ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 131–142 (December 19, 1997) S 1079-6762(97)00038-3

OPERATOR K-THEORY FOR GROUPS WHICH ACT PROPERLY AND ISOMETRICALLY ON HILBERT SPACE

NIGEL HIGSON AND GENNADI KASPAROV

(Communicated by Masamichi Takesaki)

Abstract. Let G be a countable discrete group which acts isometrically and metrically properly on an infinite-dimensional Euclidean space. We calculate the K-theory groups of the C^{∗}-algebras C_{max}^{∗}(G) and C_{red}^{∗}(G). Our result is in accordance with the Baum-Connes conjecture.

1. Introduction

Let G be a countable discrete group and denote by C_{max}^{∗}(G) and C_{red}^{∗}(G) its full and reduced group C^{∗}-algebras. This note is concerned with the computation of the K-theory of these C^{∗}-algebras, for groups G which admit the following sort of action on a Euclidean space:

1.1. Definition. Let V be a real inner product vector space which is possibly infinite-dimensional (hereafter we shall call V a Euclidean space). An affine, isometric action of a discrete group G on V is metrically proper if lim_{g→∞} ∥g ·v∥ = ∞, for every v ∈ V.

Gromov [10] calls groups which admit such an action a-T-menable. The terminology is justified by the well-known theorem that every affine, isometric action of a property T group on a Euclidean space has bounded orbits [7], and by the recent observation [3] that every countable amenable discrete group admits a metrically proper, affine, isometric action on a Euclidean space. Other examples of a-T -menable groups are proper groups of isometries of real or complex hyperbolic space, finitely generated Coxeter groups, and groups which act properly on locally finite trees. The reader is referred to the short article [3] and the monograph [12] for further information.

If G is any discrete group then denote by EG the universal proper G-space, as described in [2]. It is a paracompact and Hausdorff G-space which has a paracompact and Hausdorff quotient by G, along with the following additional properties:

(i) it is covered by open G-sets, each of which admits a continuous G-map to some coset space G/H, where H is finite; and

(ii) if X is any other G-space satisfying (i) then there is a unique-up-to-G-homotopy G-map from X into EG.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors October 25, 1997.

1991 Mathematics Subject Classification. Primary 46L20.

Key words and phrases. Baum-Connes conjecture, C^{∗}-algebras, K-theory.

The first author was partially supported by an NSF grant.

Clearly EG is unique up to G-homotopy. If G is torsion-free then EG is the universal principal space EG, whose quotient by G is the classifying space BG.

The Baum-Connes assembly maps for the C^{∗}-algebras C_{max}^{∗}(G) and C_{red}^{∗}(G) are homomorphisms of abelian groups

μ_{max} : K_{∗}^{G}(EG) → K_{∗}(C_{max}^{∗}(G)),
μ_{red} : K_{∗}^{G}(EG) → K_{∗}(C_{red}^{∗}(G)).

On the left hand side is the equivariant K-homology of EG, defined using KKtheory [2], [16]. If G is torsion-free then K_{∗}^{G}(EG) is isomorphic to K_{∗}(BG), the K-homology (with compact supports) of the classifying space for G. See [2] and Section 3 below for a further description of both equivariant K-homology and the assembly map.

We are going to outline a proof of the following result:

1.2. Theorem. If G is a discrete group which admits an affine, isometric and metrically proper action on a Euclidean space (possibly infinite-dimensional) then the Baum-Connes assembly maps μ_{max} and μ_{red} are isomorphisms.

The assertion that for any G the assembly map μ_{red} is an isomorphism is known as the Baum-Connes conjecture [2]. By an index theory argument, the injectivity of either μ_{max} or μ_{red}, for a given G, implies the Novikov higher signature conjecture [8] for manifolds with fundamental group G. Thus our theorem has the following consequence:

1.3. Corollary. If G is a discrete group which admits an affine, isometric and metrically proper action on a Euclidean space (possibly infinite-dimensional) then the Baum-Connes and Novikov conjectures hold for G. In particular, these conjectures hold for any countable amenable group.

To prove the theorem we shall use a variant of KK-theory, the E-theory of Connes and Higson [6], which appears to be more appropriate in this case. As in other instances where the K-theory of group C^{∗}-algebras may be computed, or partially computed, a key role is played by a Bott periodicity argument—in this case periodicity for infinite-dimensional Euclidean space [14]. Given periodicity, the proof that μ_{max} is an isomorphism is a relatively simple consequence of the formal machinery of E-theory. Thanks to a peculiarity of E-theory, the proof for μ_{red} is rather more complicated, unless G has the additional functional-analytic property that C_{red}^{∗}(G) is exact [22]. Conjecturally all discrete groups have this property, but in any case we shall end this note with an ad hoc proof that μ_{red} is an isomorphism for the groups under consideration here.

