∗-algebras in quitethe way we amplified it in the previous section. We need pass to homotopy:

Lemma 2.29. Let H and H ′ be graded Hilbert spaces. Any two grading-preserving isome-tries from H into H ′ induce graded ∗-homomorphisms from K(H) to K(H ′) which arehomotopic through graded ∗-homomorphisms.

Proof. Let u and v be two grading-preserving isometries from H into H ′, then we haveu(H0 ⊕H1) = u(H0)⊕ u(H1), v(H0 ⊕H1) = v(H0)⊕ v(H1), u(Hi) ⊆ H ′i and v(Hi) ⊆ H ′i.Since the unitary group in B(H ′i) is connected, we get two homotopies Tt and St on H ′0 andH ′1, respectively. So (Tt⊕St)u is a graded homotopy between u and v. It’s clear that if twograding-preserving isometries are homotopic through graded isometries, then the inducedgraded ∗-homomorphisms are homotopic through graded ∗-homomorphisms.

As a result there are canonical, up to homotopy, graded ∗-homomorphisms C→K(H)and K(H)⊗K(H)→K(H)6 such that the following diagrams

K(H)⊗K(H)⊗K(H)

1⊗β

β⊗1// K(H)⊗K(H)

β

K(H)⊗K(H)β

// K(H),

K(H)

K(H)

κ⊗1 &&

=

88

K(H)

1⊗κxx

=

ff

K(H)⊗K(H)

β

OO

commute up to homotopy.We could therefore create a stabilized homotopy category:

Definition 2.30. The stabilized homotopy category of graded C∗-algebras is the categorywhose objects are the graded C∗-algebras and for which the morphisms from A to B arethe homotopy classes of graded ∗-homomorphisms from A to B⊗K(H). Composition ofmorphisms ϕ : A→B and ψ : B→C in the stabilized homotopy category is given by thefollowing composition:

Aϕ−→ B⊗K(H)

ψ⊗1−→ C⊗K(H)⊗K(H) ∼= C⊗K(H).

The identity morphisms are A→A⊗K(H) given by a 7→ a⊗e.6The unique injective graded ∗-homomorphism in Corollary 2.14 gives an isomorphism between

K(H)⊗K(H′) and K(H⊗H′).

K-Theory 8

We could even stabilized and amplify simultaneously, and create the category in whichthe morphisms between graded C∗-algebras A and B are the homotopy classes of graded∗-homomorphisms from SA to B⊗K(H).

2.4 A Spectral Picture of K-Theory

We are going provide a “spectral” description of K-theory, which is well adapted to Fredholmindex theory and to an eventual bivariant generalization. For the rest of this section we shallfix a graded Hilbert space H whose even and odd grading-degree parts are both countablyinfinite-dimensional.

Definition 2.31. If A is a graded C∗-algebra then we define

K(A) = [S,A⊗K(H)],

where [A,B] is the set of homotopy classes of graded ∗-homomorphisms between the gradedC∗-algebras A and B.

For the moment K(A) is just a set, althought we will soon give it the structure of anabelian group. But first let us give two examples of classes in K(A):

Example 2.32. Let A = C. Let D be an odd unbounded self-adjoint operator on the gradedHilbert space H of the form

D =

(0 D−D+ 0

)and D− = D∗+. Let’s assume that D has compact resolvent7. The functional calculus

ψD : f 7−→ f(D)

defines a graded8 ∗-homomorphism ψD : S→K(H) by Proposition A.3 and hence a class inK(C).

Example 2.33. Suppose that A is unital and trivially graded, so K0(A) = [p] − [q] : p, q ∈P (A⊗K(H)). If p0 ∈ P (A⊗K(H0)) and p1 ∈ P (A⊗K(H1)), then the formula

ψp : f 7−→(f(0)p0 0

0 f(0)p1

)defines a graded ∗-homomorphism from S to A⊗K(H), where H = H0⊕H1 and p = p0⊕p1.

Remark 2.34. In fact ψD is homotopic to ψp. Indeed, if D is a self-adjoint, odd, compactresolvent operator on H, then the family

ψs : f 7−→ f(s−1D), s ∈ [0, 1]

is a homotopy from the ∗-homomorphism ψD at s = 1 to the ∗-homomorphism ψp at s = 0,where p0/p1 is the projection onto the kernel of D+/D− and p = p0 ⊕ p1 is the projectiononto ther kernel of D: Since the set f(D) : f = f ∈ S is a commuting set of self-adjoint,compact operators, it follows from the spectral theorem for compact operators that thereis an orthonormal basis ej for H and real scalars λj such that limj→∞ |λj | = ∞ andf(D)ej = f(λj)ej for all j. The vectors ej belong to dom(D) and Dej = λjej for allj. Thus when s→0 the ∗-homomorphism ψs converges in the point-norm topology to thegraded ∗-homomorphism ψp.

7For example, D is a Dirac-type operator on a compact manifold.8The algebra generated by e−x2

and xe−x2is dense in S and consists of all functions of the form p(x)e−x2

,

where p(x) is a polynomial. Hence the even/odd functions in S can be approximated by p(x)e−x2for p(x)

is a even/odd polynomial.

K-Theory 9

Lemma 2.35. K(C) ∼= Z in such a way that to the class of the ∗-homomorphism ψD ofExample 2.32 is associated the Fredholm index of D+.

Proof. Let ψ : S→K(H⊕H) be a graded ∗-homomorphism, then by the converse functionalcalculus (see Theorem 3.2 [44]) there is a self-adjoint, odd and regular operator D on H⊕Hsuch that ψ(f) = f(D) for all f ∈ S. In particular, D has compact resolvent and ψ = ψDis homotopic to ψp as in Remark 2.34. So we define a map K(C)→K0(C) given by [ψp] 7→[p0]− [p1] and the inverse map K0(C)→K(C) given by [p0]− [p1] 7→ [ψp] as in Example 2.33.

Since a self-adjoint operator D with compact resolvent is a Fredholm operator withparametrix (D + i)−1. We know that a Fredholm operator D is odd and self-adjoint, thenthe componenets D− = D∗+ are Fredholm and

Index(D+) = dimkerD+ − dimcoker(D+) = dimkerD+ − dimker(D−) = dim(p0)− dim(p1).

Since K0(C) ∼= Z by the dimension-map, we complete the proof.

Let’s define the addition on K(A): This is given by the direct sum operation whichassociates to a pair of graded ∗-homomorphisms ψ1 and ψ2 the graded ∗-homomorphism

ψ1 ⊕ ψ2 : S→A⊗K(H ⊕H) ∼= A⊗K(H),

where we identify H ⊕H with H by some grading-preserving unitary, which is well-definedby Lemma 2.29.

The zero element is the class of the zero homomorphism. The additive inverse of ψ isrepresented by the ∗-homomorphism

ψop = ψ α : S→A⊗K(Hop).

obtained by composing ψ with the grading automorphism on S and also reversing the gradingon the Hilbert space H. Hence K(A) is an abelian group and we will prove that K(A) =E(C, A) and E(A,B) is an abelian group in general.

Remark 2.36. In fact by using of the converse functional calculus we can show that K(A)is isomorphisc to KK(C, A), the Kasparov’s K-theory group (see Theorem 4.7 [44]).

Definition 2.37. Let D be a graded C∗-algebra, then by a Cayley transform for D we shallmean a unitary U in the unitalization D+ of D such that U − 1 ∈ D and α(U) = U∗, whereα is the grading automorphism on D+.

Lemma 2.38. Let D be any graded C∗-algebra, there is a bijection between the set of graded∗-homomorphisms ψ : S→D and the set of Cayley transform for D.

Proof. Let ψ : S→D be a graded ∗-homomorphism and can be uniquely extended to agraded unital ∗-homomorphism ψ+ : C(T)→D+. Using the Cayley transform

x 7−→ x− ix+ i

which is a homeomorphism from R to T − 1 and under this identification S = C0(R) isthe algebra of continuous functions on the unit circle T which vanish at 1 ∈ T. Note thatthe grading on C(T) is given by f(z) 7→ f(z) and z − 1 ∈ S. It follows that Uψ = ψ+(z)is a Cayley transform for D. Conversely, let U be a Cayley transform for D, then by thecontinuous functional calculus we get a graded unital ∗-homomorphism ψ+ : C(T)→D+

given by f 7→ f(U). To see that ψ+ respects the gradings we only need to check thegenerator z of C(T), but this follows from the fact α(U) = U∗. By the Stone-Weierstrasstheorem (see the proof of Lemma A.4) we get a graded ∗-homomorphism ψ : S→D byrestriction. It’s easy to see that we have constructed a bijection between the set of graded∗-homomorphisms ψ : S→D and the set of Cayley transform for D.

K-Theory 10

We note that Uψ = ψ+(x−ix+i ) under Cayley transform x 7−→ x−ix+i , which justifies the

Definition 2.37.

Proposition 2.39. If A is trivially graded and unital C∗-algebra, then K(A) ∼= K0(A).

Proof. Since A is trivially graded and unital, then by Corollary 2.23 A⊗K(H) ∼= M2(A ⊗K(H)) with even grading given by ε = diag(1,−1). We have already seen that K(A) is theabelian group of path components of the space of Cayley transform for A⊗K(H) (we candispense with the graded tensor product here since A is trivially graded). If U is a Cayleytransform, then εU is a self-adjoint unitary whose +1 spectral projection,

P =1

2(εU + 1),

is equal to the +1 spectral projection Pε =

(1 00 0

)of ε, modulo A⊗K(H). Conversely, if

P is a projection which is equal to Pε modulo A⊗K(H) then the formula

U = ε(2P − 1)

defines a Cayley transform for A⊗K(H). We have therefore have a new description of K(A)as the abelian group of path components of the projections which are equal to Pε, moduloA⊗K(H). The formula [P ] 7→ [P ]− [Pε] is an isomorphism from K(A) onto K0(A).

Remark 2.40. Let C1 be the C∗-algebra C⊕C with the standard odd grading. We see thatif A is trivially graded and unital then K(A⊗C1) ∼= K1(A). Indeed, as (A⊗C1)⊗K(H) ∼=(A ⊗ K(H)) ⊕ (A ⊗ K(H)) with the standard odd grading, so a Cayley transform u for(A⊗C1)⊗K(H) corresponds to (v, v∗) ∈ (A⊗K(H)⊕A⊗K(H))+ such that v ∈ (A⊗K(H))+

is an unitary and v − 1 ∈ A⊗K(H).

2.5 Long Exact Sequences

Definition 2.41. Let A be a graded C∗-algebra. Denote by K(A) the space of all graded ∗-homomorphisms from S into A⊗K(H), equipped with the topology of pointwise convergencei.e. ψα→ψ iff ψα(f)→ψ(f) in the norm topology for every f ∈ S.

The space K(A) has a natural base-point, namely the zero homomorphism from S intoA⊗K(H). It also has a natural direct sum operation

K(A)×K(A)→K(A)

which associates to a pair of ∗-homomorphisms ψ1 and ψ2 the ∗-homomorphism ψ1 ⊕ ψ2

from S into A⊗K(H ⊕H) ∼= A⊗K(H). With this operation K(A) turns to be a H-space.It’s of course this operation which gives the addition operation on the abelian group K(A) =π0(K(A)). By a general principle in homotopy theory the direct sum operation agrees withthe group operations on the higher homotopy groups πn(K(A)) for n ≥ 1.

Definition 2.42. Let A be a graded C∗-algebra. The higher K-theory groups of A are thehomotopy groups of the space K(A):

Kn(A) := πn(K(A)), n ≥ 0.

Proposition 2.43. The higher K-theory groups of A have the following properties:a) Kn(A) are abelian groups for n ≥ 0.b) Kn(A) ∼= K(C0(Rn)⊗A) for n ≥ 0.c) Kn(A) is functorial in A.

K-Theory 11

Proof. a): This follows from H-space has abelian π1.b): Since K(C0(Rn)⊗A) is homeomorphic to ΩnK(A), the nth loop space of K(A) with thecompact open topology. Hence

Kn(A) = πn(K(A)) = π0(ΩnK(A)) ∼= K(C0(Rn)⊗A)).

c): Since K(A) is functorial in A, so does Kn(A).

Theorem 2.44. If A→B is a surjective homomorphism of graded C∗-algebras, then theinduced map from K(A) to K(B) is a Serre fibration with fiber K(J), where J is the kernelof A→B. Thus there is a long exact sequence

· · ·→Kn+1(J)→Kn+1(A)→Kn+1(B)→Kn(J)→· · ·→K(B)

Proof. K(A) is the space of Cayley transforms for A⊗K(H) and thus a subspace of theunitary group. The proof that the map K(A)→K(B) is a Serre fibration is then only a smallmodification of the usual proof that the map of unitary groups corresponding to a surjectionof C∗-algebras is a Serre fibration (use the fact that the unitary group is a (deformation)retract of the invertible group and the ideas in Theorem 2.1.8 [19]). Since K(H) is an exactC∗-algebra, it follows that the fiber of K(A)→K(B) is K(J). The long exact sequence followsfrom Theorem 4.41 [16].

Corollary 2.45. (Mayer-Vietoris Sequence) Let’s consider a pull-back diagram

A

// A1

p1

A2 p2// B

with one of maps p1 and p2 is surjective, then there is long exact sequence

· · ·→Kn+1(B)→Kn(A)→Kn(A1)⊕Kn(A2)→Kn(B)→· · ·→K(A1)⊕K(A2).

Proof. Let’s consider the homotopy pullback Z:

Z

// K(A1)

(p1)∗

K(A2)(p2)∗

// K(B)

which gives a long exact sequence

· · ·→Kn+1(B)→Kn(Z)→Kn(A1)⊕Kn(A2)→Kn(B)→· · ·→K(A1)⊕K(A2).

But as one of the maps (p1)∗ and (p2)∗ is a Serre fibration by the previous theorem, thenZ is weak homotopy equivalent to the pull-back K(A1) ⊕K(B) K(A2). We see that (A1 ⊕BA2)⊗K(H) = A1⊗K(H) ⊕B⊗K(H) A2⊗K(H), thus K(A) = K(A1 ⊕B A2) = K(A1) ⊕K(B)

K(A2) (see [13] for more details).

2.6 Products

In the realm of ungraded C∗-algebras there is an associative product on K-theory groups,which culminates with the famous Kasparov product in bivariant K-theory (see [24]):

× : Ki(A)⊗Kj(B)→Ki+j(A⊗B).

K-Theory 12

A key feature of our spectral picture of K-theory is that it is very well adapted to productsand the product is defined using the “comultipliction” map ∆ that we introduced during ourdiscussion of graded C∗-algebras. Using ∆ we obtain a map of spaces

K(A)×K(B)→K(A⊗B)

by associating to a pair (ψA, ψB) the composition

S∆−→ S⊗S ψA⊗ψB−→ (A⊗K(H))⊗(B⊗K(H)) ∼= A⊗B⊗K(H).

Since this map vanishes on K(A)∨K(B), hence induces a map on smash product K(A)∧K(B).So taking homotopy groups we obtain pairings

× : Ki(A)⊗Kj(B)→Ki+j(A⊗B)

by associating to a pair ([ψA], [ψB ]) the map [ψA × ψB ], where

ψA × ψB : Si+j = Si ∧ Sj ψA∧ψB−→ K(A) ∧K(B)→K(A⊗B).

Proposition 2.46. The K-theory product has the following properties:a) It is associative.b) It is graded commutative in the sence that τ∗(x× y) = (−1)ijy×x, for all x ∈ Ki(A) andy ∈ Kj(B), where τ : A⊗B→B⊗A is the transposition isomorphism.c) It is functorial in the sense that if ϕ : A→A′ and ψ : B→B′ are graded ∗-homomorphismsthen (ϕ⊗ψ)∗(x× y) = ϕ∗(x)× ψ∗(y).d) Denote by 1 ∈ K(C) the class of the homomorphism which maps the element f ∈ S tothe element f(0)P ∈ K(H), where P is the orthogonal projection onto a one-dimensional,grading-degree zero subspace of H. If B is any graded C∗-algebra, then the class of thegenerator 1 ∈ K(C) is a two-sided unit for the product. That is

α∗(1× x) = x = α′∗(x× 1)

for all x ∈ Kj(B), where α : C⊗B→B and α′ : B⊗C→B are the canonical isomorphisms.e) If C∗-algebras are unital and ungraded, then this product on K(−) agrees with the one onK0(−) by prescription [p]⊗ [q] = [p⊗ q].

Proof. The proof of a),c) and d) is a simple calculation. To b) we need to show that thefollowing diagram

Ki(A)×Kj(B)

// Ki+j(A⊗B)

τ∗

Kj(B)×Ki(A) // Ki+j(B⊗A)

commutes up to sign (−1)ij . But this diagram can be decomposed to be two diagrams as

πi(K(A))× πj(K(B))

// πi+j(K(A) ∧K(B))

// πi+j(K(A⊗B))

πj(K(B))× πi(K(A)) // πj+i(K(B) ∧K(A)) // πj+i(K(B⊗A))

The diagram on the right hand side commutes, because

K(A) ∧K(B)

// K(A⊗B)

K(B) ∧K(A) // K(B⊗A)

K-Theory 13

commutes up to homotopy. And the diagram on the left hand side commutes up to sign(−1)ij : Indeed σ : Si ∧ Sj→Sj ∧ Si is the composition of ij transpositions of adjacentcircle factors. Such a transposition has degree −1 since it is realized as a reflection of theS2 = S1 ∧ S1 involved. Hence σ has degree (−1)ij . We complete the proof of b).e): We already know that when A is trivially graded and unital C∗-algebra, then K(A) ∼=K0(A) as in Proposition 2.39, but let’s prove it in an other way. Let x = [p]− [q] ∈ K0(A)where p and q are projections in A⊗K(H). Define a graded ∗-homomorphism

ϕx : S→M2(A⊗K(H))

by the formula

ϕx(f) =

(f(0)p 0

0 f(0)q

), f ∈ S.

Now we define a homomorphism µ : K0(A)→K(A). Conversely, given ϕ : S→A⊗K(H),then by the converse functional calculus there is a self-adjoint, odd and regular operator Don HA such that ϕ(f) = f(D). From Remark 2.34 ϕ = ϕD is homotopic to ϕx. Hence wecan define an invers to µ. Under this isomorphism we need to show the following diagramcommutes

K0(A)⊗K0(B)

// K0(A⊗B)

K(A)⊗K(B) // K(A⊗B)

where A and B are unital and ungraded. Since [ψD1] × [ψD2

] = [ψD], where D is theclosure of the essentially self-adjoint operator D1⊗1 + 1⊗D2 by Corollary A.2. We see thatD2 = D2

1⊗1 + 1⊗D22 and since the operators on the right are both positive, so

ker(D2) = ker(D21⊗1) ∩ ker(1⊗D2

2) = ker(D1)⊗ ker(D2).

It follows that ker(D) = ker(D∗D) = ker(D1)⊗ ker(D2). thus (ker(D))i = (ker(D1)⊗ ker(D2))i,for i = 0, 1. We complete the proof by noticing that A and B are ungraded.

2.7 Asymptotic Morphisms

Let B be a graded C∗-algebra. Denote by TB the graded C∗-algebra of bounded, continuousfunctions from the locally compact space [1,∞) into B and denote by T0B the ideal in TBcomprised of continuous functions which vanish in norm at infinity.

Definition 2.47. Let A and B be graded C∗-algebras. The asymptotic algebra of B is thegraded C∗-algebra

AB = TB/T0B.

An asymptotic morphism from A to B is a graded ∗-homomorphism from A into AB. Wedenote an asymptotic morphism with a dashed arrow A 99K B.

Remark 2.48. As Cb(T )⊗B → Cb(T,B) we can embed (Cb(T )/C0(T ))⊗B in AB, but ingeneral they will not coincide. In the case B = Cb(T ) they do not coincide, as can be seenby considering a function in Cb(T,Cb(T )) = Cb(T × T ) which is 1 along the diagonal and

close to 0 away from it, e.g. f(t1, t2) = e−(t1−t2)2 .

One can extract from a graded ∗-homomorphism ϕ : A→AB a family of functions

ϕtt∈[1,∞) : A→B

K-Theory 14

by composing ϕ with any set-theoretic section from AB to TB, then composing with thegraded ∗-homomorphisms from TB to B given by evaluation at t ∈ [1,∞). The family ϕtso obtained has the following properties:(i) for every a ∈ A the map t 7→ ϕt(a), from [1,∞) into B, is continuous and bounded;(ii) for every a, a′ ∈ A and λ ∈ C,

limt→∞

ϕt(a)∗ − ϕt(a∗)

ϕt(a) + λϕt(a′)− ϕt(a+ λa′)

ϕt(a)ϕt(a′)− ϕt(aa′)

α(ϕt(a))− ϕt(α(a))

= 0,

where α denotes the grading automorphism.Conversely, a family of functions ϕtt∈[1,∞) : A→B satisfying these conditions determinesan asymptotic morphism form A to B. Indeed if a ∈ A then the function t→ϕt(a) belongsto TB and by associating to a the class of this function in the quotient AB we obtain agraded ∗-homomorphism from A into AB.

Definition 2.49. Let A and B be graded C∗-algebras. An asymptotic family mapping Ato B is a family of functions

ϕtt∈[1,∞) : A→B

satisfying the conditions (i) and (ii) above. Two asymptotic families ϕt, ψt : A→B are(asymptotically) equivalent if limt→∞ ||ϕt(a)− ψt(a)|| = 0, for all a ∈ A.

Remark 2.50. The requirement that ϕt(a) be a bounded function of t follows from the otherparts of the definition of asymptotic family (see Lemma 1.2 [42]).

The following result is clear from the above discussion:

Proposition 2.51. There is a one-to-one correspondence between asymptotic morphismsfrom A to B and equivalence classes of asymptotic families ϕtt∈[1,∞) : A→B.

Remark 2.52. The proposition allows us to replace a given asymptotic morphism with anequivalent one with special properties in a number of ways (see 25.1.5 [7]):a) By elementary linear algebra, a graded ∗-homomorphism from A to AB always has a∗-linear (not neccessarily bounded) lifting to TB. Thus every asymptotic morphism fromA to B is equivalent to an asymptotic morphism ϕt, where each ϕt is ∗-linear (but notneccessarily bounded).b) By the Bartle-Graves Selection Theorem, a graded ∗-homomorphism from from A toAB always has a continuous (not neccessarily linear) lifting to TB. Thus every asymptoticmorphism from A to B is equivalent to an asymptotic morphism ϕt, where each ϕt iscontinuous (but not neccessarily linear).c) If A is graded and nuclear, then a graded ∗-homomorphism from A to AB has a completelypositive contractive lifting to TB, for any graded B. Thus every asymptotic morphism fromA to any B is equivalent to an asymptotic morphism ϕt, where each ϕt is a completelypositive linear contraction (such an asymptotic morphism is called a completely positiveasymptotic morphism).

Lemma 2.53. Every asymptotic morphism from S to a graded C∗-algebra D is equivalentto a family of graded ∗-homomorphism from S to D.

Proof. By Remark 2.52 we may asumme that each ϕt is linear, hence we have a norm contin-uous family of elements Xt in the unitalization, equal to 1 module D, which are asymptot-ically unitary and asymptotically switched to their adjoints by the grading automorphism.Replace Xt by

Yt =1

2(Xt + α(X∗t ))

K-Theory 15

Note that α(Yt) = Y ∗t and Yt is invertible for large t, so Ut = Yt(Y∗t Yt)

−1/2 is a Cayleytransform for D. Since Xt and Ut are asymptotic we complete the proof.

