Then, $K^{-1}A$ is a compact subset of $X^{+}\times X^{+}$ which does not
contain the diagonal, so there is $\delta>0$ such that all $(x,y)\in K^{-1}A$
satisfy $d(x,y)\geq\delta$. It follows for any $(x,y)\in X_{0}\times X_{0}$
with $d(x,y)<\delta$, $d(kx,ky)<\epsilon$ for all $k\in K$. We see that
$\mathrm{prop}(h(T_{t}))\to 0$ uniformly in $h\in K$ as desired. The
continuous case is proved in the same way. ∎

######  Remark 8.15.

The inclusion $\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X}\otimes B)_{\mathrm{Gcpt}}^{G}$ is never surjective
unless $G$ is finite. For example, if $G$ is compact and if $X$ is a point,
then the inclusion is identified as the inclusion

$C_{b,u}([1,\infty),\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to
C_{b,u}([1,\infty),(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G)$

which is not surjective unless $G$ is finite even if $H_{X}=\mathbb{C}$ and
$B=\mathbb{C}$.

Now we go back to the classical setting when $G$ is discrete. Here is the
promised comparison between the crossed product algebra
$RL^{*}_{u}(H_{X})\rtimes_{r}G$ and the localized equivariant Roe algebra
$C^{*}_{L,u}(H_{X})^{G}$. A generalization with coefficient $B$ is possible
but we only consider $B=\mathbb{C}$ here. The continuous case can be shown too
but with an extra effort.

######  Theorem 8.16.

Let $G$ be a countable discrete group, $X$ be a proper $G$-space which is
$G$-equivariantly homotopic to a $G$-$CW$ complex and $H_{X}$ be any ample
$X$-$G$-module. Then, the inclusion $\rho\colon
RL^{*}_{u}(H_{X})\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}$ induces an isomorphism

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}).$

In particular, when $X$ is $G$-compact, the inclusion $\rho\colon
RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}$ induces an
isomorphism

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})^{G}).$

###### Proof.

It suffices to show the case when $X$ is $G$-compact. For any $X$-$G$-module
$H_{X}$, let $L^{*}(H_{X})^{G}$ be the equivariant localization algebra as
defined in [48, Definition 6.5.1]. The $C^{*}$-algebra $L^{*}(H_{X})^{G}$ is
the norm completion of the $\ast$-algebra $\mathbb{L}[H_{X}]^{G}$ of all
bounded functions $T_{t}$ on $[1,\infty)$ to $\mathfrak{L}(H_{X})$ such that

  1. (1)

$T_{t}$ is $G$-equivariant,

  2. (2)

for any compact subset $K$ of $X$, there exists $t_{K}\geq 1$ such that for
all $t\geq t_{k}$, $\chi_{K}T_{t}$, $T_{t}\chi_{K}$ are compact, and the
functions $t\mapsto\chi_{K}T_{t}$, $t\mapsto T_{t}\chi_{K}$ are uniformly
norm-continuous when restricted to $[t_{k},\infty)$,

  3. (3)

for any open neighborhood of the diagonal in $X^{+}\times X^{+}$, there exists
$t_{U}\geq 1$ such that for all $t>t_{U}$, $\mathrm{supp}(T_{t})\subset U$.

It is proved in [48, Proposition 6.6.2] that the natural inclusion induces an
isomorphism

$K_{\ast}(C_{L,u}^{*}(H_{X})^{G})\cong K_{\ast}(L^{*}(H_{X})^{G})$

whenever $H_{X}$ is ample. Thus, to prove our claim, it suffices to show that
when $X$ is $G$-compact, for any ample $X$-$G$-module $H_{X}$, the natural
inclusion

$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}\to
L^{*}(\tilde{H}_{X})^{G}$

induces an isomorphism on K-theory. For this, we consider more generally, for
any not-necessary $G$-compact $X$, the equivariant localization algebra
$L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ with $G$-compact support to be the
completion of $\mathbb{L}[H_{X}]_{\mathrm{Gcpt}}^{G}$ which is the subalgebra
of $\mathbb{L}[H_{X}]^{G}$ consisting of $T_{t}$ that has eventually uniform
$G$-compact support, that is there is $t_{0}\geq 1$ and a $G$-compact subset
$X_{0}$ of $X$ such that for all $t\geq t_{0}$, $\mathrm{supp}(T_{t})\subset
X_{0}\times X_{0}$. Then, our claim is that the natural inclusion

$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}\to
L^{*}(\tilde{H}_{X})^{G}_{\mathrm{Gcpt}}$

induces an isomorphism on K-theory. Now, the point is that just as in the case
of our functor $\mathbb{D}_{\ast}^{G}$, the assignment

$X\mapsto
RK_{\ast}^{G}(X)=K_{\ast}(L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}),\,\,\,f\mapsto
RK_{\ast}^{G}(f)=\mathrm{Ad}_{V^{f}_{t}\ast}$

becomes a functor from $\mathcal{PR}^{G}$ to $\mathcal{GA}$, where $H_{X}$ is
a chosen ample $X$-$G$-module for a proper $G$-space $X$ and $V^{f}_{t}$ is a
chosen $G$-equivariant continuous cover of a $G$-equivariant continuous map
$f\colon X\to Y$. As in the case of $\mathbb{D}^{G}_{\ast}$ (see the proof of
Proposition 6.2), one shows the composition law for not-necessarily proper
maps by reducing it to the $G$-compact setting after showing the
representability of the functor $RK^{G}_{\ast}$. Moreover, the functor
$RK_{\ast}^{G}$ satisfies the five listed properties in Theorem 6.4. Since
this does not involve any new idea, we explain this very briefly. The first
part of the second property (2) (induction from a finite subgroup) follows
from Proposition [48, Proposition 6.5.13] and from (3) the representability of
$RK^{G}_{\ast}$. The fourth property (4) (the Mayer–Vietoris sequence) can be
shown just as in the case of $\mathbb{D}_{\ast}^{G}$ by using the quotient
$L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ of $L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$
by the ideal $L_{0}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ which is the completion of
the subalgebra of $\mathbb{L}[H_{X}]_{\mathrm{Gcpt}}^{G}$ consisting of
$T_{t}$ such that for any compact subset $K$ of $X$,
$\chi_{K}T_{t},T_{t}\chi_{K}=0$ eventually. The quotient
$L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ is a $C_{0}(X/G)$-algebra (see [48,
Lemma 6.4.18]) and the quotient map induces an isomorphism

$K_{\ast}(L^{*}(H_{X})^{G}_{\mathrm{Gcpt}})\cong
K_{\ast}(L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}})$

because $K_{\ast}(L_{0}^{*}(H_{X})^{G}_{\mathrm{Gcpt}})=0$ (see [48, Lemma
6.5.12]). For the fifth property (5) (the homotopy invariance), the proof for
$\mathbb{D}_{\ast}^{G}$ works verbatim. It remains to prove the second part of
(2), that is

$RK^{H}_{\ast}(\mathrm{point})\cong K_{\ast}(C^{*}_{r}(H))$

for any finite group $H$. Using $K_{\ast}(C_{L,u}^{*}(H_{X})^{G})\cong
K_{\ast}(L^{*}(H_{X})^{G})$, we just need to show

$K_{\ast}(C_{b,u}([1,\infty),\mathfrak{K}(l^{2}(\mathbb{N})\otimes
l^{2}(H)))^{H})\cong K_{\ast}(C^{*}_{r}(H)),$

but since $\mathfrak{K}(l^{2}(\mathbb{N})\otimes
l^{2}(H))^{H}\cong\mathfrak{K}(l^{2}(\mathbb{N}))\otimes C^{*}_{r}(H)$, this
follows because the evaluation

$\mathrm{ev}_{1}\colon
C_{b,u}([1,\infty),\mathfrak{K}(l^{2}(\mathbb{N}))\otimes
C^{*}_{r}(H))\to\mathfrak{K}(l^{2}(\mathbb{N}))\otimes C^{*}_{r}(H)$

at $t=1$ is an isomorphism on K-theory.

Now, the inclusion

$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to
L^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}$

induces a natural transformation of the functors $\mathbb{D}^{G}_{\ast}$ to
$RK_{\ast}^{G}$. We can see that this is an isomorphism when $X=G/H$ for any
finite subgroup $H$ of $G$, by considering

$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}$

instead. Indeed, at the level of K-theory, this inclusion is isomorphic to the
inclusion

$C_{b,u}([1,\infty),\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y}))\rtimes_{r}G\to
C_{b,u}([1,\infty),\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G)$

where $\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})$ is the
$G$-$C_{0}(G/H)$-algebra with fiber $\mathfrak{K}(H_{Y})$ at the coset $H$, or
more precisely

$\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})=\\{f\in
C_{0}(G,\mathfrak{K}(H_{Y}))\mid f(gh)=h^{-1}(f(g))\,\,\text{for $h\in H$}\\}$

equipped with the left translation $G$-action. It is now easy to see that the
above inclusion is an isomorphism on K-theory because both are, at the level
of K-theory, isomorphic to
$\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G$ via the evaluation at
$t=1$. Of course, we have
$K_{\ast}(\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{r}(H))$. It follows since both functors
$\mathbb{D}^{G}_{\ast}$, $RK_{\ast}^{G}$ satisfy properties (1) - (5) in
Theorem 6.4, the natural transformation

$\rho\colon\mathbb{D}^{G}_{\ast}(X)\to RK^{G}_{\ast}(X)$

is an isomorphism for all $X$ which is $G$-equivariantly homotopic to a $G$-CW
complex. Going back to the original question, we just showed that the
inclusion $\rho$ induces an isomorphism

$K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}})$

for all such $X$. ∎

##  9\. The forget-control map and the Baum–Connes assembly map

Let $G$ be a (second countable) locally compact group and $B$ be a separable
$G$-$C^{*}$-algebra. Let $X$ be a proper $G$-space and $H_{X}$ be an
$X$-$G$-module.

The evaluation map

$\mathrm{ev}_{1}\colon RL^{\ast}_{c}(H_{X}\otimes
B)\to\mathfrak{K}(H_{X})\otimes B$

at $1$, which we call the forget-control map, induces a $\ast$-homomorphism

$\mathrm{ev}_{1}\colon RL^{\ast}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes
B)\rtimes_{r}G\cong\mathfrak{K}(H_{X})\otimes(B\rtimes_{r}G).$

Here, the isomorphism on the right is obtained by trivializing the inner
$G$-action on $\mathfrak{K}(H_{X})$.

It induces a group homomorphism (the forget-control map)

(9.1) $\mathcal{F}=\mathrm{ev}_{1\ast}\colon
K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(B\rtimes_{r}G).$

######  Proposition 9.1.

The forget-control map $\mathcal{F}$ (9.1) is functorial in $X$ and in $B$ in
a sense that the following diagrams commute. For any $G$-equivariant
continuous map $f\colon X\to Y$ and for any (if exists) $G$-equivariant
continuous cover $(V_{t}\colon H_{X}\to H_{Y})_{t\in[1,\infty)}$ of $f$,

$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathrm{Ad}_{V_{t}\ast}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{Y}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G),}$

and for any $G$-equivariant $\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$,

$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\pi_{\ast}}$$\textstyle{K_{\ast}(B_{1}\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\rtimes_{r}1_{\ast}}$$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B_{2}\rtimes_{r}G).}$

In particular, $\mathcal{F}$ $\eqref{eq_forgetK}$ induces a group homomorphism

(9.2) $\mathcal{F}\colon\mathbb{D}^{B,G}_{\ast}(X)\to K_{\ast}(B\rtimes_{r}G)$

which is natural in $X$ and in $B$.

