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\controldates{17-DEC-1997,17-DEC-1997,17-DEC-1997,17-DEC-1997}
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\newtheorem*{theorem1}{1.2. Theorem}
\newtheorem*{theorem2}{1.3. Corollary}
\newtheorem*{theorem3}{2.1. Theorem}
\newtheorem*{theorem4}{3.1. Theorem}
\newtheorem*{theorem5}{3.2. Theorem}
\newtheorem*{theorem6}{4.2. Proposition}
\newtheorem*{theorem7}{4.5. Proposition}
\newtheorem*{theorem8}{4.6. Proposition}
\newtheorem*{theorem9}{4.7. Theorem}
\newtheorem*{theorem10}{5.1. Theorem}
\theoremstyle{remark}
\newtheorem*{remark1}{Remark}
\theoremstyle{definition}
\newtheorem*{definition1}{1.1. Definition}
\newtheorem*{definition2}{2.2. Definition}
\newtheorem*{definition3}{4.1. Definition}
\newtheorem*{definition4}{4.3. Definition}
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\begin{document}
\title[Operator $K$-Theory for groups which act on Hilbert space]{Operator
$K$-theory for groups which act properly and
isometrically on Hilbert space }
\author{Nigel Higson}
\address{Department of Mathematics, Pennsylvania
State University, University Park, PA 16802
}
\email{higson@math.psu.edu}
\author{Gennadi Kasparov}
\address{Institut de Math\'{e}matiques de Luminy, CNRS-Luminy-Case 930,
163 Avenue
de Luminy 13288, Marseille Cedex 9, France}
\email{kasparov@iml.univ-mrs.fr}
\subjclass{Primary 46L20}
\begin{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 $C^{*}_{
\smash{\text{red}}}(G)$. Our
result is in accordance with the Baum-Connes conjecture.\end{abstract}
\thanks{The first author was partially
supported by an NSF grant.}
\date{October 25, 1997}
\keywords{Baum-Connes conjecture, $C^{*}$-algebras, $K$-theory}
\issueinfo{3}{22}{}{1997}
\dateposted{December 19, 1997}
\pagespan{131}{142}
\PII{S 1079-6762(97)00038-3}
\def\copyrightyear{1997}
\copyrightinfo{1997}{American Mathematical Society}
\commby{Masamichi Takesaki}
\maketitle
\section*{1. Introduction }Let $G$ be a countable discrete group and denote
by $C^{*}_{\max }(G)$ and
$C^{*} _{\smash{\text{red}}}(G)$ 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:
\begin{definition1} Let $V$ be a real inner product vector
space which is possibly infinite-dimensional (hereafter we shall call
$V$ a
{\em Euclidean space\/}). An affine, isometric action of a discrete
group $G$ on $V$ is
{\em metrically proper\/} if
$\lim _{g\to \infty }\| g\cdot v\| =\infty $,
for every $v\in V$.
\end{definition1}
Gromov~\cite{10} calls groups which admit such an action {\em
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~\cite{7}, and by the recent observation~\cite{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~\cite{3} and the monograph~\cite{12} for
further
information.
If $G$ is any discrete group then denote by $\mathcal{E} G$ the universal
proper
$G$-space, as described in~\cite{2}. It is a paracompact and
Hausdorff $G$-space which has a paracompact and Hausdorff quotient by
$G$,
along with the following additional properties:
\begin{itemize}
\item[(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
\item[(ii)] if $X$ is any other $G$-space satisfying (i) then there is a
unique-up-to-$G$-homotopy $G$-map from $X$ into $\mathcal{E} G$.
\end{itemize}Clearly $\mathcal{E} G$ is unique up to $G$-homotopy. If $G$ is
torsion-free then
$\mathcal{E} G$ is the universal principal space $EG$, whose quotient by $G$
is the
classifying space $BG$.
The {\em Baum-Connes assembly maps\/} for the
$C^{*}$-algebras
$C^{*}_{\max }(G)$ and $C^{*}_{\smash{\text{red}}}(G)$ are homomorphisms of
abelian
groups
\begin{equation*}\begin{split}
\mu _{\max }\colon K_{*}^{G}(\mathcal{E} G) &\rightarrow K_{*}(C^{*}_{\max
}(G)),\\
\mu _{\smash{\text{red}}}\colon K_{*}^{G}(\mathcal{E} G) &\rightarrow
K_{*}(C^{*}_{\smash{\text{red}}}(G)).\end{split}\end{equation*}
On the left hand side is the equivariant $K$-homology of $\mathcal{E}G$,
defined using
$KK$-theory~\cite{2}, \cite{16}. If $G$ is torsion-free then $
K_{*}^{G}(\mathcal{E}
G)$ is
isomorphic to
$K_{*}(BG)$, the $K$-homology (with compact supports) of the
classifying space for
$G$. See~\cite{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:
\begin{theorem1} If $G$ is a discrete group
which admits an affine, isometric and metrically proper action
on a Euclidean space {\rm (}possibly infinite-dimensional{\rm )}
then the Baum-Connes assembly maps
$\mu _{\max }$ and $\mu _{\smash{\textup{red}}}$ are isomorphisms.\end{theorem1}
The assertion that for {\em any\/} $G$ the assembly map
$\mu _{\smash{\text{red}}}$ is an
isomorphism is known as the
{\em Baum-Connes conjecture\/}~\cite{2}. By an index theory
argument, the injectivity of either $\mu _{\max }$ or
$\mu _{\smash{\text{red}}}$, for a given $G$,
implies the Novikov higher signature conjecture~\cite{8} for
manifolds with fundamental group
$G$. Thus our theorem has the following consequence:
\begin{theorem2} If $G$ is a discrete group
which admits an affine, isometric and metrically proper action on a
Euclidean
space {\rm (}possibly infinite-dimensional{\rm )} then the
Baum-Connes and
Novikov conjectures hold for $G$. In particular, these conjectures
hold for any
countable amenable group.