Definition 2.54. A homotopy between asymptotic morphisms ϕ0t and ϕ1

t from A to Bis an asymptotic morphism ϕt from A to B[0, 1] such that (ϕt(a))(s) = ϕst (a) for s = 0, 1,all t, and all a. Denote the set of homotopy classes of asymptotic morphisms from A to Bby [[A,B]].

Note equivalent asymptotic morphisms are homotopic via the “straight line” in between,i.e. ϕst (a) = sϕt(a) + (1− s)ψt(a) for 0 ≤ s ≤ 1.

Example 2.55. A graded ∗-homomorphism ϕ from A to B defines an asymptotic morphismby setting ϕt = ϕ for all t, thus we can consider ordinary graded ∗-homomorphisms to beasymptotic morphisms in this way. A homotopy of graded homomorphisms gives a homotopyof the corresponding asymptotic morphisms, i.e. there is a natural map from the set [A,B]to [[A,B]]. This map is far from surjective in general, and can also fail to be injective.More generally, a (point-norm continuous) path ϕt of graded ∗-homomorphisms from Ato B defines an asymptotic morphism from A to B, which is homotopic to the gradedhomomorphism ϕ1 (or to ϕt0 for any t0).

From Lemma 2.53 and Example 2.55 we obtain

Proposition 2.56. If D is any graded C∗-algebra then the natural map

[S,D] −→ [[S,D]]

is a bijection.

Given an asymptotic morphism

ϕ : A⊗K(H) 99K B⊗K(H).

If ψ : S→A⊗K(H) is a graded ∗-homomorphism then the composition ϕψ is an asymptoticmorphism from S to B⊗K(H). Hence by the previous proposition an asymptotic morphismϕ : A⊗K(H) 99K B⊗K(H) induces a homomorphism ϕ∗ : K(A)→K(B) by compositionwith ϕ.

Lemma 2.57. Let D be a graded exact C∗-algebra and ϕ : A99KB be an asymptotic mor-phism between graded C∗-algebras. There is an asymptotic morphism ϕ⊗1 : A⊗D99KB⊗D.

Proof. If ϕ : A→AB is a graded ∗-homomorphism, then ϕ⊗1 : A⊗D→AB⊗D is a graded∗-homomorphism. Since D is exact ,then

AB⊗D ∼=TB⊗DT0B⊗D

⊆ T(B⊗D)

T0(B⊗D)= A(B⊗D).

We complete the proof.

Remark 2.58. This lemma is also true for any graded C∗-algebra D if we consider themaximal graded tensor product instead of the minimal one, this is because (−)⊗maxD is anexact functor for any graded D. Hence from now on the undecorated symbol ⊗ denotes themaximal graded tensor product.

The construction of homomorphisms ϕ∗ : K(A)→K(B) from asymptotic morphisms hasseveral elaborations which are quite important:a) An asymptotic morphism ϕ : A99KB determines an asymptotic morphism from A⊗K(H)to B⊗K(H) and hence a K-theory map ϕ∗ : K(A)→K(B).b) An asymptotic morphism ϕ : A99KB⊗K(H) determines an symptotic morphism fromA⊗K(H) to B⊗K(H)⊗K(H). After identifying K(H)⊗K(H) with K(H) we obtain a K-theory map ϕ∗ : K(A)→K(B).

K-Theory 16

c) An asymptotic morphism ϕ : S⊗A99KB determines an asymptotic morphism from S⊗A⊗K(H)to B⊗K(H). If ψ : S→A⊗K(H) represents a class in K(A) then by forming the composition

S∆−→ S⊗S 1⊗ψ−→ S⊗A⊗K(H)

ϕ⊗199K B⊗K(H)

we obtain a class in K(B), thus a K-theory map ϕ∗ : K(A)→K(B).d) Combining b) and c), an asymptotic morphism ϕ : S⊗A99KB⊗K(H) determines a K-theory map ϕ∗ : K(A)→K(B).

2.8 Bott Periodicity in the Spectral Picture

We are going to formulate and prove the Bott periodicity theorem using the spectral pictureof K-theory, products, and a line of argument which is due to Atiyah. We start to presentan abstract outline of the argument and then fill in the details using the theory of Cliffordalgebras to construct suitable K-theory classes and asymptotic morphisms.

Definition 2.59. We say that a graded C∗-algebra B has the rotation property if theautomorphism b1⊗b2 7→ (−1)∂b1∂b2b2⊗b1 which interchanges the two factors in the tensorproduct B⊗B is homotopic to a tensor product ∗-homomorphism 1⊗i : B⊗B→B⊗B.

Example 2.60. The trivially graded C∗-algebra B = C0(Rn) has this property with i = αas in Example 2.4. Indeed, under the identification C0(Rn) ⊗ C0(Rn) ∼= C0(Rn × Rn) theinterchange automorphism corresponds to τ : (x, y) 7→ (y, x) in Rn × Rn. GLn(R) hasexactly two path-connected components (matrices have positive determinant and matriceshave negative determinant) and τ ∈ GLn(R) has determinant (−1)n. However, 1⊗α corre-sponds to : (x, y) 7→ (x,−y) in Rn ×Rn, which also has determinant (−1)n. If n is even, wecan choose i = 1 as 1⊗1 homotopic to 1⊗α.

Theorem 2.61. Let B be a graded C∗-algebra with the rotation property. Suppose thereexists a class b ∈ K(B) and an asymptotic morphism

α : S⊗B99KK(H)

with the property that the induced K-theory homomorphism α∗ : K(B)→K(C) maps b to 1.Then for every graded C∗-algebra A the maps

α∗ : K(A⊗B)→K(A), β∗ : K(A)→K(A⊗B)

induced by α and by multiplication by the K-theory class b are inverse to one another.

Proof. We note first that α∗ : K(A⊗B)→K(A) are multiplicative in the sense that thefollowing diagram

K(C)⊗K(A⊗B)

1⊗α∗

×// K(C⊗A⊗B)

α∗

K(C)⊗K(A)×

// K(C⊗A)

commutes. This can be easily checked by considering u and v as in Lemma 2.25. It followsdirectly from the multiplicative property that α∗ is left-inverse to the map β∗:

α∗(β∗(x)) = α∗(x× b) = x× α∗(b) = x× 1 = x.

To prove that α∗ is also right-inverse to β∗ we introduce the isomorphisms σ : A⊗B→B⊗Aand τ : B⊗A⊗B→B⊗A⊗B, which interchange the first and last factors in the tensor prod-ucts. Note that σ∗(y) × z = τ∗(z × y) for all y ∈ K(A⊗B), z ∈ K(B). Since B has therotation property, τ is homotopic to the tensor product i⊗1⊗1. For z = b, we get

σ∗(y)× b = τ∗(b× y) = i∗(b)× y.

K-Theory 17

Applying α∗ we deduce that

σ∗(y) = σ∗(y)× α∗(b) = α∗(σ∗(y)× b) = α∗(i∗(b)× y) = i∗(b)× α∗(y).

Applying another flip isomorphism we conclude y = α∗(y) × i∗(b) by Proposition 2.46 b).This shows that multiplication by i∗(b) is left-inverse to α∗. Therefore α∗, being both leftand right invertible, is invertible. Moreover the right-inverse β∗ is necessarily a two-sidedinverse.

Remark 2.62. Since the theorem is true for every graded C∗-algebra A, we can choose A tobe C and y to be b and it follows that i∗(b) = b.

Definition 2.63. Let V be a finite-dimensional Euclidean vector space. Denote by C(V )the graded C∗-algebra of continuous functions, vanishing at infinity, from V into Cliff(V ).

Example 2.64. Acorrding to B.6 we have C(R1) is isomorphic to C0(R) ⊕ C0(R) with thestandard odd grading while C(R2) is isomorphic to M2(C0(R2)) with the standard evengrading.

Proposition 2.65. Let V and W be finite-dimensional Euclidean spaces. The map f1⊗f2 7→f , where f(v + w) = f1(v)f2(w) determines an isomorphism of graded C∗-algebras

C(V ⊕W ) ∼= C(V )⊗C(W ).

Proof. It follows easily by combining the isomorphism Cliff(V )⊗Cliff(W ) ∼= Cliff(V ⊕W )(Proposition B.5) with the isomorphism C0(V )⊗ C0(W ) ∼= C0(V ⊕W ).

As a consequence of this proposition we have

Proposition 2.66. Let V be a finite-dimensional Euclidean vector space. The graded C∗-algebra C(V ) has the rotation property.

Proof. Under the isomorphism C(V )⊗C(V ) ∼= C(V ⊕ V ) the flip isomorphism on the tensorproduct corresponds to the ∗-automorphism τ∗∗ of C(V ⊕V ) associated to the map τ whichexchanges the two copies of V in the direct sum V ⊕ V . But τ is homotopic, throughisomertric isomorphisms of V ⊕ V :

(v1, v2) 7→ (sin(s)v1 + cos(s)v2, cos(s)v1 − sin(s)v2),

to the map (v1, v2) 7→ (v1,−v2), and so τ∗∗ is homotopic to 1⊗i∗∗, where i : V→V ismultiplication by −1.

Definition 2.67. Denote by C : V→Cliff(V ) the function C(v) = v which includes V as areal linear subspace of self-adjoint elements in Cliff(V ).

In fact C is an isometry:

||C(v)||2 = ||C(v)2|| = ||||v||2 · 1|| = ||v||2,

hence C dose not vanish at infinity and it’s therefore not an element of C(V ). However, iff ∈ S then the function f(C) : V→Cliff(V ) defined by v 7→ f(C(v)), where f is applied to theelement C(v) ∈ Cliff(V ) in the sense of the functional calculus, does belong to C(V ): To seethis we note first that if C(v) is nonzero, then C(v) is invertible with inverse C(v)/||C(v)||2,thus ||C(v)−1||−1 = ||C(v)||. In general we have ||x−1||−1 ≤ |λ| ≤ ||x|| for λ ∈ σ(x) and x isinvertible. It follows that σ(C(v)) ⊆ ||C(v)||,−||C(v)|| for all v ∈ V , so

lim||v||→∞

||f(C(v))|| = lim||C(v)||→∞

supλ∈σ(C(v))

|f(λ)| = 0.

Since elements in C(V ) ⊆ Cliff(V ) has grading-degree one, the assignment β : f 7→ f(C) isa graded ∗-homomorphism from S to C(V ).

K-Theory 18

Definition 2.68. The Bott element b ∈ K(C(V )) is the K-theory class of the graded ∗-homomorphism β : S→C(V ) defined by β : f 7→ f(C).

We can now formulate the Bott periodicity theorem:

Theorem 2.69. For every graded C∗-algebra A and every finite-dimensional Euclideanspace V the Bott map

β : K(A)→K(A⊗C(V )),

defined by β(x) = x× b, is an isomorphism of abelian groups.

We shall prove the theorem in the next two sections by constructing a suitable asymptoticmorphism α and proving that α∗(b) = 1.

Corollary 2.70. For every graded C∗-algebra A, we havea) K(A) ∼= K(SA⊗Cliff(R));b) K2k(A) ∼= K(A) and in particular K0(S2A) ∼= K0(A) if A is trivially graded;c) Kn(A) = K(A⊗ C0(Rn)) ∼= K(A⊗Cliff(Rn)).

Proof. a): K(A) ∼= K(A⊗C(R)) ∼= K(SA⊗Cliff(R)).

b): As C(R2k) ∼= M2k(C0(R2k)) graded by

(I 00 −I

), then

K(A) ∼= K(A⊗C(R2k)) ∼= K(A⊗ C0(R2k)) = K2k(A).

c): In view of Remark B.7 we see that this follows directly from a) and b).

2.9 The Dirac Operator

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

Theorem 2.71. There exists an asymptotic morphism

α : S⊗C(V )99KK(H)

for which the induced homomorphism α∗ : K(C(V ))→K(C) maps the Bott element b ∈K(C(V )) to 1 ∈ K(C).

Let V be a finite-dimensional Euclidean vector space and we provide the finite-dimensionallinear space underlying the algebra Cliff(V ) with the Hilbert space structure as in AppendixB.

Definition 2.72. Let e, f ∈ V . Define linear operators on the finite-dimensional gradedHilbert space underlying Cliff(V ) by the formulas

e(x) = e · x,

f(x) = (−1)∂xx · f.

Remark 2.73. By using the unique even trace in Appendix B we see that the operator e isself-adjoint while the operator f is skew-adjoint. There is an important operator called thenumber operator

N =

n∑i=1

eiei,

where e1, . . . , en is an orthonormal basis of V . If i1 ≤ · · · ≤ ip, then Nei1 · · · eip = (2p −n)ei1 · · · eip .

K-Theory 19

Denote by H(V ) the infinite-dimensional complex Hilbert space of square-integrableCliff(V )-valued functions on V , thus

H(V ) = L2(V,Cliff(V )).

The Hilbert space H(V ) is a graded Hilbert space with grading inherited from Cliff(V ). AsL2(V )⊗Cliff(V ) = H(V ), thus we can consider S(V ) the dense subspace of H(V ) comprisedof Schwartz-class Cliff(V )-valued functions.

Definition 2.74. The Dirac operator of V is the unbounded operator D on H(V ), withdomain S(V ), defined by

(Df)(v) =

n∑i=1

ei(∂f

∂xi(v)).

Remark 2.75. We see that ∂∂xi

is an even operator and ei is odd, it follows that D is odd. Itfollows from integration by parts that D is formally self-adjoint on S(V ): we note first that

∂

∂xi〈f(v), g(v)〉 = lim

h→0

〈f(v + eih), g(v + eih)〉 − 〈f(v), g(v)〉h

= 〈 ∂f∂xi

(v), g(v)〉+ 〈f(v),∂g

∂xi(v)〉.

Since ei are skew-adjoint and they commute with the partial derivatives, then

〈Df, g〉 =

∫〈Df(v), g(v)〉dµ

=

∫ ∑〈 ∂∂xi

(eif(v)), g(v)〉dµ

= −∫ ∑

〈eif(v),∂g

∂xi(v)〉dµ

=

∫〈f(v), Dg(v)〉

= 〈f,Dg〉.

Lemma 2.76. Let V be a finite-dimensional Euclidean vector space. The Dirac operatoron V is essentially self-adjoint. If f ∈ S, if h ∈ C(V ) and if Mh is the operator of point-wise multiplication by h on the Hilbert space H(V ), then the product f(D)Mh is a compactoperator on H(V ).

Proof. The operator D is a constant coefficient operator acting on a Schwartz space ofvector valued functions on V ∼= Rn. It has the form D =

∑ni=1Ei

∂∂xi

, where the matricesEi are skew-adjoint. Using the Fourier transform, we see that D is unitarily equivalent tothe operator of multiplication by a polynomial D =

∑ni=1 iEiξi with domain the Schwartz-

class functions. Since iEi are self-adjoint, D is essentially self-adjoint by Theorem 6.3 andTheorem 6.4 in [15]. Moreover from the formula

D2 = (

n∑i=1

iEiξi)2 = ||ξ||,

for all ξ ∈ Rn, it follows that if say f(x) = e−x2

then f(D) is pointwise multiplication by

e−||ξ||2

, and therefore the inverse Fourier transform f(D) is convolution by e−1/4||x||2 (upto a constant). Let h ∈ C(V ) is compactly supported, then f(D)Mh is an integral operator

with kernel k(v, w) = e−1/4||v−w||2h(w) in L2, thus f(D)Mh is a Hilbert-Schmidt operatorand in particular compact. The lemma follows from this since the set of f ∈ S for whichf(D)Mh is compact, for all h, is an ideal in S, while the function e−x

2

generate S as an(closed) ideal.

K-Theory 20

Definition 2.77. Let V be a finite-dimensional Euclidean space. If h ∈ C(V ) and if t ∈[1,∞) then denote by ht ∈ C(V ) the function ht(v) = h(t−1v).

Lemma 2.78. Let V be a finite-dimensional Euclidean space with Dirac operator D. Forevery f ∈ S and h ∈ C(V ) we have

limt→∞

||[f(t−1D),Mht ]|| = 0,

where the bracket [ , ] denotes the graded commutator.

Proof. By an approximation argument involving Stone-Weierstrass theorem it suffices toconsider the cases where f(x) = (x± i)−i and where h is smooth and compactly supported.Note that

[ft(D),Mht ] = −ft(D)[t−1D,Mht ]γ∂h(ft(D)),

where γ is the grading automorphism. ft(D) is bounded as D is self-adjoint and [t−1D,Mht ]is the operator of pointwise multiplication by the function

v 7→ t−1n∑i=1

ei∂ht(v)

∂xi.

The norm of the commutator is t−2 times the supremum of the gradient of h and the proofis complete.

Proposition 2.79. There is, up to equivalence, a unique asymptotic morphism

α : S⊗C(V )99KK(H(V ))

for which, on elementary tensors,

αt(f⊗h) = f(t−1D)Mht .

Proof. It follows from Lemma 2.76 that αt is a graded ∗-linear map from algebraic tensorproduct of S and C(V ) into K(H(V )) for t ∈ [1,∞). By the universal property of themaximal tensor product ⊗ and Lemma 2.78 shows that the maps αt define a graded ∗-homomorphism from S⊗C(V )→A(K(H(V ))). Therefore we obtain an asymptotic morphismas required.

2.10 The Harmonic Oscillator

In this section we shall verify that α∗(b) = 1, which will complete the proof of the Bottperiodicity theorem. Actually we shall make a more refined computation which will berequired later on.

Definition 2.80. Let V be a finite-dimensional Euclidean vector space. The Clifford oper-ator is the unbounded odd operator on H(V ), with domain the Schwartz space S(V ), whichis given by the formula

(Cf)(v) = C(v)f(v).

If v =∑ni=1 xiei, then

(Cf)(v) =

n∑i=1

xiei(f(v)).

K-Theory 21

The Clifford operator is essentially self-adjoint on the domain S(V ) since C ± iI havedense range. So if f ∈ S we may form the bounded operator f(C) ∈ B(H(V )) by thefunctional calculus.

Lemma 2.81. If C(V ) is represented on the Hilbert space H(V ) by pointwise multiplicationoperators, then the composition

Sβ−→ C(V )

M−→ B(H(V ))

maps f ∈ S to f(C) ∈ B(H(V )).

Proof. Mβ(f)(g)(v) = f(C)(v)g(v) = f(C(V ))g(v) = f(C)(g)(v).

Definition 2.82. Let V be a finite-dimensional Euclidean vector space. Define an un-bounded odd operator B on H(V ), with domain S(V ), by the formula

(Bf)(v) =

n∑i=1

xiei(f(v)) +

n∑i=1

ei(∂f

∂xi(v)).

Thus B = C +D, where C is the Clifford operator and D is the Dirac operator.

Observe that the operator B maps the Schwartz space S(V ) into itself. So the operatorB2 is defined on S(V ).

Proposition 2.83. Let V be a finite-dimensional Euclidean vector space of dimension n.There exists within S(V ) an orhonormal basis for H(V ) consisting of eigenvector for B2,with eigenvalues 2n, each of finite multiplicity. Moreover, the eigenvalue 0 has multiplicityone and the corresponding eigenfunction is e−1/2||v||2 .

Proof. Let’ first consider the case V = R. Then

B =

(0 x− d

dx

x+ ddx 0

)if we identify H(V ) with L2(R)⊕ L2(R). Since

B2 =

(x2 − d2

dx2 − 1 0

0 x2 − d2

dx2 + 1

),

so it suffices to consider the operator H = x2− d2

dx2 . Define K = x+ ddx and L = x− d

dx , and

let f1(x) = e−12x

2

. Observe that H = KL−I = LK+I and that Kf1 = 0, so that Hf1 = f1.It follows that HL = LH + 2L and HLn = LnH + 2nLn. So if we define fn+1 = Lnf1

then Hfn+1 = (2n+1)fn+1. The functions fn+1 are orthogonal (being eigenfunctions of thesymmetric operator H with distinct eigenvalues), nonzero, and they span L2(R) (since, byinduction, fn+1 is a polynomial of degree n times f1)9. So after L2-normalization we obtainthe required basis. Note that eigenvalue 1 has multiplicity one.The general case follows from the calculation

B2 = C2 +D2 +N =

n∑i=1

x2i +

n∑i=1

− ∂2

∂x2i

+ (2p− n) on Hp(V ),

where Hp(V ) denotes the subspace of H(V ) comprised of functions V→Cliff(V ) whose valuesare combinations of the degree p monomials ei1 . . . eip . From this an eigenbasis for B2 maybe found by separation of variables.

9This implies that C ± iI have dense range, hence C is essentially self-adjoint.

K-Theory 22

Corollary 2.84. The Bott-Dirac operator B admits an orthonormal eigenbasis of Schwartz-class functions, with eigenvalues ±

√2n, each of finite multiplicity. Moreover

a) B is essentially self-adjoint;b) B has compact resolvent;

c) The kernel of (the closure of ) B is one-dimensional and generated by e−12 ||v||

2

.

Proof. Since B has an orthonormal eigenbasis of Schwartz-class functions, it is essentiallyself-adjoint on the Schwartz space. It has compact resolvent, since its eigenvalue sequenceconverges to infinity in absolute value. Finally, we note ker(B2) = ker(B∗B) = ker(B).

Theorem 2.85. Let V be a finite-dimensional Euclidean vector space. The composition

S∆−→ S⊗S 1⊗β−→ S⊗C(V )

α99K K(H(V ))

is asymptotically equivalent to the asymptotic morphism γ : S 99K K(H(V )) defined by

γt(f) = f(t−1B) (t ≥ 1).

The main tool to prove this is Mehler’s formula from quantum theory:

Proposition 2.86. Let V be a finite-dimensional Euclidean vector space and let C and Dbe the Clifford and Dirac operators for V . The operators D2, C2 and C2 +D2 are essentiallyself-adjoint on the Schwartz space S(V ), and if s > 0 then

e−s(C2+D2) = e−

12 s1C

2

e−s2D2

e−12 s1C

2

,

where s1 = (cosh(2s)− 1)/ sinh(2s) and s2 = sinh(2s)/2. In addition,

e−s(C2+D2) = e−

12 s1D

2

e−s2C2

e−12 s1D

2

.

Proof. Note that the second identity follows from the first upon taking the Fourier transformon L2(R), which interchanges the operators D2 and C2. The proof can be found in [28] or[29].

Lemma 2.87. If X is any unbounded self-adjoint operator then there are asymptotic equiv-alences

e−12 τ1X

2

∼ e− 12 t−2X2

, e−τ2X2

∼ e−t−2X2

and

t−1Xe−12 τ1X

2

∼ t−1Xe−12 t−2X2

, t−1Xe−τ2X2

∼ t−1Xe−t−2X2

where τ1 = (cosh(2t−2)− 1)/ sinh(2t−2) and τ2 = sinh(2t−2)/2.

Proof. By asymptotic equivalence we mean here that the differences between the left andright hand sides in the above relations all converges to zero, in the operator norm, as ttends to infinity. By the spectral theorem it suffices to consider the same problem with theself-adjoint operator X replaced by a real variable x and the operator norm replaced by thesupremum norm on C0(R). The lemma is then a simple calculus, based on the Taylor seriesτ1, τ2 = t−2 + o(t−2).

Lemma 2.88. If f, g ∈ S, then

limt→∞

||[f(t−1C), g(t−1D)]|| = 0.