2. E-theory

We begin by reviewing some of the foundational results in E-theory proved in [6] and [11]. Let A and B be C^{∗}-algebras. An asymptotic morphism from A to B is a family of functions

{ϕ_{t}}_{t∈[1,∞)} : A → B

OPERATOR K-THEORY FOR GROUPS WHICH ACT ON HILBERT SPACE 133

such that t ↦ ϕ_{t}(a) is bounded and norm-continuous, for every a ∈ A, and

lim_{t→∞} \{ \begin{matrix}ϕ_{t}(a_{1}a_{2}) − ϕ_{t}(a_{1})ϕ_{t}(a_{2}) \\ ϕ_{t}(a_{1} + a_{2}) − ϕ_{t}(a_{1}) − ϕ_{t}(a_{2}) \\ ϕ_{t}(αa_{1}) − αϕ_{t}(a_{1}) \\ ϕ_{t}(a_{1}^{∗}) − ϕ_{t}(a_{1})^{∗} \end{matrix}\} = 0,

for every a_{1}, a_{2} ∈ A and α ∈ ℂ. This is the definition used in [6]; it agrees with the definition in [11] if we identify those asymptotic morphisms ϕ′, ϕ′′ which are asymptotically equivalent, in the sense that ϕ′_{t}(a) − ϕ′′_{t}(a) → 0 as t → ∞ (we shall write ϕ′_{t}(a) ∼ ϕ′′_{t}(a)). If A and B are ℤ/2-graded C^{∗}-algebras then we shall consider only asymptotic morphisms which map homogeneous elements to homogeneous elements and preserve the grading degree. If A and B are equipped with actions of a countable discrete group G, then by an equivariant asymptotic morphism from A to B we shall mean an asymptotic morphism {ϕ_{t}}_{t∈[1,∞)} such that ϕ_{t}(g(a)) ∼ g(ϕ_{t}(a)), for every a ∈ A and every g ∈ G.

Two asymptotic morphisms from A to B are homotopic if there is an asymptotic morphism from A to B[0, 1] (the C^{∗}-algebra of continuous functions mapping the unit interval into B) from which the two may be recovered by evaluation at 0 and 1. Homotopy is an equivalence relation and we denote by [A, B] the set of homotopy classes of asymptotic morphisms from A to B. The same definition may be made in the graded and equivariant contexts.

Let us denote an asymptotic morphism from A to B by a broken arrow:

\xymatrix{ϕ: A \ar@{-->}[r] & B.}

It is clearly possible to compose an asymptotic morphism with a ∗-homomorphism, on either side, so as to obtain asymptotic morphisms

\xymatrix{A \ar[r] & B \ar@{-->}[r] & C}

and

\xymatrix{A \ar@{-->}[r] & B \ar[r] & C.}

Now every ∗-homomorphism from A to B determines a constant asymptotic morphism from A to B, and the starting point for the theory of asymptotic morphisms is the following result [6]:

2.1. Theorem. On the class of separable, ℤ/2-graded G-C^{∗}-algebras there is an associative composition law

[A, B] × [B, C] → [A, C]

which is compatible with the above compositions of asymptotic morphisms and ∗- homomorphisms.

(The proof in [6] is written for C^{∗}-algebras with no grading or G-action, but the argument extends to the present context with no essential change.) We shall use the following features of the homotopy category of asymptotic morphisms. Denote by A_{1} ˆ⊗A_{2} the maximal graded tensor product of A_{1} and A_{2} [4, Section 14]. There is then a tensor product functor on the homotopy category of asymptotic morphisms which associates to the asymptotic morphisms \xymatrix{ϕ_{1} : A_{1} \ar@{-->}[r] & B_{1}} and

\xymatrix{ϕ_{2} : A_{2} \ar@{-->}[r] & B_{2}} a tensor product

\xymatrix{ϕ_{1} ˆ⊗ϕ_{2} : A_{1} ˆ⊗A_{2} \ar@{-->}[r] & B_{1} ˆ⊗B_{2},}

where ϕ_{1} ˆ⊗ϕ_{2}(a_{1} ˆ⊗a_{2}) ∼ ϕ_{1}(a_{1}) ˆ⊗ϕ_{2}(a_{2}). If C_{max}^{∗}(G, B) denotes the maximal, or full crossed product C^{∗}-algebra [19, Section 7.6] then there is a descent functor on the homotopy category of equivariant asymptotic morphisms which associates to an equivariant asymptotic morphism \xymatrix{ϕ: A \ar@{-->}[r] & B} an asymptotic morphism

\xymatrix{ϕ: C_{max}^{∗}(G, A) \ar@{-->}[r] & C_{max}^{∗}(G, B),}

where ϕ_{t}(∑ a_{g}[g]) ∼ ∑ ϕ_{t}(a_{g})[g].

