The Novikov conjecture and geometry of

Banach spaces

Gennadi Kasparov and Guoliang Yu∗

Abstract: In this paper, we prove the strong Novikov conjecture for groups

coarsely embeddable into Banach spaces satisfying a geometric condition called

Property (H).

1 Introduction

An important problem in higher dimensional topology is the Novikov conjec-

ture on the homotopy invariance of higher signature. The Novikov conjecture

is a consequence of the Strong Novikov Conjecture on K-theory of group C∗-

algebras. The main purpose of this paper is to prove the Strong Novikov

conjecture (with coefficients) for any group coarsely embeddable into a Ba-

nach space satisfying a geometric condition called Property (H).

Definition 1.1. A real Banach space X is said to have Property (H) if there

exists an increasing sequence of finite dimensional subspaces {Vn} of X and

an increasing sequence of finite dimensional subspaces {Wn} of a real Hilbert

space such that

(1) V = ∪nVn is dense in X;

(2) ifW denotes ∪nWn, and S(V ), S(W ) denote respectively the unit spheres

of V , W , then there exists a uniformly continuous map ψ : S(V ) →S(W ) such that the restriction of ψ to S(Vn) is a homeomorphism (or

more generally a degree one map) onto S(Wn) for each n.

∗Both authors are partially supported by NSF.

1

As an example, let X be the Banach space lp(N) for some p ≥ 1. Let

Vn and Wn be respectively the subspaces of lp(N) and l2(N) consisting of all

sequences whose coordinates are zero after the n-th terms. We define a map

ψ from S(V ) to S(W ) by:

ψ(c1, ..., ck, ...) = (c1|c1|p/2−1, ..., ck|ck|p/2−1, · · · ).

ψ is called the Mazur map [4]. It is not difficult to verify that ψ satisfies the

conditions in the definition of Property (H). For each p ≥ 1, we can similarly

prove that the Banach space of all Schatten p-class operators on a Hilbert

space has Property (H). We can also check that uniformly convex Banach

spaces with certain unconditional bases have Property (H) (with the help of

results on the uniform homeomorphism classification of Banach space spheres

in Chapter 9 of [4]).

We also recall that a metric space Γ is said to be coarsely embeddable into

a Banach space X [7] if there exists a map h : Γ → X for which there exist

non-decreasing functions ρ1 and ρ2 from R+ = [0,∞) to R such that

(1) ρ1(d(x, y)) ≤ ∥h(x)− h(y)∥ ≤ ρ2(d(x, y)) for all x, y ∈ Γ;

(2) limr→+∞ ρi(r) = +∞ for i = 1, 2.

In the case of a countable group Γ, we endow Γ with a proper (left invariant)

length metric. If Γ is finitely generated, the word length metric is an example

of a proper length metric. The issue of coarse embeddability of a countable

group into a Banach space X is independent of the choice of the proper length

metric.

The following theorem is the main result of this paper.

Theorem 1.2. Let Γ be a countable discrete group and A any Γ-C∗-algebra.

If Γ admits a coarse embedding into a Banach space with Property (H), then

the Strong Novikov conjecture with coefficients in A holds for Γ, i.e. if EΓis the universal space for proper Γ-actions and A or Γ is the reduced crossed

product C∗-algebra, then the Baum-Connes assembly map

µ : KKΓ∗ (EΓ, A) → K∗(Aor Γ)

is injective.

2

The special case when the Banach space is the Hilbert space is proved in

[22] and [20].

If we replace the degree one condition by a nonzero degree condition in the

definition of Property (H), we say that X has rational Property (H).

Theorem 1.3. Let Γ be a countable discrete group and A any Γ-C∗-algebra.

If Γ admits a coarse embedding into a Banach space with rational Property

(H), then the rational Strong Novikov conjecture with coefficients in A holds

for Γ, i.e. if EΓ is the universal space for proper Γ-actions and Aor Γ is the

reduced crossed product C∗-algebra, then the Baum-Connes assembly map

µ : KKΓ∗ (EΓ, A)⊗Q → K∗(Aor Γ)⊗Q

is injective.

We remark that the rational Strong Novikov conjecture implies the Novikov

conjecture on homotopy invariance of higher signatures and the Gromov-

Lawson-Rosenberg conjecture regarding nonexistence of positive scalar cur-

vature metrics on closed aspherical manifolds.

It is conjectured that any countable subgroup of the diffeomorphism group

of a compact smooth manifold is coarsely embeddable into the Banach space

of all Schatten p-class operators for some p ≥ 1. If p > 2, then lp(N) is not

coarsely embeddable into a Hilbert space [13]. More generally, lp(N) does not

coarsely embed into lq(N) if p > q ≥ 2 [17]. Let C0 be the Banach space

consisting of all sequences of real numbers that are convergent to 0. It is

an open question if C0 has (rational) Property (H). By the above theorems,

a positive answer to this question would imply the Novikov conjecture since

every countable group admits a coarse embedding into C0 [5].

This paper is organized as follows. In section 2, we construct a C∗-algebra

associated to a Banach space with (rational) Property (H) and study its K-

groups. In section 3, we reformulate the Baum-Connes map and discuss its

connection with the localization algebra. In section 4, we introduce the Bott

map for K-groups. In section 5, we give a proof of the main theorem.

In this paper, K-groups of a graded C∗-algebra are defined to be the K-

3

groups of the underlying ungraded C∗-algebra obtained by forgetting the grad-

ing structure. The same comment applies to KK-groups.

The authors wish to thank Misha Gromov for inspiring discussions and the

referee for very helpful comments.

2 A C∗-algebra associated to a Banach space

with Property (H)

In this section, we construct a C∗-algebra associated to a Banach space with

(rational) Property (H) and study its K-groups.

Let ψ be as in the definition of (rational) Property (H). We extend ψ to a

map ϕ : V → W by:

ϕ(v) = ||v||ψ( v

||v||)

for any v ∈ V , where ϕ(0) should be interpreted as 0.