K-Theory 23

Proof. For any fixed f ∈ S, the set of g ∈ S for which the lemma holds is a C∗-subalgebraof S. So by the Stone-Weierstrass theorem it suffices to prove the lemmea when g is one ofthe resolvent function (x ± i)−1. If furthermore suffices to consider the case where f is asmooth and compactly supported function. In this case we have

||[f(t−1C), (t−1 ± i)−1]|| ≤ ||[f(t−1C), t−1D)]||

by the commutator identity for resolvents. But then

||[f(t−1C), t−1D)]|| = t−1||[f(t−1C), D)]|| = t−1|| 5 (f(t−1C))|| = t−2|| 5 (f(C))||.

This proves the lemma.

Proof. (Proof of Theorem 2.85). We already know that B2 = C2 +D2 +N on Hp(V ). Sinceoperator N commutes with C2 and D2 and by Mehler’s formula

e−t−2

B2 = e−12 τ1C

2

e−τ2D2

e−12 τ1C

2

e−t−2N .

It follows from Lemma 2.87 and Lemma 2.88 that

e−t−2

B2 ∼ e− 12 t−2C2

e−t−2D2

e−12 t−2C2

e−t−2N ∼ e−t

−2D2

e−t−2C2

,

since the operator N is bounded the operator e−t−2N converges in norm to the identity

operator. Now the homomorphism β : S→S⊗C(V ) maps u(x) = e−x2

to u⊗u(C), andapplying αt we obtain

αt(β(u)) = u(t−1D)u(t−1C) = e−t−2D2

e−t−2C2

But γt(u) = e−t−2

B2, and so we have shown that αt(β(u)) and γt(u) are asymptotic to one

another. A similar computation shows that if v(x) = xe−x2

then αt(β(v)) and γt(v) areasymptotic to one another. Since u and v generate S, this completes the proof.

Corollary 2.89. The homomorphism α∗ : K(C(V ))→K(C) maps the element b ∈ K(C(V ))to the element 1 ∈ K(C).

Proof. The class α∗(b) is represented by the composition of the graded ∗-homomorphismβ with the asymptotic morphism α. By Theorem 2.85, this composition is asymptoticto the asymptotic morphism γt(f) = f(t−1B). But each map γt is actually a graded ∗-homomorphism, and so γ is homotopic to the single graded ∗-homomorphism f 7→ f(B).Now denote by p the projection onto the kernel of B and then by Corrollary 2.84 c) p is aprojection onto a one-dimensional, grading-degree zero subspace of H(V ). The formula

f 7→ f(s−1B), s ∈ [0, 1]

defines a homotopy (as in Remark 2.34) proving that α∗(b) = 1.

E-Theory 24

3 E-Theory

3.1 The Asymptotic Category

The purpose of this section is to construct the asymptotic category and then define tensorproduct operation such that this category extends to an amplified asymptotic category inthe same way we constructed the amplification of the category of graded C∗-algebras.

We are going to define a notion of homotopy for asymptotic morphisms, but before doingso it is convenient to note that the correspondence B 7→ A(B) is a functor on the category ofgraded C∗-algebras. Indeed a graded ∗-homomorphism ϕ : B1→B2 gives rise to a commutingdiagram

0 // T0B1//

TB1//

AB1//

0

0 // T0B2// TB2

// AB2// 0,

in which the two leftmost vertical maps are given by composing a function T→B1 withϕ1, and the rightmost vertical map is induced from these two. Denote by An the n-foldcomposition of the functor A with itself. It’s convenient to denote by A0 the identity functor.

Definition 3.1. If I = [a, b] is a closed interval then IB := f : I→B : f is continuous.We say that two graded ∗-homomorphisms ϕ0, ϕ1 : A→AnB are n-homotopic if there is agraded ∗-homomorphism ϕ : A→AnIB from which ϕ0 and ϕ1 can recovered upon composingwith evaluation at the endpoints of I.

Lemma 3.2. There is a natural inclusion IAB → AIB such that the diagram

IAB

evt

// AIB

evt

AB =// AB

commutes.

Proof. Consider the commuting diagram

0 // IT0B //

ITB //

IAB //

0

0 // T0IB // TIB // AIB // 0,

in which the two leftmost vertical arrows map a function f : I→TB to f : T→IB definedby f(t)(s) = f(s)(t). The induced ∗-homomorphism on IAB is the one we require.

Remark 3.3. By the previous lemma homotopic graded ∗-homomorphisms into AnB aren-homotopic, but not vice-versa, in general. This is because IAB and AIB are in generalnot the same:

IAB ∼= C(I, Cb(T,B))/C(I, C0(T,B)) → Cb(I × T,B)/C0(I × T,B) = AIB.

To see this note that the fuction f(s, t) = sin(st), (s, t) ∈ I × T gives an example of a con-tinuous bounded function of two variables, which is not continuous as a function I→Cb(T ).

Remark 3.4. Let D be a graded C∗-algebra, then AnID is the quotient of T2ID correspond-ing to the C∗-seminorm

||F ||A2 = lim supt1→∞

lim supt2→∞

sups||F (t1, t2, s)|| (F ∈ T2ID).

E-Theory 25

The graded C∗-algebra A2D may be described similarly by omitting the variable s in theabove formula. The graded C∗-algebra AD is the quotient of TD corresponding to theC∗-seminorm

||f ||A = lim supt→∞

||f(t)|| (f ∈ TD).

Proposition 3.5. The relation if n-homotopy is an equivalence relation on the set of graded∗-homomorphisms from A to AnB.

To prove the proposition we shall use the following lemmas.

Lemma 3.6. A(−) is an exact functor on the category of graded C∗-algebras. In particular,a surjection of graded C∗-algebras induces a surjection of asymptotic algebras10.

Proof. This is proved by induction on n and 3× 3 Lemma, once it is shown that T(−) andT0(−) are exact functors. The only difficult point is to show that the map TB→T(B/J) issurjective, but this follows from Bartle-Graves Continuous Selection (Theorem 1.8 [4]).

Lemma 3.7. Let ϕ1 : B1→B and ϕ2 : B2→B be graded ∗-homomorphisms, one of which issurjective, and let

B1 ⊕B B2 = b1 ⊕ b2 ∈ B1 ⊕B2 : ϕ(b1) = ϕ2(b2).

The (natural) graded ∗-homomorphism

An(B1 ⊕B B2)→An(B1)⊕AnB AnB2,

induced from the projections of B1 ⊕B B2 onto B1 and B2, is an isomorphism.

Proof. According to the previous lemma, the functor A transforms surjections to surjections,so by induction it suffices to prove the present lemma for n = 1. Our surjectivity hypothesisensures that the quotient map in the sequence

0→T0B1 ⊕T0B T0B2→TB1 ⊕TB TB2→AB1 ⊕AB AB2→0

is indeed surjective, and hence a short exact sequence. Consider now the diagram

0 // T0B1 ⊕T0B T0B2//

∼=

TB1 ⊕TB TB2//

∼=

AB1 ⊕AB AB2//

∼=

0

0 // T0(B1 ⊕B B2) // T(B1 ⊕B B2) // A(B1 ⊕B B2) // 0,

where the two leftmost vertical arrow send f1 ⊕ f2 to the function f(t) = f1(t)⊕ f2(t), andthe rightmost vertical arrow is induced from the other two. The rightmost arrow is inverseto the graded ∗-homomorphism that we are asked to prove is an isomorphism.

Proof. (proof of Proposition 3.5). Reflexivity and symmetry of the n-homotopy relation arestraightforward. We concentrate on transitivity.Suppose that ϕ0 is homotopic to ϕ1 via an n-homotopy Φ1 : A→AnI1B and that ϕ1 isn-homotopic to ϕ2 via Φ2 : A→AnI2B. We can assume that I1 and I2 are consecutiveintervals on the real line, whose union is a third closed interval I. Φ1 and Φ2 determine agraded ∗-homomorphism into the pullback AnI1B ⊕AnB AnI2B, where I1B is mapped toB by evaluation at the rightmost endpoint of I1 and I2B is mapped to B by evaluation atthe leftmost endpoint. By Lemma 3.7, it is isomorphic to An(I1B ⊕B I2B), and using thefact that I1B ⊕B I2B ∼= IB we obtain a graded ∗-homomorphism from A into AnIB whichimplements an n-homotopy between ϕ0 and ϕ1.

10In fact it is trivial to show that a injection of graded C∗-algebras induces a injection of asymptoticalgebras (see Proposition 1 [35]).

E-Theory 26

Definition 3.8. Let A and B be graded C∗-algebras. Denote by [[A,B]]n the set of n-homotopy classes of graded ∗-homomorphisms from A to AnB.

Note that [[A,B]]0 is the set of homotopy classes of graded ∗-homomorphisms and[[A,B]]1 is the set of homotopy classes of asymptotic morohisms. Also, each [[A,B]]n isa pointed set with the zero map A→AnB as the basepoint.

Let αB : B→AB the graded ∗-homomorphism which associates to b ∈ B the classes inAB of the constant function t 7→ b ∈ TB. Note that α defines a natural transformation fromthe identity functor on the category of graded C∗-algebras to the functor A.

Proposition 3.9. The maps

[[A,B]]nComposition with An(αB)−−−−−−−−−−−−−−−−−→ [[A,B]]n+1

and

[[A,B]]nComposition with αAnB−−−−−−−−−−−−−−−−→ [[A,B]]n+1

are equal.

Proof. It follows from the functoriality of A and the naturality of α that the (n+1)-homotopyclass of the composition with An(αB) depends only on the n-homotopy class of ϕ ∈ [[A,B]]n.Since αAnB ϕ = A(ϕ)αA, it is easy to see that composition with αAnB maps n-homotopyclasses to (n+1)-homotopy classes. To see they are equal, it suffices to show that the graded∗-homomorphisms

An(αB), αAnB : AnB→An+1B

are (n+1)-homotopic. It follows from Lemma 3.2 and the transitivity of the (n+1)-homotopyrelation, the proof is reduced to the assertion that for any graded C∗-algebra D the graded∗-homomorphisms A(αD), αAD : AD→A2D are 2-homotopic. However, this follows fromRemark 3.4(see Proposition 2.8 [17] for more details).

We now assemble the sets [[A,B]]n, for n ∈ N, into a single set [[A,B]]∞.

Definition 3.10. Let A,B be graded C∗-algebras. Denote by [[A,B]]∞ the direct limit ofthe system of pointed sets

[[A,B]]0→[[A,B]]1→[[A,B]]2→ . . .

We are ready now to organize the set [[A,B]]∞ into the morphism sets of a category.

Proposition 3.11. Given graded ∗-homomorphisms ϕ : A→AjB and ψ : B→AkC, theconstruction

Aϕ−→ AjB

Aj(ψ)−→ Aj+kC,

defines an associative composition law

[[A,B]]∞ × [[B,C]]∞→[[A,C]]∞.

The identity map B→A0B determines an element of [[B,B]]∞ which serves as left and rightidentity elements for this composition law.

Proof. Fix a graded ∗-homomorphism ϕ : A→AjB. The construction defines a map from[[B,C]]k to [[A,C]]j+k and this map is compatible with the connecting maps, so there is a welldefined map from [[B,C]]∞ to [[A,C]]∞. If we fix a graded ∗-homomorphism ψ : B→Ak(C),

E-Theory 27

then the construction defines a map from [[A,B]]j to [[A,C]]j+k. By Proposition 3.9 thismap is compatible with the connecting maps, so there is a well defined map from [[A,B]]∞to [[A,C]]∞. Associativity is immediate, since both ways of composing a triple

ϕ : A→AjB, ψ : B→Ak(C) and θ : C→AjD

produce the same map

Aϕ−→ Aj(B)

Ajψ−→ Aj+kCAj+kθ−→ Aj+k+lD.

It’s clear that the identity map serve as identity morphisms, as required.

Definition 3.12. The asymptotic category is the category whose objects are the gradedC∗-algebras and whose morphism sets are [[A,B]]∞. The law of composition of morphismsis given by Proposition 3.11.

Let F be a covariant functor from the category of graded C∗-algebras to itself and nowwe want to develop sufficient conditions under which F determines a functor from the theasymptotic category to itself.

Definition 3.13. If B is a graded C∗-algebra and if f ∈ F (IB), then define a function

f : I→F (B) given by

f(t) = F (evt)(f),

where evt is evaluation at t. We shall say that the functor F is continuous if for every Band every f ∈ F (IB) the function f is continuous.

Example 3.14. The maximal and minimal tensor product functors B 7→ D⊗B are con-tinuous, because there are isomorphisms D⊗IB ∼= I(D⊗B) which is compatible with theevaluation maps to D⊗B. However, T(−) and A(−) are not continuous functors as wediscussed in Remark 3.3.

If F is continuous, then by associating f ∈ F (IB) the continuous function f : I→F (B)we obtain a graded ∗-homomorphism

F (IB)→IF (B).

In the same way, there are graded ∗-homomorphisms

F (TB)→TF (B), F (T0B)→T0F (B).

So if F is in addition an exact functor then we obtain an induced map from F (A(B))→A(F (B)),as indicated in the following commutative diagram:

0 // F (T0B) //

F (TB) //

F (AB) //

i

0

0 // T0F (B) // TF (B) // AF (B) // 0.

Let’s also define graded ∗-homomorphisms in : F (AnB)→AnF (B) inductively, as follows:

F (AnB)i−→ AF (An−1B)

A(in−1)−→ AnF (B).

For later purposes we note that:

E-Theory 28

Lemma 3.15. The diagram

F (Aj+kB)

ij &&

ij+k// Aj+kF (B)

AjF (AkB)

Aj(ik)

88

commutes.

Definition 3.16. From a graded ∗-homomorphism ϕ : A→AnB construct a graded ∗-homomorphism F (ϕ) : F (A)→AnF (B) by forming the composition

F (A)F (ϕ)−→ F (AnB)

in−→ AnF (B).

Our main result concerning the extension of functors to the asymptotic category is thefollowing theorem.

Theorem 3.17. For each continuous and exact functor F on the category of graded C∗-algebras there is an associated functor F from the asymptotic category to itself which mapsthe class of a graded ∗-homomorphism ϕ : A→AnB to the class of the above described graded∗-homomorphism F (ϕ) : F (A)→AnF (B) and F (B) = F (B) on objects.

Proof. We must check that the correspondence ϕ 7→ F (ϕ) is well-defined at he level of ho-motopy classes and then is descends to a well-defined map on the morphism sets [[A,B]]∞and finally that it is compatible with composition of morphisms.

Applying the F to an n-homotopy ϕ : A→AnIB we obtain F (ϕ) : F (A)→AnF (IB). Bycontinuity of F we obtain an n-homotopy F (A)→AnIF (B) as required. It follows that thecorrespondence ϕ 7→ F (ϕ) defines a map

[[A,B]]n→[[F (A), F (B)]]n.

Next we shall prove this map is compatible with connecting maps in the direct system i.e.

F (AnB)

in

F (αAnB)// F (An+1B)

in+1

AnF (B)αAn(F (B))

// An+1F (B).

However, this follows easily from the facts that α is a natural transformation and that forany graded C∗-algebra D the diagram

F (D)

=

F (αD)// F (TD)

i

F (D)αF (D)

// TF (D)

commutes. So defines a map

[[A,B]]∞→[[F (A), F (B)]]∞.

It remains to prove that the correspondence is compatible with the composition law in theasymptotic category i.e. if ϕ : A→AjB and ψ : B→AkC represent elements of [[A,B]]∞ and

E-Theory 29

[[B,C]]∞, respectively, then F (ϕ) F (ψ) = F (ϕ ψ). Hence we must to show the followingdiagram commutes:

F (A)F (ϕ)//

=

F (AjB)FAj(ψ)

//

ij

F (Aj+kC)

ij+k

F (A)F (ϕ)

// AjF (B)Aj F (ψ)

// Aj+kF (C).

But this follows from Lemma 3.15 along with naturality of ij and functoriality of Aj .

Proposition 3.18. Let F1 and F2 be continuous and exact functors and let β be a naturaltransformation from F1 to F2 (thus in particular, β is a collection of graded ∗-homomorphismsβB : F1(B)→F2(B)). If βB : F1(B)→F2(B) denotes the class in the asymptotic category ofthe graded ∗-homomorphism βB, then β is a natural transformation from F1 to F2.

Since the maximal tensor products D⊗(−) and (−)⊗D are both continuous and exact:

Proposition 3.19. There is a functor on the asymptotic category which associates to theclass of a graded ∗-homomorphism ψ : A→AnB the composition

D⊗A 1⊗ψ−→ D⊗(AnB)in−→ An(D⊗B).

Proposition 3.20. There is a functor on the asymptotic category which associates to theclass of a graded ∗-homomorphism ϕ : A→AnB the composition

A⊗D ϕ⊗1−→ (AnB)⊗D in−→ An(B⊗D).

The following lemma expresses a crucial compatibility property of left and right tensorproducts.

Lemma 3.21. If ϕ : A1→A2 and ψ : B1→B2 are morphisms in the asymptotic category,then the two compositions

A1⊗B1ϕ⊗1−→ A2⊗B1

1⊗ψ−→ A2⊗B2, A1⊗B11⊗ψ−→ A1⊗B2

ϕ⊗1−→ A2⊗B2

are equal.

Proof. Let ϕ : A1→AmA2 and ψ : B1→AnB2 be graded ∗-homomorphisms representingthe morphisms given in the lemma. The compositions in the lemma are represented by thecompositions of the graded ∗-homomorphism

ϕ⊗ψ : A1⊗B1→AmA2⊗AnB2

with two graded ∗-homomorphisms from AmA2⊗AnB2 to Am+n(A2⊗B2) obtained by takingthe two possible routes around the diagram

AmA2⊗AnB2

in

im // Am(A2⊗AnB2)

Am(in)

An(AmA2⊗B2)An(im)

// Am+n(A2⊗B2).

So it suffices to prove that the diagram commutes up to (m+n)-homotopy. By an inductionargument, it suffices to consider the case n = m = 1. However, this follows from Remark3.4 (see Lemma 4.5 [17] for more details).

E-Theory 30

Theorem 3.22. There is a functor (−)⊗(−) on the asymptotic category which is the max-imal tensor product on objects; which associates the pair of morphisms (1, ψ) the morphism1⊗ψ, as in Proposition 3.19; and which associates to the pair of morphisms (ϕ, 1) the mor-phism ϕ⊗1, as in Proposition 3.20. Let ϕ⊗ψ : A1⊗B1→A2⊗B2 to be the composition

A1⊗B1ϕ⊗1−→ A2⊗B1

1⊗ψ−→ A2⊗B2.

Proposition 3.23. Let ϕ : A→B be a morphism in the asymptotic category. After C⊗A andC⊗B are identified with A and B, respectively, the morphism 1⊗ϕ : C⊗A→C⊗B identifieswith ϕ : A→B.

Proof. This is an immediate consequence of Proposition 3.18.

Remark 3.24. This tensor product also has the usual properties of associativity, commuta-tivity, and so on.

Definition 3.25. The amplified asymptotic category is the category whose objects arethe graded C∗-algebras and for which the morphisms from A to B are the elements of[[S⊗A,B]]∞. Composition of morphisms ϕ : A→B and ψ : B→C in the amplified asymp-totic category is given by the following composition of morphisms in the asymptotic category:

S⊗A ∆⊗1−→ S⊗S⊗A 1⊗ϕ−→ S⊗B ψ−→ C.

3.2 The E-Theory Category

Definition 3.26. Let A and B be graded C∗-algebras. We shall denote by E(A,B) the setof homotopy classes of asymptotic morphism from S⊗A⊗K(H) to B⊗K(H). Thus

E(A,B) = [[S⊗A⊗K(H), B⊗K(H)]].

The sets E(A,B) come equipped with an operation of addition, given by direct sum ofasymptotic morphisms, and the zero asymptotic morphism provides a zero element for thisaddition.

Proposition 3.27. The abelian monoids E(A,B) are in fact abelian groups.

Proof. Let ϕ : S⊗A⊗K(H)99KB⊗K(H) be an asymptotic morphism and α, β and γ denotethe grading automorphism on S⊗A⊗K(H), B and H, respectively. Let u : H→Hop be thedegree one isomorphism given by transposing and define an asymptotic morphism

ϕop : S⊗A⊗K(H)99KB⊗K(Hop)

by the formula ϕopt (x) = 1⊗Adu(ϕt(α(x))). To see that ϕop is an asymptotic morphism wenote first that the grading automorphism on Hop is µ = −uγu∗ and ur = −ru, so

β⊗Adµ(ϕopt (x)) = β⊗Adµu(ϕt(α(x))) = β⊗Adγu(ϕt(α(x))) ∼ 1⊗Adγuγ(ϕt(x)) = ϕopt (α(x)).

It’s trivial to see that ϕop is almost multicative, linear and ∗-preserving. We shall showthat ϕop defines an additive inverse to ϕ in E(A,B). For a fixed scalar s ≥ 0 we see that(

0 s⊗u∗s⊗u 0

)is a self-adjoint element in M(B)⊗B(H ⊕ Hop). It’s important to use the

grading µ on Hop to make this element odd:

β⊗Adγ⊕µ((

0 s⊗u∗s⊗u 0

))= s⊗Adγ⊕µ

((0 u∗

u 0

))=

(0 −s⊗u∗

−s⊗u 0

).

E-Theory 31

So Φs : S⊗S⊗A⊗K(H)99KB⊗K(H ⊕Hop) by the formula

Φst (f⊗x) = f

(0 s⊗u∗

s⊗u 0

)(ϕt(x) 0

0 ϕopt (x)

)defines an asymptotic morphism:

β⊗Adγ⊕µ(Φst (f⊗x)) = f

(β⊗Adγ⊕µ

((0 s⊗u∗

s⊗u 0

)))(β⊗Adγ(ϕt(x)) 0

0 β⊗Adµ(ϕopt (x))

)= f

(0 −s⊗u∗

−s⊗u 0

)(β⊗Adγ(ϕt(x)) 0

0 β⊗Adµ(ϕopt (x))

)∼ Φst (ω⊗α(f⊗x)),

where ω is the grading automorphism on S. Using e−x2

and xe−x2

we see that Φs is anasymptotic morphism. By composing Φs with the comultiplication ∆ : S→S⊗S we obtainasymptotic morphisms

S⊗A⊗K(H)∆⊗1−→ S⊗S⊗A⊗K(H)

Φs

99K B⊗K(H ⊕Hop)

which constitute a homotopy parametrized by s ∈ [0,∞] connecting ϕ ⊕ ϕop to 0 by using

e−x2

and xe−x2

.

If e is a projection onto a one-dimensional, grading-degree zero subspace of H, thenby composing asymptotic morphisms with the graded ∗-homomorphism which maps theelement f⊗a ∈ S⊗A to the element f⊗a⊗e ∈ S⊗A⊗K(H) we obtain a map

[[S⊗A⊗K(H), B⊗K(H)]]→[[S⊗A,B⊗K(H)]].

Lemma 3.28. E(A,B) ∼= [[S⊗A,B⊗K(H)]]

Proof. It follows from Lemma 2.29 or the construction of stabilized homotopy category thatthe inverse is given by tensor product with the identity on K(H).

The groups E(A,B) are contravariantly functorial in A and covariantly functorial in Bon the category of graded C∗-algebras.

Corollary 3.29. The functor E(C, B) on the category of graded C∗-algebras is naturallyisomorphic to K(B).

Proof. This follows from Proposition 2.56 and Lemma 3.28.

The main technical theorem in E-theory is the following:

Theorem 3.30. Let A and B be graded C∗-algebras and assume that A is separable. Thenatural map of [[A,B]]1 into the direct limit [[A,B]]∞ is a bijection. Thus every morphismfrom A to B in the asymptotic category is represented by a unique homotopy class of asymp-totic morphisms from A to B.