Let 𝒮 = C_{0}(ℝ), graded according to even and odd functions and equipped with the trivial action of G. There are ∗-homomorphisms

\begin{matrix}Δ: 𝒮 → 𝒮 ˆ⊗𝒮, \\ Δ: f(X) ↦ f(X ˆ⊗1 + 1 ˆ⊗X), \end{matrix}  \begin{matrix}ϵ: 𝒮 → ℂ, \\ ϵ: f(X) ↦ f(0). \end{matrix}

Using them, we construct the amplified category of ℤ/2-graded C^{∗}-algebras, in which the morphisms from A to B are graded ∗-homomorphisms (equivariant, in the presence of a group G) from 𝒮A = 𝒮 ˆ⊗A to B. Similarly we can form the amplification of the homotopy category of asymptotic morphisms.

2.2. Definition. Let 𝒦(ℋ_{G}) be the C^{∗}-algebra of compact operators on the ℤ/2graded G-Hilbert space

ℋ_{G} = ℓ^{2}(G) ⊕ ℓ^{2}(G) ⊕ ⋯

(the summands are graded alternately even and odd). If A and B are separable, ℤ/2-graded G-C^{∗}-algebras then denote by E_{G}(A, B) the homotopy classes of equivariant asymptotic morphisms from 𝒮A ˆ⊗𝒦(ℋ_{G}) into B ˆ⊗𝒦(ℋ_{G}):

E_{G}(A, B) = [𝒮A ˆ⊗𝒦(ℋ_{G}), B ˆ⊗𝒦(ℋ_{G})].

When G is trivial we drop the subscript and write E(A, B).

The sets E_{G}(A, B) are in fact abelian groups. There is an associative law of composition

E_{G}(A, B) ⊗ E_{G}(B, C) → E_{G}(A, C),

derived from the composition law in the amplified homotopy category of asymptotic morphisms. There are tensor product and descent functors on the E-theory category,

E_{G}(A_{1}, B_{1}) ⊗ E_{G}(A_{2}, B_{2}) → E_{G}(A_{1} ˆ⊗A_{2}, B_{1} ˆ⊗B_{2})

and

E_{G}(A, B) → E(C_{max}^{∗}(G, A), C_{max}^{∗}(G, B)).

If G is trivial then the E-theory groups specialize in one variable to K-theory of graded C^{∗}-algebras:

E(ℂ, B) ≅ K_{0}(B).

For ungraded C^{∗}-algebras these facts are proved in [11]. The graded case will be considered elsewhere.

If ℋ is a separable, ℤ/2-graded G-Hilbert space then any equivariant asymptotic morphism from A to B ˆ⊗𝒦(ℋ), or from 𝒮A to B ˆ⊗𝒦(ℋ), determines—after tensoring with 𝒦(ℋ_{G})—an element of E_{G}(A, B). This is because ℋ ˆ⊗ℋ_{G} ≅ ℋ_{G} (cf. [18]).

Suppose that there is a continuous, one-parameter family of actions of G on a Hilbert space ℋ, and suppose that an asymptotic morphism

\xymatrix{ϕ: A \ar@{-->}[r] & B ˆ⊗𝒦(ℋ)}

is equivariant with respect to this family, in the sense that ϕ_{t}(g(a)) ∼ g_{t}(ϕ_{t}(a)). Then {ϕ_{t}}_{t∈[1,∞)} determines a class in E_{G}(A, B). The reason is that after tensoring with ℋ_{G} the one parameter family of actions on ℋ is conjugate to a constant action (cf. [18] again). We shall use this observation, and a small extension of it, in Section 4.

3. The assembly map

Let Y be a proper G-space and let B be a G-C^{∗}-algebra. We define

E_{G}(Y, B) = lim_{⟶}_{X⊂Y} E_{G}(C_{0}(X), B),

where the direct limit is over all G-compact, locally compact and second countable subsets X ⊂ Y. For each such X there is a class ϕ_{X} ∈ E(ℂ, C_{max}^{∗}(G, C_{0}(X))), determined by a basic projection in the crossed product algebra, and we define the maximal Baum-Connes assembly map to be the composition

E_{G}(C_{0}(X), B) ⟶^{descent} E_{G}(C_{max}^{∗}(G, C_{0}(X)), C_{max}^{∗}(G, B))

⟶_{ϕ_{X}∈E_{G}(ℂ,C_{max}^{∗}(G,C_{0}(X)))}^{composition with} E_{G}(ℂ, C_{max}^{∗}(G, B)).

See [11, Chapter 10]. If Y is a general proper G-space, not necessarily G-compact, then the maximal Baum-Connes assembly map for Y is the homomorphism

μ_{max} : E_{G}(Y, B) → E(ℂ, C_{max}^{∗}(G, B))

obtained as the direct limit of the assembly maps for the G-compact subsets of Y. The reduced Baum-Connes assembly map

μ_{red} : E_{G}(Y, B) → E(ℂ, C_{red}^{∗}(G, B))

is defined by composing μ_{max} with the E-theory map induced from the regular representation ρ: C_{max}^{∗}(G, B) → C_{red}^{∗}(G, B). The assembly maps are usually defined using KK-theory, as in [2], but our definitions are equivalent to these [13].