###### Proof.

The first diagram commutes since $\mathrm{Ad}_{V_{t}}$ on
$(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G$ is the identity on K-theory. The
second diagram commutes at the level of $\ast$-homomorphisms. ∎

######  Definition 9.2.

We call the group homomorphism $\mathcal{F}$ in (9.2), the forget-control map
for the functor $\mathbb{D}_{\ast}^{B,G}$.

In the rest of this section, our goal is to show that the forget-control map
$\mathcal{F}$ naturally factors through the Baum–Connes assembly map (see [6],
[47], [19])

(9.3) $\mu_{X}^{B,G}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)\to K_{\ast}(B\rtimes_{r}G)$

via a group homomorphism

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$

which we will define first. We will also show that $\rho_{X}$ is an
isomorphism for any $B$ when $G$ is a discrete group and $X$ is
$G$-equivariantly homotopy equivalent to a $G$-CW complex.

To obtain these, for technical reasons, we will use equivariant $E$-theory
$E^{G}$ of Guentner, Higson and Trout ([21], see also [24, Chapter 2]) in
place of equivariant $KK$-theory $KK^{G}$ of Kasparov [30]. For our purpose,
this is no problem because the canonical natural transformation ([32,
Appendix], see also [26, Definition 7.2])

(9.4) $KK_{\ast}^{G}(A,B)\to E_{\ast}^{G}(A,B)$

is an isomorphism when $A$ is a proper, nuclear $G$-$C^{*}$-algebra (more
generally when $A\mapsto KK_{\ast}^{G}(A,B)$ is half-exact), in particular
when $A=C_{0}(X)$ for a proper $G$-space $X$ ([32, Corollary A.3, A.4]) and
because the Baum–Connes assembly map in equivariant $E$-theory (see [21])

(9.5) $\mu^{B,G}_{X}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\to K_{\ast}(B\rtimes_{r}G)$

is known to be equivalent to the one (9.3) in $KK^{G}$ via (9.4) (see [32,
Remark A.5]).

We first recall some materials from [21]. For any (not necessarily separable)
$G$-$C^{*}$-algebras $A$ and $B$, we let

$\mathfrak{T}(B)=C_{b}([1,\infty),B)_{\mathrm{Gcont}},\,\,\,\mathfrak{T}_{0}(B)=C_{0}([1,\infty),B).$

The asymptotic algebra of $B$ is defined as

$\mathfrak{A}(B)=\mathfrak{T}(B)/\mathfrak{T}_{0}(B).$

An (equivariant) asymptotic morphism from $A$ to $B$ is an equivariant
$\ast$-homomorphism from $A$ to $\mathfrak{A}(B)$. A homotopy of asymptotic
morphisms is given by an asymptotic morphism from $A$ to $BI=B\otimes C[0,1]$.
The set of homotopy equivalence classes of asymptotic morphisms from $A$ to
$B$ is denoted by $[[A,B]]_{1}$. More generally, $[[A,B]]_{n}$ is the set of
$n$-homotopy classes of equivariant $\ast$-homomorphisms from $A$ to
$\mathfrak{A}^{n}(B)$ where $\mathfrak{A}^{n}$ is the $n$-fold composition of
the functor $\mathfrak{A}$ with itself and an $n$-homotopy is given by an
equivariant $\ast$-homomorphism from $A$ to $\mathfrak{A}^{n}(BI)$ (see [21,
Definition 2.6]). The set $[[A,B]]$ is defined as the natural inductive limit
of $[[A,B]]_{n}$ (see [21, Definition 2.7]). If $A$ is separable, the set
$[[A,B]]$ can be naturally identified as $[[A,B]]_{1}$ ([21, Theorem 2.16]).

For any $G$-$C^{*}$-algebras $A,B,C$, we have a well-defined associative
composition law

$[[A,B]]\times[[B,C]]\to[[A,C]]$

[21, Proposition 2.12]. If $A$ is separable, the composition of asymptotic
morphisms $(\phi_{t})_{t\in[1,\infty)}\colon A\to\mathfrak{A}(B)$ and
$(\psi_{t})_{t\in[1,\infty)}\colon B\to\mathfrak{A}(C)$ can be represented by
an asymptotic morphism $\psi_{s(t)}\circ\phi_{t}\colon A\to\mathfrak{A}(C)$
where $(t\mapsto s(t))$ is an increasing function on $[1,\infty)$ such that
$s(t)\to\infty$ as $t\to\infty$ sufficiently fast [15] or alternatively by an
asymptotic morphism $\psi_{t}\circ\phi_{r(t)}\colon A\to\mathfrak{A}(C)$ where
$(t\mapsto r(t))$ is a continuous function on $[1,\infty)$ such that
$r(t)\to\infty$ sufficiently slowly [21, Lemma 2.17, Claim 2.18] (both ways of
representing the compositions are homotopic).

For any $G$-$C^{*}$-algebras $A,B,C$, we have a well-defined functor given by
the maximal tensor product with the identity

$\otimes_{\mathrm{max}}\mathrm{id}_{C}\colon[[A,B]]\to[[A\otimes_{\mathrm{max}}C,B\otimes_{\mathrm{max}}C]]$

[21, Proposition 4.4] and the maximal descent

$[[A,B]]\to[[A\rtimes_{\mathrm{max}}G,B\rtimes_{\mathrm{max}}G]]$

[21, Theorem 4.12].

Let $\mathcal{H}_{G}=l^{2}(\mathbb{N})\otimes L^{2}(G)$ and
$\Sigma=C_{0}(\mathbb{R})\otimes$ be the suspension functor. For any (not
necessarily separable) $G$-$C^{*}$-algebras $A,B$, the equivariant $E$-theory
group $E^{G}(A,B)$ [21, Definition 6.8] is defined as

$E^{G}(A,B)=[[\Sigma A\otimes\mathfrak{K}(\mathcal{H}_{G}),\Sigma
B\otimes\mathfrak{K}(\mathcal{H}_{G})]].$

If $A,B$ are separable, we define for $i=0,1$,

$E_{i}^{G}(A,B)=E^{G}(\Sigma^{i}A,B).$

For any $G$-$C^{*}$-algebras $A,B$, $E^{G}(A,B)$ is an abelian group and the
composition law

$E^{G}(A,B)\times E^{G}(B,C)\to E^{G}(A,C)$

is bilinear. In this way, we have the additive category $E^{G}$ whose objects
are $G$-$C^{*}$-algebras and the morphisms groups are $E^{G}(A,B)$ [21,
Theorem 6.9]. There are natural isomorphisms $K_{0}(B)\cong E(\mathbb{C},B)$,
$K_{1}(B)\cong E(\Sigma,B)$ for any $B$ (with trivial $G$-action). [21,
Theorem 6.24].

The Bott periodicity theorem and the half-exactness of the bi-functor
$(A,B)\mapsto E^{G}(A,B)$ are only proved for separable $A,B$. On the other
hand, as we recalled above, the composition law, the (maximal) tensor product
and the (maximal) crossed product are defined for general $A,B$. We have a
bilinear map

$E^{G}(A,D_{1}\otimes_{\mathrm{max}}B)\times
E^{G}(B\otimes_{\mathrm{max}}D_{2},C)\to
E^{G}(A\otimes_{\mathrm{max}}D_{2},D_{1}\otimes_{\mathrm{max}}C)$

given by the maximal tensor product with the identity on $\mathrm{id}_{D_{2}}$
on the first slot and with the identity on $\mathrm{id}_{D_{1}}$ on the second
slot, followed by the composition law. The maximal decent defines a group
homomorphism

$j^{G}_{\mathrm{max}}\colon E^{G}(A,B)\to
E(A\rtimes_{\mathrm{max}}G,B\rtimes_{\mathrm{max}}G)$

which is functorial in both variables. When $A$ is a proper algebra (more
generally, if $A\rtimes_{r}G=A\rtimes_{\mathrm{max}}G$), we define

$j^{G}_{r}\colon E^{G}(A,B)\to E(A\rtimes_{r}G,B\rtimes_{r}G)$

by the composition of $j^{G}_{\mathrm{max}}$ and the map
$E(A\rtimes_{r}G,B\rtimes_{\mathrm{max}}G)\to E(A\rtimes_{r}G,B\rtimes_{r}G)$
induced by the quotient map $B\rtimes_{\mathrm{max}}G\to B\rtimes_{r}G$.

For any separable $G$-Hilbert space $H$, any asymptotic morphism $\phi\colon
A\to\mathfrak{A}(B\otimes\mathfrak{K}(H))$ defines an element in $E^{G}(A,B)$.
This is given by the suspension of the tensor product

$\phi\otimes\mathrm{id}_{\mathfrak{K}(\mathcal{H}_{G})}\colon
A\otimes\mathfrak{K}(\mathcal{H}_{G})\to\mathfrak{A}(B\otimes\mathfrak{K}(H)\otimes\mathfrak{K}(\mathcal{H}_{G}))\cong\mathfrak{A}(B\otimes\mathfrak{K}(\mathcal{H}_{G}))$

using any isomorphism $\mathcal{H}\otimes\mathcal{H}_{G}\cong\mathcal{H}_{G}$.

With these in mind, we do some preparation for constructing a group
homomorphism

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B).$

######  Lemma 9.3.

For any $X$-$G$-module $H_{X}$ and for any $T\in C_{L,c}^{*}(H_{X}\otimes
B)_{\mathrm{Gcpt}}^{G}$

$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)),\,\,[\phi,T]\in
C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$

holds for any $\phi\in C_{0}(X)$.