Proof. This follows from the fact that the map

[[A,B]]n→[[A,B]]n+1

is bijective for any n ≥ 1, when A is separable (see Theorem 2.16 [17] for more details).

It follows that the abelian group E(A,B) (for A separable) may be identified with theset of morphisms in the amplified asymptotic category from A⊗K(H) to B⊗K(H). As aresult we obtain the E-theory category:

E-Theory 32

Theorem 3.31. The E-theory groups E(A,B) = [[S⊗A⊗K(H), B⊗K(H)]] are the mor-phism groups in an additive category whose objects are the separable graded C∗-algebras, andwhose associative and bilinear composition law

E(A,B)⊗ E(B,C) −→ E(A,C)

is the composition law in the amplified aymptotic category. There is a functor from thehomotopy category of graded separable C∗-algebras into the E-theory category which is theidentity on objects and associates to a graded ∗-homomorphism ϕ : A→B the 1-homotopyclass of the graded ∗-homomorphism

S⊗A⊗K(H)η⊗ϕ⊗1−−−−−→ B⊗K(H)

αB⊗K(H)−−−−−−→ A(B⊗K(H)).

Corollary 3.32. Let ϕ : A→B be a graded ∗-homomorphism such that the tensor productϕ⊗1 : A⊗K(H)→B⊗K(H) is homotopy equivalent to a graded ∗-isomorphism. Then ϕdetermines an invertible element of E(A,B). In particular, A ∼= A⊗K(H) ∼= K(H)⊗A andA ∼= A⊗Mn(C) ∼= Mn(C)⊗A in the E-theory category.

Proof. If ϕ⊗1 is homotopic to a graded ∗-isomorphism Φ : A⊗K(H)→B⊗K(H), then[ϕ⊗1] = [Φ] ∈ E(A⊗K(H), B⊗K(H)) is an invertible element. So is [ϕ] ∈ E(A,B) un-der E(A⊗K(H), B⊗K(H)) ∼= E(A,B).

Remark 3.33. If ϕ : A→B be a graded ∗-homomorphism and if ψ : B99KC is an asymptoticmorphism then ϕ and ψ determine elements [ϕ] ∈ E(A,B) and [ψ] ∈ E(B,C), where [ψ] isrepresented by

S⊗B⊗K(H)η⊗ψ⊗1−−−−−→ A(C)⊗K(H)

i−→ A(C⊗K(H)).

The naive composition ψ ϕ is an asymptotic morphism from A to C, and so defines anelement [ψ ϕ] ∈ E(A,C). We can show that [ψ ϕ] = [ψ] [ϕ]. The same applies tocomposition of graded ∗-homomorphisms and asymptotic morphisms the other way round,and also to compositions in the amplified category of separable graded C∗-algebras.

The E-theory category plays an important role in the computation of C∗-algebra K-theory groups, as follows. To compute the K-theory of a C∗-algebra A one can, on occasion,find a C∗-algebra B and elements of E(A,B) and E(B,A) whose compositions are the iden-tity morphisms in E(A,A) and E(B,B). Composition with these two elements of E(A,B)and E(B,A) now gives a pair of mutually inverse maps between E(C, A) and E(C, B). Butas we noted that E(C, A) and E(C, B) are the K-theory groups K(A) and K(B). It there-fore follows that K(A) ∼= K(B). Therefore, assuming that K(B) can be computed, so canK(A). This is the main strategy for computing the K-theory of group C∗-algebras.

Comparing Remark 2.28 the tensor product functor on the asymptotic category extendsto the amplified asymptotic category, and we obtain a tensor product in E-theory:

Theorem 3.34. There is a maximal tensor product (−)⊗(−) on the E-theory categoty whichis compatible with the tensor product on C∗-algebras via the functor from the homotopycategory of graded separable C∗-algebras into the the E-theory category i.e. if ϕ : A1→B1

and ψ : A2→B2 are graded ∗-homomorphisms, then [ϕ⊗ψ] = [ϕ]⊗[ψ] ∈ E(A1⊗A2, B1⊗B2).Moreover, the maximal tensor prodict extends the product on K-theory in the following sense

K(A)⊗K(B)

∼=

×// K(A⊗B)

∼=

E(C, A)⊗ E(C, B)⊗

// E(C, A⊗B).

E-Theory 33

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

Theorem 3.35. Let B be a separable, graded and exact C∗-algebra. There is a functorA 7→ A⊗minB on the E-theory category. In particular, if A1 and A2 are isomorphic in theE-theory category then A1⊗minB and A2⊗minB are isomorphic in the E-theory category.

Our proof of Bott periodicity in the spectral picture of K-theory may be recast as acomputation in E-theory.

Definition 3.36. Let V be a finite-dimensional Euclidean vector space. Denote by β ∈K(C(V )) = E(C, C(V )) the E-theory class of the graded ∗-homomorphism β : S→C(V ) asin Definition 2.68. Denote by α ∈ [[S⊗C(V ),K(H(V ))]] = E(C(V ),C) the E-theory class ofthe asymptotic morphism α : S⊗C(V )99KK(H(V )) as in Proposition 2.79.

Proposition 3.37. The composition C β−→ C(V )α−→ C in the E-theory category is the

identity morphism on C.

Proof. Remark 3.33 and Theorem 2.85, as in the proof of Corollary 2.89.

A small variation of Theorem 2.61 now proves the following basic result:

Theorem 3.38. The morphisms α : C(V )→C and β : C→C(V ) in the E-theory categoryare mutual inverses.

Proof. Let C be any graded C∗-algebra and let f ∈ E(C, C) and g ∈ E(C,C), then f⊗g ∈E(C⊗C,C⊗C) corresponds to f g ∈ E(C,C) by Proposition 3.23. So it suffices to showthat the maps

α∗ : E(C(V ), C(V ))→E(C(V ),C) given by f 7→ α fβ∗ : E(C(V ),C)→E(C(V ), C(V )) given by g 7→ β⊗g

are inverse to one another.

It follows from Proposition 3.23 that α∗β∗(g) = (α⊗1C)∗(β⊗g) = α∗(β)⊗g = 1C⊗g = g.Consider the isomorphisms

σ : C⊗C(V )→C(V )⊗C, τ : C(V )⊗C⊗C(V )→C(V )⊗C⊗C(V ).

Note that τ∗(z⊗y) = σ∗(y)⊗z for y ∈ E(C(V ),C⊗C(V )), z ∈ E(C, C(V )). Since C(V ) hasthe rotation property, τ is homotopic to i⊗1⊗1. Therefore, setting z = β, we get

i∗(β)⊗y = τ∗(β⊗y) = σ∗(y)⊗β.

Applying α∗ we deduce that i∗(β)⊗α∗(y) = α∗(i∗(β)⊗y) = α∗(σ∗(y)⊗β) = σ∗(y). Applyinganother flip isomorphism we conclude that y = α∗(y)⊗i∗(β). So α∗, being both left and rightinvertible, is invertible. Moreover the right inverse β∗ is necessarily a two-sided inverse.

Remark 3.39. It follows that the map E(C(V ),C)→E(C(V ), C(V )) given by g 7→ g⊗i∗(β)equals β∗. In particular, i∗(β) α = i∗(β)⊗α = α⊗i∗(β) = β⊗α = β α = 1C(V ). It followsthat i β = β in E(C, C(V )).

Let A be a C∗-algebra. The suspension of A is the tensor product of A with Σ = C0(0, 1).If A is graded then so is ΣA (Σ is given by the trivial grading).

Theorem 3.40. For all separable graded C∗-algebras A and B the suspension map

E(A,B) −→ E(ΣA,ΣB)

is an isomorphism. Moreover there are natural isomorphisms

E(A,B) ∼= E(Σ2A,B) and E(A,B) ∼= E(A,Σ2B).

E-Theory 34

Proof. Σ2 is isomorphic to C in the E-theory category, and this proves the second partof the theorem. With the periodicity isomorphisms available, we obtain an inverse to thesuspension map by simply suspending a second time, because by Theorem 3.38 there is anatural equivalence of functors from Σ2 on the E-theory category to the identity functor.

Now we want to prove the six-term exact sequences in E-theory. First, let’s begin theconstruction of the asymptotic morphism associated to a short exact sequence of C∗-algebras:

Proposition 3.41. Given a short exact sequence of separable graded C∗-algebras

0 −→ J −→ Bπ−→ A −→ 0

there is a norm-continuous family utt∈T of degree-zero elements in J such that(i) 0 ≤ ut ≤ 1;(ii) limt→∞ ||utj − j|| = 0, for all j ∈ J ;(iii) limt→∞ ||utb− but|| = 0, for all b ∈ B.If s : A→B is any set-theoretic section (not necessarily graded) of the quotient map π, thenthe formula

σt(f ⊗ a) = f(ut)s(a)

defines an asymptotic morphism from ΣA into J .

Proof. From separablity we have an quasicentral approximate unit vnn∈N for the pairJ ⊆ B. Let β be the grading automorphism on B, then un = 1/2(vn + β(vn)) is still anquasicentral approximate unit vnn∈N for the pair J ⊆ B. We finish the first part by linearinterpolation:

un+s = (1− s)un + sun+1

for s ∈ [0, 1]. The proof of the second part based on the following lemma:

Lemma 3.42. Let f be a continuous complex valued function on the unit interval such thatf(0) = 0.(i) If b ∈ B, then limt→∞ ||f(ut)b− bf(ut)|| = 0;(ii) If in addition f(1) = 0 then limt→∞ ||f(ut)j|| = 0, for every j ∈ J .

Let’s consider the first item. By the Weierstrass approximation theorem it suffice toprove for the single function f(x) = x, which is clearly true. The set of f for which the limitis zero, for all j ∈ J , is an ideal in the C∗-algebra of continuous function f which vanish at0 and 1. So it suffies to prove (ii) for the function f(x) = x(1− x), but

limt→∞

||ut(1− ut)j|| ≤ limt→∞

||(1− ut)j|| = 0.

It follows from Lemma 3.42 (i) that σt is asymptotic multiplicative and Lemma 3.42 (ii)implies that σt is asymptotic graded and σ is independent of the choice of section s.

Lemma 3.43. The homotopy class of the asymptotic morphism σ : ΣA→AJ in Proposition3.41 depends only on the short exact sequence and not on approximate unit ut.

Proof. Let vt be a second approximate unit for J ⊆ B and denote by I the unit interval.The functions wt(s) = sut+(1−s)vt constitute an approximate unit for IJ ⊆ IB. If B ⊆ IBdenotes the C∗-algebra of continuous functions from I to B which are constant modulo Jthen the asymptotic morphism ΣA→AIJ associated to the short exact sequence

0 −→ IJ −→ B −→ A −→ 0,

constructed using the approximate unit wt, is a homotopy of asymptotic morphisms con-necting those constructed from ut and vt.

E-Theory 35

We note in passing the following interesting fact. It plays an important role in theproblem of characterizing E-theory, but we shall not use it in what follows. The contentof the theorem is that, up to suspension, every asymptotic morphism is associated to someshort exact sequence, which is a way of demonstrating the non-triviality if our construction.

Theorem 3.44. Let A and B be separable graded C∗-algebras and let ϕ : A→AB be anasymptotic morphism. There is a short exact sequence

0 −→ ΣB −→ E −→ A −→ 0

whose associated asymptotic morpshim σ : ΣA→AΣB is homotopic to Σϕ.

Proof. Let C be the graded C∗-algebra of continuous and bounded functions from the interval(0, 1] into B which vanish at 1, and let

E = a⊕ f ∈ A⊕ CB : f(s) ∼ ϕs−1(a).

There is then a short exact sequence in which the first map is the inclusion f 7→ 0⊕ f andthe second is the projection a⊕ f 7→ a (see Theorem 5.12 [17]).

Definition 3.45. Let θ : B→A be a graded ∗-homomorphism. The mapping cone Cθ of θis the graded C∗-algebra defined by

Cθ = b⊕ f ∈ B ⊕ CA : θ(b) = f(0).

Define graded ∗-homomorphisms

α : Cθ→B and β : ΣA→Cθ

by α(b⊕ f) = b and β(f) = 0⊕ f .

Let π : B→A be a surjective graded ∗-homomorphism of separable graded C∗-algebras.Let J be kernel of π, then J embeds as an ideal in the mapping cone Cπ via τ : b 7→ b⊕ 0.There is a short exact sequence

0 −→ ΣJ −→ CBπ1−→ Cπ −→ 0,

where π1 : f 7→ f(0)⊕ π f . From it we obtain an asymptotic morpshim σ : ΣCπ→AΣJ .

Proposition 3.46. The inclusion ∗-homomorphism Στ : ΣJ→ΣCπ and the asymptoticmorphism σ : ΣCπ→AΣJ determine mutually inverse morphisms in the asymptotic category.In particular in the E-theory category.

Proof. See Proposition 5.14 [17] for a proof.

Corollary 3.47. J ∼= Cπ in the E-theory category.

Proof. It follows from Theorem 3.40 and the maximal tensor product is compatible withcomposition in the E-theory category.

Proposition 3.48. Let π : B→C be a graded ∗-homomorphism. For every graded C∗-algebra A there is a long exact sequence of pointed sets

. . . −→ [[A,ΣB]] −→ [[A,ΣC]] −→ [[A,Cπ]] −→ [[A,B]] −→ [[A,C]].

Thus for every graded C∗-algebra A there is a functorial six-term exact sequence

E(A,Cπ) // E(A,B) // E(A,C)

E(A,ΣC)

OO

E(A,ΣB)oo E(A,ΣCπ)oo

E-Theory 36

Proof. We have a sequence of graded ∗-homomorphisms

. . . −→ ΣB −→ ΣC −→ Cπ −→ B −→ C,

where the composition of two successive maps are null-homotopic11. Hence the compositionof any two successive maps of the sequence in the proposition is trivial. To see exactness werefer to Proposition 5.16 in [17]. By using the second natural isomorphism in Theorem 3.40we complete the proof.

It follows from Proposition 3.48 and Corollary 3.47 we obtain:

Theorem 3.49. Let A be a separable graded C∗-algebra and let

0→J→B→B/J→0

be a short exact sequence of separable graded C∗-algebras. There is a functorial six-termexact sequence

E(A, J) // E(A,B) // E(A,B/J)

E(A,ΣB/J)

OO

E(A,ΣB)oo E(A,ΣJ)oo

This theorem is also true in first variable (see Theorem 6.20 [17]):

Theorem 3.50. Let B be a separable graded C∗-algebra and let

0→I→A→A/I→0

be a short exact sequence of separable graded C∗-algebras. There is a functorial six-termexact sequence

E(A/I,B) // E(A,B) // E(I,B)

E(I,ΣB)

OO

E(A,ΣB)oo E(A/I,ΣB)oo

3.3 The Equivariant E-Theory Category

We are now going to define an equivariant version of E-theory which will be particularlyuseful for computing the K-theory of group C∗-algebras.

Definition 3.51. Let G be a countable discrete group and let A and B be graded G-C∗-algebras (that is, graded C∗-algebras equipped with action of G by grading-preserving∗-automorphisms). An equivariant asymptotic family from A to B is an asymptotic familyϕtt∈[1,∞) : A→B such that

ϕt(g · a)− g · (ϕt(a))→0, as t→∞,

for all a ∈ A and all g ∈ G.

It should be noted that the grading-preserving action of G on B passes in a natural wayto an action by grading-preserving ∗-automorphisms on the asymptotic algebra AB.

Definition 3.52. An equivariant asymptotic morphism from one graded G-C∗-algebra Ato another one B is an equivariant graded ∗-homomorphism from A to AB.

11See Lemma 4.1 and Theorem 4.2 in [4] for details and note that maps and homotopies in Lemma 4.1 [4]are all graded.

E-Theory 37

Proposition 3.53. Let A and B be graded G-C∗-algebras. There is a one-to-one corre-spondence between equivariant asymptotic morphisms from A to B and equivalence classesof equivariant asymptotic families.

Thanks to this proposition it is a straightforward matter to define an equivariant versionof the asymptotic category. Since the higher asymptotic algebras An(B) are also gradedG-C∗-algebras, we define [[A,B]]Gn to be the set of n-homotopy classes of equivariant graded∗-homomorphisms from A to AnB; and we define

[[A,B]]G∞ = lim−→

[[A,B]]Gn .

These are the morphism sets of the equivariant asymptotic category, using the compositionlaw described in Proposition 3.11, and this category may be amplified as in Definition 3.25.Finally, if A is separable (and G is countable), then the canonical map gives an isomorphism

[[A,B]]G1∼=−→ [[A,B]]G∞.

To define the equivariant E-theory groups it remains to introduce a stabilization operationwhich is appropriate to the equivariant context.

Definition 3.54. Let G be a countable discrete group. A graded G-Hilbert space is aseparable graded Hilbert space equipped with unitary representations of G on its even andodd grading-degree summands. The standard graded G-Hilbert space HG is the countabledirect sum of Hilbert spaces

HG =

∞⊕n=0

l2(G),

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

Remark 3.55. We can write HG as a tensor product HG∼= l2(G) ⊗ H0, where l2(G) is a

trivially graded G-Hilbert space equipped with the left regular representation of G and H0

is a separable graded Hilbert space equipped with the trivial G-action.

The standard graded G-Hilbert space has the following universal property:

Lemma 3.56. If H 6= 0 is any graded G-Hilbert space, then H⊗HG∼= HG via a grading-

preserving G-equivariant unitary isomorphism of Hilbert spaces.

Proof. Let K be a (ungraded) G-Hilbert space and denote K0 the Hilbert space K equippedwith the trivial G-action. The formular v⊗ δg 7→ (g−1 · v)⊗ δg defines a equivariant unitaryisomorphism from K ⊗ l2(G) to K0 ⊗ l2(G) and from it we obtain an equivariant gradedunitary isomorphism H⊗HG

∼= H0⊗HG, where H0 is the Hilbert space H with the trivialG-action. By separability and Remark 3.55 H0⊗HG

∼= HG via a grading-preserving G-equivariant unitary isomorphism, hence H⊗HG

∼= HG as required.

Definition 3.57. Let G be a countable discrete group and let A and B be graded G-C∗-algebras. The equivariant E-theory group EG(A,B) is defined by

EG(A,B) = [[S⊗A⊗K(HG), B⊗K(HG)]]G.

Remark 3.58. The virtue of working with the Hilbert space HG is that if H 6= 0 is any gradedG-Hilbert space and if ϕ : S⊗A99KB⊗K(H) is an equivariant asymptotic morphism, thenϕ determines an element of EG(A,B) by tensoring K(HG) and applying Lemma 3.56.

E-Theory 38

Remark 3.59. The construction described in the previous remark has a generalization whichwill be important in the sequel. Suppose that H is a separable, graded Hilbert space whichis equipped with a continuous family of unitary G-actions, parametrized by t ∈ [1,∞).The continuity is pointwise strong continuity, so that if g ∈ G and k ∈ K(H) then g.tk isnorm-continuous in t. Suppose now that A and B are graded G-C∗-algebras and that

ϕ : S⊗A99KB⊗K(H)

is an asymptotic morphism which is equivariant with respect to the given family of G-actions,in the sense that

limt→∞

||ϕt(g.x)− g.t(ϕt(x))|| = 0,

for all g ∈ G and x ∈ S⊗A. Then ϕ determines an element of EG(A,B) too. Indeed,after we tensor with HG the family of actions on H is conjugate to a constant action by afamily of equivariant unitaries H⊗HG

∼= H0⊗HG parametrized by t ∈ [1,∞) (see the proofof Lemma 3.56). Hence we obtain an asymptotic morphism into B⊗K(H0⊗HG) which isequivariant in the usual sense for the single, fixed action of G on H0⊗HG

∼= HG.

By comparing Theorem 3.31 we immediately obtain the following result:

Theorem 3.60. The EG-theory groups EG(A,B) are the morphism groups of an addi-tive category whose objects are the separable graded G-C∗-algebras. There is a functorfrom the homotopy category of graded separable G-C∗-algebras and graded G-equivariant∗-homomorphisms into the equivariant E-theory category which is the identity on objects.

Remark 3.61. The equivariant E-theory category has a maximal tensor product and more-over there are six-term exact sequences of the EG-theory groups associated to short exactsequences of G-C∗-algebras. The proofs are only minor modifications of what we saw in thenon-equivariant case, and we shall omit them here (see [17]).

In order to apply equivariant E-theory to the problem of computing C∗-algebra K-theoryone must first apply a descent operation which transfers computations in equivariant E-theory to computations in the non-equivariant E-theory. This involves the notion of crossedproduct C∗-algebras.Let A be a graded G-C∗-algebra, then the full crossed product C∗-algebra C∗(G,A) hasa natural grading. An equivariant graded ∗-homomorphism ϕ : A→B induces a graded∗-homomorphism ϕ = C∗(G,ϕ) : C∗(G,A)→C∗(G,B) so that the full crossed product is afunctor from graded G-C∗-algebras to graded C∗-algebras.

Lemma 3.62. The functor B 7→ C∗(G,B) is exact.

Proof. Given a short exact sequence

0→J→B→B/J→0,

the induced map C∗(G, J)→C∗(G,B) is injective by virtue of the fact that every covariantrepresentation of J extends to a covariant representation of B. As a ∗-homomorphismhas closed range, C∗(G,B)→C∗(G,B/J) is surjective. The quotient C∗(G,B)/C∗(G, J) isisomorphic to C∗(G,B/J) by virtue of the facts that the map

Cc(G,B)

Cc(G, J)−→ C∗(G,B)

C∗(G,B)

is a dense embedding and the universal property of C∗(G,B/J).

Lemma 3.63. The functor B 7→ C∗(G,B) is continuous.

E-Theory 39

Proof. There is an isomorphism

C∗(G, IB) ∼= IC∗(G,B)

which is compatibel with the evaluation maps to C∗(G,B). This is a special case of the factthat if G acts trivially on A then there is a canonical isomorphism

C∗(G,A⊗B) ∼= A⊗C∗(G,B).

Proposition 3.64. There is a descent functor from the equivariant asymptotic category tothe asymptotic category, which associates to the class of an equivariant graded ∗-homomorphismϕ : A→AnB the class of the composition

C∗(G,A)ϕ−→ C∗(G,AnB)

in−→ AnC∗(G,B).

In order to obtain a corresponding functor in E-theory we need the following lemma:

Lemma 3.65. Let G be a discrete group, let B be a graded G-C∗-algebra and H be a gradedG-Hilbert space, then

C∗(G,B⊗K(H)) ∼= C∗(G,B)⊗K(H).

Proof. The group element g ∈ G acts on H as the unitary operator Ug : H→H. The maps

ϕ :Cc(G,BK(H))→Cc(G,B)K(H) given by ϕ(∑g∈G

(bg⊗kg) · g) =∑g∈G

(bg · g)⊗kgUg

ψ :Cc(G,B)K(H)→Cc(G,BK(H)) given by ψ((∑g∈G

bg · g)⊗k) =∑g∈G

(bg⊗kU∗g ) · g

are inverse to one another. Using the definition of the norms for the maximal tensor productand full crossed product we see that ϕ extends to an isomorphism of graded C∗-algebras.

Theorem 3.66. There is a descent functor from the equivariant E-theory category to theE-theory category which maps the graded G-C∗-algebra A to the graded full crossed productC∗(G,A), and which maps the class of a graded G-equivariant ∗-homomorphism ϕ : A→Bto the class of the induced graded ∗-homomorphism from C∗(G,A) to C∗(G,B).