Let us say that a G-C^{∗}-algebra D is proper if there exists a second countable, locally compact, proper G-space Z, and an equivariant ∗-homomorphism from C_{0}(Z) into the center of the multiplier algebra of D, such that C_{0}(Z)D is dense in D (cf. [16, Definition 1.5]). The main theorem in [11], and a key tool for us here, is the following result:

3.1. Theorem ([11, Theorem 14.1]). Let G be a countable discrete group and let D be a proper G-C^{∗}-algebra. The assembly map

μ_{max} : E_{G}(EG, D) → E(ℂ, C^{∗}(G, D))

is an isomorphism.

We will deduce our main result on the Baum-Connes conjecture, Theorem 1.2, from Theorem 3.1 (in our first proof of Theorem 1.2 we used instead of Theorem 3.1 a longer argument based on Poincaré duality for manifolds modelled on Hilbert space). As explained in [11], Theorem 3.1 may be viewed as a generalization of a theorem of Green [9] and Julg [15] which identifies equivariant K-theory (in the sense of Atiyah and Segal [20]) with the C^{∗}-algebra K-theory of crossed product algebras. An immediate consequence is the following result:

3.2. Theorem (see [11, Theorem 15.1] and also [21, Theorem 2.1]). Let G be a discrete group and suppose that there is a proper G-C^{∗}-algebra D, and elements β ∈ E_{G}(ℂ, D) and α ∈ E_{G}(D, ℂ), whose composition is α ∘ β = 1 ∈ E_{G}(ℂ, ℂ). Then for any G-C^{∗}-algebra B the assembly map

μ_{max} : E_{G}(EG, B) → E(ℂ, C^{∗}(G, B))

is an isomorphism.

Proof. The hypotheses imply that the assembly map for B = B ⊗ ℂ may be viewed as a direct summand of the assembly map for the proper C^{∗}-algebra B ⊗ D, and by Theorem 3.1 the latter is an isomorphism.

4. The C^{∗}-algebra of a Euclidean space

Let G be a discrete group which admits an affine, isometric and metrically proper action on a Euclidean space. Then G is countable and it admits such an action on a countably infinite-dimensional Euclidean space. With this in mind, let us fix such a Euclidean space V.

The following definitions are adapted from [14, Section 3]. Denote by V_{a}, V_{b}, and so on, the finite-dimensional, affine subspaces of V. In addition, denote by V_{a}^{0} the finite-dimensional linear subspace of V comprised of differences of elements in V_{a}. Let 𝒞(V_{a}) be the ℤ/2-graded C^{∗}-algebra of continuous functions from V into the complexified Clifford algebra of V_{a}^{0} which vanish at infinity (the grading on 𝒞(V_{a}) is inherited from the grading on the Clifford algebra). Let 𝒜(V_{a}) be the graded tensor product of 𝒮 = C_{0}(ℝ) and 𝒞(V_{a}).

If V_{a} ⊂ V_{b} then there is a canonical decomposition V_{b} = V_{ba}^{0}+V_{a}, where V_{ba}^{0} is the orthogonal complement of V_{a}^{0} in V_{b}^{0}. We shall write elements of V_{b} as v_{b} = v_{ba} + v_{a} in accordance with this decomposition. Every function h on V_{a} may be extended to a function h̃ on V_{b} by the formula h̃(v_{b}) = h(v_{a}).

4.1. Definition. If V_{a} ⊂ V_{b} then denote by C_{ba} the Clifford algebra-valued function on V_{b} which maps v_{b} to v_{ba} ∈ V_{ba}^{0} ⊂ Cliff(V_{b}^{0}). Define a ∗-homomorphism

β_{ba}: 𝒜(V_{a}) → 𝒜(V_{b})

by the formula

β_{ba}(f ˆ⊗h) = f(X ˆ⊗1 + 1 ˆ⊗C_{ba})(1 ˆ⊗h̃).

Remark. The element X ˆ⊗1 + 1 ˆ⊗C_{ba} is regarded as an unbounded multiplier of 𝒜(V_{b}); see for example [5, Section 6.5]. The elements f(X ˆ⊗1 + 1 ˆ⊗C_{ba}) and 1 ˆ⊗h̃ are then bounded multipliers of 𝒜(V_{b}), and their product lies in 𝒜(V_{b}).