\end{theorem2}
To prove the theorem we shall use a variant of $KK$-theory, the
$E$-theory of
Connes and Higson~\cite{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~\cite{14}. Given periodicity, the proof that
$\mu _{\max }$ 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
$\mu _{\smash{\text{red}}}$ is rather more complicated, unless $G$ has
the additional
functional-analytic property that $C^{*}_{\smash{\text{red}}}(G)$ is
exact~\cite{22}. Conjecturally
all discrete groups have this property, but in any case we shall end
this
note with an {\em ad hoc\/} proof that $\mu _{\smash{\text{red}}}$ is an
isomorphism for the
groups under consideration here.
\section*{2. $E$-theory }
We begin by reviewing some of the foundational
results in $E$-theory proved in~\cite{6} and~\cite{11}. Let
$A$ and $B$ be $C^{*}$-algebras. An {\em asymptotic morphism\/} from $A$
to $B$ is a
family of functions
\begin{equation*}\{\phi _{t}\}_{t\in [1,\infty )}\colon A\to B\end{equation*}
such that $ t\mapsto \phi _{t}(a )$ is bounded and
norm-continuous, for every $a\in A$, and
\begin{equation*}\lim _{t\to \infty } \left \{ \begin{gathered}
\phi _{t}(a_{1}a_{2}) -\phi _{t}(a_{1})\phi _{t}(a_{2}) \\
\phi _{t}(a_{1}+a_{2}) -\phi _{t}(a_{1})-\phi _{t}(a_{2}) \\
\phi _{t}(\alpha a_{1}) -\alpha \phi _{t}(a_{1}) \\
\phi _{t}(a_{1}^{*}) -\phi _{t}(a_{1})^{*} \\
\end{gathered}\right \} = 0,\end{equation*}
for every $a_{1},a_{2}\in A$ and $\alpha \in \mathbb{C}$.
This is the definition used in~\cite{6}; it agrees with the definition
in~\cite{11} if we
identify those asymptotic morphisms $\phi '$, $\phi ''$ which are {\em
asymptotically
equivalent,\/} in the sense that $\phi '_{t}(a)-\phi ''_{t}(a)\to 0$ as
$t\to \infty $ (we
shall write $\phi '_{t}(a)\sim \phi _{t}''(a)$). If $A$ and
$B$ are
$\mathbb{Z}/2$-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 {\em equivariant\/}
asymptotic morphism from $A$ to $B$ we shall mean an
asymptotic morphism $\{\phi _{t}\}_{t\in [1,\infty )}$ such that
$\phi _{t}(g(a))\sim g(\phi _{t}(a))$, for every $a\in A$ and every $g\in G$.
Two asymptotic morphisms from $A$ to
$B$ are {\em 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:
\begin{equation*}\xy \xymatrix { \phi \colon A\ar @{-->}[r] &
B.}
\endxy \end{equation*} It is clearly possible to compose an asymptotic
morphism with a
$*$-homomorphism, on either side, so as to obtain asymptotic morphisms
\begin{equation*}\xy \xymatrix { A\ar [r] &
B\ar @{-->}[r]& C}\endxy \end{equation*}
and
\begin{equation*}\xy \xymatrix { A\ar @{-->}[r] &
B\ar [r]& C.}\endxy \end{equation*}
Now every $*$-homomorphism from $A$ to $B$ determines a {\em constant\/}
asymptotic morphism from $A$ to $B$, and the starting point for the
theory of
asymptotic morphisms is the following result~\cite{6}:
\begin{theorem3} On the class of separable, $\mathbb{Z}/2$-graded
$G$-$C^{*}$-algebras there is an associative composition law
\begin{equation*}[ A, B] \times [ B, C] \to [ A,C]\end{equation*}
which is compatible with the above compositions of asymptotic morphisms
and
$*$-homomorphisms.\end{theorem3}
(The proof in~\cite{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
$A_{1}\hotimes A_{2}$ the {\em maximal\/} graded tensor product of $A_{1}$ and
$A_{2}$~\cite[Section~14]{4}.
There is then a tensor product functor on the homotopy category
of asymptotic morphisms which associates to the asymptotic morphisms
$\xy \xymatrix {\phi _{1}\colon A_{1} \ar @{-->}[r] &
B_{1}}\endxy $ and
$\xy \xymatrix {\phi _{2}\colon A_{2} \ar @{-->}[r] & B_{2}}\endxy $ a
tensor product
\begin{equation*}\xy \xymatrix {\phi _{1}\hotimes \phi _{2} \colon
A_{1}\hotimes A_{2} \ar @{-->}[r] &
B_{1} \hotimes B_{2},}
\endxy \end{equation*}
where $\phi _{1}\hotimes \phi _{2} (a_{1}\hotimes a_{2}) \sim \phi
_{1}(a_{1})\hotimes \phi _{2}(a_{2})$.