Corollary 3.67. Let G be a countable discrete group. Suppose that A and B are separablegraded G-C∗-algebras and that A and B are isomorphic in the equivariant E-theory category.Then K(C∗(G,A)) is isomorphic to K(C∗(G,B)).

We also wish to apply equivariant E-theory to the computation of K-theory for thereduced crossed products. Like the full crossed product, the reduced crossed product is afunctor from graded G-C∗-algebras to graded C∗-algebras. However, unlike the full crossedproduct the reduced crossed product is not exact for every G. This prompts us to make thefollowing definition:

Definition 3.68. A discrete group G is exact if the functor A 7→ C∗λ(G,A) is exact.

Lemma 3.69. The functor B 7→ C∗λ(G,B) is continuous.

Proof. The proof is similar to that of Lemma 3.63, but uses the canonical isomorphism

C∗λ(G,A⊗minB) ∼= A⊗minC∗λ(G,B),

valid for C∗-algebras A with trivial G-action. We apply this to A = C[I].

E-Theory 40

Hence we obtain a reduced descent functor from the equivariant asymptotic categoryto the asymptotic category as in Proposition 3.64. Using a counterpart of Lemma 3.65 wearrive at the following result:

Theorem 3.70. Let G be an exact, countable, discrete group. There is a reduced descentfunctor from the equivariant E-theory category to the E-theory category which maps thegraded G-C∗-algeba A to the graded reduced crossed product C∗λ(G,A), and which maps theclass of a graded G-equivariant ∗-homomorphism ϕ : A→B to the class of the induced graded∗-homomorphism from C∗λ(G,A) to C∗λ(G,B).

Corollary 3.71. Let G be an exact, countable, discrete group. Suppose that A and B areseparable graded G-C∗-algebras and that A and B are isomorphic in the equivariant E-theorycategory. Then K(C∗λ(G,A)) is isomorphic to K(C∗λ(G,B)).

The Baum-Connes Conjecture 41

4 The Baum-Connes Conjecture

4.1 Proper Actions

Definition 4.1. Let G be a countable discrete group. A G- space is a topological spaceX equipped with actions of G by homeomorphisms such that the spaces X and X/G areparacompact and Hausdorff.

Definition 4.2. A G-space X is proper if for every x ∈ X there is an open neighborhoodU of x with g.u ∈ U for all (g, u) ∈ G× U , a finite subgroup H of G, and a G-map from Uto G/H.

Example 4.3. If H is a finite subgroup of G, then the discrete homogeneous space G/His a proper G-space. Moreover if Y is any H-space, then G ×H Y is a proper G-space.Recall that G ×H Y is the orbit space of the H-space G × Y , where H acts on G × Yby h(g, y) = (gh−1, hy). Furthermore, there is a natural action of G on G ×H Y given byg′[g, y] = [g′g, y], where [g, y] denotes the H-orbit of (g, y) ∈ G× Y . Since there is a G-mapG×H Y→G/H given by [g, y] 7→ gH, then G×H Y is a proper G-space.

In fact every proper G-space is locally induced from a finite group action:

Lemma 4.4. A G-space X is proper if and only if for every x ∈ X there is a G-invariantopen subset U ⊆ X containing x, a finite subgroup H of G, an H-space Y , and a G-homeomorphism from U to G×H Y .

Proof. One direction follows easily from Example 4.3 that there is a G-map G×H Y→G/H.Now assume that X is proper G-space, then there is a G-map ρ : U→G/H. Let Y :=ρ−1(eH), then Y is an H-space. The map G ×H Y→U given by [g, y] 7→ gy is a G-homeomorphism by Lemma 3.5 in [1] or [14].

Lemma 4.5. A locally compact G-space X is proper if and only if the map from the mapfrom G×X to X×X which takes (g, x) to (g.x, x) is a proper map of locally compact spaces(meaning that the inverse image of every compact set is compact).

Proof. See [6] or [14].

Definition 4.6. A proper G-space EG is universal if for every proper G-space X thereexists a G-map f : X→EG, and any two G-maps from X to EG are G-homotopic.

Remark 4.7. It’s clear from the definition that any two universal proper G-spaces are G-homotopy equivalent and f is unique up to G-homotopy.

Proposition 4.8. A metrizable proper G-space X with X/G paracompact is universal ifand only if the two following conditions hold:(i) For every finite subgroup H of G there is an x ∈ X stabilized by H (that is, H.x = x);(ii) The two projection maps p1, p2 : X ×X→X are G-homotopy.

Proof. See Proposition 1.8 [10].

Remark 4.9. If G is finite, then every G-space is proper, so we may take EG = ∗.Proposition 4.10. Let G be a countable discrete group. There exists a universal properG-space.

Proof. One defines a G-space

EG = f : G→[0, 1] : f has finite support and∑x∈G

f(x) = 1

with left G-action (g.f(x) = f(g−1.x)). The topology of EG is given by the metric

d(f1, f2) = supx∈G|f1(x)− f2(x)|.

See Lemma 1 [34] for more details.

The Baum-Connes Conjecture 42

Definition 4.11. A proper G-space X is said to be G-compact if X/G is compact.

Lemma 4.12. Let X be a proper G-space. Then X/G is locally compact iff X is locallycompact.

Proof. See Corollary 1.15 [1].

Proposition 4.13. A proper G-space X is G-compact iff there is a compact subset K ⊆ Xsuch that X = GK.

Proof. If X = GK for a compact subset K ⊆ X, then there is a surjective map K→X/G,hence X/G is compact. Conversely, assume that X/G is compact, then by previous lemmaX is locally compact. We want to show that X = GK for some compact subset K ⊆ X.For each x ∈ X let Ux be an open neighborhood of x with compact cloure. If π : X→X/Gdenotes the quotient map then the collection π(Ux) cover X/G. We know that X/G iscompact and so get a finite subcover π(Ux1), . . . , π(Uxn). Now let K = ∪ni=1Uxi . This isa compact set with GK = X.

Definition 4.14. Let X be a G-compact proper G-space12. A cut-off function for X is acontinuous and compactly supported function θ : X→[0, 1] such that∑

g∈Gθ(g−1x)2 = 1, for all x ∈ X.

We observe that the sum is finite by properness. In fact, every G-compact proper G-space admit a cut-off function: we know already X is locally compact and X = GS forsome compact subset S ⊆ X. Take an open neighborhood V of S with compact closure.Since every paracompact Hausdorff space is normal, then by the Urysohn Lemma there is acontinuous function f : X→[0, 1] which is equal to 1 on S and zero on the complement ofV , hence compactly supported. Now for any x ∈ X the function

fx : G→[0, 1], fx(g) = f(g−1x)

is continuous. Because GS = X we may find a g ∈ G with g−1x ∈ S. Recall f = 1 on Sand so fx(g) = 1. Thus we may conclude

∑g∈G fx(g) > 0. The action of G on the locally

compact space is proper so the set

g ∈ G : g.x ∩ supp(f) 6= ∅

is finite for each x ∈ X. This implies fx is finitely supported and so∑g∈G fx(g) < ∞.

Thanks to this we may define

c(x) =f(x)∑

g∈G fx(g)

This is compacted supported and has the properties thatG.supp(c) = X and∑g∈G c(g

−1x) =

1 for each x ∈ X. Clearly θ =√c is a cut-off function on X. This is also true for locally

compact topological group G (see Proposition 3.4 [33]).

Moreover any two cut-off function are, in a sense, homotopic if θ0 and θ1 are cut-offfunctions then the functions

θt =√tθ2

1 + (1− t)θ20, t ∈ [0, 1]

are all cut-off functions.

12Note that X is locally compact and X/G is compact.

The Baum-Connes Conjecture 43

Lemma 4.15. Let θ be a cut-off function for the G-compact proper G-space X. The formular

p(g)(x) = θ(g−1x)θ(x)

defines a projection in Cc(G,Cc(X)), and hence in C∗(G,X)13. The K-theory class of thisprojection is independent of the choice of cut-off function.

Proof. Recall that C∗(G,X) is the universal C∗-completion of Cc(G×X) with convolutionproduct and involution given by

f1 ∗ f2(g, x) =∑h∈G

f1(h, x)f2(h−1g, h−1x)

f∗(g, x) = f(g−1, g−1x)

We see that

p ∗ p(g, x) =∑h∈G

p(h, x)p(h−1g, h−1x)

=∑h∈G

θ(h−1x)θ(x)θ(g−1x)θ(h−1x)

= θ(x)θ(g−1x)

= p(g, x)

It’s clear that p∗ = p, hence p is a compactly supported projection. The homotopy of cut-offfunctions θt defined above gives a homotopy of projections pt, hence the K-theory class ofthis projection is independent of the choice of cut-off function.

Definition 4.16. Let X be a G-compact, locally compact, second countable proper G-space (from now on we shall just say ”G-compact proper G-space”). We will call the uniqueK-theory class of projections associated to cut-off functions the unit class

[p] ∈ K(C∗(G,X)) = E(C, C∗(G,X)).

Remark 4.17. A cut-off function exists also for a locally compact, second countable, properG-space X, which is not necessarily G-compact. In this case the cut-off function is notcompactly supported, but the intersection of support of cut-off function with any G-compactset in X is compact (see 3.2 [27]14). Such a proper G-space X is amenable in the sense ofDefinition 2.1 in [3]: Indeed, there is a continuous non-negative function h on X such that∑t∈G h(t−1x) = 1. Define gi(x, t) = h(t−1x) and the conditions in Proposition 2.2 (2) [3] are

satisfied. However, the converse is not true. If G is amenable and infinite, then the actionof G on a point is amenable but not proper, since the trivial action of G on a one-point setis amenable if and only if G is amenable.

We are going to study a class of G-C∗-algebras which extends the class of locally compactproper G-spaces.

Definition 4.18. A graded G-C∗-algebra B is proper if there exists(i) a second countable, locally compact, proper G-space X;(ii) an equivariant ∗-homomorphism ϕ from C0(X) into the grading-degree zero part of thecenter of multiplier algebra of B such that ϕ(C0(X))B is norm-dense in B.

Remark 4.19. C0(X) is a trivially graded G-C∗-algebra with (g.f)(x) = f(g−1x). We shallsay that B, as in the Definition, is proper over X. However, there is usually some freedomin the choice of X. Indeed if f : X→Y is a continuous and equivariant map of second

13We write C∗(G,X) for the full crossed product C∗(G,C0(X)).14Note that a second countable locally compact space is σ-compact.

The Baum-Connes Conjecture 44

countable, locally compact, proper G-spaces then any B which is proper over X becomes,via f , a C∗-algebra which is proper over Y . This is because f induces a ∗-homomorphismf∗ : C0(Y )→Cb(X), while the given structure map from C0(X) into the grading-degree zeropart of the center of the multiplier algebra of B extends to Cb(X). One checks easily thatϕf∗C0(Y )B is dense in B. Note that to carry out this construction we do not need to requirethat the map f : X→Y be proper.

Example 4.20. If G is finite then every graded G-C∗-algebra is proper over the one-pointspace. If X is a second countable, locally compact, proper G-space then C0(X) is clearlya proper G-C∗-algebra over X. Furthermore, if B is proper over X and D is any gradedG-C∗-algebra then B⊗D is proper over X as well. In particular, C0(X)⊗D is proper overX for any graded G-C∗-algebra D.

It follows from Remark 4.17 and Theorem 5.3 in [3] we obtain:

Proposition 4.21. For a proper graded G-C∗-algebra B we have

C∗(G,B) ∼= C∗λ(G,B)

In particular, C∗(G,X) ∼= C∗λ(G,X) for X is locally compact, second countable, properG-space.

Remark 4.22. As a result of above proposition, we can obviously defines a unit class inK(C∗λ(G,X)) too.

4.2 The Assembly Map

Paul Baum and Alain Connes have defined a C∗-algebraic assembly map which relatesequivariant K-homology to the K-theory of (reduced) group C∗-algebras. The main reasonfor studying proper G-C∗-algebras is that they appear to play an important role in theanalysis of this assembly map.

Definition 4.23. Let X be a G-compact proper G-space and let D be any separable gradedG-C∗-algebra. The assembly map15

µ : EG(X,D)→K(C∗(G,D))

is the composition

EG(X,D)descent−→ E(C∗(G,X), C∗(G,D))

[p]−→ E(C, C∗(G,D))

where the second map is the composition with the unit class [p] ∈ E(C, C∗(G,X)).

Definition 4.24. Let G be a countable discrete group and let D be a separable gradedG-C∗-algebra. The topological K-theory of G with coefficients in D is defined by

Ktop(G,D) = lim−→

EG(X,D),

where the limit is taken over the collection ofG-invariant andG-compact subspacesX ⊆ EG.

Remark 4.25. All such spaces in the definition are proper G-spaces, since they are G-invariant. By Lemma 4.12 they are locally compact. In fact they are also second countable,since EG is metrizable. To explain the direct limit, note that if X1 ⊆ X2 ⊆ EG are G-compact proper G-spaces, then X1 is a closed subset of X2. This follows from the fact thatG-invariant and G-compact subspaces are closed, since their image in X/G is compact. Theproperness of the action is needed to imply that X/G is Hausdorff, so compact subsets areclosed. Thus the inclusion is proper and restriction of functions defines a G-equivariant∗-homomorphism from C0(X2) to C0(X1). This induces a homomorphism from EG(X1, D)to EG(X2, D).

15We write EG(X,D) for EG(C0(X), D).

The Baum-Connes Conjecture 45

If X ⊆ Y ⊆ EG are G-compact proper G-spaces then under the restriction map fromE(C, C∗(G, Y )) to E(C, C∗(G,X)) the unit class for Y maps to the unit class for X; con-sequently the assembly maps for the various G-compact subsets of EG are compatible andpass to the direct limit:

Definition 4.26. The (full) Baum-Connes assembly map with coefficients in a separablegraded G-C∗-algebra D is the map

µ : Ktop(G,D)→K(C∗(G,D))

which is obtained as the direct limit of the assembly maps of Definition 4.23 for the G-compact subspaces of X ⊆ EG.

Definition 4.27. The reduced Baum-Connes assembly map with coefficients in a separablegraded G-C∗-algebra D is the map

µλ : Ktop(G,D)→K(C∗λ(G,D))

obtained by composing the full Baum-Connes assembly map µ with the map fromK(C∗(G,D))to K(C∗λ(G,D)) induced from the natural ∗-homomorphism C∗(G,D)→C∗λ(G,D).

Remark 4.28. If G is exact and X is a G-compact proper G-space then there is a reducedassembly map

µλ : EG(X,D)→K(C∗λ(G,D))

defined by the composition

EG(X,D)descent−→ E(C∗λ(G,X), C∗λ(G,D))

[p]−→ E(C, C∗λ(G,D))

involving the reduced descent functor. The reduced Baum-Connes assembly map µλ maythen be equivalently defined as a direct limit of such maps.

The following is known as the Baum-Connes conjecture with coefficients:

Conjecture 4.1. Let G be a countable discrete group. The reduced assembly map

µλ : Ktop(G,D)→K(C∗λ(G,D))

is an isomorphism of abelian groups for every separable graded G-C∗-algebra D.

Unfortunately, thanks to some recent constructions of Gromov (see [5]), the Baum-Connes conjecture with coefficients appears to be false:

Theorem 4.29. Let G be a Gromov group in the sense of Definition 5.6 in [23]. There is aseparable, commutative G-C∗-algebra D for which the reduced Baum-Connes assembly map

µλ : Ktop(G,D)→K(C∗λ(G,D))

fails to be an isomorphism.

Proof. See Theorem 5.6 [23]

The theorem just shows that the reduced assembly map fails to be surjective for somespaces, or else it fails to be injective for another. On the other hand, the mapping conetrick in section 1 [30] allows one to construct from our counterexample a second-countable,locally compact Hausdorff space W such that

K(C∗(G,W )) = 0 and K(C∗λ(G,W )) 6= 0.

The Baum-Connes Conjecture 46

For such a space the reduced assembly map of course fails to be surjective (although it isinjective since in this case the topological K-theory is zero). N. Ozawa has found a counterex-ample similar to ours, but with a trivial action on a non-commutative C∗-algebra instead ofa non-trivial action of G on a commutative C∗-algebra (see Remark 13 in [30]).

In the next conjecture, which is the official Baum-Connes conjecture for discrete groups,the coefficient D is specialized to C and C0(0, 1). We shall use the notations Ktop

∗ (G) andK∗(C

∗λ(G)) to denote topological and C∗-algebra K-theory in these two cases.

Conjecture 4.2. Let G be a countable discrete group. The reduced assembly map

µλ : Ktop∗ (G)→K∗(C∗λ(G))

is an isomorphism of abelian groups.

Our formulation of the conjecture, which uses E-theory, is equivalent to the formulationin [10] which uses KK-theory according to [20] and [43]. Indeed there is a natural transfor-mation from KK to E which determines an isomorphism from the KK-theoretic left-handside of the Baum-Connes conjecture to its E-theoretic counterpart (see Corollary A.4 andRemarks A.5 b) in [25]). In many cases the conjecture can be reduces to a statement inK-theory, independent of both E-theory and KK-theory by Poincare Duality (see Theorem4.11 [23] or the last section).

4.3 The Green-Julg Theorem

We begin our investigation of the assembly map by considering the rather easy case of finitegroups. It is basically equivalent to a well-known result of Green and Julg which identifiesequivariant K-theory and the K-theory of crossed product algebras for finite groups:

Theorem 4.30 (Green-Julg). Let G be a finite group. The assembly map

µ : Ktop(G,D)→K(C∗(G,D))

is an isomorphism for every graded G-C∗-algebra D.

Remark 4.31. Since G is amenable, then C∗(G,D) = C∗λ(G,D) and µ = µλ.

Proof. Since G is finite group, then EG can be taken to be one-point space. So the assemblymap becomes a homomorphism

µ : EG(C, D)→E(C, C∗(G,D)).

The theorem is proved by defining an inverse to the assembly map µ. For this purpose wenote that C∗(G,D) may be identified with a fixed-point algebra (see Proposition 11.2 [17]),

C∗(G,D)∼=−→ D ⊗K(l2(G))G,

by mapping d to∑g∈G g.d⊗ pg (where pg is the projection onto the functions supported on

g) and by mapping g to 1 ⊗ ρ(g), where ρ is the right regular representation (the fixed-point algebra is computed using the left regular representation). From this isomorphism weget a graded ∗-homomorphism

ψ : C∗(G,D)→D ⊗K(l2(G)).

Furthermore if we give the crossed product C∗-algebra the trivial G-action then ψ is equivari-ant. Define κ : D→D⊗K(l2(G)) by κ(d) = d⊗p, where p ∈ K(l2(G)) is the orthogonal pro-jection onto the constant functions in l2(G). It turns out that κ⊗1HG is homotopic, throughequivariant graded ∗-homomorphisms, to a graded ∗-isomorphism, where HG is the standard

The Baum-Connes Conjecture 47

graded G-Hilbert space, hence determines an invertible element in EG(D,D⊗K(l2(G))) (seeProposition 9.5 and Corollary 9.6 in [17]). Consider now the following diagram,

EG(C, C∗(G,D))ψ∗ // EG(C, D ⊗K(l2(G)))

E(C, C∗(G,D))

OO

ν // EG(C, D)

∼= κ

OO

where the bottom map ν is chosen so as to make the square commutative. In fact, ν invertsthe assembly map µ (see Theorem 11.1 [17]).

We are now ready to tackle our generalization of the Green-Julg theorem.

Theorem 4.32. Let G be a countable discrete group and let B be a proper graded G-C∗-algebra. The assembly map

µ : Ktop(G,B)→K(C∗(G,B))

is an isomorphism.

Remark 4.33. Thanks to Proposition 4.21, the reduced assembly map µλ into K(C∗λ(G,B))is an isomorphism as well.

To prove this theorem we need a lemma:

Lemma 4.34. Let H be a finite subgroup of a countable discrete group G and let W be alocally compact H-space. If D is any graded G-C∗-algebra there is a natural isomorphism

EH(C0(W ), D) ∼= EG(C0(G×H W ), D),

where on the left hand side D is viewed as an H-C∗-algebra by restriction of the G-action.

Proof. The space W is included into G×HW as the open set e×W , and as a result thereis an H-equivariant map from C0(W ) into C0(G×HW ). Composition with this map definesa restriction homomorphism

EG(C0(G×H W ), D)Res−→ EH(C0(W ), D)

To construct an inverse, the important observation to make is that every H-equivariantasymptotic morphism from C0(D) into D extends uniquely to a G-equivariant asymptoticmorphism from C0(G×H W ) into D ⊗K(l2(G/H)), hence we obtain an inverse map

EH(C0(W ), D)→EG(C0(G×H W ), D)

as required. This lemma is a special case of Lemma 12.11 [17], because we observe thatA := C0(G ×H W ) is proper over G ×H W→G/H as in Remark 4.19 and ComG

HA :=pC0(G ×H W ) = C0(W ), where p ∈ C0(G/H) is the characteristic function of the identitycoset in G/H.

Proof. (Proof of Theorem 4.32) It follows from Theorem 11.1, Lemma 12.3, 13.2 and 13.3,along with Proposition 12.9, that if B is proper over G/H, where H is a finite subgroup ofG, then assembly map is an isomorphism. Thus if B is proper over Y , and if Y maps to someG/H, then assembly is an isomorphism for B. Suppose next that B is proper over a spaceY which has a cover by a finite number, n, of G-invariant open sets, each of which mapsto some G/H, by the long exact sequence in E-theory, combined with the five lemma andinduction on n, shows that assembly is an isomorphism for B. Observe that this argumentapplies to any B which is proper over a G-compact proper G-space.

The Baum-Connes Conjecture 48

If B is general proper algebra then it is a direct limit of an increasing sequence of C∗-subalgebras Bn, each of which is proper over a G-compact proper G-space. Since K-theoryand the full crossed product functor commute with direct limits, so it suffices to prove thefunctor D 7→ Ktop(G,D) also commutes with direct limits. We shall use the fact that EGmay be represented as a G-CW -complex (see Appendix in [45]), so by passing to directlimit it suffices to prove that if Z is a G-finite proper G-simplicial complex then the functorD 7→ EG(C0(Z), D) commutes with direct limits. This we shall do by induction on thedimension of Z. We notice that if Z is zero dimensional then it is a disjoint union of cosetspaces G/H and that Zn\Zn−1 is a disjoint union of open n-simplices and C0(Zn\Zn−1) isproper over the zero-dimensional complex Z0 formed of the barycenters of the n-simplices,hence the proof of this reduces to the case where Z is a proper homogeneous space G/H.But here we have a sequence of isomorphisms

EG(C0(G/H), D) ∼= EH(C, D) ∼= K(C∗(H,D)),

the first by Lemma 4.34 and the second by Theorem 4.30. We complete the proof, sinceK-theory and full crossed product functor commute with direct limits (see Theorem 13.1[17] for more details).

4.4 The Dirac and Dual Dirac Method

The following simple theorem provides a strategy for attacking the Baum-Connes conjecturefor general coefficient algebras. The theorem, or its extensions and relatives, is invokedin nearly all approaches to the Baum-Connes conjecture, for example it is central to theargument in [22] that the assembly map is an isomorphism for discrete groups which actisometrically and metrically properly on an infinite dimensional Euclidean space.

Theorem 4.35. Let G be a countable discrete group. Suppose there exists a separable propergraded G-C∗-algebra B and morphisms α ∈ EG(B,C) and β ∈ EG(C, B) in the equivariantE-theory category such that α β = 1 ∈ EG(C,C). Then the assembly map

µ : Ktop(G,D)→K(C∗(G,D))

is an isomorphism for every separable graded G-C∗-algebra D. If in addition G is exact,then the reduced assembly map µλ is an isomorphism.