###### Proof.

The first condition follows since $T_{t}$ is locally compact for each $t$. The
second one holds because for any $T$ in the dense subalgebra
$\mathbb{C}_{L,c}(H_{X}\otimes B)_{\mathrm{Gcpt}}^{G}$ which satisfies the
second condition (2) in Definition 8.10, we have $\lVert[\phi,T_{t}]\rVert\to
0$ as $t\to\infty$. This follows from Lemma 2.6. ∎

An important example of asymptotic morphisms is obtained by the (maximal)
tensor product of two asymptotically commuting $\ast$-homomorphisms:

######  Lemma 9.4.

Let $A_{1},A_{2}$ and $B$ be $G$-$C^{*}$-algebras and let

$\phi_{i}\colon A_{i}\to C_{b}([1,\infty),M(B))$

(i=1, 2) be equivariant $\ast$-homomorphisms such that

$\phi_{1}(a_{1})\phi_{2}(a_{2})\in
C_{b}([1,\infty),B),\,\,[\phi(a_{1}),\phi(a_{2})]\in C_{0}([1,\infty),B)$

for any $a_{1}\in A$ and $a_{2}\in A_{2}$. Then, there is a (unique)
equivariant asymptotic morphism

$\phi_{1}\otimes\phi_{2}\colon
A_{1}\otimes_{\mathrm{max}}A_{2}\to\mathfrak{A}(B)$

such that the image of $a_{1}\otimes a_{2}$ is represented by

$\phi_{1}(a_{1})\phi_{2}(a_{2})\in C_{b}([1,\infty),B).$

###### Proof.

This is trivial. We just note here that the image of any equivariant
$\ast$-homomorphism from a $G$-$C^{*}$-algebra consists of $G$-continuous
elements. ∎

######  Lemma 9.5.

(c.f. [16, Section 5], [48, Construction 6.7.5]) For any $X$-$G$-module
$H_{X}$, the natural map

$\pi_{X}\colon C_{0}(X)\to C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$

and the inclusion

$\iota\colon C_{L,c}^{*}(H_{X}\otimes B)_{\mathrm{Gcpt}}^{G}\subset
C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$

induce an asymptotic morphism

$\pi_{X}\otimes\iota\colon C_{0}(X)\otimes C_{L,c}^{*}(H_{X}\otimes
B)_{\mathrm{Gcpt}}^{G}\to\mathfrak{A}(\mathfrak{K}(H_{X}\otimes B))$

such that the image of $\phi\otimes T$ is represented by

$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)).$

###### Proof.

This follows from the previous two lemmas. ∎

We recall from the previous section that the right-regular representation

$\rho\colon C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes
B))_{\mathrm{Gcont}}\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))$

restricts to

$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{L,c}^{*}(\tilde{H}_{X}\otimes B)_{\mathrm{Gcpt}}^{G}.$

Thus, we obtain the following asymptotic morphism.

######  Proposition 9.6.

For any $X$-$G$-module $H_{X}$, the natural map

$\pi_{X}\colon C_{0}(X)\to C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes
B))$

and the right-regular representation

$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))$

induce an asymptotic morphism

$\pi_{X}\otimes\rho\colon C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\to\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes B))$

such that the image of $\phi\otimes T$ is represented by

$\phi\rho(T)\in C_{b}([1,\infty),\mathfrak{K}(\tilde{H}_{X}\otimes B)).$

######  Definition 9.7.

For any $X$-$G$-module $H_{X}$, the element

$[\pi_{X}\otimes\rho]\in E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B)$

is defined by the asymptotic morphism $\pi_{X}\otimes\rho$ in Proposition 9.6.

######  Definition 9.8.

We define a group homomorphism

$\rho_{X}\colon K_{i}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{i}^{G}(C_{0}(X),B)$

for $i=0,1$, by sending

$K_{i}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\cong
E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$

to $E_{i}^{G}(C_{0}(X),B)$ by the composition with the class
$[\pi_{X}\otimes\rho]$ under the composition law

$E^{G}(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\times
E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G),B)$ $\to
E^{G}(\Sigma^{i}C_{0}(X),B).$

######  Lemma 9.9.

The group homomorphism $\rho_{X}$ is natural with respect to any
$G$-equivariant $\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$ in a sense that
the following diagram commutes

$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B_{2}).}$

###### Proof.

This follows because the following diagram commutes

$\textstyle{C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}\otimes\rho}$$\scriptstyle{\mathrm{id}_{C_{0}(X)}\otimes\pi}$$\textstyle{\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes
B_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}\otimes\rho}$$\textstyle{\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes
B_{2})).}$

∎

Now suppose that a $G$-equivariant continuous map $f\colon X\to Y$ is proper,
so that we have $f^{\ast}\colon C_{0}(Y)\to C_{0}(X)$. It defines

$f_{\ast}=(f^{\ast})^{\ast}\colon E_{\ast}^{G}(C_{0}(X),B)\to
E_{\ast}^{G}(C_{0}(Y),B).$

Let $H_{X}$ and $H_{Y}$ be an $X$-$G$-module and a $Y$-$G$-module respectively
and suppose that there is an equivariant continuous cover $(V_{t}\colon
H_{X}\to H_{Y})_{t\in[1\infty)}$ of $f$.

######  Lemma 9.10.

The following diagram commutes for any proper $f\colon X\to Y$ and for any
equivariant continuous cover $(V_{t}\colon H_{X}\to H_{Y})_{t\in[1\infty)}$ of
$f$:

$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{V_{t},\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{K_{\ast}(RL^{*}_{c}(H_{Y}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E_{\ast}^{G}(C_{0}(Y),B).}$

###### Proof.

Recall that $V_{t}\colon H_{X}\to H_{Y}$ is a cover of $f$ if and only if it
is a cover of the identity when we view $H_{X}$ as a $Y$-module $(H_{X})_{Y}$
via $f^{\ast}$. It is clear from the definition of $\rho$ that the diagram
commutes when $V_{t}$ is the identity map $H_{X}\to(H_{X})_{Y}$. Thus, it is
enough to show the claim when $X=Y$ and $(V_{t}\colon H^{0}_{X}\to
H^{1}_{X})_{t\in[1\infty)}$ is a cover of the identity map on $X$. For
$i=0,1$, we have

$\pi_{X}^{i}\otimes\rho^{i}\colon C_{0}(X)\otimes RL^{*}_{c}(H^{i}_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}^{i}\otimes B)),$

defined by the two asymptotically commuting $\ast$-homomorphisms

$\pi^{i}_{X}\colon C_{0}(X)\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{i}_{X}\otimes B)),$ $\rho^{i}\colon
RL^{*}_{c}(H^{i}_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{i}_{X}\otimes B)).$

We consider $H_{X}^{0}\oplus H_{X}^{1}$ and
$\tilde{H}_{X}^{0}\oplus\tilde{H}_{X}^{1}$. We show that two asymptotic
morphisms

$\pi_{X}^{0}\otimes\rho^{0},\,\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})\colon
C_{0}(X)\otimes RL^{*}_{c}(H^{0}_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{A}(\mathfrak{K}((\tilde{H}_{X}^{0}\otimes
B\oplus\tilde{H}_{X}^{1}\otimes B))$

are homotopic. Note that $V_{t}\colon H^{0}_{X}\to H^{1}_{X}$ defines an
$G$-equivariant isometry $\tilde{V}_{t}=V_{t}\otimes
1\colon\tilde{H}^{0}_{X}\to\tilde{H}^{1}_{X}$ in
$C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B))$ which conjugates $\rho^{0}$ to
$\rho^{1}\circ\mathrm{Ad}_{V_{t}}$. Furthermore, since $V_{t}$ is a cover of
the identity on $X$, it satisfies

$\pi_{X}^{1}(\phi)\tilde{V}_{t}-\tilde{V}_{t}\pi_{X}^{0}(\phi)\in
C_{0}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B)),$

and so

$\pi_{X}^{1}(\phi)\tilde{V}_{t}\tilde{V}_{t}^{\ast}-\tilde{V}_{t}\pi_{X}^{0}(\phi)\tilde{V}_{t}^{\ast}\in
C_{0}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B)).$

From this, we see that the isometry $\tilde{V}_{t}$ asymptotically conjugates
$\pi_{X}^{0}\otimes\rho^{0}$ to
$\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})=\pi_{X}^{1}\otimes(\tilde{V}_{t}\rho^{0}\tilde{V}_{t}^{\ast})$
in a sense that

$\mathrm{Ad}_{\tilde{V}_{t}}(\pi_{X}^{0}\otimes\rho^{0})=\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})$

in $\mathfrak{A}(\mathfrak{K}((\tilde{H}_{X}^{0}\otimes
B\oplus\tilde{H}_{X}^{1}\otimes B))$. It follows that the two asymptotic
morphisms are homotopic. The claim follows from this.

∎

Thanks to the previous lemma, and since $X\to E^{G}(C_{0}(X),B)$ is functorial
for proper $f\colon X\to Y$, the following definition is well-defined.

######  Definition 9.11.

For any $G$-compact proper $G$-space $X$, we define a group homomorphism

$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to E_{\ast}^{G}(C_{0}(X),B)$

by

$\rho_{X}\colon K_{\ast}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{\ast}^{G}(C_{0}(X),B)$

where $H_{X}$ is the chosen universal $X$-$G$-module. More generally for any
proper $G$-space $X$, we define a group homomorphism

$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$

so that the following diagram commutes:

$\textstyle{\mathbb{D}^{B,G}_{\ast}(X)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}\mathbb{D}^{B,G}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)}$$\textstyle{\mathbb{D}^{B,G}_{\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E_{\ast}^{G}(C_{0}(Y),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$

######  Theorem 9.12.

The group homomorphisms

$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$

define a natural transformation of functors from the category
$\mathcal{PR}^{G}$ of (second countable, locally compact) proper $G$-spaces to
the category $\mathcal{GA}$ of graded abelian groups. Furthermore, the
transformation is natural with respect to a $G$-equivariant
$\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$ in a sense that the following
diagram commutes

$\textstyle{\mathbb{D}^{B_{1},G}_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{\mathbb{D}^{B_{2},G}_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B_{2}).}$

###### Proof.

The first assertion follows from Lemma 9.10. The second assertion follows from
Lemma 9.9. ∎

The next goal is to show the following:

######  Theorem 9.13.

The forget-control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ factors through the Baum–Connes assembly map
$\mu_{X}^{B,G}$ via $\rho_{X}$ in a sense that the following diagram commutes:

$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{B,G}_{X}}$

For this, we need some preparation. Recall that for any $G$-Hilbert $B$-module
$\mathcal{E}$, a Hilbert $B\rtimes_{r}G$-module $\mathcal{E}\rtimes_{r}G$ is
defined [30, Defintion 3.8]. For simplicity (for our purpose, this is enough)
we will only consider $\mathcal{E}$ of the form $H\otimes B$ where $H$ is a
$G$-Hilbert space and in this case $\mathcal{E}\rtimes_{r}G=H\otimes
B\rtimes_{r}G$ is canonically identified as the Hilbert $B\rtimes_{r}G$-module
$H\otimes(B\rtimes_{r}G)$. Kasparov defined for any $G$-equivariant
$\ast$-homomorphism $\pi\colon A\to\mathfrak{L}(\mathcal{E})$, its crossed
product $\pi\rtimes_{r}1\colon
A\rtimes_{r}G\to\mathfrak{L}(\mathcal{E}\rtimes_{r}G)$. In the case when
$\mathcal{E}=H\otimes B$, the $\ast$-homomorphism $\pi\rtimes_{r}1$ is defined
by sending $a\in A$ to

$\pi(a)\in\mathfrak{L}(\mathcal{E})\subset\mathfrak{L}(\mathcal{E}\rtimes_{r}G)$

and $g\in G$ to

$(g\otimes u_{g}\colon v\otimes f\mapsto gv\otimes
u_{g}f)\in\mathfrak{L}(H\otimes B\rtimes_{r}G)$

where $u_{g}\in M(B\rtimes_{r}G)$ is the unitary corresponding to $g$.