If $C^{*}_{\max }(G,B)$
denotes the {\em maximal,\/} or {\em full\/} crossed product
$C^{*}$-algebra~\cite[Section~7.6]{19} then there is a {\em descent
functor\/} on the homotopy
category of
equivariant asymptotic morphisms which associates to an equivariant
asymptotic
morphism
$\xy \xymatrix {\phi \colon A\ar @{-->}[r] & B}\endxy $ an asymptotic morphism
\begin{equation*}\xy \xymatrix {\phi \colon C^{*}_{\max }(G,A)\ar @{-->}[r]
& C^{*}_{\max }(G,B),}
\endxy \end{equation*}
where $\phi _{t} (\sum a_{g} [g])\sim \sum \phi _{t} (a_{g})[g]$.
Let $\mathscr{S}=C_{0}(\mathbb{R})$, graded according to even and
odd functions and equipped with the trivial action of $G$. There are
$*$-homomorphisms
\begin{equation*}\begin{gathered}
\Delta \colon \mathscr{S}\to \mathscr{S}\hotimes \mathscr{S},\\
\Delta \colon f(X) \mapsto f(X\hotimes 1 + 1\hotimes X),\end{gathered}\qquad
\qquad \qquad \begin{gathered}
\epsilon \colon \mathscr{S}
\to \mathbb{C}, \\ \epsilon \colon f(X) \mapsto
f(0).\end{gathered}\end{equation*}
Using them, we construct the {\em amplified category of $\mathbb{Z}/2$-graded
$C^{*}$-algebras,\/} in which the morphisms from $A$ to $B$ are graded
$*$-homomorphisms (equivariant, in the presence of a group $G$) from
$\mathscr{S} A = \mathscr{S} \hotimes A$ to $B$. Similarly we can form the
amplification
of the
homotopy category of asymptotic morphisms.
\begin{definition2} Let $\mathscr{K} (\mathscr{H}_{G})$ be the
$C^{*}$-algebra of
compact operators on the $\mathbb{Z}/2$-graded $G$-Hilbert space
\begin{equation*}\mathscr{H}_{G} =
\ell ^{2}(G)\oplus \ell ^{2} (G)\oplus \cdots \end{equation*} (the summands
are graded alternately even and odd).
If $A$ and $B$ are separable, $\mathbb{Z}/2$-graded $G$-$C^{*}$-algebras then
denote
by
$E_{G}(A,B)$ the homotopy classes of equivariant asymptotic morphisms from
$\mathscr{S} A\hotimes \mathscr{K} (\mathscr{H}_{G})$ into $ B\hotimes
\mathscr{K} (\mathscr{H}_{G})$:
\begin{equation*}E_{G}(A,B) = [ \mathscr{S} A\hotimes \mathscr{K}
(\mathscr{H}_{G}), B\hotimes \mathscr{K}
(\mathscr{H}_{G})].\end{equation*}
When $G$ is trivial we drop the subscript and write
$E(A,B)$.\end{definition2}
The sets $E_{G}(A,B)$ are in fact abelian groups. There is an
associative law of
composition
\begin{equation*}E_{G}(A, B)\otimes E_{G}(B,C)\to E_{G}(A,C),\end{equation*}
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,
\begin{equation*}E_{G}(A_{1},B_{1})\otimes E_{G}(A_{2}, B_{2}) \to
E_{G}(A_{1}\hotimes A_{2}, B_{1}\hotimes B_{2})\end{equation*}
and
\begin{equation*}E_{G}(A,B)\to E(C^{*}_{\max }(G,A), C^{*}_{\max
}(G,B)).\end{equation*}
If $G$ is trivial then the $E$-theory groups specialize in one variable
to $K$-theory
of graded $C^{*}$-algebras:
\begin{equation*}E(\mathbb{C}, B)\cong K_{0}(B).\end{equation*}
For ungraded $C^{*}$-algebras these facts are proved in~\cite{11}. The
graded case
will be considered elsewhere.
If $\mathscr{H}$ is a separable, $\mathbb{Z}/2$-graded $G$-Hilbert space
then
any equivariant
asymptotic morphism from
$A$ to
$B\hotimes \mathscr{K}(\mathscr{H})$, or from
$\mathscr{S} A$ to
$B\hotimes \mathscr{K}(\mathscr{H})$, determines---after tensoring with
$\mathscr{K}(\mathscr{H}_{G})$---an element of
$E_{G}(A,B)$. This is because
$\mathscr{H}\hotimes \mathscr{H}_{G}\cong \mathscr{H}_{G}$
(cf.~\cite{18}).
Suppose that there is a
continuous, one-parameter family of actions of $G$ on a Hilbert space
$\mathscr{H}$, and
suppose that an asymptotic morphism \begin{equation*}\xy \xymatrix { \phi
\colon A\ar @{-->}[r]&B\hotimes \mathscr{K}(\mathscr{H})}\endxy
\end{equation*} is equivariant with respect
to this family, in the
sense that $\phi _{t}(g(a))\sim g_{t}(\phi _{t}(a))$. Then
$\{\phi _{t}\}_{t\in [1,\infty )}$
determines a class in $E_{G}(A,B)$. The reason is that after tensoring
with
$\mathscr{H}_{G}$ the one parameter family of actions on $\mathscr{H}$ is
conjugate to a constant action
(cf.~\cite{18} again). We shall use this observation, and a
small
extension of it, in Section~4.