Proof. Consider the following commutative diagram:

Ktop(G,C⊗D)

β∗

µ// K(C∗(G,C⊗D)

β∗

Ktop(G,B⊗D)

α∗

∼=

µ// K(C∗(G,B⊗D)

α∗

Ktop(G,C⊗D)µ// K(C∗(G,C⊗D).

The horizontal maps are the assembly maps and the vertical maps are induced from E-theory classes α⊗1 ∈ EG(B⊗D,C⊗D) and β⊗1 ∈ EG(C⊗D,B⊗D). Since B is proper, sois the tensor product B⊗D (see Example 4.20) and therefore by Theorem 4.32 the middlehorizontal map is an isomorphism. By assumption, the compositions of the vertical mapson the left and right hand side are the identity. A trivial diagram chase now shows that theassembly map for D, which appears both at the top and the bottom of the diagram, is alsoan isomorphism.

Remark 4.36. α ∈ EG(B,C) is called Dirac element and β ∈ EG(C, B) is called dual-Diracelement. The above theorem is so-called ”the Dirac and dual Dirac method”.

The Baum-Connes Conjecture 49

Since the injectivity of the assembly map µ suffices for most of the applications of theBaum-Connes theory to topology and geometry (for example injectivity of µ for discretegroup G implies the Novikov higher signature conjecture for G (compare [26])), it is ofinterest to ask when the assembly map is injective.

Theorem 4.37. Let G be a countable discrete group. Suppose there exists a separable propergraded G-C∗-algebra B and elements α ∈ EG(B,C) and β ∈ EG(C, B) such that for everyfinite subgroup H of G the composition γ = α β ∈ EG(C,C) restricts to the identity inEH(C,C). Then for every separable graded G-C∗-algebra D the assembly map

µ : Ktop(G,D)→K(C∗(G,D))

is split injective. If in addition G is exact, then µλ is split injective.

Proof. We begin by considering the same diagram we introduced in the proof of Theorem4.35: The middle assembly map is still an isomorphism, of course, since B⊗D is proper. Wewant to show that the top assembly map is split injective, and for this it suffices to showthat the top left-hand vertical map β∗ : Ktop(G,D)→Ktop(G,D⊗B) is split injective. Forthis we shall show that the composition

Ktop(G,C⊗D)β∗−→ Ktop(G,B⊗D)

α∗−→ Ktop(G,C⊗D)

is an isomorphism. In view of the definition of Ktop it suffices to show that if Z is aG-compact proper G-space then the map

γ∗ := α∗ β∗ : EG(Z,D)→EG(Z,D)

is an isomorphism. The proof of this is an induction argument on the number n ofG-invariantopen sets U needed to cover Z, each of which admits a map to a proper homogeneous spaceG/H. If n = 1, so that Z itself admits such a map, then Z = G ×H W , where W is acompact space equipped with an action of H (see Lemma 10 [34]). By Lemma 4.34 there isthen a commuting diagram of restriction isomorphisms

EG(G×H W,D)

Res ∼=

γ∗ // EG(G×H W,D)

∼= Res

EH(W,D)γ∗

// EH(W,D),

and the bottom map is an isomorphism (in fact the identity) since γ = 1 in EH(C, C).If n > 1 then choose a G-invariant open set U ⊆ Z which admits a map to a properhomogeneous space, and for which the space Z1 = Z\U may be covered by n−1 G-invariantopen sets, each admitting a map to a proper homogeneous space. By induction we mayassume that the map γ∗ is an isomorphism for Z1. Applying the five lemma to the diagram

· · · // EG(Z1, D) //

γ∗ ∼=

EG(Z,D) //

γ∗

EG(U,D) //

γ∗ ∼=

· · ·

· · · // EG(Z1, D) // EG(Z,D) // EG(U,D) // · · · ,

we conclude that γ∗ is an isomorphism for Z too.

We conclude this section by introducing an important notion from KK-theory:

Proposition 4.38. Let G be a countable discrete group and let γ1 and γ2 be two elementsof EG(C,C) such that:

The Baum-Connes Conjecture 50

(i) γ1 and γ2 are compositions C β1−→ B1α1−→ C and C β2−→ B2

α2−→ C where B1 and B2 areproper graded G-C∗-algebras;(ii) if H is any finite group of G then under the restriction homomorphism

EG(C,C)→EH(C,C)

both γ1 and γ2 map to 1 ∈ EH(C,C). Then γ1 = γ1γ2 = γ2.

Proof. See Proposition 14.5 [17].

Thus if an element γ ∈ EG(C,C) as in the proposition exists then it is unique and is anidempotent. This is what in KK-theory is called the ”gamma-element” for G (compare [27]section 5). To summarize the theorems in this section, the existence of the gamma-elementimplies split injectivity of the (full) assembly map, while isomorphism of the (full) assemblymap follows from the assertion γ = 1.

4.5 Conjecture for the Free Abelian Group G = Zn

In this section we shall apply the approach outlined in the previous section to just about thesimplest example possible beyond finite groups: the free abelian group G = Zn. Let G = Znact by translations on Rn in the usual way and then let G act on the graded C∗-algebraC(Rn) that we introduced in Definition 2.63 by (g.f)(v) = f(g.v).

Lemma 4.39. The graded G-C∗-algebra C(Rn) is proper over Rn.

Proof. Note first that Rn is a second countable, locally compact, proper G-space by Lemma4.5. By Remark B.7 there are two cases: If n = 2k is even, then C(R2k) = M2k(C0(R2k))and we define an equivariant ∗-homomorphism ϕ : C0(R2k)→(Z(M2k(Cb(R2k))))0 by f 7→diag(f); If n = 2k + 1 is odd, then C(R2k+1) = M2k(C0(R2k+1))⊕M2k(C0(R2k+1)) and wedefine an equivariant ∗-homomorphism ϕ : C0(R2k+1)→(Z(M(C(R2k+1))))0 given by f 7→(diag(f),diag(f)). The norm density follows by using an approximate unit in C0(Rn).

Definition 4.40. Denote by β : S→C(Rn) the graded ∗-homomorphism in Definition 2.68,and for t ≥ 1 denote by βt : S→C(Rn) the graded ∗-homomorphism βt(f) = β(ft), whereft(x) = f(t−1x).

Thus βt(f)(v) = f(t−1C(v)), where C : Rn→Cliff(Rn) is the canonical inclusion.

Lemma 4.41. The asymptotic morphism β : S99KC(Rn) given by the above family of graded∗-homomorphisms βt : S→C(Rn) is Zn-equivariant.

Proof. Since S is equipped with trivial Zn-action, we must show that if f ∈ S and g ∈ Zn

limt→∞

||βt(f)− g.(βt(f))||∞ = limt→∞

supv∈Rn

||f(t−1C(v))− f(t−1C(g.v))||Cliff(Rn) = 0.

Since the set of all f ∈ S for which this holds (for all g ∈ Zn) is a C∗-subalgebra of S itsuffices to prove the limit formula for the generators f(x) = (x ± i)−1 of S. We completethe proof by the resolvent identity and C is an isometry:

||f(t−1C(v))− f(t−1C(g.v))||Cliff(Rn)

=||(t−1C(v)± i)−1 − (t−1C(g.v)± i)−1||≤||(t−1C(v)± i)−1||||t−1C(g.v)− t−1C(v)||||(t−1C(g.v)± i)−1||≤||f ||2∞t−1||C(g.v)− C(v)||Cliff(Rn)

=t−1||g||Rn .

The Baum-Connes Conjecture 51

Definition 4.42. Denote by β ∈ EZn(C, C(Rn)) the class of the equivariant asymptoticmorphism β : S99KC(Rn).

Definition 4.43. If g ∈ Zn and v ∈ Rn, and if s ∈ [0, 1], then denote by g.sv the translationof v by sg ∈ Rn.

To define the class α ∈ EZn(C(Rn),C) that we require we shall use the asymptoticmorphism α : S⊗C(Rn)99KK(H(Rn)) that we defined in Proposition 2.79, but we shallinterpret it as an equivariant asymptotic morphism in the following way:

Lemma 4.44. If f⊗h ∈ S⊗C(Rn), g ∈ Zn, and t ∈ [1,∞) then

limt→∞

||αt(f⊗g.h)− g.t−1αt(f⊗h)|| = 0.

Proof. The Dirac operator D is translation invariant, and so g.t−1f(t−1D) = f(t−1D) forall t. But g.t−1Mht = M(g.h)t for all t, hence we complete the proof.

Definition 4.45. Denote by α ∈ EZn(C(Rn),C) the class of the equivariant asymptoticmorphism α : S⊗C(Rn)99KK(H(Rn)), where K(H(Rn)) is equipped with the family ofactions (g, k) 7→ g.t−1k (see Remark 3.59).

Proposition 4.46. Continuing with the notation above, α β = 1 ∈ EZn(C,C).

Proof. Let s ∈ [0, 1] and denote by Cs(Rn) the C∗-algebra C(Rn), but with the scaled Zn-action (g, h) 7→ g.sh. The algebras Cs(Rn) form a continuous field of Zn-C∗-algebras overthe unit interval (since the algebras are all the same this just means that the Zn-actionsvary continuously). Denote by C[0,1](Rn) the Zn-C∗-algebra of continuous sections of thisfield (namely the continuous functions from [0, 1] into C(Rn), equipped with the Zn-action(g.h)(s) = g.sh(s)). In a similar way, form the continuous field of Zn-C∗-algebrasKs(H(Rn))and denote by K[0,1](H(Rn)) the Zn-C∗-algebra of continuous sections. With this notation,what we want to prove is that the composition

C β−→ C1(Rn)α−→ C

is the identity in equivariant E-theory.

The asymptotic morphism α induces an asymptotic morphism

α⊗1 = α : S⊗C[0,1](Rn)99KK[0,1](H(Rn)),

and similarly the asymptotic morphism β determines an asymptotic morphism

β : S99KC[0,1](Rn)

by forming the tensor product of β with the identity on C[0, 1] and then composing withthe inclusion S ⊆ S⊗C[0, 1] as constant functions. Consider then the commutative diagramof equivariant E-theory morphisms (see Remark 3.58)

Cβ//

=

C[0,1](Rn)α //

εs

C[0, 1]

εs

Cβ// Cs(Rn)

α// C

where εs denotes the element induced from evaluation at s ∈ [0, 1]. Observe that εs isinvertible in equivariant E-theory, for every s, because εs, considered as a ∗-homomorphism,is an equivariant homotopy equivalence. For s = 0, in this case the bottom composition

The Baum-Connes Conjecture 52

is the identity element of EZn(C,C). This is because when s = 0 the action of Zn onRn is trivial and the asymptotic morphism β : S99KC0(Rn) is homotopic to the (triviallyequivariant) ∗-homomorphism β : S→C(Rn) of Definition 2.68. So α β = 1 follows fromProposition 3.37. Since the bottom composition in the diagram is the identity it follows thatε0 α β = 1. Since ε0 is homotopic to ε1, we have ε1 α β = 1. This implies the bottomcomposition is the identity too for s = 1 and the proof is complete.

Since Zn is an amenable group, in particular exact. It follows from Theorem 4.35 thatthe Baum-Connes conjecture with coefficients holds for Zn.

4.6 Conjecture for the Groups with the Haagerup Property

Definition 4.47. An affine Euclidean space is a set E equipped with a simply-transitive(i.e, free and transitive) action of the additive group underlying a Euclidean vector space V .An affine subspace of E is an orbit of a point in E by a vector subspace of V . A subset Xof E generates E if the smallest affine subspace of E which contains X is E itself.

We note that every Euclidean space is of course an affine Euclidean space over itself. IfE is an affine Euclidean space over the Euclidean vector space V . If e1, e2 ∈ E, then thereexists a unique vector v ∈ V such that e1 + v = e2. We define the distance between e1 ande2 to be d(e1, e2) = ||v||. Suppose that a group G acts on E by isometries, then there is alinear representation π of G by orthogonal transformations on V such that

g.(e+ v) = g.e+ π(g)v,

for all g ∈ G, all e ∈ E, and all v ∈ V (see section 1 [2]).

Definition 4.48. Let G be a countable discrete group. An isometric action of G on anaffine Euclidean space E is metrically proper if for some (and hence for every) point e of E,

limg→∞

d(e, g.e) =∞.

In other words, an action is metrically proper if for every R > 0 there are only finitely mangeg ∈ G such that d(e, g.e) ≤ R.

Definition 4.49. A countable discrete group G has the Haagerup property if it admits ametrically proper isometric action on an affine Euclidean space.

Remark 4.50. Groups with the Haagerup property are also called a-T-menable, this is be-cause every countable amenable group has the Haagerup property and a discrete group withproperty (T) and the Haagerup property is finite.

Remark 4.51. The class of groups having the Haagerup property is closed under takingsubgroups, direct products, free products and increasing unions (see section 12.2 [9]). Variousclasses of discrete groups are known to have the Haagerup property: Amenable groups,groups act properly on locally finite trees16, Coxeter groups, discrete subgroups of SO(n, 1)and SU(n, 1) and Thompson’s groups.

The main objective of this section is to prove the following theorem:

Theorem 4.52. Let G be a countable discrete group with the Haagerup property. Thenthere exists a proper graded G-C∗-algebra B and EG-theory elements α ∈ EG(B,C) andβ ∈ EG(C, B) such that α β = 1 ∈ EG(C,C).

16In particular, free groups and SL(2,Z) have the Haagerup property. Indeed, the group SL(2,Z) =(Z/4Z) ∗Z/2Z (Z/6Z) acts properly on the Bass-Serre tree. Since every finitely generated free group actsproperly on its Cayley graph, which is a tree. Infinite generated free groups also have the Haagerup property,because they are increasing unions of finitely generated free groups.

The Baum-Connes Conjecture 53

Corollary 4.53. Let G be a countable discrete group with the Haagerup property and letD be a separable graded G-C∗-algebra. The assembly map with coefficients in D is anisomorphism. Moreover if G is exact then the reduced assembly map with coefficients in Dis also an isomorphism.

In fact the final conclusion is known to hold whether or not G is exact in the followingsense (see Theorem 5.1 [22]):

Theorem 4.54. If G is a countable discrete group with the Haagerup property, then forevery separable graded G-C∗-algebra D the regular representation

C∗(G,D)→C∗λ(G,D)

determines an invertible morphism in E(C∗(G,D), C∗λ(G,D)).

In connection with the last theorem it is perhaps worth noting that the following problemremains unsolved:

Problem: Is every countable discrete group with the Haagerup property exact?

Remark 4.55. The above theorems and corollary are also true for every second countable,locally compact topological group with the Haagerup property (see [21]).

From here on we shall fix an affine Euclidean space E over a Euclidean vector space Vand E equipped with a metrically proper, isometric action of a countable discrete groupG. We denote Ea and Eb for finite-dimensional affine subspaces of E and Va for the vectorsubspace of V corresponding to the finite-dimensional affine subspace Ea. If Ea ⊆ Eb thenwe shall denote by Vba the orthogonal complement of Ea in Eb and this is the orthogonalcomplement of Va in Vb. Note that

Ea = Vba + Ea,

i.e. every point of Eb has a unique decomposition eb = vba + ea.

Definition 4.56. Let Ea be a finite-dimensional affine Euclidean subspace of E, then wedefine C(Ea) = C0(Ea,Cliff(Va)).

Here is the counterpart of Proposition 2.65:

Lemma 4.57. Let Ea ⊆ Eb be finite-dimensional subspaces of E. The correspondenceh↔ h1⊗h2, where h(v+ e) = h1(v)h2(e) determines an isomorphism of graded C∗-algebras

C(Eb) ∼= C(Vba)⊗C(Ea).

Definition 4.58. Let Va be a finite-dimensional linear subspace of V , we define a graded∗-homomorphism

βa : S→S⊗C(Va)

by the composition

S∆−→ S⊗S 1⊗β−→ S⊗C(Va),

where β : S→C(Va) is the graded ∗-homomorphism β(f) = f(Ca) of Definition 2.68.

We are going to construct a proper graded G-C∗-algebra A(E) as a direct limit of gradedC∗-algebras S⊗C(Ea) associated to finite-dimensional affine subspaces Ea of E.

The Baum-Connes Conjecture 54

Definition 4.59. Let Ea ⊆ Eb be finite-dimensional affine subspaces of E. Define a graded∗-homomorphism

βb,a : S⊗C(Ea)→S⊗C(Eb)

by the formula

S⊗C(Ea) 3 f⊗g 7→ βba(f)⊗g ∈ S⊗C(Vba)⊗C(Ea) ∼= S⊗C(Eb),

where βba : S→S⊗C(Vba) is the graded ∗-homomorphism of Definition 4.58.

Lemma 4.60. Let Ea ⊆ Eb ⊆ Ec be finite-dimensional affine subspaces of E. We haveβc,b βb,a = βc,a.

Proof. Compute using the generators u(x) = e−x2

and v(x) = xe−x2

of S.

As a result the graded C∗-algebras S⊗C(Ea), where Ea ranges over the finite-dimensionalaffine subspaces of E, form a directed system and we denote A(E) for the direct limit. Anaction of G by isometries on E makes A(E) into a graded G-C∗-algebra. Too see this, firstdefine graded ∗-isomorphisms

g∗∗ : C(Ea)→C(gEa)

by (g∗∗f)(e) = g∗(f(g−1e)), where g∗ : Cliff(Va)→Cliff(gVa) is induced from the linearisometry of V associated to g : E→E. By using the generators u and v of S, we see thatthe following diagram commutes:

S⊗C(Ea)

1⊗g∗∗

βb,a// S⊗C(Eb)

1⊗g∗∗

S⊗C(gEa)βgb,ga

// S⊗C(gEb).

Hence the maps 1⊗g∗∗ are compatible with the maps in the directed system which is usedto define A(E). Consequently, we obtain a map g∗∗ on the direct limit and in this way A(E)is made into a graded G-C∗-algebra as required.

Theorem 4.61. Let E be an affine Euclidean space equipped with a metrically proper,isometric action of a countable discrete group G. Then A(E) is proper.

Proof. Denote by Z(Ea) the grading-degree zero part of the center of S⊗C(Ea), whichis isomorphic to the algebra of continuous functions, vanishing at infinity, on the locallycompact space [0,∞)×Ea. The connecting map βb,a embeds Z(Ea) into Z(Zb), and so wecan form the direct limit Z(E), which is the grading-degree zero part of the center of A(E).The C∗-subalgebra Z(E) has the property that Z(E) · A(E) is dense in A(E) (see Lemma4.39). The Gelfand spectrum of Z(E) is the locally compact space Z = [0,∞) × E, whereE is the metric space completion of E and Z is given by the weakest topology for whichthe projection to E is weakly continuous17 and the function t2 + d(e0, e)

2 is continuous, forsome (hence any) fixed e0 ∈ E. If G acts metrically properly on E, then the induced actionon the locally compact space Z is proper.

Since we have already proved the Baum-Connes conjecture for finite groups, then we mayassume that G is infinite countable discrete group with the Haagerup property. It followsfrom the proof of Theorem 12.2.4 in [9] that G will act metrically properly and isometrically

17Observe that E is affine space over the Hilbert space V and by identifying E as an orbit of V we cantransfer the weak topology of the Hilbert space V to E.

The Baum-Connes Conjecture 55

on a countably infinite-dimensional affine Euclidean space E. We shall notice that in the restof the proof we do not need the action to be metrically proper, which is used to show A(E)is proper. We shall begin with the construction of β ∈ EG(C,A(E)), and for this purposewe fix a point e0 ∈ E. This point is an affine subspace of E and for each finite-dimensionalaffine subspace Ea which contains e0, then e − e0 ∈ Va for all e ∈ Ea. We define a graded∗-homomorphism

βa : S→C(Ea) by f 7→ f(Ca,0),

where Ca,0 : Ea→Cliff(Va) by Ca,0(e) = Ca(e − e0). We get a graded ∗-homomorphismβ : S→A(E) by the composition

S∆−→ S⊗S 1⊗βa−→ S⊗C(Ea)→A(E).

Lemma 4.62. If β : S99KA(E) is the asymptotic morphism defined by

βt(f) = β(ft),

where ft(x) = f(t−1x), then β is well-defined and G-equivariant.

Proof. We start to show that β is well-defined. We must show that if e0 and e1 are twopoints in a finite-dimensional affine space Ea, then for every f ∈ S,

limt→∞

||f(t−1Ca,0)− f(t−1Ca,1)|| = 0,

where Ca,i(e) = e − ei. It suffices to compute the limit for the functions f(x) = (x ± i)−1

and one has

||f(t−1Ca,0)− f(t−1Ca,1)|| ≤ t−1||Ca,0 − Ca,1|| = t−1d(e0, e1).

This shows that β is independent of the base point e0 ∈ E, hence well-defined. By the sameproof of Lemma 4.41 β is G-equivariant.

Definition 4.63. The element β ∈ EG(C,A(E)) is the E-theory class of the equivariantasymptotic morphism β : S99KA(E) defined by βt : f 7→ β(ft).

For defining α : A(E)99KK(H(E)) we need associate a Hilbert space H(E) to the infinite-dimensional affine Euclidean space E.

Definition 4.64. Let Ea be a finite-dimensional affine subspace of E, with associatedlinear subspace Va. We define H(Ea) = L2(Ea,Cliff(Va)). This is a graded Hilbert spacewith grading inherited from Cliff(Va).

Lemma 4.65. Let Ea ⊆ Eb be finite-dimensional subspaces of the affine space E and let Vbabe the orthogonal complement of Ea in Eb. The correspondence h↔ h1⊗h2, where h(v+e) =h1(v)h2(e) determines an isomorphism of graded Hilbert spaces H(Eb) ∼= H(Vba)⊗H(Ea).

Definition 4.66. If W is a finite-dimensional Euclidean vector space, then the basic vectorfW ∈ H(W ) is defined by

fW (w) = π−1/4dim(W )e−1/2||w||2 .

Remark 4.67. Note that fW maps w ∈ W to the multiple π−1/4dim(W )e−1/2||w||2 of theidentity element in Cliff(W ) and ||fW || = 1.

The next step is to assemble the Hilbert spaces H(Ea) into a directed system by usingthe basic vectors fba ∈ H(Vba):

The Baum-Connes Conjecture 56

Definition 4.68. If Ea ⊆ Eb, then define an isometry of graded Hilbert spaces Vba :H(Ea)→H(Eb) by

H(Ea) 3 f 7→ fba⊗f ∈ H(Vba)⊗H(Ea) ∼= H(Eb).

Let Ea ⊆ Eb ⊆ Ec be finite-dimensional affine subspaces of E, then Vca = VcbVba. As aresult the graded Hilbert spaces H(Ea), where Ea ranges over the finite-dimensional affinesubspaces of E, form a directed system in the category of Hilbert spaces and graded iso-metric inclusions and we denote H(E) for the direct limit. If G acts isometrically on Ethen H(E) is equipped with a unitary representation of G, just as A(E) is equipped with aG-action.

We are now almost ready to begin the definition of the asymptotic morphism α :A(E)99KK(H(E)). What we are going to do is construct a family of asymptotic mor-phisms αa : S⊗C(Ea)99KK(H(E)), one for each finite-dimensional subspace of E, and thenprove that if Ea ⊆ Eb then the diagram

S⊗C(Ea)

βb,a

αa // K(H(E))

=

S⊗C(Eb)αb // K(H(E))

is asymptotically commutative. Once we have done that we shall obtain a asymptoticmorphisn defined on the direct limit lim−→S⊗C(Ea), as required. Suppose for a moment thatE is itself a finite-dimensional space. Fix a point in E and call it 0 ∈ E and use it toidentity E with its underlying linear space V and use this identification to define scalingmaps e 7→ t−1e on E for t ≥ 1 with the common fixed point 0 ∈ E. If h ∈ C(Ea) and 0 ∈ Eathen define ht ∈ C(Ea) by the usual formula ht(e) = h(t−1e).