######  Lemma 9.14.

(c.f. [38, Proposition 5.2]) Let $X$ be a $G$-compact proper $G$-space. Let
$H_{X}$ be an $X$-$G$-module and let $\tilde{H}_{X}$ be the $X$-$G$-module as
in Definition 8.4. Consider

$\pi_{X}\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B),$

the structure map for the $X$-$G$-module $\tilde{H}_{X}\otimes B$ and consider
its Kasparov’s crossed product (see the explanation right before the lemma)

$\pi_{X}\rtimes_{r}1\colon
C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G).$

Let $p_{c}(g)=\Delta(g)^{-1/2}cg(c)$ be the cut-off projection in
$C_{0}(X)\rtimes_{r}G$ for a cut-off function $c\in C_{c}(X)$ and let

$\rho\colon\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes
B)\subset\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)$

be the right-regular representation as in Definition 8.5. There is an
adjointable isometry

$U_{c}\colon H_{X}\otimes B\rtimes_{r}G\to\tilde{H}_{X}\otimes B\rtimes_{r}G$

of Hilbert $B\rtimes_{r}G$-modules such that

$U_{c}U_{c}^{\ast}=(\pi_{X}\rtimes_{r}1)(p_{c})$

and such that the c.c.p. map

$U_{c}^{\ast}\rho U_{c}\colon\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G\to\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$

is identified as the c.c.p. map on $\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G$ (naturally viewed as a subalgebra of
$\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$ via Kasparov’s crossed product)
defined by the $G$-equivariant c.c.p. map

$\mathfrak{L}(H_{X}\otimes B)\ni T\mapsto T^{\prime}=\int_{g\in
G}g(c)Tg(c)d\mu_{G}(g)\in\mathfrak{L}(H_{X}\otimes B).$

###### Proof.

An isometry $U_{c}\colon H_{X}\otimes B\rtimes_{r}G\to\tilde{H}_{X}\otimes
B\rtimes_{r}G$ is defined by sending $v\in H_{X}\otimes B\rtimes_{r}G$ to

$(G\ni h\mapsto\Delta(h)^{-1/2}(\pi_{X}\rtimes_{r}1)(c)(h\otimes u_{h})v\in
H_{X}\otimes B\rtimes_{r}G)$

in $L^{2}(G)\otimes H_{X}\otimes B\rtimes_{r}G$. Its adjoint $U_{c}^{\ast}$ is
defined by sending

$(G\ni h\mapsto v_{h}\in H_{X}\otimes B\rtimes_{r}G)\in L^{2}(G)\otimes
H_{X}\otimes B\rtimes_{r}G$

to

$\int_{h\in G}\Delta(h)^{-1/2}(h^{-1}\otimes
u_{h^{-1}})((\pi_{X}\rtimes_{r}1)(c)v_{h})d\mu_{G}(h)$

which converges weakly in $H_{X}\otimes B\rtimes_{r}G$. We can first see that
$U_{c}$ is well-defined and $\lVert U_{c}\rVert=1$ (as an a-priori not
necessarily adjointable map) and from which the weak convergence of the
formula of $U_{c}^{\ast}$ can be deduced. The equalities $U_{c}^{\ast}U_{c}=1$
and $U_{c}U_{c}^{\ast}=(\pi_{X}\rtimes_{r}1)(p_{c})$ can be checked directly.

Now, the following equalities can be checked directly,

$U_{c}^{\ast}\rho(T)U_{c}=\int_{h\in
G}h(c)Th(c)d\mu_{G}(h)\in\mathfrak{L}(H_{X}\otimes
B)\subset\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$

for any $T\in\mathfrak{L}(H_{X}\otimes B)$ and

$U_{c}^{\ast}\rho_{g}U_{c}=g\otimes u_{g}\in\mathfrak{L}(H_{X}\otimes
B\rtimes_{r}G)$

for any $g\in G$. ∎

######  Lemma 9.15.

Let $X$ be a $G$-compact proper $G$-space. Let $H_{X}$ be any $X$-$G$-module.
Consider the element

$[\pi_{X}\otimes\rho]\in E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B).$

Let

$j^{G}_{r}([\pi_{X}\otimes\rho])\in
E(C_{0}(X)\rtimes_{r}G\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B\rtimes_{r}G)$

be its reduced crossed product (which is defined since the first variable is a
proper $G$-$C^{*}$-algebra). Let

$[p_{c}]\in E(\mathbb{C},C_{0}(X)\rtimes_{r}G)$

be the element corresponding to the cut-off projection $p_{c}$ in
$C_{0}(X)\rtimes_{r}G$ for a cut-off function $c$ on $X$. Then, the class

$j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]\in E(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G,B\rtimes_{r}G)$

is equal to the class associated to the natural inclusion map

$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)\rtimes_{r}G).$

###### Proof.

The crossed product $j^{G}_{r}([\pi_{X}\otimes\rho])$ is represented by the
product of the two asymptotically commuting representations

$\pi_{X}\rtimes_{r}1\colon
C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)\subset
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G))$

and

$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))\subset
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)).$

The class $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is represented by the
asymptotic morphism from $RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G$ to
$\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes B\rtimes_{r}G))$ which is
represented by the c.c.p. map

$\pi_{X}\rtimes_{r}1(p_{c})\rho\pi_{X}\rtimes_{r}1(p_{c})\colon
RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(\tilde{H}_{X}\otimes B\rtimes_{r}G)).$

In view of Lemma 9.14, this map can be identified as the c.c.p. map

$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B\rtimes_{r}G))$

defined as the natural inclusion

$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B\rtimes_{r}G))$

preceded by the c.c.p. map on $RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G$ induced
by the $G$-equivariant c.c.p. map

$T_{t}\mapsto\int_{h\in G}h(c)T_{t}h(c)d\mu_{G}(h)$

on $RL_{c}^{\ast}(H_{X}\otimes B)$. Thus, to prove our claim, we just need to
show that for any $T\in RL^{*}_{c}(H_{X}\otimes B)$,

$\lVert T_{t}-\int_{h\in G}h(c)T_{t}h(c)d\mu_{G}(h)\rVert\to 0$

as $t\to\infty$. We may assume $T$ has uniform compact support in $X$. In this
case, $h(c)T_{t}=0$ for $h\in G\backslash K$ for some compact subset $K$ of
$G$ and $\lVert[h(c),T_{t}]\rVert\to 0$ uniformly in $h\in K$ as $t\to\infty$.
We see the last assertion holds so we are done. ∎

######  Lemma 9.16.

Let $X$ be a $G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module.
The forget-control map $\mathcal{F}\colon K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(B\rtimes_{r}G)$ factors through
$E_{\ast}^{G}(C_{0}(X),B)$ via $\rho_{X}$. That is, the following diagram
commutes,

$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\mu^{G}_{X}}$

###### Proof.

Let $[\phi]\in K_{i}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)=E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$. Using
the functoriality of the crossed product functor and the composition law, we
see that the assembly map $\mu^{G}_{X}$ sends $\rho_{X}([\phi])$ to the
composition of $[\phi]$ with the element

$j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]\in E(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G,B\rtimes_{r}G)$

under the composition law

$E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\times
E(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G,B\rtimes_{r}G)\to
E(\Sigma^{i},B\rtimes_{r}G).$

By Lemma 9.15, the element $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is
represented by the natural inclusion

$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)\rtimes_{r}G).$

That is $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is represented by a
continuous family $(\mathrm{ev}_{t})_{t\in[1,\infty)}$ of $\ast$-homomorphisms
(evaluation at $t$). On the other hand, such a continuous family of
$\ast$-homomorphisms is homotopic (as an asymptotic morphism) to the constant
one $\mathrm{ev}_{1}$. It is now clear that the element
$\mu^{G}_{X}\circ\rho_{X}([\phi])$ coincides in
$E_{\ast}(\mathbb{C},B\rtimes_{r}G)$ with the one represented by the
composition of $\phi$ with the evaluation map $RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{K}(H_{X})\otimes B\rtimes_{r}G$ at $t=1$ so we are
done. ∎

###### Proof of Theorem 9.13.

The theorem follows from Lemma 9.16. ∎

As the last thing in this section, we prove that the natural transformation

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$

is an isomorphism when $G$ is discrete and if $X$ is $G$-equivariantly
homotopic to a $G$-CW complex. Recall that for any open subgroup $H$ of $G$
and for any proper $H$-space $Y$, if $X$ is the balanced product
$G\times_{H}Y$, we have a natural isomorphism (see Theorem 7.4 (2))

$\mathbb{D}_{\ast}^{B,G}(X)\cong\mathbb{D}_{\ast}^{B,H}(Y).$

We also have a natural isomorphism ([21, Lemma 12.11], [13, Proposition 5.14])

$E_{\ast}^{G}(C_{0}(X),B)\cong E_{\ast}^{H}(C_{0}(Y),B).$

The rightward map is obtained by the restriction to the $H$-$C^{*}$-subalgebra
$C_{0}(Y)\subset C_{0}(X)$.

######  Lemma 9.17.

The natural transformation $\rho_{X}$ commutes with the induction. That is,
for any open subgroup $H$ of $G$, for any proper $H$-space $Y$ and for
$X=G\times_{H}Y$, the following diagram commutes.

$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho_{X}}$$\textstyle{E^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{D}_{\ast}^{B,H}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E^{H}(C_{0}(Y),B).}$

###### Proof.

Recall that the isomorphism
$\mathbb{D}_{\ast}^{B,H}(Y)\cong\mathbb{D}_{\ast}^{B,G}(X)$ is obtained by the
canonical inclusion

$RL^{*}_{c}(H_{Y}\otimes B)\rtimes_{r}H\to RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G.$

With this in mind, the claim is direct to check. ∎

######  Theorem 9.18.

Let $G$ be a countable discrete group and $X$ be a proper $G$-space which is
$G$-equivariantly homotopy equivalent to a $G$-CW complex. Then, the group
homomorphism

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$

is an isomorphism for any separable $G$-$C^{*}$-algebra $B$.

###### Proof.

Since both functors satisfy the axioms (1)-(5) in Theorem 7.4 and since
$\rho_{X}$ is a natural transformation which commutes with the induction from
a finite subgroup (more generally from an open subgroup), it is enough to
prove that $\rho_{X}$ is an isomorphism when $G$ is a finite group $H$ and $X$
is a point. On the other hand, we have the following commutative diagram

$\textstyle{\mathbb{D}_{\ast}^{B,H}(\mathrm{point})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}H)}$$\textstyle{E_{\ast}^{H}(\mathbb{C},B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{r}^{B,H}}$

We know that the assembly map $\mu_{r}^{B,H}$ is an isomorphism for any finite
group $H$. We also know that the forget-control map is an isomorphism for any
finite group $H$ when $X$ is a point (see Proposition 6.5). Thus, $\rho_{X}$
is an isomorphism. Of course, we may also directly show that $\rho_{X}$ is an
isomorphism. ∎

######  Theorem 9.19.

Let $G$ be a countable discrete group $G$ and $X$ be a proper $G$-space which
is $G$-equivariantly homotopy equivalent to a $G$-CW complex. The forget-
control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ is naturally equivalent to the Baum–Connes assembly
map $\mu_{X}^{B,G}$ for any separable $G$-$C^{*}$-algebra $B$.