Let Clifford(Wn) be the complex Clifford algebras of Wn, satisfying the

relation w2 = ||w||2 for all w ∈ Wn. We define the complex Clifford alge-

bra Clifford(W ) to be the C∗-algebra inductive limit of Clifford(Wn). Let

C0(V,Clifford(W )) be the graded C∗-algebra of all bounded and uniformly

continuous functions on V with values in the Clifford algebra

Clifford(W ) which vanish at infinity, where the grading is given by the nat-

ural grading structure on the Clifford algebra. Let S = C0(R) be the graded

C∗-algebra of all complex-valued continuous functions on R vanishing at in-

finity (graded by even and odd functions). We define S⊗C0(V,Clifford(W ))

to be the graded C∗-algebra tensor product of S and C0(V,Clifford(W )).

For any f ∈ C0(R), we can define an element

f((s, ϕ(v))) ∈ S⊗C0(V,Clifford(W ))

by:

f((s, ϕ(v))) = f(s⊗1 + 1⊗ϕ(v)),

4

where s and ϕ(v) should be respectively viewed as unbounded degree one

multipliers of S and C0(V,Clifford(W )), (s, ϕ(v)) is defined to be s⊗1 +

1⊗ϕ(v) as an unbounded degree one multiplier of S⊗C0(V,Clifford(W )),

and f(s⊗1 + 1⊗ϕ(v)) is defined using functional calculus.

More concretely, f((s, ϕ(v))) can be defined as follows:

(1) if f(t) = g(t2) for some g ∈ C0(R), we define a scalar valued function

f((s, ϕ(v))) of the variable (s, v) ∈ R× V by:

(s, v) → g(s2 + ||ϕ(v)||2) = g(s2 + ||v||2)

for every (s, v) ∈ R× V ;

(2) if f(t) = tg(t2) ∈ C0(R) for some g ∈ C0(R), we define an element

f((s, ϕ(v))) ∈ S⊗C0(V,Clifford(W )) by:

f((s, ϕ(v))) = g(s2 + ||ϕ(v)||2)(s⊗1 + 1⊗ϕ(v))

= g(s2 + ||v||2)(s⊗1 + 1⊗ϕ(v))

for every s ∈ R, v ∈ V , here s⊗1 + 1⊗ϕ(v) should viewed as an unbounded

degree one multiplier of S⊗C0(V,Clifford(W ));

(3) the general definition of f((s, ϕ(v))) follows using approximation of f

by linear combinations of special functions of the above two types in C0(R).

Now we are ready to define a C∗-algebra asscociated to a Banach space

with (rational) Property (H).

Definition 2.1. Let X be a Banach space with (rational) Property (H). Let ϕ

be as above. We define A(X) to be the graded C∗-subalgebra of

S⊗C0(V,Clifford(W )) generated by all f((s, ϕ(v−v0))) for s ∈ R , all v0 ∈ V

and f ∈ C0(R).

It is not difficult to compute K∗(A(X)) when X is an lp-space for some

1 ≤ p < ∞ (cf. [12] for p=2). In general, it is an open question how to

compute K∗(A(X)). The following result provides a partial solution.

Proposition 2.2. Let X be a Banach space with Property (H) and let A(X)

be the C∗-algebra associated to X. If B is a (graded) C∗-algebra, then the

homomorphism from S to A(X): f(s) → f((s, ϕ(v)), induces an injection:

K∗(S⊗B) → K∗(A(X)⊗B).

5

Proof. Let A(X,Vn) be the C∗-subalgebra of A(X) generated by all elements

f((s, ϕ(v − v0))) for f ∈ C0(R) and s ∈ R, v0 ∈ Vn. Note that A(X)⊗B is the

inductive limit of A(X,Vn)⊗B. It suffices to prove that the homomorphism

βn:

K∗(S⊗B) → K∗(A(X,Vn)⊗B)

is injective for each n, where βn is induced by the homomorphism from S to

A(X,Vn): f(t) → f((s, ϕ(v)).

Let C(Vn) be the graded C∗-algebra of all continuous functions on Vn with

values in the Clifford algebra Clifford(Wn) which vanish at infinity. Define

A(Vn) to be the graded C∗-algebra tensor product S⊗C(Vn). Let rn be the

restriction homomorphism from A(X,Vn)⊗B to A(Vn)⊗B. By the definition

of Property (H), such restriction homomorphism is well defined. By Bott

periodicity (cf. [12]) and the degree one condition in the definition of Property

(H), we observe that the composition (rn)∗◦βn is an isomorphism. Proposition

2.2 follows from this observation.

The following result follows from Proposition 2.2 and the Green-Julg the-

orem.

Corollary 2.3. Let X be a Banach space with Property (H) and let A(X) be

the C∗-algebra associated to X. If H is a compact topological group and B is

a (graded) H-C∗-algebra, then we have a natural injective homomorphism:

KH∗ (S⊗B) → KH

∗ (A(X)⊗B),

where H acts on A(X) trivially.

We can prove the following results using essentially the same argument.

Proposition 2.4. Let X be a Banach space with rational Property (H) and

let A(X) be the C∗-algebra associated to X. If B is a (graded) C∗-algebra,

then we have a natural injective homomorphism

K∗(S⊗B)⊗Q → K∗(A(X)⊗B)⊗Q.

6

Corollary 2.5. Let X be a Banach space with rational Property (H) and let

A(X) be the C∗-algebra associated to X. If H is a compact topological group

and B is a (graded) H-C∗-algebra, then we have a natural injection:

KH∗ (S⊗B)⊗Q → KH

∗ (A(X)⊗B)⊗Q.

Let B be a graded C∗-algebra, let K be the graded C∗-algebra of all com-

pact operators on a graded separable and infinite dimensional Hilbert space.

Let Cc(R)⊗alg(B⊗K) be the algebraic graded tensor product of Cc(R) with

B⊗K, where Cc(R) is considered as an algebra graded by even and odd func-

tions.

We define S∞b (B) to be the graded C∗-algebra of all bounded sequences of

uniformly equi-continuous functions {fk} in S⊗B⊗K such that for each ϵ > 0,

there exists R > 0 for which there exists a bounded sequence of uniformly equi-

continuous functions {gk} in Cc(R)⊗alg(B⊗K) satisfying

diameter(support(gk)) < R and ||fk − gk|| < ϵ for all k, where the support

of gk is a subset of R. Let S∞0 (B) be the graded C∗-subalgebra of S∞

b (B)

consisting of all sequence (fk) in S∞b (B) which are convergent to 0 in the sup

norm. Define S∞(B) to be the quotient graded C∗-algebra S∞b (B)/S∞

0 (B).