If V_{a} ⊂ V_{b} ⊂ V_{c} then β_{cb} ∘ β_{ba} = β_{ca}. In view of this we can construct the C^{∗}-algebra

𝒜(V) = lim_{⟶} 𝒜(V_{a}),

where the limit is over the directed set of all V_{a} ⊂ V. If G acts on V by affine isometries then there is an induced action of G on 𝒜(V) by ∗-automorphisms.

4.2. Proposition. If G acts metrically properly on V then 𝒜(V) is a proper G-C^{∗}-algebra.

Proof. The following elegant argument, which much improves our original proof, is due to G. Skandalis. The center 𝒵(V_{a}) of the C^{∗}-algebra 𝒜(V_{a}) contains the algebra of continuous functions, vanishing at infinity, on the locally compact space [0, ∞)×V_{a} (if V_{a} is even-dimensional and nonzero this is the full center). The linking map β_{ba} takes 𝒵(V_{a}) into 𝒵(V_{b}), and so we can form the direct limit 𝒵(V). It has the property that 𝒵(V)·𝒜(V) is dense in 𝒜(V). Its Gelfand spectrum is the locally compact space Z = [0, ∞) × \overline{V}, where \overline{V} is the Hilbert space completion of V and Z is given the weakest topology for which the projection to \overline{V} is weakly continuous and the function t^{2} + ∥v∥^{2} is continuous, and if G acts metrically properly on V then the induced action on the locally compact space Z is proper, in the ordinary sense of the term.

In view of Proposition 4.2 and Theorem 3.2, to prove Theorem 1.2 it suffices to exhibit classes α ∈ E_{G}(𝒜(V), ℂ) and β ∈ E_{G}(ℂ, 𝒜(V)) such that α ∘ β = 1 ∈ E_{G}(ℂ, ℂ). This is what we shall now do. We shall follow the article [14], which is in turn an adaptation of Atiyah’s elliptic operator proof [1] of the Bott periodicity theorem. The construction of β is quite simple:

4.3. Definition. Denote by β: 𝒮 = 𝒜(0) → 𝒜(V) the ∗-homomorphism associated to the inclusion of the zero-dimensional linear space 0 into V. Denote by β_{t}: 𝒮 → 𝒜(V) the ∗-homomorphism β_{t}(f) = β(f_{t}), where f_{t}(x) = f(t^{−1}x). The family of ∗- homomorphisms {β_{t}}_{t∈[1,∞)} is asymptotically equivariant and so is an equivariant asymptotic morphism from 𝒮 to 𝒜(V). Denote by β ∈ E_{G}(ℂ, 𝒜(V)) its E-theory class.

As in Atiyah’s paper [1], to construct α ∈ E_{G}(𝒜(V), ℂ) we shall need some elliptic operator theory. In the present context the operators must be defined on the infinite-dimensional space V. Denote by ℋ_{a} the Hilbert space of square integrable functions from V_{a} into Cliff(V_{a}^{0}). If V_{a} ⊂ V_{b} then there is a canonical isomorphism

ℋ_{b} ≅ ℋ_{ba} ˆ⊗ℋ_{a},

where ℋ_{ba} denotes the Hilbert space associated to V_{ba}^{0}. We define a unit vector ξ_{0} ∈ ℋ_{ba} by

ξ_{0}(v_{ba}) = π^{−n_{ba}/4} exp(−\frac{1}{2}∥v_{ba}∥^{2}),

where n_{ba} = dim(V_{ba}^{0}), and we regard ℋ_{a} as included in ℋ_{b} via the isometry ξ ↦ ξ_{0} ˆ⊗ξ. We define

ℋ = lim_{⟶} ℋ_{a}

and denote by 𝔰 = lim_{⟶} 𝔰_{a} the direct limit of the Schwartz subspaces 𝔰_{a} ⊂ ℋ_{a}. If V_{a} ⊂ V is a finite-dimensional affine subspace then the Dirac operator D_{a}, an unbounded operator on ℋ with domain 𝔰, is defined by

D_{a}ξ = ∑_{i=1}^{n}(−1)^{deg(ξ)} \frac{∂ξ}{∂x_{i}} v_{i},

where {v_{1}, …, v_{n}} is an orthonormal basis for V_{a}^{0}, and {x_{1}, …, x_{n}} are the dual coordinates to {v_{1}, …, v_{n}}. If V_{a} is a linear subspace then we also define the Clifford operator by

C_{a}ξ = ∑_{i=1}^{n} x_{i}v_{i}ξ

(cf. Definition 4.1).

4.4. Definition. Fix an algebraic, direct sum decomposition

V = V_{0} ⊕ V_{1} ⊕ V_{2} ⊕ ⋯,

where each V_{j} is a finite-dimensional linear subspace of V. For each n, define an unbounded operator B_{n,t} on ℋ, by the formula

B_{n,t} = t_{0}D_{0} + t_{1}D_{1} + ⋯ + t_{n−1}D_{n−1}

+t_{n}(D_{n} + C_{n}) + t_{n+1}(D_{n+1} + C_{n+1}) + ⋯,

where t_{j} = 1 + t^{−1}j.