Lemma 4.69. Let Ea be an affine subspace of a finite-dimensional affine Euclidean spaceE. Denote by Da the Dirac operator for Ea and denote by Ba⊥ = Ca⊥ +Da⊥ the Clifford-plus-Dirac operator for E⊥a . The formula

αat : f⊗h 7→ ft(Ba⊥⊗1 + 1⊗Da)(1⊗Mht)

defines an asymptotic morphism αa : S⊗C(Ea)99KK(H(E)).

Proof. Since the operator Ba⊥ is odd essentially self-adjoint and has compact resolvent byCorollary 2.84, so we can define graded ∗-homomorphisms γt : S→K(H(E⊥a )) by γt(f) =ft(Ba⊥). Moreover the formula

αt : f⊗h 7→ ft(Da)Mht

defines an asymptotic morphism α : S⊗C(Ea)99KK(H(Ea)) by Proposition 2.79. The for-mula for αa in the statement of the lemma is the composition

S⊗C(Ea)∆⊗1−→ S⊗S⊗C(Ea)

γ⊗α99K K(H(E⊥a ))⊗K(H(Ea)) ∼= K(H(E)).

So αa is an asymptotic morphism, as required.

Lemma 4.70. Let Ea ⊆ Eb be affine subspaces of a finite-dimensional affine Euclideanspace E. Denote by Da and Db the Dirac operators for Ea and Eb, and denote by

αa : S⊗C(Ea)99KK(H(E)) and αb : S⊗C(Eb)99KK(H(E))

The Baum-Connes Conjecture 57

the asymptotic morphisms of Lemma 4.69. The diagram

S⊗C(Ea)

βb,a

αa // K(H(E))

=

S⊗C(Eb)αb // K(H(E))

is asymptotically commutative.

Proof. Denote by Eba the orthogonal complement of Ea in Eb, so that E = E⊥b ⊕Eba ⊕Eaand H(E) ∼= H(E⊥b )⊗H(Eba)⊗H(Ea). Notice that under the isomorphism of Hilbert spacesH(Eb) ∼= H(Eba)⊗H(Ea) the Dirac operator Db corresponds to Dba⊗1 + 1⊗Da. SimilarlyBa⊥ corresponds to Bb⊥⊗1 + 1⊗Bba under the isomorphism H(E⊥a ) ∼= H(E⊥b )⊗H(Eba).Hence by making these identifications of Hilbert spaces we get (see Corollary A.2 or [28]Appendix A)

exp(−t−2D2b ) = exp(−t−2D2

ba)⊗ exp(−t−2D2a)

and

exp(−t−2B2a⊥) = exp(−t−2B2

b⊥)⊗ exp(−t−2B2ba).

Now, applying αat to the element u⊗h ∈ S⊗C(Ea) we get

exp(−t−2B2b⊥)⊗ exp(−t−2B2

ba)⊗ exp(−t−2D2a)Mht

in K(H(E⊥b ))⊗K(H(Eba))⊗K(H(Ea)), while applying αbt βb,a to u⊗h we get

exp(−t−2B2b⊥)⊗ exp(−t−2D2

ba) exp(−t−2C2ba)⊗ exp(−t−2D2

a)Mht .

But we saw in the proof of Theorem 2.85 that the two families of operators exp(−t−2B2ba)

and exp(−t−2D2ba) exp(−t−2C2

ba) are asymptotic to one another as t→∞. It follows thatαat (u⊗h) is asymptotic to αbt(βb,a(u⊗h)), as required. The calculation for v⊗h ∈ S⊗C(Ea)is similar.

Turning to the infinite-dimensional case, it is clear that the major problem is to constructa suitable operator Ba⊥ . We begin by assembling some preliminary facts. Suppose that wefix for a moment a finite-dimensional affine subspace Ea of E and denote by E⊥a its orthogonalcomplement in E. This is an infinite-dimensional subspace of V , but in particular it is aEuclidean space in its own right, and we can form the direct limit Hilbert space H(E⊥a )as before. Since the connecting maps in the directed system are isometric inclusions, thefunctor (−)⊗H(Ea) commutes with the direct limit. The following lemma is a consequenceof Lemma 4.65:

Lemma 4.71. Let Ea be a finite-dimensional affine subspace of E and let E⊥a be its orthog-onal complement in E, then

H(E) ∼= H(E⊥a )⊗H(Ea).

Definition 4.72. Let Ea be a finite-dimensional subspace of an affine Euclidean space E.The Schwartz space S(Ea) of Ea is Schwartz-class Cliff(Va)-valued functions on Ea.

The Schwartz spaces S(Ea) together with the inclusions Vba : S(Ea)→S(Eb), whereEa ranges over the finite-dimensional affine subspaces of E, form a directed system andwe denote S(E) for the direct limit. We now want to define a suitable operator Ba⊥ on

The Baum-Connes Conjecture 58

H(E⊥a ) with domain S(E⊥a ). If V ⊆ E⊥a is a finite-dimensional subspace then the operatorBV = CV +DV acts on every Schwartz space S(W ), where V ⊆W : just use the formula

(BV f)(w) =

n∑i=1

xiei(f(w)) +

n∑i=1

ei(∂f

∂xi(w)),

where e1, . . . , en is an orthonormal basis for V and x1, . . . , xn are the dual coordinates onV , extended to coordinates on W by orthogonal projection. The operators on the Schwartzspaces S(W ) are compatible with the inclusions used to define the direct limit S(E⊥a ) =lim−→S(W ), and we obtain an unbounded odd operator B⊥a on H(E⊥a ) with domain S(E⊥a ).Let’s now make the following key observation:

Lemma 4.73. Suppose that E⊥a is decomposed as a direct sum of pairwise orthogonal, finite-dimensional subspaces, E⊥a = V0 ⊕ V1 ⊕ V2 ⊕ · · · . If f ∈ S(E⊥a ) then the sum

Ba⊥f = B0f +B1f +B2f + · · · ,

where Bj = Cj + Dj is the Clifford-Dirac operator on Vj, has only finitely many nonzeroterms. The operator defined by the sum is essentially self-adjoint on S(E⊥a ) and is indepen-dent of the direct sum decomposition of E⊥a used in its construction.

Proof. Observe that S(E⊥a ) = lim−→S(V0⊕ · · · ⊕ Vn). Therefore if f ∈ S(E⊥a ) then f belongs

to some S(V0 ⊕ · · · ⊕ Vn) and its image in S(V0 ⊕ · · · ⊕ Vn+k) under the connecting map inthe directed system is a function of the form

fk(v0 + · · ·+ vn+k) = constant · f(v0 + · · ·+ vn)e−1/2||vn+1||2 · · · e−1/2||vn+k||2 .

The rest of the proof follows from Corollary 2.84. Since e−1/2||vn+k ||2 is in the kernel of Bn+k

we see that Bn+kf = 0 for all k ≥ 1. This proves the first part of the lemma. If ξj ∈ S(Vj)is an eigenfunction for B2

j with eigenvalue λj , then ξ ∈ S(V0 ⊕ · · · ⊕ Vn) defined by

ξ(v0 + v1 + · · ·+ vn) = ξ0(v0)⊗ξ1(v1)⊗ · · · ⊗ξn(vn)

is an eigenfunction for B2a⊥ with eigenvalue λ = λ0 + λ1 + · · · + λn. It follows that B2

a⊥ ,and hence Ba⊥ has an orthonormal eigenbasis within S(E⊥a ), which proves the essentialself-adjointness. The fact that Ba⊥ is independent of the choice of direct sum decompositionfollows from the formula

Ba⊥f = BW f if f ∈ S(W ) ⊆ S(E⊥a ),

which in turn follows from the formula BW1+BW2

= BW1⊕W2in finite dimensions.

Unfortunately, the operator Ba⊥ above does not have compact resolvent. Indeed, sincethe eigenvalue 0 occurs with multiplicity 1, then each positive integer is an eigenvalue of B2

a⊥

of infinite multiplicity. Because Ba⊥ fails to have compact resolvent we cannot immediatelyfollow Lemma 4.69 to obtain our asymptotic morphism αa. Instead we first have to perturbthe operators Ba⊥ in a certain way: fix an increasing sequence E0 ⊆ E1 ⊆ E2 ⊆ · · · of finite-dimensional affine subspaces of E whose union is E. We shall denote by Vn the orthogonalcomplement of En−1 in En (and write V0 = E0), so that there is an orthogonal direct sumdecomposition

E = V0 ⊕ V1 ⊕ V2 ⊕ · · · . (1)

Definition 4.74. Let Ea be a finite-dimensional affine subspace of E.(1) An orthogonal direct sum decomposition E⊥a = W0⊕W1⊕ · · · is standard if it’s of form

E⊥a = Va ⊕ Vn ⊕ Vn+1 ⊕ · · · ,

The Baum-Connes Conjecture 59

for some finite-dimensional linear space Va and some n ≥ 1, where the space Vn are themembers of the fixed decomposition of E given above.(2) An orthogonal direct sum decomposition E⊥a = Z0 ⊕ Z1 ⊕ · · · into finite-dimensionallinear subspaces is acceptable if there is a standard decomposition

E⊥a = W0 ⊕W1 ⊕ · · ·

so that W0 ⊕ · · · ⊕Wn ⊆ Z0 ⊕ · · · ⊕ Zn ⊆W0 ⊕ · · · ⊕Wn+1 for all sufficiently large n.

We are now going to define perturbed operators Ba⊥,t which depend on a choice ofacceptable decomposition, as well as on a parameter t ∈ [1,∞).

Definition 4.75. Let Ea be a finite-dimensional affine subspace of E and E⊥a = Z0⊕Z1⊕· · ·be an acceptable decomposition of E⊥a as an orthogonal direct sum of finite-dimensionallinear spaces. For each t ≥ 1 define an unbounded odd operator Ba⊥,t on H(E⊥a ) withdomain S(E⊥a ) by the formula

Ba⊥,t = t0B0 + t1B1 + t2B2 + · · ·

where tj = 1 + t−1j and Bn = Cn +Dn on the finite-dimensional spaces Zn.

Lemma 4.76. The unbounded odd operator Ba⊥,t defined above is essentially self-adjointand has compact resolvent.

Proof. The proof of self-adjointness follows the same argument as the proof of Lemma 4.73.For compactness of the resolvent the formula

B2a⊥,t = t20B

20 + t21B

21 + t22B

22 + · · ·

implies that the eigenvalues of B2a⊥,t are the sums λ = t20λ0 + t21λ1 + · · · , where λj is an

eigenvalue for B2j and where almost all λj are zero. Since the only accumulation point of

the eigenvalue sequence is infinity, the operator B2a⊥,t, and hence also the operator Ba⊥,t

has compact resolvent.

We can now define the asymptotic morphism αa : S⊗C(Ea)99KK(H(E)) that we need.Fix a point 0 ∈ E and use it to define scaling automorphisms h 7→ ht on each C(Ea) forwhich 0 ∈ Ea.

Proposition 4.77. Let Ea be a finite-dimensional affine subspace of E for which 0 ∈ Ea andlet Ba⊥,t be the operator associated to some acceptable decomposition of E⊥a . The formula

αat : f⊗h 7→ ft(Ba⊥,t⊗1 + 1⊗Da)(1⊗Mht)

defines an asymptotic morphism αa : S⊗C(Ea)99KK(H(E⊥a ))⊗K(H(Ea)) ∼= K(H(E)).

Proof. By using Lemma 4.71 this is proved in exactly the same way as in Lemma 4.69.

It should be pointed out that operator Ba⊥,t depend on the choice of acceptable decom-position, but the situation improves in the limit as t→∞:

Lemma 4.78. Let Ea be finite-dimensional affine subspace of E and denote by Bt = Ba⊥,tand B′t = B′a⊥,t be the operators associated to two acceptable decomposition of E⊥a . Thenfor every f ∈ S,

limt→∞

||f(Bt)− f(B′t)|| = 0.

The Baum-Connes Conjecture 60

Proof. It suffices to show the case that if the summands in the acceptable decompositionsare Zn and Z ′n, and if Z0 ⊕ · · · ⊕ Zn ⊆ Z ′0 ⊕ · · · ⊕ Z ′n ⊆ Z0 ⊕ · · · ⊕ Zn+1 for all n, thenlimt→∞ f(Bt)− f(B′t) = 0.

Denote by Xn the orthogonal complement of Zn in Z ′n, and by Yn the orthogonalcomplement of Z ′n−1 in Zn (set Y0 = Z0). There is then a direct sum decompositionE⊥a = Y0 ⊕ X0 ⊕ Y1 ⊕ X1 ⊕ · · · , with respect to which the operators Ba⊥,t and B′a⊥,tcan be written as infinite sums

Bt = t0BY0+ t1BX0

+ t1BY1+ t2BX1

+ · · ·

and

B′t = t0BY0+ t0BX0

+ t1BY1+ t1BX1

+ · · · .

Since tj − tj−1 = t−1 it follows that Bt − B′t = t−1BX0 + t−1BX1 + · · · . Since tj ≥ 1 itfollows that ||(Bt −B′t)g|| ≤ t−1||Btg|| for every g ∈ S(E⊥a ). So if f(x) = (x± i)−1 then

||f(Bt)− f(B′t)|| = ||(B′t ± i)−1(B′t −Bt)(Bt ± i)−1|| ≤ ||(B′t −Bt)(Bt ± i)−1|| ≤ t−1.

The proof is complete by the Stone-Weierstrass theorem.

Remark 4.79. If we repeat the proof with sBt in place of Bt, then we have in fact that

limt→∞

||f(sBt)− f(sB′t)|| = 0,

uniformly in s ∈ [1,∞).

It follows from Lemma 4.78 that our definition of the asymptotic morphism αa is inde-pendent, up to asymptotic equivalence, of the choice of acceptable decomposition of E⊥a ,but αa does depend on the choice of initial direct sum decomposition as in (1).

Proposition 4.80. The diagram

S⊗C(Ea)

βb,a

αa // K(H(E))

=

S⊗C(Eb)αb // K(H(E))

is asymptotically commutative.

Proof. As we did in the proof of Lemma 4.70, we see that αb βb,a is asymptotic to theasymptotic morphism

f⊗h 7→ ft(B′t⊗1 + 1⊗Da)(1⊗Mht),

where αb is computed using the acceptable decomposition

E⊥b = Z0 ⊕ Z1 ⊕ · · · ,

then B′t is the operator of Definition 4.75 associated to the decomposition

E⊥a = (Eba ⊕ Z0)⊕ Z1 ⊕ · · · .

But this is an acceptable decomposition for E⊥a , and so αb βb,a is asymptotic to αa.

The Baum-Connes Conjecture 61

It follows that the asymptotic morphisms αa form a single asymptotic morphism

α : A(E)99KK(H(E)).

Our definition of the class α ∈ EG(A(E),C) is therefore almost complete. It remains only todiscuss the equivariance of α: suppose that the countable discrete group G acts isometricallyon E and using the point 0 ∈ E we identify E with its underlying Euclidean vector spaceV , and define a family of isometric actions on E, parametrized by s ∈ [0, 1] by

g.se = s(g.0) + π(g)v, (0 + v = e).

Thus the action g.1e is the original action, while g.0e has a global fixed point 0 ∈ E.

Lemma 4.81. There exists a direct sum decomposition E =⊕∞

0 Vj as in (1) in such a waythat for every g ∈ G there is an N ∈ N for which

n > N ⇒ g.

n⊕0

Vj ⊆n+1⊕

0

Vj .

Proof. We just choose an increasing sequence E0 ⊆ E1 ⊆ E2 ⊆ · · · ⊆ E of finite-dimensionalaffine subspaces of E such that

⋃nEn = E and such that if g ∈ G then gEn ⊆ En+1, for

all large enough n. Note that we are assuming E is countably infinite dimensional, so it ispossible to arrange for the union

⋃nEn to be all of E, not just dense in E.

Proposition 4.82. If the direct sum decomposition E =⊕∞

0 Vj is chosen as in Lemma4.81 then the asymptotic morphism α : A(E)99KK(H(E)) is equivariant in the sense that

limt→∞

||αt(g.a)− g.t−1αt(a)|| = 0,

for all a ∈ A(E) and all g ∈ G.

Proof. Examining the definitions, we see that on S⊗C(Ea) the asymptotic morphism a 7→g−1.t−1αt(g.a) is given by exactly same formula used to define αat except for the choice ofacceptable direct sum decomposition of E⊥a . But we already noted that different choicesof acceptable direct sum decomposition give rise to asymptotically equivalent asymptoticmorphisms, so the proposition is proved.

Definition 4.83. Denote by α ∈ EG(A(E),C) the class of the equivariant asymptoticmorphism α : A(E)99KK(H(E)), where K(H(E)) is equipped with the family of actions(g, k) 7→ g.t−1k

Lemma 4.84. Suppose that the action of G on the affine Euclidean space E has a fixedpoint. Then the composition

C β−→ A(E)α−→ C

in equivariant E-theory is the identity morphism on C.

Proof. We choose a point which is fixed for the action of G on E to define β by Lemma4.62, so each graded ∗-homomorphism βt(f) = β(ft) in the asymptotic morphism β is indi-vidually G-equivariant. It follows that the equivariant asymptotic morphism β : S99KA(E)is equivariantly homotopy to the equivariant graded ∗-homomorphism β : S→A(E). Thecomposition of the asymptotic morphism α with the ∗-homomorphism β is asymptotic toγ : S99KK(H(E)), where γt(f) = ft(Bt) and Bt is the operator of Definition 4.75 associatedto any acceptable decomposition. This in turn is homotopic to the asymptotic morphism

The Baum-Connes Conjecture 62

f 7→ f(Bt). Finally this is homotopic to the asymptotic morphism defining 1 ∈ EG(C,C)by the homotopy

f 7→

f(sBt) s ∈ [1,∞)

f(0)P s =∞,

where P is the projection onto the kernel of Bt (note that all the Bt have the same 1-dimensional, G-fixed kernel).

Theorem 4.85. The composition α β = 1 ∈ EG(C,C).

Proof. The proof is exactly the same as the proof of Proposition 4.46. Let s ∈ [0, 1] anddenote by As(E) the C∗-algebra A(E), but with the scaled G-action (g, h) 7→ g.sh. Thealgebras As(E) form a continuous field of G-C∗-algebras over the unit interval. Denoteby A[0,1](E) the G-C∗-algebra of continuous sections of this field. In a similar way, formthe continuous field of G-C∗-algebras Ks(H(E)) and denote by K[0,1](H(E)) the G-C∗-algebra of continuous sections. The asymptotic morphism α : A(E)99KK(H(E)) induces anasymptotic morphism α : A[0,1](E)99KK[0,1](H(E)) and similarly the asymptotic morphism

β : S99KA(E) determines an asymptotic morphism β : S99KA[0,1](E). From the diagram ofequivariant E-theory morphisms

Cβ//

=

A[0,1](E)α //

εs

C[0, 1]

εs∼=

Cβ// As(E)

α// C,

where εs denotes the element induced from evaluation at s ∈ [0, 1], we see that if the bottomcomposition is the identity for some s ∈ [0, 1], then it is the identity for all s ∈ [0, 1]. But byLemma 4.84 the composition is the identity when s = 0 since the action (g, e) 7→ g.0e has afixed point 0 ∈ E. It follows that the composition is the identity when s = 1, which is whatwe wanted to prove.

4.7 Conjecture for the infinite Property (T) Groups

In this section we will discuss the Baum-Connes conjecture for infinite property T groups,in particular for SL(3,Z) and uniform lattices in Sp(n, 1).

Definition 4.86. Let (π,H) and (ρ,K) be unitary representations of a discrete group G.We say that π is weakly contained in ρ (we write this for π ≺ ρ) if every ξ in H, every finitesubset F of G and every ε > 0, there exist η1, . . . , ηn in K such that, for all x ∈ F ,

|〈π(x)ξ, ξ〉 −n∑i=1

〈ρ(x)ηi, ηi〉| < ε.

Definition 4.87. The set of equivalence classes of irreducible unitary representations of Gis called the unitary dual of G and is denoted by G. We say a net (πi)i converges to π in Gwith respect to Fell’s topology, if π ≺ ⊕jπj for every subnet (πj)j of (πi)i.

Definition 4.88. A discrete group G has property T if the trivial representation is anisolated point in the unitary dual of G.

Theorem 4.89. Let G be a discrete group. The following are equivalent:(a) G has property T.(b) Every isometric action of G on an affine Euclidean space has a fixed point.(c) There is a unique nonzero central projection p ∈ C∗(G) with the property that in any

The Baum-Connes Conjecture 63

unitary representation of G, on a Hilbert space H the operator p acts as the orthogonal pro-jection onto the G-fixed vectors in H.(d) Every finite dimensional irreducible unitary representation of G is isolated in G.(e) whenever a finite dimensional irreducible unitary representation π0 of G is weakly con-tained in a unitary representation π of G, then π0 is contained in π.

Proof. See [12]

The projection p ∈ C∗(G) in the theorem will be called the Kazhdan projection associatedwith the trivial representation:

Definition 4.90. If G has property T and the representation π : C∗(G)→Mn(C) is irre-ducible, then the central cover c(π) is called the Kazhdan projection associated with π.

One of the remarkable consequences of property T is that all Kazhdan projections actu-ally live in the center of C∗(G) (not just C∗(G)∗∗). In particular, C∗(G) has at least onenontrivial projection, coming from the trivial representation if |G| > 1 (Compare with thefact that C∗(F2) has tons of finite dimensional representations, yet no nontrivial projections![11] Theorem VII 6.6).

Theorem 4.91. (Structure theorem for property T groups). Let G be a discrete group withproperty T . For each finite-dimensional irreducible representation π : C∗(G)→Mn(C), theKazhdan projection c(π) is a central projection in C∗(G) and π(c(π)) = 1.

Proof. Let’s consider an essential representation σ : C∗(G)→B(H) such that (1) π ⊕ σ :C∗(G)→B(Cn ⊕ H) is faithful and (2) σ contains no subrepresentation which is unitarilyequivalent to π. One can start with a faithful representation of the algebra (1−c(π))C∗(G) ⊆C∗(G)∗∗ and inflate, if necessary, to force the essential part and standard theory of centralcovers shows that such a σ has no subrepresentation unitarily equivalent to π.Note that such a representation σ can’t be faithful if it were the Voiculescu’s Theoremwould imply that σ is approximately unitarily equivalent to σ⊕π. In other words, σ weaklycontains π and thus, by the generalization of Schur’s Lemma, actually contains π. Thiscontradicts our assumption (2).Thus J = ker(σ) is a nontrivial ideal in C∗(G). But assumption (1) implies that π|J mustbe faithful, since σ|J = 0. Hence J is finite-dimensional and has a unit p which is necessarilya central projection in C∗(G). We will show p = c(π): Since π is irreducible, then π(p) = 1and J ∼= Mn(C). Hence we may identify the representations A→pA and π. Thus they havethe same central covers-i.e. p = c(π) as in the proof of Theorem C.5 (see [8], [9] and [24] formore details).

The above two theorems have the following consequences:

Corollary 4.92. If G is a discrete group with property T and if G has in addition theHaagerup property, then G is finite.