The proof of the following result is straightforward and is therefore omitted.

Proposition 2.6. If H is a finite group and B is a graded H-C∗-algebra, then

KH∗ (S∞(B)) is naturally isomorphic to (

∏KH

∗+1(B))/(⊕KH∗+1(B)).

We remark that by the Green-Julg theorem the equivariant case of the

above proposition follows from the non-equivariant case.

Let C(Vk) be the graded C∗-algebra of all continuous functions on Vk with

values in the Clifford algebra Clifford(Wk) which vanish at infinity. Define

A(Vn) to be the graded C∗-algebra tensor product S⊗C(Vn). Endow R × Vk

with the product metric of the standard metric on R and the Banach norm

metric on Vk.

We define A∞b (X,B) to be the graded C∗-algebra of all bounded sequences

(ak) such that

7

(1) ak ∈ A(Vk)⊗B⊗K and (ak) is uniformly equi-continuous in {(s, vk)} in the

sense that for any ϵ > 0 there exists a δ > 0 (independent of k) such that

||ak(s, vk)− ak(s′, v′k)|| < ϵ

if d((s, vk), (s′, v′k)) < δ;

(2) for each ϵ > 0, there exists R > 0 for which there exists a sequence {gk}in A(Vk)⊗B⊗K satisfying

diameter(support(gk)) < R,

||ak(s, vk)− gk(s, vk)|| < ϵ

for all k and (s, vk) ∈ R× Vk, where the support of gk is a subset of R× Vk.

Let A∞0 (X,B) be the graded C∞-subalgebra of A∞

b (X,B) consisting of all

sequences which are convergent to 0 in norm. Define A∞(X,B) to be the

quotient graded C∗-algebra A∞b (X,B)/A∞

0 (X,B).

Let βk be the homomorphism from S to A(Vk) defined by:

(βk(f))(s, vk)) = f((s, ϕ(vk)))

for each f ∈ S, s ∈ R and vk ∈ Vk.

βk induces a homomorphism (still denoted by βk) from S⊗B⊗K to

A(Vk)⊗B⊗K. We define a homomorphism

β : S∞(B) → A∞(X,B)

by:

β([(f1, · · · , fk, · · · )]) = [(β1(f1), · · · , βk(fk), · · · )]

for all [(f1, · · · , fk, · · · )] ∈ S∞(B).

Theorem 2.7. If X is a Banach space with Property (H), then the homomor-

phism β induces an isomorphism:

β∗ : K∗(S∞(B)) → K∗(A∞(X,B)).

8

Proof. For each k, endow Vk with a Euclidean metric and let Dk be the cor-

responding Dirac operator on Vk. Let αk be the asymptotic morphism from

A(Vk) to S⊗K associated to Dk:

αk(t) : f⊗h→ f(t−1(s⊗1 + 1⊗Dk))π(h)

for all f ∈ S and h ∈ C(Vk), where C(Vk) is as in the defintion of A(Vk), Wk (in

the definition of C(Vk)) is identified with Vk as Euclidean vector spaces, and

π(h) is the multiplication operator associated to h. αk induces an asymptotic

morphism (still called αk) from A(Vk)⊗B⊗K to S⊗B⊗K. This asymptotic

morphism was first introduced by Higson-Kasparov-Trout in [12].

For each t ∈ [1,∞), we define a map:

α(t) : A∞(X,B) → S∞(B)

by:

(α(t))([(a1, · · · , ak, · · · )]) = [((α1(t))(a1), · · · , (αk(t))(ak), · · · )].

After rescaling αk(t) for each k, α is an asymptotic morphism from A∞(X,B)

to S∞(B), where the rescaling constant for each k can be chosen as the Lip-

schitz constant of the Lipschitz equivalence between the Euclidean norm on

Vk and the Banach norm on Vk. α is called the Dirac morphism. More pre-

cisely, let Ck be the Lipschitz constant of the Lipschitz equivalence between

the Euclidean norm || · ||E on Vk and the Banach norm on || · || on Vk, i.e. Ck

is a positive constant satisfying C−1k ||v||E ≤ ||v|| ≤ Ck||v||E for all v ∈ Vk. For

each k, the scaled asymptotic morphism is defined to be αk(Ckt) (still denoted

by αk).

Let A∞(X × X,B) be defined as A∞(X,B) using the sequence of finite

dimensional subspaces Vn × Vn and Wn ×Wn.

We define a homomorphism β′k from A(Vk) to A(Vk × Vk) by:

(β′k(f⊗g))(s, vk, v′k) = f((s, ϕ(v′k)))⊗g(vk),

where f ∈ S, g ∈ C0(Vk, Clifford(Wk)), s ∈ R, (vk, v′k) ∈ Vk × Vk, and

f((s, ϕ(v′k))) is defined to be f(s⊗1 + 1⊗ϕ(v′k)).

9

β′k induces a homomorphism (still denoted by β′

k) from A(Vk)⊗B⊗K to

A(Vk × Vk)⊗B⊗K. Using {β′k} we can define a homomorphism β′ from

A∞(X,B) to A∞(X × X,B). Similarly we define the Dirac morphism α′

from A∞(X ×X,B) to A∞(X,B) using the Dirac operator on the first copy

of Vk in Vk × Vk for each k.

We now apply Atiyah’s rotation trick [1]. For any θ ∈ [0, π2], we define the

rotation Rθ by:

Rθ(v, w) = (cosθv − sinθw, sinθv + cosθw).

Rθ induces an automorphism (still denoted by Rθ) on the algebra A∞(X ×X,B). In particular, for any sequence of vectors {(ck, c′k)} such that (ck, c

′k) ∈

Vk × Vk, we have

Rθ : [(fk((s, (ϕ(vk − ck), ϕ(v′k − c′k)))] →

[(fk((s, (cosθϕ(cosθvk + sinθv′k − ck)− sinθϕ(−sinθvk + cosθv′k − c′k),

sinθϕ(cosθvk + sinθv′k − ck) + cosθϕ(−sinθvk + cosθv′k − c′k)))))],

where fk ∈ S, {fk} is equi-continuous, and s ∈ R, (vk, v′k) ∈ Vk × Vk.