\section*{3. The assembly map }
Let $Y$ be a proper $G$-space and let $B$ be a
$G$-$C^{*}$-algebra. We define
\begin{equation*}E_{G}(Y, B) = \underset {{X\subset Y}}{\lim
_{{}\longrightarrow {}}}\,
E_{G}(C_{0}(X),B),\end{equation*}
where the direct limit is over all $G$-compact, locally compact and
second
countable subsets $X\subset Y$.
For each such $X$ there is a class $\phi _{X}\in E(\mathbb{C}, C^{*}_{\max
}(G,C_{0}(X)))$, determined by a basic projection in the crossed product
algebra,
and we define the
{\em maximal Baum-Connes assembly
map\/}
to be the composition
\begin{multline*}
E_{G}(C_{0}(X),B) \xrightarrow{\text{descent}}
E_{G}(C^{*}_{\max}(G,C_{0}(X)), C^{*}_{\max }(G,B))\\
\xrightarrow[\phi_{X}\in E_{G}(\mathbb{C},
C^{*}_{\max}(G,C_{0}(X)))]{\text{composition with}}
E_{G}(\mathbb{C},C^{*}_{\max}(G,B)).
\end{multline*}
See~\cite[Chapter~10]{11}.
If $Y$ is a general proper $G$-space, not necessarily $G$-compact, then
the maximal Baum-Connes assembly map for
$Y$ is the homomorphism
\begin{equation*}\mu _{\max }\colon E_{G}(Y, B)\to E(\mathbb{C} ,
C^{*}_{\max }(G,B))
\end{equation*}
obtained as the direct limit of the assembly maps for the $G$-compact
subsets of $Y$.
The {\em reduced\/} Baum-Connes assembly map
\begin{equation*}\mu _{\smash{\text{red}}}\colon E_{G}(Y, B)\to E(\mathbb{C},
C^{*}_{\smash{\text{red}}}(G,B))
\end{equation*}
is defined by composing $\mu _{\max }$ with the $E$-theory map induced
from the
regular representation $\rho \colon C^{*}_{\max }(G, B) \to
C^{*}_{\smash{\text{red}}} (G, B)$.
The assembly maps are usually defined using
$KK$-theory, as in~\cite{2}, but our definitions are equivalent to
these~\cite{13}.
Let us say that a $G$-$C^{*}$-algebra $D$ is {\em 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.~\cite[Definition~1.5]{16}).
The main theorem
in~\cite{11}, and a key tool for us here, is the following result:
\begin{theorem4}[{\cite[Theorem 14.1]{11}}] Let $G$ be a
countable discrete group and let
$D$ be a proper
$G$-$C^{*}$-algebra. The assembly map
\begin{equation*}\mu _{\max }\colon E_{G}(\mathcal{E} G , D) \rightarrow
E(\mathbb{C} , C^{*}(G,D))\end{equation*}
is an isomorphism.\end{theorem4}
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 Poincar\'{e}
duality for manifolds modelled on Hilbert space). As explained in
\cite{11}, Theorem 3.1 may be viewed as a generalization of a theorem of
Green~\cite{9} and Julg~\cite{15} which identifies equivariant
$K$-theory (in the
sense of Atiyah and Segal~\cite{20}) with the
$C^{*}$-algebra $K$-theory of crossed product algebras.
An immediate consequence is the following result:
\begin{theorem5}[{see~\cite[Theorem 15.1]{11} and also
\cite[Theorem 2.1]{21}}] Let $G$ be a \break discrete group and suppose that
there is
a proper
$G$-$C^{*}$-algebra $D$, and elements
$\beta \in E_{G}(\mathbb{C} , D)$ and $\alpha \in E_{G}(D,\mathbb{C})$, whose
composition is $\alpha \circ \beta =1\in E_{G}(\mathbb{C} ,\mathbb{C})$.
Then for
any
$G$-$C^{*}$-algebra $B$ the assembly map
\begin{equation*}\mu _{\max }\colon E_{G}(\mathcal{E} G, B) \to E(\mathbb{C}
, C^{*}(G,B))\end{equation*}
is an isomorphism.
\end{theorem5}
\begin{proof} The hypotheses imply that the assembly map for $B=B\otimes
\mathbb{C}$
may be viewed as a direct summand of the assembly map for the proper
$C^{*}$-algebra
$B\otimes D$, and by Theorem~3.1 the latter is an isomorphism.\end{proof}
\section*{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 {\em countably infinite-dimensional\/} Euclidean space.
With this in
mind, let us fix such a Euclidean space $V$.
The following definitions are adapted from~\cite[~ Section~3]{14}.
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 $\mathscr{C}(V_{a})$ be the $\mathbb{Z}/2$-graded
$C^{*}$-algebra of continuous functions from $V$ into the complexified
Clifford
algebra of $V^{0}_{a}$ which vanish at infinity (the grading on $\mathscr{C}
(V_{a})$ is inherited
from the grading on the Clifford algebra). Let $\mathscr{A}(V_{a})$ be the
graded
tensor product of $\mathscr{S} = C_{0}(\mathbb{R})$ and $\mathscr{C}(V_{a})$.