###### Proof.

This follows from Theorem 9.13 and Theorem 9.18. ∎

Let $RL^{0}_{c}(H_{X}\otimes B)$ be the kernel of the evaluation map
$\mathrm{ev}_{1}$ on $RL^{*}_{c}(H_{X}\otimes B)$. The short exact sequence

$0\to RL^{0}_{c}(H_{X}\otimes B)\to RL^{*}_{c}(H_{X}\otimes
B)\to\mathfrak{K}(H_{X})\otimes B\to 0$

admits a $G$-equivariant c.c.p. splitting (by extending constantly and by
multiplying a bump function) and thus it descends to the short exact sequence

$0\to RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G\to RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to 0.$

######  Corollary 9.20.

For any countable discrete group $G$, the Baum–Connes assembly map
$\mu^{B,G}_{r}$ is an isomorphism if and only if

$K_{\ast}(RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G)=0$

for a universal (or ample) $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.

We also have the following short exact sequence

$0\to(RL^{0}_{c}(H_{X})\otimes B)\rtimes_{r}G\to(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to 0.$

Recall that the natural transformation $\mathbb{D}^{\otimes
B,G}_{\ast}(X)\to\mathbb{D}^{B,G}_{\ast}(X)$ is an isomorphism if $G$ is
discrete and $X$ is $G$-equivariantly homotopy equivalent to a proper $G$-CW
complex (Theorem 7.9). Hence, we also have the following:

######  Theorem 9.21.

Let $G$ be a countable discrete group $G$ and $X$ be a proper $G$-space which
is $G$-equivariantly homotopy equivalent to a $G$-CW complex. The forget-
control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{\otimes B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ is naturally equivalent to the Baum–Connes assembly
map $\mu_{X}^{B,G}$ for any separable $G$-$C^{*}$-algebra $B$.

######  Corollary 9.22.

For any countable discrete group $G$, the Baum–Connes assembly map
$\mu^{B,G}_{r}$ is an isomorphism if and only if

$K_{\ast}((RL^{0}_{c}(H_{X})\otimes B)\rtimes_{r}G)=0$

for a universal (or ample) $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.

##  10\. $\rho_{X}$ is an isomorphism, part I

We begin our proof of the following:

######  Theorem.

Let $G$ be a locally compact group. The natural transformation

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$

is an isomorphism for any proper $G$-space $X$ and for any separable
$G$-$C^{*}$-algebra $B$.

The proof will be given over the next four sections. We will entirely focus on
the case when $X$ is $G$-compact. Note that the general case follows from
this. Here is a rough idea of the proof. Let $X$ be a $G$-compact proper
$G$-space. We first show (in this section) that the right-regular
representation induces an isomorphism

$\rho\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$

whenever $H_{X}$ is a universal $X$-$G$-module. Then, we prove an isomorphism

$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\cong
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$

following the idea of [16].

Let $X$ be a $G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module. We
recall that the equivariant Roe algebra $C^{*}(H_{X}\otimes
B)^{G}=C^{*}(H_{X}\otimes B)^{G}_{\mathrm{Gcpt}}$ is defined as the completion
of the $\ast$-algebra $\mathbb{C}(H_{X}\otimes B)^{G}$ consisting of
$G$-equivariant, locally compact operators in $\mathfrak{L}(H_{X}\otimes B)$
that are properly supported. Note if $X_{0}$ is any compact subset of $X$ such
that $GX_{0}=X$ then a $G$-equivariant operator $T$ on $H_{X}\otimes B$ is
properly supported if and only if there is a compact subset $X_{1}$ of $X$ so
that $\chi_{X_{0}}T=\chi_{X_{0}}T\chi_{X_{1}}$ and
$T\chi_{X_{0}}=\chi_{X_{1}}T\chi_{X_{0}}$. We also recall that the localized
equivariant Roe algebra $C^{*}_{L,u}(H_{X}\otimes
B)^{G}=C^{*}_{L,u}(H_{X}\otimes B)^{G}_{\mathrm{Gcpt}}$ is the norm completion
of the $\ast$-subalgebra $\mathbb{C}_{L,u}(H_{X}\otimes B)^{G}$ of
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ consisting of
$\mathbb{C}(H_{X}\otimes B)^{G}$-valued functions $T_{t}$ with
$\mathrm{prop}(T_{t})\to 0$ with respect to a (any) fixed metric $d$ on
$X^{+}$.

Let $H_{X}$ be any $X$-$G$-module. We introduce several $C^{*}$-subalgebras of
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ containing the localized
equivariant Roe algebra. Let

$\pi\colon C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$

be the structure map for the $X$-$G$-module $H_{X}\otimes B$. We let
$\mathcal{C}(\pi)^{G}$ to be the $C^{*}$-subalgebra of
$\mathfrak{L}(H_{X}\otimes B)$ consisting of $G$-equivariant, locally compact
operators. We have an inclusion

$C^{*}(H_{X}\otimes B)^{G}\subset\mathcal{C}(\pi)^{G}.$

  * •

$\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ consisting of $T_{t}$ such
that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ for any $\phi\in C_{0}(X)$.

  * •

$\mathcal{C}_{L}(\pi)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),\mathcal{C}(\pi)^{G})$ consisting of functions $T_{t}$
such that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ for any $\phi\in
C_{0}(X)$.

We have inclusions

$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}\subset\mathcal{C}_{L}(\pi)^{G}.$

Let $c$ be a cut-off function on $X$. A $G$-equivariant u.c.p. map $\psi_{c}$
on $\mathfrak{L}(H_{X}\otimes B)$ defined as

$\psi_{c}\colon T\mapsto\int_{g\in G}g(c)Tg(c)d\mu_{G}(g)$

extends to a $G$-equivariant u.c.p. map on
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$. Note that $\psi_{c}$ sends
any $G$-equivariant, locally compact operator to a $G$-equivariant, locally
compact operator which is properly supported. Thus, $\psi_{c}$ maps
$\mathcal{C}(\pi)^{G}$ to $\mathbb{C}(H_{X}\otimes B)^{G}\subset
C^{*}(H_{X}\otimes B)^{G}$. Moreover, $\psi_{c}$ sends
$\mathcal{C}_{L}(\pi)^{G}$ to $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$.

######  Lemma 10.1.

Let $T\in\mathcal{C}_{L}(\pi)^{G}$. The following are equivalent:

  1. (1)

$\lim_{t\to\infty}\lVert T_{t}-\psi_{c}(T_{t})\rVert=0$ for any cut-off
function $c$ on $X$.

  2. (2)

$\lim_{t\to\infty}\lVert T_{t}-\psi_{c}(T_{t})\rVert=0$ for some cut-off
function $c$ on $X$.

  3. (3)

There is $S$ in $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ such that
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ and $S$ is properly supported in
a sense that for any compact subset $A$ of $X$, there is a compact subset $B$
of $X$ such that $\chi_{A}S=\chi_{A}S\chi_{B}$ and
$S\chi_{A}=\chi_{B}S\chi_{A}$.

  4. (4)

There is $S$ in $\mathcal{C}_{L}(\pi)^{G}$ such that $\lim_{t\to\infty}\lVert
T_{t}-S_{t}\rVert=0$ and $S$ is properly supported.

###### Proof.

(1) $\implies$ (2): Obvious. (2) $\implies$ (3): This is because $\psi_{c}(T)$
is properly supported for any $T$. (3) $\implies$ (4): Obvious. (4) $\implies$
(1): It is enough to show that if $S$ in $\mathcal{C}_{L}(\pi)^{G}$ is
properly supported, then $\lim_{t\to\infty}\lVert
S_{t}-\psi_{c}(S_{t})\rVert=0$ for any cut-off function $c$ on $X$. Since $S$
is properly supported, there is $\chi\in C_{c}(X)$ such that $c=c\chi$ and
$cS=cS\chi$. We have

$S_{t}-\psi_{c}(S_{t})=\int_{g\in G}g(c)^{2}S_{t}-g(c)S_{t}g(c)d\mu_{G}(g)$
$=\int_{g\in G}g(c)^{2}S_{t}g(\chi)-g(c)S_{t}g(c)g(\chi)d\mu_{G}(g)=\int_{g\in
G}g(c)[g(c),S_{t}]g(\chi)d\mu_{G}(g).$

We have
$\lim_{t\to\infty}\lVert[g(c),S_{t}]\rVert=\lim_{t\to\infty}\lVert[c,S_{t}]\rVert=0$
(uniformly in $g$ in $G$). It follows $\lim_{t\to\infty}\lVert
S_{t}-\psi_{c}(S_{t})\rVert=0$. ∎

It is easy to see that the conditions (3) and (4) are preserved by taking
sums, products and adjoint and that the conditions (1) and (2) pass to the
norm limit. We define some $C^{*}$-subalgebras of
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ and $\mathcal{C}_{L}(\pi)^{G}$ as
follows:

  * •

$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ consisting of $T$
satisfying the four equivalent conditions in Lemma 10.1.

  * •

$\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}$ is the $C^{*}$-subalgebra of
$\mathcal{C}_{L}(\pi)^{G}$ consisting of $T$ satisfying the four equivalent
conditions in Lemma 10.1.

We have inclusions

$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}.}$

We have the following commutative diagram of short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L,Q}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}([1,\infty),\mathcal{C}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L,Q}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

In particular, the inclusion induces an isomorphism

$K_{\ast}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})\cong
K_{\ast}(\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}).$

######  Proposition 10.2.

The two $C^{*}$-subalgebras $C^{*}_{L,u}(H_{X}\otimes B)^{G}$ and
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ in
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ are identical.

###### Proof.

$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$: Take any $T\in
C_{b,u}([1,\infty),\mathbb{C}(H_{X}\otimes B)^{G})$ such that
$\mathrm{prop}(T_{t})\to 0$ with respect to a fixed metric $d$ on $X^{+}$. Let
$X_{0}$ be a compact subset of $X$ such that $GX_{0}=X$ and $X_{1}$ be any
compact neighborhood of $X_{0}$ in $X$. Since $\mathrm{prop}(T_{t})\to 0$,
there is $t_{0}\geq 1$ such that for any $t>t_{0}$,
$\chi_{X_{0}}T_{t}=\chi_{X_{0}}T_{t}\chi_{X_{1}}$ and
$T_{t}\chi_{X_{0}}=\chi_{X_{1}}T_{t}\chi_{X_{0}}$. This implies that $T$
satisfies the condition (3) of Lemma 10.1. We already know
$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ so
we have $C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$.