By homotopy invariance, we have

(Id)∗ = (R0)∗ = (Rπ2)∗,

where (Rθ)∗ is the automorphism on K∗(A∞(X ×X,B)) induced by Rθ.

For any x ∈ K∗(S∞(B)), we have:

α∗(β∗(x)) = x.

For any y ∈ K∗(A∞(X,B)), we have

β∗(α∗(y)) = α′∗(β

′∗(y)) = α′

∗(Rπ2)∗(β

′∗(y)) = y,

where the map y → y is induced by the map: (vk) → (−vk).The above two identities imply that β and α are isomorphisms, inverse to

each other, and y → y is the identity map.

10

The following result follows from Proposition 2.7 and the Green-Julg the-

orem.

Corollary 2.8. Let H be a finite group and let B be a graded H-C∗-algebra.

If X is a Banach space with Property (H), then β induces an isomorphism:

β∗ : KH∗ (S∞(B)) → KH

∗ (A∞(X,B)).

Proposition 2.9. If X is a Banach space with rational Property (H), then

the homomorphism β induces an isomorphism:

β∗ : K∗(S∞(B))⊗Q → K∗(A∞(X,B))⊗Q.

Corollary 2.10. Let H be a finite group and let B be a graded H-C∗-algebra.

If X is a Banach space with rational Property (H), then β induces an isomor-

phism:

β∗ : KH∗ (S∞(B))⊗Q → KH

∗ (A∞(X,B))⊗Q.

3 The Baum-Connes map and localization

In this section, we briefly recall the Baum-Connes map and its relation to the

localization algebra. Our reformulation of the Baum-Connes map follows the

work of Roe [19] and uses a localization technique introduced in [21]. This

reformulation will be useful in the next section.

Let Γ be a countable discrete group. Let ∆ be a locally compact metric

space with a proper and cocompact isometric action of Γ. Let C0(∆) be

the algebra of all complex valued continuous functions on ∆ which vanish at

infinity. Let B be a Γ-C∗-algebra.

The following definition is due to John Roe [19].

Definition 3.1. Let H be a Hilbert module over B and let φ be a ∗-homomorphism

from C0(∆) to B(H), the C∗-algebra of all bounded (adjointable) operators on

H. Let T be an operator in B(H).

(1) The support of T is defined to be the complement (in ∆×∆) of the set of

all points (x, y) ∈ ∆×∆ for which there exists f ∈ C0(∆) and g ∈ C0(∆)

satisfying φ(f)Tφ(g) = 0 and f(x) = 0 and g(y) = 0;

11

(2) The propagation of T is defined to be: sup{d(x, y) : (x, y) ∈ Supp(T )};

(3) T is said to be locally compact if φ(f)T and Tφ(f) are in K(H) for all

f ∈ C0(∆), where K(H) is defined to be the operator norm closure of all

finite rank operators on the Hilbert module H.

Let H be a (countably generated) Γ-Hilbert module over B and let φ be

a ∗-homomorphism from C0(∆) to B(H) which is covariant in the sense that

φ(γf)h = (γ(φ(f))γ−1)h for all γ ∈ Γ, f ∈ C0(∆) and h ∈ H. Such a triple

(C0(∆),Γ, φ) is called a covariant system.

Definition 3.2. We define the covariant system (C0(∆),Γ, φ) to be admissible

if

(1) the Γ-action on ∆ is proper and cocompact;

(2) there exist a Γ-Hilbert space H∆ and a separable and infinite dimensional

Γ-Hilbert space E such that

(a) H is isomorphic to H∆ ⊗ E ⊗B as Γ-Hilbert modules over B;

(b) φ = φ0 ⊗ I for some Γ-equivariant ∗-homomorphism φ0 from C0(∆)

to B(H∆) such that φ0(f) is not in K(H∆) for any nonzero function

f ∈ C0(∆) and φ0 is nondegenerate in the sense that {φ0(f)H∆ : f ∈C0(∆)} is dense in H∆,

(c) for each x ∈ ∆, E is isomorphic to l2(Γx)⊗Hx as Γx-Hilbert spaces

for some Hilbert space Hx with a trival Γx action, where Γx is the finite

isotropy subgroup of Γ at x.

In the above definition, the Γx-action on l2(Γx) is regular, i.e. (γξ)(z) =

ξ(γ−1z) for every γ ∈ Γx, ξ ∈ l2(Γx), and z ∈ Γx, B is the Γ-Hilbert module

over B with the inner product < a, b >= a∗b, and I is the identity operator

on E ⊗B.

We remark that such an admissible covariant system always exists (for

example we can choose E to be l2(Γ) with a regular action Γ). We point out

that condition (2) implies that E contains all unitary representations of the

12

finite isotropy groups and this point is important for Proposition 3.4 of this

section.

Definition 3.3. Let (C0(∆),Γ, φ) be an admissible covariant system. We

define C(Γ,∆, B) to be the algebra of Γ-invariant locally compact operators

in B(H) with finite propagation. The C∗-algebra C∗r (Γ,∆, B) is the operator

norm closure of C(Γ,∆, B).

The following result is essentially due to John Roe.

Proposition 3.4. If (C0(∆),Γ, φ) is an admissible covariant system, then

C∗r (Γ,∆, B) is ∗-isomorphic to (B or Γ) ⊗ K, where B or Γ is the reduced

crossed product C∗-algebra and K is the algebra of all compact operators on a

separable and infinite dimensional Hilbert space.

Proof. Proposition 3.4 follows from the definitions of the admissible covariant

system, C∗r (Γ,∆, B), and the reduced crossed product C∗-algebra.

Next we will describe the Baum-Connes map.

Let H be a Γ-Hilbert module over B, let F be an operator in B(H), let φ

be a ∗-homomorphism from C0(∆) to B(H) such that F is Γ-equivariant, i.e.

γFγ−1 = F for all γ ∈ Γ, φ(f)F −Fφ(f), φ(f)(FF ∗ − I) and φ(f)(F ∗F − I)

are in K(H) for all f ∈ C0(∆).