If $V_{a}\subset V_{b}$ then there is a canonical decomposition
$V_{b} = V^{0}_{ba} + V_{a}$,
where $V^{0}_{ba}$ 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
$\tilde h$ on
$V_{b}$ by the formula $\tilde h(v_{b}) = h(v_{a})$.
\begin{definition3} If $V_{a}\subset V_{b}$ then denote by
$C_{ba}$ the Clifford
algebra-valued function on $V_{b}$ which maps $v_{b}$ to
$v_{ba}\in V_{ba}^{0} \subset \Cliff (V_{b}^{0})$. Define a
$*$-homomorphism
\begin{equation*}\beta _{ba} \colon \mathscr{A}(V_{a}) \to
\mathscr{A}(V_{b}) \end{equation*} by the formula
\begin{equation*}\beta _{ba} (f\hotimes h) = f( X\hotimes 1 + 1 \hotimes
C_{ba})(1\hotimes \tilde h).\end{equation*}
\end{definition3}
\begin{remark1} The element $ X\hotimes 1 + 1 \hotimes C_{ba}$
is regarded as an {\em unbounded multiplier\/} of $\mathscr{A}(V_{b})$; see
for
example~\cite[Section~6.5]{5}. The elements
$f(X\hotimes 1 + 1 \hotimes C_{ba})$ and $1\hotimes \tilde h$ are
then bounded
multipliers of $\mathscr{A}(V_{b})$, and their product lies in
$\mathscr{A}(V_{b})$.\end{remark1}
If $V_{a}\subset V_{b} \subset V_{c}$ then
$\beta _{cb}\circ \beta _{ba} = \beta _{ca}$. In view of this we can
construct the
$C^{*}$-algebra
\begin{equation*}\mathscr{A} (V) = \lim _{{}\longrightarrow
{}}\,\mathscr{A} (V_{a}),\end{equation*} where
the limit is over the
directed set of all $V_{a}\subset V$.
If $G$ acts on $V$ by affine isometries then there is an induced action
of $G$ on
$\mathscr{A} (V)$ by $*$-automorphisms.
\begin{theorem6} If $G$ acts metrically properly on $V$
then $\mathscr{A} (V)$ is
a proper $G$-$C^{*}$-algebra.\end{theorem6}
\begin{proof} The following elegant argument, which much improves our
original
proof, is due to G.~Skandalis.
The center $\mathscr{Z}(V_{a})$ of the $C^{*}$-algebra $\mathscr{A}
(V_{a})$ contains
the algebra
of continuous functions, vanishing at infinity, on the locally compact
space
$[0,\infty )\times V_{a}$ (if $V_{a}$ is even-dimensional and nonzero
this is the full center). The
linking map
$\beta _{ba}$ takes $\mathscr{Z}(V_{a})$ into $\mathscr{Z}(V_{b})$, and so
we can form the
direct limit
$\mathscr{Z}(V)$. It has the property that $\mathscr{Z}(V)\cdot \mathscr{A}
(V)$ is dense in
$\mathscr{A} (V)$. Its
Gelfand spectrum is the locally compact space $Z=[0,\infty )\times
\overline{V}$,
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.\end{proof}
In view of Proposition~4.2 and Theorem~3.2, to prove
Theorem~1.2 it suffices to exhibit classes
$\alpha \in E_{G}(\mathscr{A} (V), \mathbb{C})$ and $\beta \in E_{G}
(\mathbb{C} ,
\mathscr{A} (V))$
such that $\alpha \circ \beta =1\in E_{G}(\mathbb{C}, \mathbb{C})$. This is
what
we shall now
do. We shall follow the article~\cite{14}, which is in turn an
adaptation of Atiyah's
elliptic operator proof~\cite{1} of the Bott periodicity theorem. The
construction of
$\beta $ is quite simple:
\begin{definition4} Denote by $\beta \colon \mathscr{S}=\mathscr{A}(0) \to
\mathscr{A} (V)$ the
$*$-homomorphism associated to the inclusion of the zero-dimensional
linear space
$0$ into $V$. Denote by $\beta _{t} \colon \mathscr{S}\to \mathscr{A} (V)$
the
$*$-homomorphism
$\beta _{t} (f) = \beta (f_{t})$, where $f_{t}(x) = f(t^{-1}x)$.
The family of $*$-homomorphisms
$\{\beta _{t}\}_{t\in [1,\infty )}$ is asymptotically equivariant and so
is an equivariant
asymptotic morphism from $\mathscr{S}$ to
$\mathscr{A} (V)$. Denote by $\beta \in E_{G}(\mathbb{C}, \mathscr{A}(V))$
its
$E$-theory
class.\end{definition4}
As in Atiyah's paper~\cite{1}, to construct
$\alpha \in E_{G}(\mathscr{A} (V),
\mathbb{C})$ we shall need some elliptic operator theory. In the present
context the operators must be defined on the infinite-dimensional space
$V$.
Denote by $\mathscr{H}_{a}$ the Hilbert space of square
integrable functions from
$V_{a}$ into $\Cliff (V_{a}^{0})$. If $V_{a}\subset V_{b}$ then there is a
canonical
isomorphism \begin{equation*}\mathscr{H}_{b}\cong \mathscr{H}_{ba}\hotimes
\mathscr{H}_{a},\end{equation*} where
$\mathscr{H}_{ba}$ denotes the
Hilbert space associated to $V_{ba}^{0}$. We define a unit vector
$\xi _{0}\in \mathscr{H}_{ba}$
by
\begin{equation*}\xi _{0}(v_{ba}) = \pi ^{- n_{ba}/{4} } \exp
(-\tfrac{1}{2}{\|v_{ba}\|^{2}}),\end{equation*}
where $n_{ba}=\dim (V_{ba}^{0})$, and we regard $\mathscr{H}_{a}$ as
included in
$\mathscr{H}_{b}$ via the
isometry
$\xi \mapsto \xi _{0}\hotimes \xi $. We define
\begin{equation*}\mathscr{H} = \lim _{{}\longrightarrow
{}}\mathscr{H}_{a}\end{equation*} and denote by
$\mathfrak{s}=\lim \limits _{{}\longrightarrow {}} \mathfrak{s}_{a}$ the
direct limit of the
Schwartz subspaces $\mathfrak{s}_{a}\subset \mathscr{H}_{a}$.