The infinite sum is well defined because when B_{n,t} is applied to any vector in the Schwartz space 𝔰 the resulting infinite series has only finitely many nonzero terms. This is because the function exp(−\frac{1}{2}∥v_{j}∥^{2}) ∈ s_{j}, used in the definition of the direct limit 𝔰, lies in the kernel of D_{j} + C_{j}.

If h ∈ 𝒞(V_{0} ⊕ ⋯ ⊕ V_{n}) then the function h_{t}(v) = h(t^{−1}v) acts as a bounded operator π(h_{t}) on ℋ by pointwise multiplication. With this notation in hand, the main technical results concerning the operators B_{n,t} are as follows.

4.5. Proposition (cf. [14, Lemma 2.9 and Proposition 4.2]). The operators B_{n,t} are essentially selfadjoint. The formula

α_{t}^{n}: f ˆ⊗h ↦ f_{t}(B_{t})π(h_{t}) (f ∈ 𝒮, h ∈ 𝒞(V_{0} ⊕ ⋯ ⊕ V_{n}))

defines an asymptotic morphism from 𝒜(V_{0} ⊕⋯⊕V_{n}) into 𝒦(ℋ), and the diagram

[figure: 𝒜(V_{0} ⊕ ⋯ ⊕ V_{n}); α^{n}; β_{n+1}; 𝒦(ℋ); =; 𝒜(V_{0} ⊕ ⋯ ⊕ V_{n+1}); α^{n+1}; 𝒦(ℋ)]

is asymptotically commutative.

It follows from the second part of the proposition that the asymptotic morphisms α^{n} combine to form a single asymptotic morphism

\xymatrix{α: 𝒜(V) \ar@{-->}[r] & 𝒦(ℋ).}

Now let G act on the Euclidean space V according to the affine, isometric action

g · v = π(g)v + κ(g),

where π is a linear, isometric representation of G on V. For s ≥ 0 form the scaled G-action

g_{s} · v = π(g)v + sκ(g),

and denote the induced actions on 𝒜(V) and 𝒦(ℋ) in the same way.

4.6. Proposition (cf. [14, Lemma 5.15]). Suppose that the direct sum decompostion V = ⨁_{0}^{∞} V_{j} is chosen in such a way that for every g ∈ G there is an N ∈ ℕ such that if n > N then g[⨁_{0}^{n} V_{j}] ⊂ ⨁_{0}^{n+1} V_{j}. Then for every a ∈ 𝒜(V) and every g ∈ G, α_{t}(g(a)) ∼ g_{t}(α_{t}(a)). Consequently the asymptotic morphism {α_{t}}_{t∈[1,∞)} determines a class α ∈ E_{G}(𝒜(V), ℂ).

We are ready to prove the following result:

4.7. Theorem. The composition α ∘ β ∈ E_{G}(ℂ, ℂ) is the identity.

Proof. Let s ∈ [0, 1] and denote by 𝒜_{s} the C^{∗}-algebra 𝒜 = 𝒜(V), but with the scaled G-action (g, a) ↦ g_{s}(a). The algebras 𝒜_{s} form a continuous field of G-C^{∗}- algebras over the unit interval, and we shall denote by 𝒜[0, 1] the G-C^{∗}-algebra of continuous sections. In a similar way, form a continuous field of G-C^{∗}-algebras 𝒦_{s} = 𝒦(ℋ) and denote by 𝒦[0, 1] the G-C^{∗}-algebra of continuous sections.

The asymptotic morphism \xymatrix{α: 𝒜 \ar@{-->}[r] & 𝒦} induces an asymptotic morphism

\xymatrix{ᾱ: 𝒜[0, 1] \ar@{-->}[r] & 𝒦[0, 1]} ,

and similarly the asymptotic morphism \xymatrix{β: 𝒮 \ar@{-->}[r] & 𝒜} determines an asymptotic morphism

\xymatrix{β̄: 𝒮 \ar@{-->}[r] & 𝒜[0, 1]}

by forming the tensor product \xymatrix{𝒮[0, 1] \ar@{-->}[r] & 𝒜[0, 1]} and composing with the inclusion 𝒮 ⊂ 𝒮[0, 1] as constant functions. Consider now the diagram

𝒮[figure]_{=}  [figure]^{β̄} [figure] 𝒜[0, 1] [figure]_{ϵ_{s}}  [figure] ᾱ [figure] 𝒦[0, 1] [figure]_{ϵ_{s}} 𝒮  [figure]_{β} [figure] 𝒜_{s}  [figure]_{α} [figure] 𝒦_{s} ,

where ϵ_{s} denotes evaluation at s ∈ [0, 1]. All the asymptotic morphisms determine classes in E_{G}-theory, and the ∗-homomorphism ϵ_{s}: 𝒦[0, 1] → 𝒦_{t} defines an isomorphism, so to prove the theorem it suffices to show that when s = 0 the bottom composition