We denote the group KKΓ0 (C0(∆), B) by KKΓ

0 (∆, B). (H,φ, F ) gives a

KK-cycle representing a class in KKΓ0 (∆, B). It is not difficult to prove that

every class in KKΓ0 (∆, B) is equivalent to (H,φ, F ) such that (C0(∆),Γ, φ)

is an admissible covariant system. This can be seen as follows. We define a

new KK-group KKΓ

∗ (∆, B) using KK-cycles (H,φ, F ) such that (C0(∆),Γ, φ)

is an admissible covariant system. By the proof of Proposition 5.5 in [15], we

can show that there exists a Γ-Hilbert space H∆ satisfying the conditions of

Definition 3.2 such that H⊕(H∆⊗l2(Γ)⊗B) is isomorphic to H∆⊗l2(Γ)⊗B as

Γ-Hilbert modules overB. Using this stabilization result, we can prove that the

natural homomorphism from KKΓ

∗ (∆, B) to KKΓ0 (∆, B) is an isomorphism.

For any ϵ > 0, let {Ui}i∈I be a locally finite and Γ-equivariant open cover of

∆ satisfying diameter(Ui) < ϵ for all i. Let {ψi} be a Γ-equivariant partition

13

of unity subordinate to {Ui}i∈I . We define

Fϵ =∑i∈I

φ(√ψi)Fφ(

√ψi),

where the convergence is in the strict topology.

Note that Fϵ has propagation ϵ and (H,φ, Fϵ) is equivalent to (H,φ, F ) in

KKΓ0 (∆, B).

Fϵ is a multiplier of C∗r (Γ,∆, B). Fϵ is invertible modulo C∗

r (Γ,∆, B).

Let ∂ be the boundary map in K-theory:

K1(M(C∗r (Γ,∆, B))/C∗

r (Γ,∆, B)) → K0(C∗r (Γ,∆, B)),

whereM(C∗r (Γ,∆, H)) is the multiplier algebra of C∗

r (Γ,∆, H). We can define

the Baum-Connes map

µ : KKΓ0 (∆, B) → K0(C

∗r (Γ,∆, B)) ∼= K0(B or Γ)

by:

µ([(H,φ, F )]) = ∂([Fϵ]).

More precisely the Baum-Connes map can be implemented as follows.

Let pϵ be the idempotent:

FϵF∗ϵ + (I − FϵF

∗ϵ )FϵF

∗ϵ Fϵ(I − F ∗

ϵ Fϵ) + (I − FϵF∗ϵ )Fϵ(I − F ∗

ϵ Fϵ)

(I − F ∗ϵ Fϵ)F

∗ϵ (I − F ∗

ϵ Fϵ)2

.

Observe that the propagation of pϵ is at most 5ϵ.

Let

p0 =

I 0

0 0

.

We have

µ([(H,φ, F )]) = [pϵ]− [p0].

Similarly we can define the Baum-Connes map:

µ : KKΓ1 (∆, B) → K1(C

∗r (Γ,∆, B)) ∼= K1(B or Γ).

14

This induces the Baum-Connes map:

µ : KKΓ∗ (EΓ, B) → K∗(B or Γ),

where KKΓ∗ (EΓ, B) is defined to be the inductive limit of KKΓ

∗ (∆, B) over

all Γ-invariant and cocompact subspaces ∆ of EΓ which are finite dimensional

simplicial polyhedra. Here we choose a model of EΓ so that EΓ is equal to

the union of Γ-invariant and cocompact subspaces ∆ of EΓ which are finite

dimensional simplicial polyhedra (the existence of such model follows from

the construction of EΓ in the proof of Proposition 1.7 in [3] which is based on

Milnor’s join construction).

Let ∆ be a locally compact and finite dimensional simplicial polyhedron.

We endow ∆ with the simplicial metric. Let (C0(∆),Γ, φ) be an admissible

covariant system as before, where φ is a ∗-homomorphism from C0(∆) to B(H)

for some Hilbert module H over B.

Definition 3.5. (1) The algebraic localization algebra CL(Γ,∆, B) is de-

fined to be the algebra of all bounded and uniformly continuous functions

f : [0,∞) → C(Γ,∆, B) such that the propagation of f(t) goes to 0 as

t→ ∞, where C(Γ,∆, B) is as in Definition 3.3.

(2) The localization algebra C∗L(Γ,∆, B) is the norm closure of CL(Γ,∆, B)

with respect to the following norm:

||f || = supt∈[0,∞)||f(t)||.

It is not difficult to prove that, up to a ∗-isomorphism, CL(Γ,∆, B) and

C∗L(Γ,∆, B) are independent of the choice of the admissible covariant system

(C0(∆),Γ, φ). The localization algebra is an equivariant analogue of the alge-

bra introduced in [21].

Any class in KKΓ0 (∆, B) can be represented by (H,φ, F ) such that the

covariant system (C0(∆),Γ, φ) is admissible, where H is a Γ-Hilbert module

over B, F is an operator in B(H), φ is a ∗-homomorphism from C0(∆) to

B(H) such that F is Γ-equivariant, and φ(f)F − Fφ(f), φ(f)(FF ∗ − I) and

φ(f)(F ∗F − I) are in K(H) for all f ∈ C0(∆).

15

For each natural number n, we let F 1nbe defined as above. We define an

operator valued function F (t) on [0,∞) by:

F (t) = (t− n+ 1)F 1n+ (t− n)F 1

n+1

for all t ∈ [n, n+ 1].

F (t) is a multiplier of C∗L(Γ,∆, B) and is invertible modulo C∗

L(Γ,∆, B).

We define the local Baum-Connes map:

µL : KKΓ0 (∆, B) → K0(C

∗L(Γ,∆, B)),

by

µL[H,φ, F )] = ∂[F (t)],

where

∂ : K1(M(C∗L(Γ,∆, B))/C∗

L(Γ,∆, B)) → K∗0(C

∗L(Γ,∆, B)),

is the boundary map in K-theory andM(C∗L(Γ,∆, B)) is the multiplier algebra

of C∗L(Γ,∆, B).

Similarly we can define the local Baum-Connes map:

µL : KKΓ1 (∆, B) → K1(C

∗L(Γ,∆, B)).

We remark that the local Baum-Connes map is very much in the spirit of

the local index theory of elliptic differential operators.

Theorem 3.6. Let B be a Γ-C∗-algebra. The local Baum-Connes map µL is an

isomorphism from KKΓ∗ (∆, B) to K∗(C

∗L(Γ,∆, B)) if ∆ is a finite dimensional

simplicial polyhedron with the Γ-invariant simplicial metric.