If $V_{a}\subset V$ is a finite-dimensional affine
subspace then
the {\em Dirac operator\/}
$D_{a}$, an unbounded operator on $\mathscr{H}$ with domain $\mathfrak{s}$,
is defined
by
\begin{equation*}D_{a} \xi = \sum _{i=1}^{n} (-1)^{\deg (\xi )}
{\frac{\partial \xi }{\partial x_{i}}}v_{i} ,\end{equation*}
where $\{v_{1}, \dots ,v_{n}\}$ is an orthonormal basis for $V_{a}^{0}$, and
$\{x_{1},\dots ,
x_{n}\}$ are the dual coordinates to $\{v_{1},\dots , v_{n}\}$. If $V_{a}$
is a
{\em linear\/}
subspace then we also define the {\em Clifford operator\/} by
\begin{equation*}C_{a} \xi = \sum _{i=1}^{n} x_{i}
v_{i} \xi \end{equation*}
(cf.~Definition~4.1).
\begin{definition5} Fix an algebraic, direct sum decomposition
\begin{equation*}V = V_{0}\oplus V_{1}\oplus V_{2}\oplus \cdots ,
\end{equation*}
where each $V_{j}$ is a finite-dimensional linear subspace of $V$.
For each $n$,
define an unbounded
operator
$B_{n,t}$ on
$\mathscr{H} $, by the formula
\begin{multline*}
{B_{n,t} = t_{0} D_{0} + t_{1} D_{1} + \cdots + t_{n-1}D_{n-1}}\\{
+t _{n}(D_{n} +
C_{{n}}) + t _{n+1}(D_{{n+1}} + C_{{n+1}})
+ \cdots ,}\end{multline*} where $t _{j}= 1+t ^{-1} j$.
\end{definition5}
The infinite sum is well defined because when $B_{n,t} $
is applied to any vector
in the Schwartz space $\mathfrak{s}$ the resulting infinite series
has only finitely
many nonzero terms. This is because the function
$\exp (-\frac{1}{2}\|v_{j}\|^{2})\in s_{j}$, used in the definition of the
direct limit $\mathfrak{s}$, lies in the
kernel of
$D_{j}+C_{j}$.
If
$h\in \mathscr{C} (V_{0}\oplus \dots \oplus V_{n})$ then the function
$h_{t}(v) =
h(t^{-1}v)$ acts
as a bounded operator
$\pi (h_{t})$ on
$\mathscr{H}$ by pointwise multiplication. With this notation in hand,
the main technical results concerning the operators $B_{n,t}$ are as
follows.
\begin{theorem7}[{cf.~\cite[Lemma 2.9 and
Proposition~4.2]{14}}] The operators
$B_{n,t}$ are essentially selfadjoint. The formula
\begin{equation*}\alpha ^{n} _{t}\colon f\hotimes h\mapsto f_{t}(B_{t})\pi (
h_{t})\qquad (f\in \mathscr{S},\ h\in \mathscr{C}(V_{0}\oplus \dots
\oplus V_{n}))\end{equation*}
defines an asymptotic morphism from $\mathscr{A}(V_{0}\oplus \dots \oplus
V_{n})$ into
$\mathscr{K}(\mathscr{H})$, and the diagram
\begin{equation*}\xy \xymatrix { \mathscr{A}(V_{0}\oplus \dots \oplus V_{n})
\ar @{-->}[r]^-{\alpha^{n}}
\ar [d]_{\beta _{n+1}} & \mathscr{K}(\mathscr{H}) \ar [d]^{=} \\
\mathscr{A}(V_{0}\oplus \dots \oplus V_{n+1}) \ar @{-->}[r]_-{\alpha^{n+1}} &
\mathscr{K}(\mathscr{H}) }\endxy \end{equation*}
is asymptotically commutative.\end{theorem7}
It follows from the second part of the proposition that the asymptotic
morphisms
$\alpha ^{n}$ combine to form a single asymptotic morphism
\begin{equation*}\xy \xymatrix { \alpha \colon \mathscr{A}(V) \ar @{-->}[r]
& \mathscr{K}(\mathscr{H}).}
\endxy \end{equation*}
Now let $G$ act on the Euclidean space $V$ according to the
affine, isometric action
\begin{equation*}g\cdot v = \pi (g) v + \kappa (g),\end{equation*}
where $\pi $ is a linear, isometric representation of $G$ on $V$. For
$s\ge 0$
form the scaled $G$-action
\begin{equation*}g_{s}\cdot v = \pi (g) v + s \kappa (g),\end{equation*}
and denote the induced actions on $\mathscr{A}(V)$ and
$\mathscr{K}(\mathscr{H})$
in the same way.
\begin{theorem8}[{cf.~\cite[Lemma~5.15]{14}}]
Suppose that the direct sum decompostion $V =
\bigoplus_{0} ^{\infty }V_{j}$ is chosen in such a way that for every $g\in G$
there is an
$N\in \mathbb{N}$ such that if $n>N$ then $g [\bigoplus _{0}^{n} V_{j}]\subset
\bigoplus _{0}^{n+1} V_{j}$. Then for every $a\in \mathscr{A}(V)$ and every
$g\in G$,
$\alpha _{t}(g(a))\sim g_{t}(\alpha _{t}(a))$. Consequently the asymptotic
morphism
$\{\alpha _{t}\}_{t\in [1,\infty )}$ determines a class $\alpha \in
E_{G}(\mathscr{A}(V),
\mathbb{C})$.\end{theorem8}
We are ready to prove the following result:
\begin{theorem9} The composition $\alpha \circ \beta \in E_{G}(\mathbb{C} ,
\mathbb{C})$ is the identity.