$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}\subset
C^{*}_{L,u}(H_{X}\otimes B)^{G}$: Let $T\in\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. It is enough to show that $\psi_{c}(T)\in
C^{*}_{L,u}(H_{X}\otimes B)^{G}$ for a cut-off function $c$ on $X$ since both
algebras contain the common ideal $C_{0}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})$. Since $\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ for any $\phi\in
C_{0}(X)$, there is $S\in C_{b,u}([1,\infty),\mathfrak{L}(H_{X}))$ such that
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ and such that
$\mathrm{prop}(S_{t})\to 0$ as $t\to\infty$ for some (any) fixed metric $d$ on
$X^{+}$. We can arrange it so that $S_{t}$ a $G$-continuous, locally compact
operator in $\mathfrak{L}(H_{X})$ for each $t\geq 1$. In this case, since
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ with $T$ $G$-equivariant, we see
that $S$ is $G$-continuous in $C_{b,u}([1,\infty),\mathfrak{L}(H_{X}))$. Now
consider $\tilde{S}\in C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$
defined (weakly) by

$\tilde{S}_{t}=\int_{g\in G}g(c)g(S_{t})g(c)d\mu_{G}(g).$

Note this is uniformly continuous in $t$. Indeed $S\mapsto\tilde{S}$ is a
u.c.p. map on $C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ ($S$ does not
have to be $G$-continuous for this to be well-defined). We have

$\psi_{c}(T_{t})-\tilde{S}_{t}=\int_{g\in
G}g(c)(T_{t}-g(S_{t}))g(c)d\mu_{G}(g).$

Together with $\lim_{t\to\infty}\lVert
T_{t}-g(S_{t})\rVert=\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ (uniformly
in $g$ in $G$), we see that
$\lim_{t\to\infty}\lVert\psi_{c}(T_{t})-\tilde{S}_{t}\rVert=0$. Since
$\tilde{S}_{t}\in\mathbb{C}(H_{X}\otimes B)^{G}\subset C^{*}(H_{X}\otimes
B)^{G}$ for any $t\geq 1$, to show $\psi_{c}(T)\in C^{*}_{L,u}(H_{X}\otimes
B)^{G}$ it is enough to show $\tilde{S}\in C^{*}_{L,u}(H_{X}\otimes B)^{G}$.
For this, it suffices to show $\mathrm{prop}(\tilde{S}_{t})\to 0$ as
$t\to\infty$ with respect to the fixed metric $d$ on $X^{+}$. It suffices to
show $\mathrm{prop}(\chi_{X_{0}}\tilde{S}_{t})\to 0$ and
$\mathrm{prop}(\tilde{S}_{t}\chi_{X_{0}})\to 0$ as $t\to\infty$ for any
compact subset $X_{0}$ of $X$. On the other hand,

$\chi_{X_{0}}\tilde{S}_{t}=\int_{g\in
G}\chi_{X_{0}}g(c)g(S_{t})g(c)d\mu_{G}(g)$

and $\chi_{X_{0}}g(c)=0$ for any $g\in G\backslash K$ for some compact subset
$K$ of $G$. The claim $\mathrm{prop}(\chi_{X_{0}}\tilde{S}_{t})\to 0$ now
follows since $\mathrm{prop}(g(S_{t}))\to 0$ uniformly in $g\in K$. We also
have $\mathrm{prop}(\tilde{S}_{t}\chi_{X_{0}})\to 0$ for the same reason, so
we are done. ∎

Note we have a natural inclusion ($\ast$-homomorphism)

$\iota\colon\mathcal{C}_{L}(\pi)^{G}\to M(RL^{*}_{u}(H_{X}\otimes B))\subset
M(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$

just because $T\in\mathcal{C}_{L}(\pi)^{G}$ satisfies
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$. Moreover, for any
$T\in\mathcal{C}_{L}(\pi)^{G}$, $\phi$ in $C_{0}(X)$, we have

$\phi T\in RL^{*}_{u}(H_{X}\otimes B)$

because $T_{t}$ is locally compact and
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$. We let

$\pi\rtimes_{r}1\colon C_{0}(X)\rtimes_{r}G\to M(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)$

be the crossed product of the structure map $\pi\colon C_{0}(X)\to
M(RL^{*}_{u}(H_{X}\otimes B))$. Let $c\in C_{c}(X)$ be a cut-off function on
$X$ and $p_{c}\in C_{0}(X)\rtimes_{r}G$ be the associated cut-off projection.
We have a c.c.p. map

$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})\colon\mathcal{C}_{L}(\pi)^{G}\to
RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G$

which sends $T\in\mathcal{C}_{L}(\pi)^{G}$ to the compression
$(\pi\rtimes_{r}1)(p_{c})\iota(T)(\pi\rtimes_{r}1)(p_{c})$ in
$RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G$. For any
$T\in\mathcal{C}_{L}(\pi)^{G}$, we have

$[\iota(T),(\pi\rtimes_{r}1)(p_{c})]\in RL^{*}_{0}(H_{X}\otimes
B)\rtimes_{r}G.$

Thus, after passing to the quotient $RL^{*}_{u,Q}(H_{X}\otimes
B)\rtimes_{r}G$, the c.c.p. map
$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})$ becomes a
$\ast$-homomorphism

$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})\colon\mathcal{C}_{L}(\pi)^{G}\to
RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G.$

Recall that the quotient map induces an isomorphism

$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G).$

######  Definition 10.3.

We define a group homomorphism

$\iota_{p_{c}}\colon K_{\ast}(\mathcal{C}_{L}(\pi)^{G})\to
K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)$

as the composition of the group homomorphism

$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})_{\ast}\colon
K_{\ast}(\mathcal{C}_{L}(\pi)^{G})\to
K_{\ast}(RL^{*}_{u,Q}(H_{X})\rtimes_{r}G)$

and the inverse of the isomorphism

$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G)$

induced by the quotient map. If $A$ is any of the subalgebras

$\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(H_{X}\otimes
B)^{G}$

of $\mathcal{C}_{L}(\pi)^{G}$, we set

$\iota_{p_{c}}\colon K_{\ast}(A)\to K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)$

be the restriction of $\iota_{p_{c}}$ to the subalgebra $A$.

Now, for an $X$-$G$-module $H_{X}$, let $\tilde{H}_{X}=H_{X}\otimes L^{2}(G)$
be the $X$-$G$-module as before (see Definition 8.4). Let

$\tilde{\pi}\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$

be the structure map. We have the right-regular representation (see
Proposition 8.14)

$\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$

and the inclusions

$\textstyle{C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{\pi})^{G}.}$

######  Definition 10.4.

If $A$ is any of the algebras

$\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G},\mathcal{C}_{L}(\tilde{\pi})^{G},$

we set

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(A)$

to be the group homomorphism induced by the right-regular representation

$\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$

followed by the inclusion from $C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$ to
$A$.

We are now interested in computing the compositions

$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$

and

$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(H_{X}\otimes
B)^{G})\to K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}).$

Let us first compute the first one. We remind that a proper $G$-space $X$ has
been assumed to be $G$-compact from the beginning of this section.

######  Lemma 10.5.

For any (not-necessarily separable) $G$-$C^{*}$-algebra $A$ equipped with a
non-degenerate representation of the $G$-$C^{*}$-algebra $C_{0}(X)$ to $M(A)$,
the right regular representation

$\rho\colon A\rtimes_{r}G\to M(A\otimes\mathfrak{K}(L^{2}(G))),\,\,A\ni
a\mapsto(g(a))_{g\in G},\,\,G\ni g\mapsto\rho_{g},$

is an isomorphism onto the $C^{*}$-algebra
$M(A\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$ defined as the completion of the
$\ast$-algebra consisting of $G$-equivariant, locally compact, properly
supported operators in
$M(A\otimes\mathfrak{K}(L^{2}(G)))\cong\mathfrak{L}(A\otimes L^{2}(G))$.

###### Proof.

The proof of Proposition 8.7 works verbatim. ∎

######  Remark 10.6.

The algebra $M(A\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$ is known as a
generalized fixed point algebra and this lemma is proved in a more general
setting ([11, Corollary 3.25, Remark 3.26] see also [10] [12] for a more
general theory).

######  Lemma 10.7.

The composition $\iota_{p_{c}}\circ\rho_{\ast}\colon
K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$ coincides with the
map induced by the strict cover $V_{c}\colon H_{X}\to\tilde{H}_{X}$ of the
identity map on $X$ defined as

$V_{c}\colon H_{X}\ni v\mapsto(g\mapsto g(c)v)\in\tilde{H}_{X}=H_{X}\otimes
L^{2}(G).$

###### Proof.

A c.c.p. map

$\iota^{1}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G\to(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G$

is defined as the compression of the right-regular representation

$RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\ni T\to\rho(T)\in
M(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))$

by the projection $p^{(1)}_{c}$ which is the image of the cut-off projection
$p_{c}$ in $C_{0}(X)\rtimes_{r}G$ by the map

$\pi^{1}\rtimes_{r}1\colon C_{0}(X)\rtimes_{r}G\to M((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)$

induced by the natural $G$-equivariant representation

$\pi^{1}\colon C_{0}(X)\to M(RL^{*}_{u}(H_{X}))\to M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))).$

Passing to the quotient, the c.c.p. map $\iota^{1}_{p_{c}}$ becomes a
$\ast$-homomorphism

$\iota^{1}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G\to(RL^{*}_{u,Q}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G.$

Composing its induced map on K-theory with the inverse of the isomorphism (the
quotient map)

$K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)\cong
K_{\ast}((RL^{*}_{u,Q}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G),$

we obtain a group homomorphism

$\iota^{1}_{p_{c}}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G).$

Here, we used that the reduced crossed product preserves the exact sequence

$0\to RL^{*}_{0}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to
RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))$ $\to
RL^{*}_{u,Q}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to 0$

since the quotient is a proper $G$-$C^{*}$-algebra.

The composition $\iota_{p_{c}}\circ\rho_{\ast}$ factors through the natural
inclusion

$(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G\to
RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G$

via the group homomorphism $\iota^{1}_{p_{c}}$.

We now consider

$\mathrm{Ad}_{V_{c}\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G).$

Let
$p_{c}^{(2)}=V_{c}V_{c}^{\ast}\in\mathfrak{L}(\tilde{H}_{X})\subset\mathfrak{L}(\tilde{H}_{X}\otimes
B)$. The $G$-equivariant projection $p^{(2)}_{c}$ is the image of $p_{c}\in
C_{0}(X)\rtimes_{r}G$ by the right-regular representation

$C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X})=\mathfrak{L}(H_{X}\otimes
L^{2}(G)),C_{0}(X)\ni\phi\mapsto(g(\phi))_{g\in G},\,\,G\ni g\mapsto\rho_{g}.$

We claim that the $\ast$-homomorphism

$\mathrm{Ad}_{V_{c}}\colon RL^{*}_{u}(H_{X}\otimes B)\ni T\mapsto
V_{c}TV_{c}^{\ast}\in RL^{*}_{u}(\tilde{H}_{X}\otimes B)$

and the $G$-equivariant c.c.p. map

$\iota^{2}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes B)\ni T\mapsto
p^{(2)}_{c}(T)_{g\in G}p^{(2)}_{c}\in RL^{*}_{u}(\tilde{H}_{X}\otimes B)$

coincide after passing to the quotient $RL^{*}_{u,Q}(\tilde{H}_{X}\otimes B)$.
Indeed, for $S$ in $\mathfrak{L}(H_{X}\otimes B)$, we have

$V_{c}^{\ast}(S)_{g\in G}V_{c}=\int_{g\in G}g(c)Sg(c)d\mu_{g}(g).$

Thus, for any $T\in RL^{*}_{u}(H_{X}\otimes B)$, we have

$\lim_{t\to\infty}\lVert T_{t}-V_{c}^{\ast}(T_{t})_{g\in G}V_{c}\rVert=0.$

We can see this for example by considering $T$ which has uniform compact
support.