Proof. This theorem is a consequence of the Mayer-Vietoris and five lemma

argument (essentially similar to the proof of the non-equivariant analogue in

[21]).

Let C∗L(Γ, EΓ, B) be the C∗-algebra inductive limit of C∗

L(Γ,∆, B), where

the limit is taken over all Γ-invariant and cocompact subspaces ∆ of EΓ which

are finite dimensional simplicial polyhedra.

16

The above local Baum-Connes map induces a map (still called the local

Baum-Connes map):

µL : KKΓ∗ (EΓ, B) → K∗(C

∗L(Γ, EΓ, B)).

Corollary 3.7. The local Baum-Connes map µL is an isomorphism from

KKΓ∗ (EΓ, B) to K∗(C

∗L(Γ, EΓ, B)).

Next we shall discuss the relation between the Baum-Connes map and

an evaluation map. This connection will be useful in the proof of the main

theorem in Section 5.

Let e be the evaluation map:

C∗L(Γ,∆, B) → C∗

r (Γ,∆, H) ∼= (B or Γ)⊗K

defined by:

e(f) = f(0)

for all f ∈ C∗L(Γ,∆, B).

The above evaluation maps induce an evaluation homomorphism (still de-

noted by e):

C∗L(Γ, EΓ, B) → (B or Γ)⊗K.

We have

µ = e∗ ◦ µL.

4 The Bott map

In this section, we introduce a Bott map for K-groups. The Bott map will

play an essential role in the proof of the main result of this paper.

Let X be a Banach space with (rational) Property (H). Let Γ be a count-

able group with a left invariant proper length metric. Let h : Γ → X be a

coarse embedding. Without loss of generality we can assume that the image of

h is contained in V , where V is as in the definition of the (rational) Property

(H).

17

For each v ∈ V,w ∈ W, and γ ∈ Γ, we define bounded functions ξv,γ and

ηv,w,γ on Γ by:

ξv,γ(x) = ||ϕ(v + h(x)− h(xγ))|| = ||v + h(x)− h(xγ)||,

ηv,w,γ(x) =< ϕ(v + h(x)− h(xγ)), w >

for all x ∈ Γ, where ϕ is defined as in Section 2. The boundedness of ξv,γ and

ηv,w,γ follows from the fact that h is a coarse embedding.

Let c0(Γ) be the algebra of all functions on Γ vanishing at infinity. We

define the Γ action on l∞(Γ) as follows: (γ(η))(x) = η(xγ) for each η ∈l∞(Γ), γ ∈ Γ, x ∈ Γ. Let Y be the spectrum of the unital commutative Γ-

invariant C∗-subalgebra of l∞(Γ) generated by c0(Γ), all constant functions on

Γ, all ξv,γ and ηv,w,γ, and their translations by group elements of Γ. Notice

that Y is a separable compact space and is a quotient space of βΓ, the Stone

Cech compactification of Γ.

Let A(X) be the C∗-algebra associated to X (as defined in Section 2). If

A is a Γ-C∗-algebra, then Γ acts on the C∗-algebra

C(Y )⊗A(X)⊗A ⊆ l∞(Γ)⊗A(X)⊗A

as follows:

γ(η(x)⊗f((s, ϕ(v − v0)))⊗a)

= η(xγ)⊗f((s, ϕ(v − v0 + h(x)− h(xγ)))⊗γ(a)

for each γ ∈ Γ, η ∈ C(Y ) ⊆ l∞(Γ), x ∈ Γ, s ∈ R, f ∈ C0(R), v ∈ V, v0 ∈ V and

a ∈ A, where f((s, ϕ(v − v0))) is as in the definition of A(X).

The above Γ action on C(Y )⊗A(X)⊗A has its origin in [22] and [20]. We

can see that the Γ action on C(Y )⊗A(X)⊗A is well defined as follows. Recall

that, for any net {wi}i∈I in a Hilbert space, if wi converges to a vector w of the

Hilbert space in weak topology and ||wi|| converges to ||w||, then wi converges

to w in norm. Using this fact and the definition of Y , we know that ϕ(v−v0+h(x)−h(xγ)) can be extended to a norm continuous function on Y with values

in H. It follows that f((s, ϕ(v − v0 + h(x)− h(xγ))) can be identified with a

18

norm continuous function on Y with values in A(X)⊗A. This, together withthe compactness of Y , implies that γ(η(x)⊗f((s, ϕ(v− v0)))⊗a) is an element

in C(Y )⊗A(X)⊗A. Let D be the subalgebra of C(Y )⊗A(X)⊗A consisting of

all linear combinations of products of elements of the type η(x)⊗f((s, ϕ(v −v0)))⊗a. Note that, by the definition of A(X), D is dense in the C∗-algebra

C(Y )⊗A(X)⊗A. By extending linearly and multiplicatively, for each d ∈ D,

we can define γ(d) as an element of C(Y )⊗A(X)⊗A. By the definition of Y ,

Γ is a dense subset of Y . This, together with the definition of γ(d), implies

that ||γ(d)|| = ||d|| for each d ∈ D. Finally the Γ action extends continuously

to C(Y )⊗A(X)⊗A.Let Z be the space of all probability measures on Y with the weak topol-

ogy. Note that Z is a convex and compact topological space with the weak∗

topology. The idea of using the space of probability measures is due to Nigel

Higson [10]. The action of Γ on C(Y )⊗A(X) ⊗ A induces an action of Γ on

C(Z)⊗A(X)⊗A by:

γ(u(µ)⊗f((s, ϕ(v − v0)))⊗a)

= u(γ(µ))⊗f((s, ϕ(v − v0 +

∫Y

(h(y)− h(yγ))dµ)))⊗γ(a)

for each γ ∈ Γ, u ∈ C(Z), µ ∈ Z, f ∈ C0(R), s ∈ R, v ∈ V, v0 ∈ V, a ∈ A,

where the Γ action on Z is induced by the Γ action on Y . The assumption

that h is a coarse embedding implies that C(Z)⊗A(X)⊗A is a Γ-proper C∗-

algebra. This can be seen as follows: let C(X) be the abelian C∗-subalgebra of

C(Z)⊗A(X) generated by all elements b⊗g(s2 + ||v − v0||2), where b ∈ C(Z),

g ∈ C0(R) and v0 ∈ V . C(X) is isomorphic to a commutative C∗-algebra

C0(T ) for some locally compact space T . Notice that T is a proper Γ-space (as

a consequence of the fact that h is a coarse embedding), C(X) is in the center

of the multiplier algebra of C(Z)⊗A(X)⊗A, and C(X)(C(Z)⊗A(X)⊗A) is

dense in C(Z)⊗A(X)⊗A.For each f ∈ S, let ft ∈ A(X) be defined by

ft((s, v)) = f(t−1(s, ϕ(v)))

for all s ∈ R, v ∈ V .