[figure: 𝒮; β; 𝒜_{0}; α; 𝒦_{0}]

determines the identity element of E_{G}(ℂ, ℂ). When s = 0 the action of G on V is linear and the asymptotic morphism \xymatrix{β: 𝒮 \ar@{-->}[r] & 𝒜_{0}} is homotopic to the equivariant ∗-homomorphism β: 𝒮 → 𝒜_{0} of Definition 4.3. The composition of the ∗-homomorphism β with the asymptotic morphism α is the asymptotic morphism

\begin{matrix}γ: 𝒮 & [figure] & [figure] & 𝒦, \\ γ_{t}: f ↦ f_{t}(B_{0,t}) & . \end{matrix}

Replacing B_{0,t} with s^{−1}B_{0,t}, for 0 < s ≤ 1 we obtain a homotopy to the ∗-homomor-phism γ_{0} : f ↦ f(0)P, where P is the projection onto the kernel of B_{0,t}. Since the kernel is one-dimensional and G-invariant, γ_{0} determines the identity in E_{G}(ℂ, ℂ).

5. Reduced group C^{∗}-algebras

If C_{red}^{∗}(G) is an exact C^{∗}-algebra [22] then there is a reduced descent functor on the homotopy category of asymptotic morphisms which associates to an equivariant asymptotic morphism \xymatrix{ϕ: A \ar@{-->}[r] & B} an asymptotic morphism

\xymatrix{ϕ: C_{red}^{∗}(G, A) \ar@{-->}[r] & C_{red}^{∗}(G, B).}

We can therefore define a descent functor in E-theory and the arguments of the previous sections then apply equally well to the reduced Baum-Connes assembly map. In the absence of exactness the following ad hoc argument proves that μ_{red} is an isomorphism for the groups we are considering.

5.1. Theorem. If G is a discrete group which admits an affine, isometric and metrically proper action on a Euclidean space then for every G-C^{∗}-algebra B the regular representation

ρ: C_{max}^{∗}(G, B) → C_{red}^{∗}(G, B)

determines an invertible morphism ρ ∈ E(C_{max}^{∗}(G, B), C_{red}^{∗}(G, B)).

Proof. Let us consider the case where B = ℂ. We shall define a class σ ∈ E(C_{red}^{∗}(G), C_{max}^{∗}(G)) which is inverse to ρ ∈ E(C_{max}^{∗}(G), C_{red}^{∗}(G)). Suppose given an equivariant asymptotic morphism \xymatrix{ϕ: 𝒮 \ar@{-->}[r] & A} , where A is any G-C^{∗}-algebra. Both A and C_{red}^{∗}(G) embed in the multiplier algebra of C_{red}^{∗}(G, A), and C_{red}^{∗}(G) asymptotically commutes with the image of the asymptotic morphism ϕ (this is because ϕ is asymptotically equivariant). It follows that ϕ induces an asymptotic morphism

\xymatrix{ϕ_{red} : 𝒮 ˆ⊗C_{red}^{∗}(G) \ar@{-->}[r] & C_{red}^{∗}(G, A).}

In particular, β induces an asymptotic morphism

\xymatrix{β_{red} : 𝒮 ˆ⊗C_{red}^{∗}(G) \ar@{-->}[r] & C_{red}^{∗}(G, 𝒜).}

Now since 𝒜 is a proper C^{∗}-algebra the regular representation C_{max}^{∗}(G, 𝒜) → C_{red}^{∗}(G, 𝒜) is an isomorphism. We define σ ∈ E(C_{red}^{∗}(G), C_{max}^{∗}(G)) to be the E-theory class of the composition

[figure: 𝒮 ˆ⊗C_{red}^{∗}(G); β_{red}; C_{red}^{∗}(G, 𝒜); C_{max}^{∗}(G, 𝒜); ≅; ρ; α; C_{max}^{∗}(G, 𝒦); ≅; C_{max}^{∗}(G) ˆ⊗𝒦.]

It follows from Theorem 4.7 that σ ∘ ρ = 1 ∈ E_{G}(C_{max}^{∗}(G), C_{max}^{∗}(G)). To calculate the reverse composition, we note that since C_{max}^{∗}(G, 𝒜) ≅ C_{red}^{∗}(G, 𝒜) the asymptotic morphism \xymatrix{α: 𝒜 \ar@{-->}[r] & 𝒦} induces an asymptotic morphism

\xymatrix{α_{red} : C_{red}^{∗}(G, 𝒜) \ar@{-->}[r] & C_{red}^{∗}(G, 𝒦).}

The composition ρ ∘ σ ∈ E(C_{red}^{∗}(G), C_{red}^{∗}(G)) is given by the composition

[figure: 𝒮 ˆ⊗C_{red}^{∗}(G); β_{red}; C_{red}^{∗}(G, 𝒜); α_{red}; C_{red}^{∗}(G, 𝒦).]