The $G$-equivariant c.c.p. map $\iota^{2}_{p_{c}}$ naturally factors through
the inclusion

$RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to
RL^{*}_{u}(\tilde{H}_{X}\otimes B).$

Overall, in order to show
$\iota_{p_{c}}\circ\rho_{\ast}=\mathrm{Ad}_{V_{c}\ast}$ on
$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$, it suffices to show that
the two group homomorphisms

$\iota^{1}_{p_{c}},\iota^{2}_{p_{c}}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)$

induced by the c.c.p. maps $\iota^{1}_{p_{c}},\iota^{2}_{p_{c}}$ (which become
$\ast$-homomorphisms after passing to the quotient) coincide. On the other
hand, by viewing

$(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G\cong
M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$

as in Lemma 10.5, we can see that the two map $\iota^{1}_{p_{c}}$ and
$\iota^{2}_{p_{c}}$ are conjugate by the unitary $U$ in the multiplier algebra
of

$M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$

defined as the flip on $G\times G$, i.e. $U\in\mathfrak{L}(L^{2}(G\times G))$
defined by $Uf(g_{1},g_{2})=f(g_{2},g_{1})$ for $f\in L^{2}(G\times G)$ so we
are done. ∎

######  Corollary 10.8.

The composition

$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$

is an isomorphism whenever $H_{X}$ is a universal $X$-$G$-module.

We now study the other composition.

######  Lemma 10.9.

The composition

$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(H_{X}\otimes
B)^{G})\to K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$

coincides with the map induced by the strict cover $V_{c}\colon
H_{X}\to\tilde{H}_{X}$ of the identity map on $X$ defined as

$V_{c}\colon H_{X}\ni v\mapsto(g\mapsto g(c)v)\in\tilde{H}_{X}=H_{X}\otimes
L^{2}(G).$

###### Proof.

We can directly see that the composition is induced by the c.c.p. map (which
becomes a $\ast$-homomorphism after passing to the quotient),

$p_{c}\cdot p_{c}\colon C^{*}_{L,u}(H_{X}\otimes B)^{G}\ni T\mapsto
p_{c}(T)_{g\in G}p_{c}\in C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$

where the cut-off projection $p_{c}$ is represented by the right-regular
representation of $C_{0}(X)\rtimes_{r}G$:

$C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X})=\mathfrak{L}(H_{X}\otimes
L^{2}(G)),C_{0}(X)\ni\phi\mapsto(g(\phi))_{g\in G},\,\,G\ni g\mapsto\rho_{g}.$

As before, more precisely, this c.c.p. map induces the map on K-theory as the
composition of the $\ast$-homomorphism

$p_{c}\cdot p_{c}\colon C^{*}_{L,u}(H_{X}\otimes B)^{G}\to
C^{*}_{L,Q}(\tilde{H}_{X}\otimes B)^{G}=C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}/C_{0}([1,\infty),C^{*}(\tilde{H}_{X}\otimes B)^{G})$

and the inverse of the isomorphism (the quotient map)

$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(C^{*}_{L,Q}(\tilde{H}_{X}\otimes B)^{G}).$

We have $p_{c}=V_{c}V_{c}^{\ast}$ and as before, for
$S\in\mathfrak{L}(H_{X}\otimes B)$,

$V_{c}^{\ast}(S)_{g\in G}V_{c}=\int_{g\in G}g(c)Sg(c)d\mu_{g}(g).$

To show that $p_{c}\cdot p_{c}$ and $\mathrm{Ad}_{V_{c}\ast}$ coincide on
$K_{\ast}(C^{*}_{L,u}(H_{X}\otimes B)^{G})$, it is enough to show that for any
$T\in C^{*}_{L,u}(H_{X}\otimes B)^{G}$, we have $\lim_{t\to\infty}\lVert
T_{t}-V^{\ast}_{c}(T_{t})_{g\in G}V_{c}\rVert=0$ but this is precisely the
condition (2) of Lemma 10.1, which is satisfied for any $T$ in
$\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}$ in particular for any $T\in
C^{*}_{L,u}(H_{X}\otimes B)^{G}=\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. ∎

The same proof shows the following:

######  Lemma 10.10.

The composition

$\rho_{\ast}\circ\iota_{p_{c}}\colon
K_{\ast}(\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}})\to
K_{\ast}(\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}})$

coincides with the map induced by the strict cover $V_{c}\colon
H_{X}\to\tilde{H}_{X}$ of the identity map on $X$.

Before jumping to the conclusion, we make an important remark here. The
equivariant Roe algebra $C^{*}(H_{X}\otimes B)^{G}$, the localized equivariant
Roe algebra $C^{*}_{L,u}(H_{X}\otimes B)^{G}$ as well as all the other
algebras such as $\mathcal{C}(\pi)^{G}$ and $\mathcal{C}_{L}(\pi)^{G}$ are not
well-behaved with respect to an arbitrary equivariant continuous cover
$(V_{t}\colon H_{X}\to H_{Y})_{t\in[1,\infty)}$ of a $G$-equivariant
continuous map $f\colon X\to Y$ even if $f$ is the identity map on a
$G$-compact space. This is because a continuous cover $V_{t}$ may not be
properly supported in general and hence the conjugation by $V_{t}$ may not
preserve locally compact operators. On the other hand, they are functorial
with respect to a strict cover $V$, if exists, of the identity map.

More relevantly, if we focus on $X$-$G$-modules of the form
$\tilde{H}_{X}\otimes B$ for an $X$-$G$-module $H_{X}$, the equivariant Roe
algebra $C^{*}(\tilde{H}_{X}\otimes B)^{G}$, the localized equivariant Roe
algebra $C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}_{\mathrm{proper}}$ as well
as $\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}$ are functorial with respect
to a $G$-equivariant continuous cover of the identity map of the form
$\tilde{V}=V\otimes 1\colon\tilde{H}_{X}\to\tilde{H}^{\prime}_{X}$ for a
$G$-equivariant continuous cover $V\colon H_{X}\to H_{X}^{\prime}$. To see
this, first, we have the following commutative diagram

$\textstyle{C^{*}(\tilde{H}_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{\tilde{V}}}$$\textstyle{C^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}}$$\textstyle{\mathfrak{K}(H_{X}\otimes
B)\rtimes_{r}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{V}}$$\scriptstyle{\cong}$$\scriptstyle{\rho}$$\textstyle{\mathfrak{K}(H^{\prime}_{X}\otimes
B)\rtimes_{r}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho}$

for any $G$-equivariant isometry $V\colon H_{X}\to H_{X}^{\prime}$, which in
particular says that $\mathrm{Ad}_{\tilde{V}}$ maps
$C^{*}(\tilde{H}_{X}\otimes B)^{G}$ to $C^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}$. Now, if $(V_{t}\colon H_{X}\to H^{\prime}_{X})_{t\in[1,\infty)}$ is a
$G$-equivariant continuous cover of the the identity map on $X$, we can see
that $\mathrm{Ad}_{\tilde{V}_{t}}$ maps $\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}$ to $\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes B)^{G}$ and maps
$C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}$ to $C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}_{\mathrm{proper}}$. To see the latter, let $X_{0}$ be a compact subset
of $X$ with $X=GX_{0}$. If $S\in\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}$
is properly supported, then
$S^{\prime}_{t}=\tilde{V}_{t}S_{t}\tilde{V}_{t}^{\ast}$ satisfies

$\chi_{X_{0}}S^{\prime}_{t}=\chi_{X_{0}}S^{\prime}_{t}\chi_{X_{1}},\,\,\,S^{\prime}_{t}\chi_{X_{0}}=\chi_{X_{1}}S^{\prime}_{t}\chi_{X_{0}}$

for some compact subset $X_{1}$ of $X$ for large enough $t$. This already
implies that $S^{\prime}$ satisfies the condition (3) of Lemma 10.1.
Similarly, for any two covers $(V_{i,t}\colon H_{X}\to
H^{\prime}_{X})_{t\in[1,\infty)}$ for $i=1,2$,
$\tilde{V}_{1,t}\tilde{V}_{2,t}^{\ast}$ multiplies
$C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}_{\mathrm{proper}}$. Using this, we see that the induced map

$\mathrm{Ad}_{\tilde{V}_{t}\ast}\colon
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes B)^{G})$

on $K$-theory is independent of the covers and it is an isomorphism whenever
$H_{X}$ and $H_{X}^{\prime}$ are universal $X$-$G$-modules.

We say that an $X$-$G$-module $H_{X}$ is of infinite multiplicity if
$H_{X}\cong H^{0}_{X}\otimes l^{2}(\mathbb{N})$ as an $X$-$G$-module for some
$X$-$G$-module $H^{0}_{X}$.

######  Lemma 10.11.

For any universal $X$-$G$-module $H_{X}$ of infinite multiplicity, the right-
regular representation $\rho$ induces an isomorphism

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}).$

###### Proof.

By Corollary 10.8, we already know that the composition

$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$

is an isomorphism. It thus suffices to show that the composition

$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G})\to K_{\ast}(C^{*}_{L,u}(\tilde{H}^{(2)}_{X}\otimes B)^{G})$

is injective (isomorphism) where $\tilde{H}^{(2)}_{X}=\tilde{H}^{\prime}_{X}$
for $H^{\prime}_{X}=\tilde{H}_{X}$. By Lemma 10.9, this composition is induced
by the isometry $V_{c}\colon H_{X}^{\prime}\to\tilde{H}_{X}^{\prime}$ which is
a strict cover of the identity map. On the other hand, $H_{X}^{\prime}\cong
H^{0}_{X}\otimes l^{2}(\mathbb{N})\otimes L^{2}(G)$ and
$\tilde{H}_{X}^{\prime}=H_{X}^{\prime}\otimes L^{2}(G)$ are isomorphic as
$X$-$G$-modules. From this, we see that
$\rho_{\ast}\circ\iota_{p_{c}}=\mathrm{Ad}_{V_{c}\ast}$ is an isomorphism. ∎

######  Theorem 10.12.

For any universal $X$-$G$-module $H_{X}$, the right-regular representation
$\rho$ and the natural inclusion induce isomorphisms

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\cong
K_{\ast}(\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}}).$

###### Proof.

The first isomorphism follows from the previous lemma using the functoriality
of both $K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$ and
$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$ with respect to a
$G$-equivariant continuous cover $(V_{t}\colon H_{X}\to
H^{\prime}_{X})_{t\in[1,\infty)}$ of the identity map on $X$, i.e. we can
assume that the universal $X$-$G$-module $H_{X}$ is of infinite-multiplicity.
We always have the second isomorphism. ∎

##  11\. $\rho_{X}$ is an isomorphism, part II

In this section, we study a $G$-equivariant analogue of [16]. We let $X$ be a
$G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module which will be
assumed to be of infinite-multiplicity at some places. We recall that
$\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=C^{*}_{L,u}(H_{X}\otimes B)^{G}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$
consisting of functions $T$ such that
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ and such that $T-\psi_{c}(T)\in
C_{0}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ for some (any) cut-off function
$c$ on $X$, where

$\psi_{c}(T)=\int_{g\in G}g(c)Tg(c)d\mu_{G}(g).$

We introduce the following $C^{*}$-algebras (c.f. [16, Section 3]).

  * •

$D^{*}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$\mathfrak{L}(H_{X}\otimes B)$ generated by $G$-equivariant, properly
supported, pseudo-local operators. Here, an operator $T$ in
$\mathfrak{L}(H_{X}\otimes B)$ is pseudo-local if
$[\phi,T]\in\mathfrak{K}(H_{X}\otimes B)$ for any $\phi$ in $C_{0}(X)$.

  * •

$\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$ consisting of functions $T$
such that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$.