19

We define the Bott map

β : K∗((C(Z)⊗S⊗A)or Γ) → K∗((C(Z)⊗A(X)⊗A)or Γ)

to be the homomorphism induced by the following asymptotic morphism from

(C(Z)⊗S⊗A)or Γ to (C(Z)⊗A(X)⊗A)or Γ:

βt((u⊗f⊗a)γ) = (u⊗ft⊗a)γ

for t ∈ [1,∞), u ∈ C(Z), f ∈ S, a ∈ A, γ ∈ Γ.

The fact that βt is an asymptotic morphism follows from the identity

ϕ(cv) = |c|ϕ(v) for any scalar c and the assumption that the restriction of

ϕ to the sphere of V is uniformly continuous.

Let ∆ be a locally compact metric space with a proper and cocompact

isometric action of Γ. We define an asymptotic morphism

βt : C∗L(Γ,∆, C(Z)⊗S⊗A) → C∗

L(Γ,∆, C(Z)⊗A(X)⊗A)

induced by the homomorphism from S to A(X): f → ft, where

ft((s, v)) = f(t−1(s, ϕ(v)))

for all f ∈ S and s ∈ R, v ∈ V .

More precisely βt is defined as follows. Let (C0(∆),Γ, φ) be an admissible

covariant system, where φ is a ∗-homomorphism from C0(∆) to B(H) for some

Hilbert module H over C(Z)⊗S⊗A. By the definition of admissible covariant

system, we have

K(H) ∼= C(Z)⊗S⊗A⊗K,

where K(H) is the operator norm closure of all finite rank operators on the

Hilbert module H and K is the graded C∗-algebra of all compact operators

on a graded separable and infinite dimensional Hilbert space. Let β′t be the

asymptotic morphism from C(Z)⊗S⊗A⊗K to C(Z)⊗A(X)⊗A⊗K induced

by the homomorphism from S to A(X): f → ft, where ft is defined as in the

previous paragraph.

Let c be a Γ cut-off function on ∆, i.e. c is a compactly supported non-

negative continuous function on ∆ satisfying∑γ∈Γ

c(γ−1x) = 1

20

for all x ∈ ∆. The existence of such cut-off function follows from properness

and cocompactness of the Γ action on ∆.

For each T ∈ CL(Γ,∆, C(Z)⊗S⊗A), we define βt(T ) in

CL(Γ,∆, C(Z)⊗A(X)⊗A) by:

βt(T ) =∑γ∈Γ

β′t(φ(γ(c))γTγ

−1),

where β′t is the asymptotic morphism defined above, (γ(c))(x) = c(γ−1x) for

all x ∈ ∆, and the sum converges in the strong operator topology because T

has finite propagation.

Note that βt(T ) is Γ-invariant and is an element of

CL(Γ,∆, C(Z)⊗A(X)⊗A). It is not difficult to see that βt can be extended

to an asymptotic morphism:

βt : C∗L(Γ,∆, C(Z)⊗S⊗A) → C∗

L(Γ,∆, C(Z)⊗A(X)⊗A).

By Corollary 3.7, the above asymptotic morphism induces a homomor-

phism (called the Bott map):

β : KKΓ(EΓ, C(Z)⊗S⊗A) → KKΓ(EΓ, C(Z)⊗A(X)⊗A).

Proposition 4.1. The Bott map

β : KKΓ∗ (EΓ, C(Z)⊗S⊗A) → KKΓ

∗ (EΓ, C(Z)⊗A(X)⊗A)

is injective, where EΓ is the universal space for proper Γ-actions.

Proof. Let A∞(X,A) be as defined in Section 2. For any fixed v0 ∈ Vk ⊆ V ,

we note that the following translation by v0 on A∞(X,A) is well defined:

[(f1(s, v1), · · · , fk−1(s, vk−1), fk(s, vk), fk+1(s, vk+1), · · · )]

→ [(0, · · · , 0, fk(s, vk + v0), fk+1(s, vk+1 + v0), · · · )]

for all [(f1, · · · , fk−1, fk, fk+1 · · · )] ∈ A∞(X,A).

By the uniform equi-continuity condition in the definition of

A∞(X,A), this translation operator is norm-continuous in v0. Hence a Γ

21

action on C(Z)⊗A∞(X,A) can be defined exactly in the same way as the Γ

action on C(Z)⊗A(X)⊗A.Now we can define the Bott map:

β∞ : KKΓ∗ (EΓ, C(Z)⊗S∞(A)) → KKΓ

∗ (EΓ, C(Z)⊗A∞(X,A)),

in a way similar to the definition of the Bott map:

β : KKΓ∗ (EΓ, C(Z)⊗S⊗A) → KKΓ

∗ (EΓ, C(Z)⊗A(X)⊗A),

where S∞(A) is defined in Section 2.

The Bott map β∞ is induced by the asymptotic homomorphism from

C(Z)⊗S∞(A) to C(Z)⊗A∞(X,A):

[u⊗(f1⊗b1, · · · , fk⊗bk, · · · )] → [u⊗((f1)t⊗b1, · · · , (fk)t⊗bk, · · · )],

where u ∈ C(Z), fk ∈ S, bk ∈ A, and (fk)t(s, vk) = fk(t−1(s, ϕ(vk))) for all

(s, vk) ∈ R× Vk.

We claim that the Bott map:

β∞ : KKΓ∗ (EΓ, C(Z)⊗S∞(A)) → KKΓ

∗ (EΓ, C(Z)⊗A∞(X,A))

is an isomorphism.