\end{theorem9}
\begin{proof}
Let $s\in [0,1]$ and denote by
$\mathscr{A}_{s}$ the $C^{*}$-algebra $\mathscr{A}=\mathscr{A}(V)$, but with
the scaled
$G$-action
$(g,a)\mapsto g_{s}(a)$.
The algebras $\mathscr{A}_{s}$ form a continuous field of
$G$-$C^{*}$-algebras
over the
unit interval, and we shall denote by
$\mathscr{A}[0,1]$ the $G$-$C^{*}$-algebra of continuous sections.
In a similar way, form a continuous field of $G$-$C^{*}$-algebras
$\mathscr{K}_{s}=\mathscr{K}(\mathscr{H})$ and denote by
$\mathscr{K}[0,1]$ the
$G$-$C^{*}$-algebra of continuous
sections.
The asymptotic
morphism
$\xy \xymatrix {\alpha \colon \mathscr{A}
\ar @{-->}[r] &
\mathscr{K}}\endxy $ induces an asymptotic morphism
\begin{equation*}\xy \xymatrix {\bar \alpha \colon \mathscr{A}[0,1]\ar
@{-->}[r]& \mathscr{K}[0,1]}\endxy,\end{equation*} and similarly
the asymptotic morphism $\xy \xymatrix {\beta \colon \mathscr{S}
\ar @{-->}[r] &
\mathscr{A}}\endxy $ determines an asymptotic morphism
\begin{equation*}\xy \xymatrix {\bar \beta \colon \mathscr{S}\ar @{-->}[r]&
\mathscr{A}[0,1]}\endxy \end{equation*}
by forming the tensor product $\xy \xymatrix { \mathscr{S}[0,1] \ar
@{-->}[r]&
\mathscr{A}[0,1]}\endxy $
and composing with the inclusion $\mathscr{S} \subset \mathscr{S}[0,1]$ as
constant
functions. Consider now the diagram
\begin{equation*}\xy \xymatrix {
\mathscr{S}\ar [d]_{=} \ar @{-->}[r]^{\bar \beta } & \mathscr{A}[0,1]
\ar [d]^{\epsilon _{s}} \ar @{-->}[r]^{\bar \alpha } &
\mathscr{K}[0,1] \ar [d]^{\epsilon _{s}}
\\
\mathscr{S}\ar @{-->}[r]_{ \beta } & \mathscr{A}_{s} \ar @{-->}[r]_{ \alpha
}& \mathscr{K}_{s}},
\endxy \end{equation*}
where $\epsilon _{s}$ denotes evaluation at $s\in [0,1]$.
All the asymptotic morphisms determine classes in $E_{G}$-theory, and the
$*$-homomorphism
$\epsilon _{s}
\colon \mathscr{K}[0,1]\to \mathscr{K}_{t}$ defines an isomorphism, so to
prove the theorem it suffices
to show that when $s=0$ the bottom composition
\begin{equation*}\xy \xymatrix {\mathscr{S} \ar @{-->}[r]^{\beta } &
\mathscr{A}_{0} \ar @{-->}[r]^{\alpha }&
\mathscr{K}_{0}}\endxy \end{equation*}
determines the identity element of $E_{G}(\mathbb{C} , \mathbb{C})$. When
$s=0$ the action of $G$ on $V$ is linear and the asymptotic morphism
$\xy \xymatrix { \beta \colon \mathscr{S}\ar @{-->}[r]&
\mathscr{A}_{0}}\endxy $ is homotopic to the equivariant
$*$-homomorphism
$\beta \colon \mathscr{S}\to \mathscr{A}_{0}$ of Definition~4.3.
The composition of the $*$-homomorphism $\beta $ with the asymptotic
morphism $\alpha $ is the asymptotic morphism
\begin{equation*}\begin{split}
\xy \xymatrix { \gamma \colon \mathscr{S} \ar @{-->}[r]& \mathscr{K},}
\endxy &\\
\gamma _{t} \colon f\mapsto f_{t}(B_{0,t})&.\end{split}\end{equation*}
Replacing $B_{0,t}$ with $s^{-1}B_{0,t}$, for $0}[r] & B}\endxy $ an asymptotic morphism
\begin{equation*}\xy \xymatrix {\phi \colon
C^{*}_{\smash{\textup{red}}}(G,A)\ar @{-->}[r] &
C^{*}_{\smash{\textup{red}}}(G,B).}\endxy \end{equation*}
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 {\em ad hoc\/}
argument proves that
$\mu _{\smash{\text{red}}}$ is an isomorphism for the groups we are
considering.
\begin{theorem10} 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
\begin{equation*}\rho \colon C^{*}_{\max }(G,B)\to
C^{*}_{\smash{\textup{red}}} (G,B)\end{equation*}
determines an invertible morphism $\rho \in E(C^{*}_{\max }(G,B),
C^{*}_{\smash{\textup{red}}}(G,B))$.