The composition α_{red} ∘ β_{red} is the asymptotic morphism

\xymatrix{γ_{red} : 𝒮 ˆ⊗C_{red}^{∗}(G) \ar@{-->}[r] & C_{red}^{∗}(G, 𝒦)}

induced, according to the remark made above, by the equivariant asymptotic morphism γ: 𝒮 → 𝒦 which is the composition of β and α and is defined by the formula: γ_{t}(f) = f_{t}(B_{0,t}). (This follows from a similar fact for full crossed products.) It is now possible to deform the action of G on V to a linear action, as in the proof of Theorem 4.7, and verify that γ_{red} = 1 ∈ E(C_{red}^{∗}(G), C_{red}^{∗}(G)).

References

1. M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford 19 (1968), 113–140. MR 37:3584

2. P. Baum, A. Connes, and N. Higson, Classifying space for proper G-actions and K-theory of group C^{∗}-algebras, Contemp. Math. 167 (1994), 241–291. MR 96c:46070

3. M. E. B. Bekka, P.-A. Cherix, and A. Valette, Proper affine isometric actions of amenable groups, Novikov conjectures, Index theorems and rigidity, vol. 2, S. Ferry, A. Ranicki and J. Rosenberg, editors, Cambridge University Press, Cambridge, 1995, pp. 1–4. MR 97e:43001

4. B. Blackadar, K-theory for operator algebras, MSRI Publication Series 5, Springer-Verlag, New York-Heidelberg-Berlin-Tokyo, 1986. MR 88g:46082

5. A. Connes, An analogue of the Thom isomorphism for crossed products, Advances in Math. 39 (1981), 31–55. MR 82j:46084

6. A. Connes and N. Higson, Déformations, morphismes asymptotiques et K-théorie bivariante, C. R. Acad. Sci. Paris 311 Série 1 (1990), 101–106. MR 91m:46114

7. P. Delorme, 1-cohomologie des représentations unitaries des groupes de Lie semi-simples et résolubles. Produits tensoriels continus et représentations, Bull. Soc. Math. France 105 (1977), 281–336. MR 58:28272

8. S. Ferry, A. Ranicki and J. Rosenberg, A history and survey of the Novikov conjecture, Novikov conjectures, Index theorems and rigidity, vol. 1, S. Ferry, A. Ranicki and J. Rosenberg, editors, Cambridge University Press, Cambridge, 1995, pp. 7–66. MR 97f:57036

9. P. Green, Equivariant K-theory and crossed product C^{∗}-algebras, Proceedings of Symposia in Pure Mathematics 38 Part 1 (1982), 337–338. MR 83j:46004a

10. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, G. A. Niblo and M. A. Roller, editors, Cambridge Univeristy Press, Cambridge, 1993, pp. 1–295. MR 95m:20041

11. E. Guentner, N. Higson, and J. Trout, Equivariant E-theory, Preprint, 1997.

12. P. de la Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. de France, 1989. MR 90m:22001

13. N. Higson and G. Kasparov, A note on the Baum-Connes conjecture in KK-theory and E-theory, In preparation.

14. N. Higson, G. Kasparov, and J. Trout, A Bott periodicity theorem for infinite-dimensional Euclidean space, Advances in Math. (to appear).

15. P. Julg, K-théorie equivariante et produits croisés, C. R. Acad. Sci. Paris 292 Série 1 (1981), 629–632. MR 83b:46090

16. G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Math. 91 (1988), 147–201. MR 88j:58123

17. E. Kirchberg and S. Wassermann, In preparation.

18. J. Mingo and W. Phillips, Equivariant triviality theorems for Hilbert modules, Proc. Amer. Math. Soc. 91 (1984), 225–230. MR 85f:46111

19. G. K. Pedersen, C^{∗}-algebras and their automorphism groups, Academic Press, London–New York–San Francisco, 1979. MR 81e:46037

20. G. Segal, Equivariant K-theory, Publ. Math. IHES 34 (1968), 129–151. MR 38:2769

21. J.-L. Tu, The Baum-Connes conjecture and discrete group actions on trees, Preprint.

22. S. Wassermann, Exact C^{∗}-algebras and related topics, Res. Inst. of Math. Lecture Note Series 19, Seoul National University, Seoul, South Korea, 1994. MR 95b:46081

Department of Mathematics, Pennsylvania State University, University Park, PA 16802

E-mail address: higson@math.psu.edu

Institut de Mathématiques de Luminy, CNRS-Luminy-Case 930, 163 Avenue de Luminy 13288, Marseille Cedex 9, France

E-mail address: kasparov@iml.univ-mrs.fr