Proof. If an isometric action has a fixed point it cannot be metrically proper, unless G isfinite.

This corollary shows that we can not apply Corollary 4.53 to infinite property T groups.In fact situation is much worse (compare with Theorem 4.54):

Corollary 4.93. If G is an infinite property T group then the quotient mapping from C∗(G)onto C∗λ(G) does not induce an isomorphism in K-theory.

Proof. We may write C∗(G) = pC∗(G)⊕ (1− p)C∗(G) = C⊕ (1− p)C∗(G), where p is theKazhdan projection associated to the trivial representation 1G. Since 1G(p) = 1 and [1] isthe canonical generator of K0(C) = Z, we see that [p] generates a cyclic direct summandof K(C∗(G)). On the other hand, it is mapped to zero in K(C∗λ(G)), since l2(G) does notexist a non-zero G-fixed vector under the left regular representation, unless G is finite.

The Baum-Connes Conjecture 64

It follows immediately that if G is an infinite property T group then the Baum-Connesassembly maps into K(C∗(G)) and K(C∗λ(G)) cannot both be isomorphism. In fact it is theassembly map into K(C∗(G)) which is problem. To see the problem we need some facts:

Corollary 4.94. If G has property T , then it has at most countably many nonequivalentfinite-dimensional irreducible representations.

Proof. Since property T groups are finitely generated, hence countable. A separable C∗-algebra C∗(G) can have at most countably many orthogonal projections (see PropositionC.3), since it can be represented on a separable Hilbert space.

Remark 4.95. In fact if G is a (discrete) property T group, then for m ∈ N there are at mostfinitely many equivalence classes of irreducible representations of G of dimension m. This isproved by Wassermann (see Corollary 3.13 [46]).

Proposition 4.96. If G is an infinite, residually finite group with property T , for exam-ple SL(n,Z), n ≥ 3, then G has countably infinitely many equivalence classes of finite-dimensional irreducible representations.

Proof. Since G is an infinite, residually finite group, then G has infinitely many equivalenceclasses of finite-dimensional irreducible representations.

Corollary 4.97. If G is an infinite, residually finite group with property T , then

Z∞ ⊆ K(C∗(G))

Proof. C∗(G) contains countably infinitely many orthogonal nonzero central projectionsc(πi)∞i=1 such that c(πi)C

∗(G) ∼= Mni(C) by restriction of πi : C∗(G)→Mni(C).

Remark 4.98. Since πi(c(πi)) = 1 ∈Mni(C), then the class [c(πi)] in K(Mni(C)) ∼= Z is notthe generator [1], but ni · [1].

On the other hand we already know the topological K-theory of SL(3,Z) from the com-putation of its Bredon homology with respect to finite subgroups and coefficients in therepresentation ring:

Theorem 4.99 (Ruben Sanchez-Garcia [39]).

Ktopi (SL(3,Z)) =

Z8 i = 0,

0 i = 1.

It follows that the full assembly map into K(C∗(SL(3,Z))) can not be surjective, butthe full/reduced assembly map into K(C∗(SL(n,Z)))/K(C∗λ(SL(n,Z))) is known to be in-jective, in general for all closed subgroups of Lie group with a finite number of connectedcomponents [10, §7]. All Kazhdan projections in fact map to zero in C∗λ(G) under λ18, thisis the reason why C∗λ(G) is used in place of C∗(G).

Unfortunately the Dirac and dual Dirac method we have applied to prove cases of theBaum-Connes conjecture treats the full and reduced C∗-algebra more or less equally. Henceproperty T causes the method to fail:

Proposition 4.100. If G is an exact, infinite property T group, then G does not satisfy thehypotheses of Theorem 4.35.

Proof. If G did satisfy the hypotheses then by Theorem 4.35 the quotient mapping fromK(C∗(G)) onto K(C∗λ(G)) would be an isomorphism.

18Otherwise, there is a finite-dimensional irreducible unitary representation which is weakly contained inλ (Theorem F.4.4 [12]), hence contained in λ, but l2(G) has no finite-dimensional G-invariant subspacesunder λ whenever G is infinite.

The Baum-Connes Conjecture 65

The above proposition indicates that our basic strategy for proving the Baum-Connesconjecture for a group G, which involves proving an identity in equivariant E-theory, willnot work for infinite property T groups (at least if these groups are exact). In fact, it isnot possible to prove the Baum-Connes conjecture for certain groups (for example uniformlattices in Sp(n, 1)) by working purely within E-theory (or for that matter within KK-theory). To explain this we need Poincare Duality:

Theorem 4.101. Let G be a countable exact group and let A be a separable proper gradedG-C∗-algebra. Suppose that there is a class α ∈ EG(A,C) such that for every finite subgroupsH of G the restricted class α|H ∈ EH(A,C) is invertible. Then the reduced assembly map

µλ : Ktop(G,D)→K(C∗λ(G,D))

is an isomorphism for a given separable graded G-C∗-algebra D if and only if the map

α∗ : K(C∗λ(G,A⊗D))→K(C∗λ(G,D))

induced from α is an isomorphism.

Proof. Consider the commutative diagram

Ktop(G,A⊗D)

α∗ ∼=

µλ

∼=// K(C∗λ(G,A⊗D)

α∗

Ktop(G,D)µλ// K(C∗λ(G,D)).

It has an important application to groups which act isometrically on Riemannian man-ifold: we denote by C(M) the C∗-algebra of sections of the bundle of Clifford algebrasCliff(TxM) associated to the tangent space of M . There is a Dirac operator on M (anunbounded self-adjoint operator acting on the Hilbert space of L2-sections of the Cliffordalgebra bundle on M), and it defines a class α ∈ E(C(M),C). Moreover if a group G actsisometrically on M then the Dirac operator defines an equivariant class

α ∈ EG(C(M),C).

Now if M happens to be a universal proper G-space then the hypotheses of the abovetheorem are met:

Theorem 4.102. Let M be a complete Riemannian manifold and suppose that a countablegroup G acts on M by isometries. Assume further that M is a universal proper G-space.Then the Dirac operator on M defines an equivariant E-theory class α ∈ EG(C(M),C),which, restricting from G to any finite subgroup H ⊆ G, determines invertible elements

α|H ∈ EH(C(M),C).

The proposition applies for example when G is a lattice in a semisimple group (take M tobe the associated symmetric space), and in this case the conjecture reduces to a statementwhich can be formulated purely within K-theory. So the official Baum-Connes conjecture(D is specialized to C and C0(0, 1)) for G is equivalent to the assertion that the map

α∗ : K∗(C∗λ(G, C(M)))→K∗(C∗λ(G))

is an isomorphism. One might hope that in fact the descended class

α ∈ E(C∗λ(G, C(M)), C∗λ(G))

is invertible. This is not always the case, as the following theorem of Skandalis [40] shows:

The Baum-Connes Conjecture 66

Theorem 4.103. Let G be an infinite, hyperbolic property T group. Then C∗λ(G) is notequivalent in E-theory to any nuclear C∗-algebra.

Since the C∗-algebra C∗λ(G, C(M)) is easily proved to be nuclear, then we obtain

Corollary 4.104. Let G be an infinite, hyperbolic, property T group and assume that Gacts on a complete Riemannian manifold M by isometries. The Dirac operator class

α ∈ E(C∗λ(G, C(M)), C∗λ(G))

is not invertible.

The corollary applies to discrete, cocompact subgroups of the Lie groups Sp(n, 1) (M isquaternionic hyperbolic space). Despite this, it follows from the work of Lafforgue [31] thatin this case α as above does induce an isomorphism on K-theory. This shows that E-theoryis not a perfect weapon with which to attach the Baum-Connes conjecture.19

19Exactly the same remarks apply here to KK-theory.

Appendix A: Self-adjoint Operators 67

Appendix A: Self-adjoint Operators

Let D be an operator20. Elements of ∩∞n=1dom(Dn) ⊆ H are called the smooth vectors forD. We say that ξ ∈ H is an analytical vector for D if ξ is smooth and

∞∑n=0

||Dnξ||n!

tn <∞

for some t > 0.

The domain of a self-adjoint operator always contains a dense set of analytical vectors.Indeed, if D is self-adjoint, then any ξ ∈ 1[−M,M ](D)H,M > 0 is an analytical vetor for D,since ||Dnξ|| ≤ Mn||ξ|| by the spectral theorem, and ∪M>01[−M,M ](D)H is dense in H. Inthe converse direction, we have the following result.

Theorem A.1. (Nelson). Suppose that the analytical vectors of a densely defined symmetricoperator form a dense subset inside the domain. Then the operator is essentially self-adjoint.

Corollary A.2. Let D1 and D2 be odd self-adjoint operators. Then D1×algD2 := D1⊗1 +1⊗D2 with domain dom(D1)⊗algdom(D2) is essentially self-adjoint. Let D1×D2 denote theclosure. Then

e−(D1×D2)2 = e−D21 ⊗e−D

22 .

Proof. See Lemma A.1.9. [36].

We say that a self-adjoint operator D has compact resolvent if bounded operators (D ±i)−1 are compact operators. The following is well-known and easy to show:

Proposition A.3. Let D be a self-adjoint operator. The following are equivalent:(1) The operator D has compact resolvent.(2) The resolvent Rλ = (λ−D)−1 is compact for any λ /∈ σ(D).(3) The operator D is diagonalizable and the diagonal entries converging to infinity in ab-solute value.(4) f(D) is a compact operator for all f ∈ C0(R).

This proposition is a consequence of the following lemma:

Lemma A.4. Let D be a self-adjoint operator on H and let a ∈ B(H) be given. If theoperator a · f(D) ∈ B(H) is compact for some non-vanishing function f ∈ C0(R), thena · f(D) is compact for all f ∈ C0(R). Similarly for the operators f(D) · a.

Proof. Let C denote the set of all f ∈ C0(R) such that a ·f(D) is compact. Then C is clearlya closed ideal in C0(R). But any non-vanishing element of C0(R) generates a dense ideal

in C0(R). Indeed, if f ∈ C0(R) is non-vanishing, then fe−x2

, fxe−x2 generates a dense

subalgebra in C0(R) by the Stone-Weierstrass theorem. Hence C = C0(R).

20We refer to [36] for more details.

Appendix B: Clifford Algebras 68

Appendix B: Clifford Algebras

Let V be a Euclidean vector space (that is, a real vector space equipped with a positive-definite inner product), a Clifford map on V we shall mean a real-linear map f : V→B intoa unital associative complex algebra B such that f(v)2 = ||v||21 for all v ∈ V .

Definition B.1. A complex Clifford algebra over V is a unital associative complex algebraA together with a Clifford map ϕ : V→A satisfying the universal property: if f : V→Bis any Clifford map, then there exists a unique complex algebra homomorphism F : A→Bsuch that F ϕ = f .

As we now proceed to show, V always carries a complex Clifford algebra and any twocomplex Clifford algebras over V are naturally isomorphic: Denote by V C = C ⊗R V thecomplexification of V and T (V ) stands for the full tensor algebra over V C

T (V ) =

∞⊕r=0

T r(V ),

where T 0(V ) = C and T r(V ) = V C ⊗C . . .⊗C VC for r > 0. Let I(V ) be the bilateral ideal

of T (V ) gnerated by the subset

v ⊗ v − (v, v)1 : v ∈ V ⊆ V C.

Let A be the quotient algebra T (V )/I(V ) and ϕ : V→A given by v 7→ v + I(V ). It’splain both that A is a unital associative complex algebra and that ϕ is a Clifford map.Now, let f : V→B be a Clifford map and extend to f : V C→B by z ⊗ v 7→ zf(v). Theuniversal property of the tensor algebra guarantees that f extends uniquely to an algebramap T (f) : T (V )→B. The assumption that f is a Clifford map ensures that T (f) vanisheson the ideal I(V ). Consequently, there exists a unique algebra map F : A→B such thatF ϕ = f . The uniqueness of complex Clifford algebras follows as usual from the univer-sal property. Thus, we may fix one and refer to it as the complex Clifford algebra C(V ) of V.

Notice that the Clifford property, ϕ(v)2 = ||v||21, satisfied by the Clifford map ϕ :V→C(V ) implies that ϕ is necessarily injective.

Proposition B.2. C(V ) is genrated by its real subspace V satisfying the Clifford relations

x, y ∈ V ⇒ xy + yx = 2(x, y)1.

Proof. It’s clear that V C ∼= V ⊕ iV , so the tensor algebra T (V ) is generated by its realsubspace V . As a result, the quotient algebra C(V ) is generated by its own copy of V . TheClifford relation follows from the Clifford property, replacing v by x+ y.

The Clifford relations just established have as a particular consequence the following factthat vectors in V are orthogonal if and only if they anticommute as elements of C(V ). Alinear map g : V→V ′ is said to be isometric if (gx, gy)′ = (x, y). By universal property ofC(V ) every linear isometry g : V→V ′ extends uniquely to an algebra map θg : C(V )→C(V ′).In particular, we have a group homomorphism

θ : O(V )→Aut(C(V )),

where O(V ) comprises all isometric real-linear automorphisms of V . We refer to γ = θ−Ias the grading automorphism of C(V ) sending each element of V to its negative. Usingthe oppositie algebra C(V )op of C(V ) and the Clifford map V→C(V )op, we get a uniquealgebra homomorphism α : C(V )→C(V )op restricting to V as the identity. In fact, α isan antiautomorphism of C(V ) as α2 = 1C(V ). Using the conjugate algebra C(V ) of C(V )

Appendix B: Clifford Algebras 69

and the Clifford map V→C(V ), we get a unique algebra homomorphism κ : C(V )→C(V )restricting to V as the identity. In fact κ is an antilinear ring automorphism of C(V ) asκ2 = 1C(V ). It’s clear that α and κ commute and their product is the unique antilinearantiautomorphism of C(V ) restricting to V as the identity. Thus, α κ = κ α is aninvolution of the complex Clifford algebra. In this way, C(V ) becomes a ∗-algebra and itsatifies a further universal property:

Proposition B.3. If B is an associative unital complex ∗-algebra and f : V→B is a self-adjoint (i.e. f(v)∗ = f(v) for all v ∈ V ) Clifford map, then the unique algebra map F :C(V )→B such that F |V = f is involution-preserving.

Proof. If follows simply that the set a ∈ C(V ) : F (a)∗ = F (a∗) is a subalgebra of C(V )containing V and recall that V generates C(V ) as a complex algebra.

In this regard, θg is involution-preserving and hence an ∗-automorphism of C(V ); more-over, θg commutes with γ, α and κ.

Now let V be a finite-dimensional Euclidean vector space with v1, . . . , vm as a specificorthonormal basis. If S = s1 < · · · < sp is a subset of m = 1, . . . ,m, then we put vS =vs1 · · · vsp and by convention v∅ = 1. Notice that vS is a unitary element whenever S ⊆ m.It turns out that vS : S ⊆m is a basis for C(V ) as a complex vector space with dimension2m (see Theorem 1.1.6 [38]). It’s clear that vS : S ⊆ m spans C(V ) by Proposition B.2.However, the linear independence follows from the fact the vT vS = (−1)|S||T |+|S∩T |vSvT(see Theorem 1.1.4 [38]).

Proposition B.4. C(V) possesses a unique normalized even central linear functional, itstrace τ defined by

τ(vS) =

1 S = ∅0 S 6= ∅

Proof. See Theorem 1.1.7 [38].

Note that τ(a∗) = τ(a), since σ(a) := τ(a∗) is normalized, even and central and henceσ = τ . It turns out that vS : S ⊆ m is an orthonormal basis for C(V ) relative to thecanonical positive-definite Hermitian inner product defined by

〈ε, η〉 = τ(η∗ε) for ε, η ∈ C(V )

and the norm comes from the inner product is given by

||∑S⊆m

µSvS ||2 =∑S⊆m

|µS |2

for any collection µS : S ⊆m of complex coefficients.Let Hτ be the Hilbert space (C(V ), 〈 , 〉) and define the left regular representation λ :C(V )→B(Hτ ) by λ(a)ξ = aξ for a ∈ C(V ), ξ ∈ Hτ . It’s clear that λ is a faithful ∗-representation, hence we define a C∗-norm on C(V ) by

||a||λ := ||λ(a)|| for a ∈ C(V ).

With this C∗-norm C(V ) becomes a C∗-algebra and denote it by C[V ]. Note that thecanonical embedding V→C[V ] is isometric:

||v||2λ = ||v∗v||λ = ||v2||λ = ||||v||2 · 1||λ = ||v||2.

By using the left regular representation λ and uniqueness of the trace τ we see that theinner product defined above does not depend on the choice of basis:

τ(a) :=Tr(λ(a))

dim(C(V )).

Appendix B: Clifford Algebras 70

Proposition B.5. Let V and W be finite-dimensional Euclidean vector spaces, then thereis a graded ∗-isomorphism

C(V )⊗C(W ) ∼= C(V ⊕W ).

Proof. Define a map f : V ⊕ W→C(V )⊗C(W ) by (v, w) 7→ v⊗1 + 1⊗w. In fact f is aself-adjoint Clifford map:

f(v, w)2 = v2⊗1 + 1⊗w2 = (||v||2 + ||w||2)1⊗1 = ||(v, w)||21⊗1.

Hence by universal property f extends to a ∗-homomorphism F : C(V ⊕W )→C(V )⊗C(W ).It’s clear that F θ−IV⊕W = θ−IV ⊗θ−IW F . The image of F is a subalgebra which

contains C(V )⊗1 and 1⊗C(W ). Therefore, F is surjective. Let ϕ : C(V )→C(V ⊕W ) andψ : C(W )→C(V ⊕W ) be the graded ∗-homomorphisms induced by the inclusions V→V ⊕Wand W→V ⊕W . As vectors in V ⊕W are orthogonal if and only if they anticommute aselements of C(V ⊕W ), we see that ϕ and ψ are graded-commuting, hence induces a graded ∗-homomorphism G : C(V )⊗C(W )→C(V ⊕W ) by a⊗b 7→ ϕ(a)ψ(b). It is clear that GF = 1,hence F is injective.

From now on we write Cliff(Rn) for the complex Clifford algebra over Rn in order todistinguish from the complex-valued continuous functions on Rn.

Corollary B.6. As graded ∗-algebrasa) Cliff(R0)∼= C with the trivial grading;b) Cliff(R) ∼= C⊕ C with the standard odd grading;c) Cliff(R2) ∼= M2(C) with the standard even grading;d) Cliff(Rn)⊗ Cliff(Rm) ∼= Cliff(Rn+m).

Proof. a) The zero map R0→C is a self-adjoint Clifford map and extends to a unital ∗-homomorphism from Cliff(R0) to C. In fact this ∗-homomorphism is bijective, as it mapsthe basis 1 in Cliff(R0) to the basis 1 ∈ C. Since the grading automorphism is unital, henceC is trivial graded.b) Define a self-adjoint Clifford map f : R→C ⊕ C by f(e1) = (1,−1), where e1 is theorthonormal basis for R. Hence we get a unital ∗-homomorphism F : Cliff(R)→C ⊕ C andF is bijective as (1, 1) and (1,−1) form a basis for C⊕C. Since (x, y) = (x+ y)/2 · (1, 1) +(x− y)/2 · (1,−1) and the grading automorphism γ on Cliff(R) maps 1 to 1 and e1 to −e1,we see that C⊕ C has the standard odd grading.c) Define a self-adjoint Clifford map f : R2→M2(C) by

f(e1) =

(0 11 0

)f(e2) =

(0 i−i 0

)Hence we get a unital ∗-homomorphism F : Cliff(R2)→M2(C). In fact F is bijective, asF (1), F (e1), F (e2) and F (e1e2) form a basis for M2(C). The inner grading is given by theself-adjoint unitary grading operator ε = ie1e2: Adε(ei) = εeiε = −ei = γ(ei) for i = 1, 2.

We complete c) by noticing that F (ε) =

(1 00 −1

).

d) It follows from Proposition B.5.

Remark B.7. More generally, each even Clifford algebra Cliff(R2k) is a matrix algebra

M2k(C), graded by ε = ike1 · · · e2k =

(I 00 −I

); each odd Clifford algebra Cliff(R2k+1)

is a direct sum M2k(C) ⊕M2k(C), graded by the automorphism which switches the sum-mands. This fact follows from the previous corollary by induction or see Corollary 2.1.21 in[36]. The classification for the real Clifford algebra can be found in [32].

Appendix C: The Central Cover 71

Appendix C: The Central Cover

In this appendix, we develop the general theory of central covers, which will be used in thediscussion of the Baum-Connes conjecture. We refer to [9], [37] and [41] for details.Recall that the universal representation of a C∗-algebra A is

πu =⊕

ϕ∈S(A)

πϕ : A→B

⊕ϕ∈S(A)

L2(A,ϕ)

= B(Hu).

By definition, the enveloping von Neumann algebra of A is the double commutant πu(A)′′.Thanks to the next result, we need not distinguish between the double dual A∗∗ and theenveloping von Neumann algebra.

Theorem C.1. The enveloping von Neumann algebra of A is isometrically isomorphic tothe double dual A∗∗. Hence the ultraweak topology on πu(A)′′(= A∗∗) restricts to the weaktopology on A.

Since every representation can be decomposed as a direct sum of cyclic representations(i.e., GNS representations), it is easily seen that A∗∗ enjoys the following universal property:for each nondegenerate representation π : A→B(H) there exists a unique normal extensionπ : A∗∗→B(H) such that π|A = A and π(A∗∗) = π(A)′′. The kernel of π is weakly closed(by normality); hence it’s von Neumann algebra. As such, it has a unit eπ which is a centralprojection in A∗∗.

Definition C.2. Let π : A→B(H) be a nondegenerate representation. The central cover ofπ, denoted c(π), is defined to be e⊥π = 1A∗∗ − eπ.

It is clear that c(π) is a projection in the centre of A∗∗ and

c(π)A∗∗ = c(π)A∗∗c(π) ∼= π(A∗∗) = π(A)′′.

Proposition C.3. If π1 and π2 are irreducible representations, the following are equivalent:(1) c(π1)c(π2) 6= 0;(2) c(π1) = c(π2);(3) π1 and π2 are unitarily equivalent.

Proposition C.4. For two representations π : A→B(H) and ρ : A→B(K), the followingare equivalent:(1) c(π)c(ρ) = 0;(2) (π ⊕ ρ)(A)′′ = π(A)′′ ⊕ ρ(A)′′.

Two representations π : A→B(H) and ρ : A→B(K) are said to be quasi-equivalentif there exists an isomorphism θ : π(A)′′→ρ(A)′′ such that θ(π(a)) = ρ(a) for all a ∈ A.Of course, unitarily equivalent representations are quasi-equivalent in this sense, but theconverse is false. Here are two simple facts.

Theorem C.5. Two representations (π1, H1) and (π2, H2) of a C∗-algebra A are quasi-equivalent if and only if c(π1) = c(π2), and the map (π,H)→c(π) gives a bijective cor-respondence between quasi-equivalence classes of representations of A and nonzero centralprojections in A∗∗.

Proof. For each central projection p 6= 0 in A∗∗ the map x 7→ px, x ∈ A is a representationof A on pHu with central cover p, since its normal extension is x 7→ px, x ∈ A∗∗. If (π,H) isa representation of A then (π,H) is quasi-equivalent to the representation ρ : x 7→ c(π)x onc(π)Hu, with the restriction π to c(π)A∗∗ as the intertwining isomorphism between ρ(A)′′

and π(A)′′. From this the theorem follows.

Proposition C.6. The representation π is quasi-equivalent to a subrepresentation of ρ ifand only if c(π) ≤ c(ρ).

References 72

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