\end{theorem10}
\begin{proof} Let us consider the case where $B=\mathbb{C}$.
We shall define a class $\sigma \in
E(C^{*}_{\smash{\text{red}}}(G),C^{*}_{\max }(G))$
which is inverse to $\rho \in E(C^{*}_{\max }(G),
C^{*}_{\smash{\text{red}}}(G))$.
Suppose given an equivariant asymptotic morphism $\xy \xymatrix {\phi \colon
\mathscr{S}
\ar @{-->}[r]& A}\endxy $, where $A$ is any $G$-$C^{*}$-algebra. Both $A$
and
$C^{*}_{\smash{\text{red}}}(G)$ embed in the multiplier algebra of
$C^{*}_{\smash{\text{red}}}(G,A)$, and
$C^{*}_{\smash{\text{red}}}(G)$ asymptotically commutes with the image
of the asymptotic
morphism
$\phi $ (this is because
$\phi $ is asymptotically equivariant). It follows that
$\phi $ induces an asymptotic morphism
\begin{equation*}\xy \xymatrix { \phi _{\smash{\text{red}}}\colon
\mathscr{S}\hotimes C^{*}_{\smash{\text{red}}}(G)\ar @{-->}
[r] & C^{*}_{\smash{\text{red}}}(G,A).}\endxy \end{equation*}
In particular, $\beta $ induces an asymptotic morphism
\begin{equation*}\xy \xymatrix { \beta _{\smash{\text{red}}}\colon
\mathscr{S}\hotimes C^{*}_{\smash{\text{red}}}(G)\ar @{-->}
[r] & C^{*}_{\smash{\text{red}}}(G,\mathscr{A}).}\endxy \end{equation*}
Now since $\mathscr{A}$ is a proper $C^{*}$-algebra the regular
representation
$C^{*}_{\max }(G,\mathscr{A})\to C^{*}_{\smash{\text{red}}}(G,\mathscr{A})$
is an
isomorphism.
We define $\sigma \in E(C^{*}_{\smash{\text{red}}}(G),C^{*}_{\max }(G)) $ to
be the $E$-theory class of
the composition
\begin{equation*}\xy \xymatrix { \mathscr{S}\hotimes
C^{*}_{\smash{\text{red}}}(G)
\ar @{-->}[r]^{\beta _{\smash{\text{red}}}} &
C^{*}_{\smash{\text{red}}}(G,\mathscr{A})\\
& C^{*}_{\max }(G,\mathscr{A}) \ar [u]^{\cong }_{\rho } \ar
@{-->}[r]^{\alpha } &
C^{*}_{\max }(G,\mathscr{K})\ar [r]_{\cong }& C^{*}_{\max }(G)\hotimes
\mathscr{K}.}\endxy \end{equation*}
It follows from Theorem~4.7 that $\sigma \circ \rho =1\in E_{G}(C^{*}_{\max
}(G),C^{*}_{\max }(G))$. To calculate the reverse
composition,
we note that since
$C^{*}_{\max }(G,\mathscr{A})\cong
C^{*}_{\smash{\text{red}}}(G,\mathscr{A})$ the
asymptotic morphism $\xy \xymatrix { \alpha \colon \mathscr{A}\ar @{-->}[r]&
\mathscr{K}}\endxy $ induces an
asymptotic morphism
\begin{equation*}\xy \xymatrix {\alpha _{\smash{\text{red}}} \colon
C^{*}_{\smash{\text{red}}}(G,\mathscr{A})\ar @{-->}[r] &
C^{*}_{\smash{\text{red}}}(G,\mathscr{K}).}\endxy \end{equation*}
The composition $\rho \circ \sigma \in E(C^{*}_{\smash{\text{red}}}(G),
C^{*}_{\smash{\text{red}}}(G))$
is given by the composition
\begin{equation*}\xy \xymatrix {\mathscr{S}\hotimes
C^{*}_{\smash{\text{red}}} (G)
\ar @{-->}[r]^{\beta _{\smash{\text{red}}}} &
C^{*}_{\smash{\text{red}}}(G,\mathscr{A})\ar @{-->}[r]^{\alpha
_{\smash{\text{red}}}} &
C^{*}_{\smash{\text{red}}}(G,\mathscr{K}).}\endxy \end{equation*}
The
composition $\alpha _{\smash{\text{red}}} \circ \beta
_{\smash{\text{red}}}$ is the
asymptotic morphism
\begin{equation*}\xy \xymatrix {\gamma _{\smash{\text{red}} }\colon
\mathscr{S} \hotimes C^{*}_{\smash{\text{red}}}(G)\ar @{-->}
[r] & C^{*}_{\smash{\text{red}}}(G,\mathscr{K})}\endxy \end{equation*}
induced, according to the
remark made above, by the
equivariant asymptotic morphism
$\gamma \colon \mathscr{S} \rightarrow \mathscr{K} $ which is the
composition of
$\beta $ and $\alpha $ and is defined by the formula:
$\gamma _{t}(f) = f_{t}(B_{0,t})$. (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
$\gamma _{\smash{\text{red}}}=1\in
E(C^{*}_{\smash{\text{red}}}(G),C^{*}_{\smash{\text{red}}}(G))$.\end{proof}
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