This claim follows from the standard Mayer-Vietoris and five lemma argu-

ment, Corollary 2.8, and the fact that, for any finite subgroup H of Γ, Z is

H-equivariantly homotopy equivalent to a point µ0 ∈ Z fixed by H (using the

linear homotopy). The point µ0 can be obtained by averaging the H-orbit of

a point in Z using the assumption that H is a finite group and the fact that

Z is a convex and compact topological space. Then

g[u(µ)⊗(fk((s, ϕ(vk − vk0)))⊗ak)]

evaluated at µ = µ0 is equal to

[u(µ)⊗(fk((s, ϕ(vk − vk0)))⊗g(ak))]

evaluated at µ = µ0 for all g ∈ H, u ∈ C(Z), fk ∈ S, vk0 ∈ Vk, s ∈ R, vk ∈Vk, ak ∈ A, and [(fk⊗ak)] ∈ S∞(A). The last equation follows from the

definition of the Γ action on C(Z)⊗A∞(X,A).

We now consider the following commutative diagram:

22

KKΓ∗ (EΓ,S⊗A)

β−→ KKΓ∗ (EΓ, C(Z)⊗A(X)⊗A)

↓ σ ↓ σ′

KKΓ∗ (EΓ, C(Z)⊗S∞(A))

β∞−→ KKΓ

∗ (EΓ, C(Z)⊗A∞(X,A)),

where σ is induced by the homomorphism S⊗A → C(Z)⊗S∞(A) mapping

each element f to 1⊗[(f⊗p0)] (here 1 is the constant 1 function on Z, [(f⊗p0)]is the element represented by the constant sequence consisting of f⊗p0, andp0 is a rank one projection of grading degree 0 in K) and σ′ is induced by the

homomorphism from C(Z)⊗A(X)⊗A to C(Z)⊗A∞(X,A) which maps each

element u⊗f to the element represented by the sequence u⊗(fk⊗p0) (here

u ∈ C(Z), f ∈ A(X)⊗A, and fk is the restriction of f to Vk).

We observe that σ is injective. This can be seen as follows. Let σ be the

homomorphism:

KKΓ∗ (EΓ,S⊗A) → KKΓ

∗ (EΓ,S∞(C(Z)⊗A))

obtained by composing σ with the homomorphism fromKKΓ∗ (EΓ, C(Z)⊗S∞(A))

to KKΓ∗ (EΓ,S∞(C(Z)⊗A)) induced by the inclusion homomorphism from

C(Z)⊗S∞(A) to S∞(C(Z)⊗A). It suffices to prove that σ is injective. There

exists a natural homomorphism:∏KKΓ

∗+1(EΓ, C(Z)⊗A) → KKΓ∗ (EΓ,S∞(C(Z)⊗A)),

where∏KKΓ

∗+1(EΓ, C(Z)⊗A) is defined to be the inductive limit of∏KKΓ

∗+1(∆, C(Z)⊗A) over all Γ-cocompact subsets ∆ of EΓ. This homo-

mophism induces a homomorphism (denoted by τ):

(∏

KKΓ∗+1(EΓ, C(Z)⊗A))/(⊕KKΓ

∗+1(EΓ, C(Z)⊗A))

→ KKΓ∗+1(EΓ,S∞(C(Z)⊗A)).

Proposition 2.6, together with a standard Mayer-Vietoris and five lemma ar-

gument, implies that τ is an isomorphism. There is also a natural isomorphism

(denoted by θ) from KKΓ∗ (EΓ,S⊗A) to KKΓ

∗+1(EΓ, A). Let ς be the homo-

morphism:

KKΓ∗+1(EΓ, A) → (

∏KKΓ

∗+1(EΓ, C(Z)⊗A))/(⊕KKΓ∗+1(EΓ, C(Z)⊗A))

23

obtained by composing the homomorphism from KKΓ∗+1(EΓ, A) to

KKΓ∗+1(EΓ, C(Z)⊗A) induced by the inclusion map from A to C(Z)⊗A with

the group homomorphism mapping each element z ∈ KKΓ∗+1(EΓ, C(Z)⊗A)

to the element in (∏KKΓ

∗+1(EΓ, C(Z)⊗A))/(⊕KKΓ∗+1(EΓ, C(Z)⊗A)) repre-

sented by the constant sequence consisting of z. By contractibility of Z, we

know that ς is injective. We have

σ = τ ◦ ς ◦ θ.

Now the injectivity of σ follows from the injectivity of ς and the fact that τ

and θ are isomorphisms.

Finally Proposition 4.1 follows from the injectivity of σ, the claim that β∞

is an isomorphism, and the commutative diagram in this proof.

5 Proof of the main result

In this section, we give the proof of Theorem 1.2, the main result of this paper.

The proof of Theorem 1.3 is essentially similar.

Proof. Let A be any Γ-C∗-algebra. We consider the following commutative

diagram:

KKΓ∗ (EΓ,S⊗A)

µ1−→ K∗((S⊗A)or Γ)

↓ β ↓ β′

KKΓ∗ (EΓ, C(Z)⊗A(X)⊗A) µ2−→ K∗((C(Z)⊗A(X)⊗A)or Γ),

where µ1 and µ2 are the Baum-Connes assembly maps, β and β′ are respec-

tively the homomorphisms induced by the inclusion: C → C(Z) composed

with the Bott maps defined in Section 4. The commutativity of the above

diagram follows from the definitions of the Bott maps and the relation be-

tween the Baum-Connes map and the evaluation map (cf. the discussion after

Corollary 3.7 at the end of Section 3).

Notice that the inclusion map C → C(Z) induces an isomorphism

KKΓ∗ (EΓ,S⊗A) → KKΓ

∗ (EΓ, C(Z)⊗S⊗A).

24

This, together with Proposition 4.1, implies that β is injective. The fact that

C(Z)⊗A(X)⊗A is a Γ-proper C∗-algebra implies that µ2 is an isomorphism

(it can be seen by the Mayer-Vietoris and five lemma argument [9]). The

above facts, together with commutativity of the above diagram, imply that µ1

is injective. It follows that the Baum-Connes assembly map µ in Theorem 1.2

is injective.

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Department of Mathematics, 1326 Stevenson Center,

Vanderbilt University, Nashville, TN 37240, USA

e-mail: guoliang.yu@vanderbilt.edu and gennadi.kasparov@vanderbilt.edu

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