  * •

$\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ consisting of
functions $T$ such that $T-\psi_{c}(T)\in C_{0}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ for some (any) cut-off function $c$ on $X$. Similarly to Lemma 10.1,
the second condition is equivalent to that there is a properly supported
function $S$ in $\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ such that $T-S\in
C_{0}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$.

  * •

$\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$
generated by functions $T$ which are properly supported.

  * •

$\mathcal{D}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$
generated by functions $T$ which are properly supported.

We have inclusions (each vertical map is an inclusion as an ideal)

$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}},}$

as well as inclusions (horizontal maps are inclusions as constant functions)

$\textstyle{C^{*}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{*}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}.}$

######  Lemma 11.1.

We have

$\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}.$

###### Proof.

Let $T\in\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. Then, we have $T\in
C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ and
$\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ and $\lim_{t\to\infty}\lVert
T_{t}-\psi_{c}(T_{t})\rVert=0$ for any cut-off function $c$ on $X$. Thus,
$T\in\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. The converse is
trivial. ∎

######  Lemma 11.2.

(c.f. [16, Lemma 4.1]) Let $D\subset C_{b,u}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ be a separable subset. Assume that the set $D$ is properly supported
in a sense that for any compact subset $A$ of $X$, there is a compact subset
$B$ so that $\chi_{A}T=\chi_{A}T\chi_{B}$ (and $T\chi_{A}=\chi_{B}T\chi_{A}$)
for any $T\in D$. There is $x\in C_{b,u}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})$ which is properly supported such that

  * •

$[x,\phi]\in C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$ for any $\phi\in
C_{0}(X)$,

  * •

$(1-x)[T,\phi]\in C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$ for any
$\phi\in C_{0}(X)$ and $T\in D$.

###### Proof.

Let $c$ be a cut-off function on $X$ and $X_{0}$ be the support of $c$ which
is a compact subset of $X$ with $GX_{0}=X$. Since $D$ is properly supported,
there is $\chi\in C_{c}(X)$ such that $c=c\chi$ and $cT=cT\chi$ for any $T\in
D$. Let $X_{1}$ be the support of $\chi$. Let $K$ be a compact subset of $G$
so that $X_{0}\cap gX_{1}=\emptyset$ for $g\in G\backslash K$. Let
$(\phi_{n})_{n}$ be a countable subset of $C_{0}(X)$ such that
$\mathrm{supp}(\phi_{n})\subset X_{0}$ for all $n$ and such that
$G(\phi_{n})_{n}$ is total in $C_{0}(X)$. Let $(T^{(m)})_{m\geq 1}$ be a
countable dense subset of $D$.

Let $y_{n}$ be an approximate unit in $\mathfrak{K}(H_{X}\otimes B)$ quasi-
central with respect to $C_{0}(X)$ so that we have
$\lim_{n\to\infty}\lVert[\phi,y_{n}]\rVert=0$ for any $\phi\in C_{0}(X)$ and
such that

$\lVert(1-y_{n})c[T^{(m)}_{t},g^{-1}(\phi_{k})]\rVert<1/n$

for any $g\in K$, $t\in[1,n+1]$, $1\leq m\leq n$, and $1\leq k\leq n$. Let

$x_{n}=\int_{g\in G}g(c)g(y_{n})g(c)d\mu_{G}(g)\in C^{*}(H_{X}\otimes B)^{G}.$

Then, we have $(n\mapsto[\phi,x_{n}])\in
C_{0}(\mathbb{N},\mathfrak{K}(H_{X}\otimes B))$ for any $\phi$ in $C_{0}(X)$.
Take any $\phi$ in $C_{0}(X)$ with support contained in $X_{0}$. For any $T\in
D$, we have

$(1-x_{n})[T_{t},\phi]=\int_{g\in
G}g(c)(1-g(y_{n}))g(c)[T_{t},\phi]g(\chi)d\mu_{G}(g)$ $=\int_{g\in
K}g(c)(1-g(y_{n}))g(c)[T_{t},\phi]g(\chi)d\mu_{G}(g),$

and

$\lVert(1-g(y_{n}))g(c)[T_{t},\phi]\rVert=\lVert(1-y_{n})c[T_{t},g^{-1}(\phi)]\rVert.$

In particular, we have

$\lVert(1-g(y_{n}))g(c)[T^{(m)}_{t},\phi_{k}]\rVert\leq 1/n$

for any $g\in K$, $t\in[1,n+1]$, $1\leq m\leq n$ and $1\leq k\leq n$ and so

$\lVert(1-x_{n})[T^{(m)}_{t},\phi_{k}]\rVert\leq C/n$

for any $t\in[1,n+1]$, $1\leq m\leq n$ and $1\leq k\leq n$, where the constant
$C>0$ only depends on fixed functions $c$ and $\chi$. A desired function $x$
can be obtained by

$x_{t}=(n+1-t)x_{n}+(t-n)x_{n+1}$

for $t\in[n,n+1]$.

∎

######  Lemma 11.3.

(c.f. [39, Proposition 2.3], [16, Proposition 4.2]) The natural map
(inclusion)

$\eta\colon\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ \to\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$

is bijective (isomorphism).

###### Proof.

It is enough to show that for any $T\in C_{b,u}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ which is properly supported, there is
$S\in\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ such that
$T-S\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.

We let $x$ as given by Lemma 11.2 for $D=\\{T\\}$. Set $S=(1-x)T$. Then, we
have

$T-S=xT\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$

since $x\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. We claim
that $S\in\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. Since $x$
and $T$ are properly supported, $S=(1-x)T$ is properly supported. Thus, it is
enough to show that $\lim_{t\to\infty}\lVert[\phi,S_{t}]\rVert=0$ but this
follows from

$[\phi,S]=-[\phi,x]T+(1-x)[\phi,T]$

and from the property of $x$. ∎

######  Lemma 11.4.

(c.f. [39, Proposition 3.5], [16, Proposition 4.3]) Assume that $H_{X}$ is of
infinite multiplicity. We have $K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$. Hence the boundary map

$\partial\colon K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\to K_{\ast+1}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})$

is an isomorphism.

###### Proof.

We have $H_{X}\cong H^{(0)}(X)\otimes l^{2}(\mathbb{N})$ and let $U_{n}$ be
isometries on $l^{2}(\mathbb{N})$ such that

$\sum_{n\geq 0}U_{n}U_{n}^{\ast}=1.$

The following map is well-defined on $\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$,

$\alpha\colon T\mapsto\sum_{n\geq 0}U_{n}T_{t+n}U_{n}^{\ast}.$

Indeed, it maps the ideal $C_{0}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$ to
itself and maps properly supported functions in $\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ to themselves. Once we have $\alpha$ well-defined,
it is routine to show that $K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$. ∎

######  Lemma 11.5.

(c.f. [39, Proposition 3.6], [16, Proposition 4.3]) Assume that $H_{X}$ is of
infinite multiplicity. The evaluation map at $t=1$ induces an isomorphism

$\mathrm{ev}_{1\ast}\colon K_{\ast}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\cong K_{\ast}(D^{*}(H_{X}\otimes
B)^{G}/C^{*}(H_{X}\otimes B)^{G}).$

###### Proof.

Let $\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ be the kernel
of the evaluation map $\mathrm{ev}_{1}$ on $\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ and let $\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. We have a split short exact sequence

$0\to\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ $\to\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\to D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G}\to 0.$

Thus, it suffices to show that $K_{\ast}(\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$ and $K_{\ast}(\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$.

Let $U_{n}$ be isometries on $H_{X}\otimes B$ as in the proof of Lemma 11.4.
The following map is well-defined on $\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ and on $\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$,

$\alpha\colon T\mapsto\sum_{n\geq 0}U_{n}T_{t-n}U_{n}^{\ast}$

where functions $T$ on $[1,\infty)$ vanishing at $t=1$ are constantly extended
to the left by zero. Indeed, it maps properly supported functions in
$\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, resp. in
$\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, vanishing at
$t=1$ to themselves and it is not to hard to see that they form a dense
subalgebra of $\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$,
resp. of $\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.

Once we have $\alpha$ well-defined, it is routine to show that
$K_{\ast}(\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}})=0$ and
$K_{\ast}(\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}})=0$. ∎

Let $\pi\colon C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$ be the structure map
for the $X$-$G$-module $H_{X}\otimes B$. We define:

  * •

$C^{*}(\pi)^{G}$ is the $C^{*}$-subalgebra of $\mathfrak{L}(H_{X}\otimes B)$
consisting of $G$-equivariant, locally compact operators.

  * •

$D^{*}(\pi)^{G}$ is the $C^{*}$-subalgebra of $\mathfrak{L}(H_{X}\otimes B)$
consisting of $G$-equivariant, pseudo-local operators.

The following is a standard fact (c.f. [27, Lemma 6.2], [40, Lemma 5.8], [28,
Lemma 12.3.2]).

######  Lemma 11.6.

The natural map

$\eta_{0}\colon D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G}\to
D^{*}(\pi)^{G}/C^{*}(\pi)^{G}$

is bijective (isomorphism).

###### Proof.

Surjectivity: Let $T\in D^{*}(\pi)^{G}$. For a cut-off function $c$ on $X$,
consider

$T^{\prime}=\int_{g\in G}g(c)Tg(c)d\mu_{G}(g).$

Then, we have $T^{\prime}\in D^{*}(H_{X}\otimes B)^{G}$ and $T-T^{\prime}\in
C^{*}(\pi)^{G}$. Injectivity: we need to show that $D^{*}(H_{X}\otimes
B)^{G}\cap C^{*}(\pi)^{G}=C^{*}(H_{X}\otimes B)^{G}$. Note that this is not a
trivial consequence of the definitions. We first claim that $T-T^{\prime}\in
C^{*}(H_{X}\otimes B)^{G}$ for any $T\in D^{*}(H_{X}\otimes B)^{G}$. In fact,
if $T$ is properly supported, this is easy to see since then, $T-T^{\prime}$
is $G$-equivariant, locally compact and properly supported. The general case
follows from the continuity of the map $T\mapsto T-T^{\prime}$. Let $T\in
D^{*}(H_{X}\otimes B)^{G}\cap C^{*}(\pi)^{G}$. Then, $T\in C^{*}(H_{X}\otimes
B)^{G}$ follows from $T^{\prime}\in C^{*}(H_{X}\otimes B)^{G}$ which is easy
to see since $T^{\prime}$ is $G$-equivariant, locally compact and properly
supported. ∎

######  Proposition 11.7.

Assume that $H_{X}$ is of infinite multiplicity. We have the following
sequence of isomorphisms.

$\textstyle{K_{\ast}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}^{-1}}$$\textstyle{K_{\ast}(D^{*}(H_{X}\otimes
B)^{G}/C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$$\textstyle{K_{\ast}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$

where $\iota$ is the natural inclusion as constant functions.

###### Proof.

Combine Lemma 11.3, Lemma 11.4, Lemma 11.5 and Lemma 11.6. ∎

##  12\. $\rho_{X}$ is an isomorphism, part III

Let us consider, in general, an $X$-$G$-Hilbert $B$-module $\mathcal{E}$, that
is a (countably generated) $G$-Hilbert $B$-module equipped with a non-
degenerate representation of the $G$-$C^{*}$-algebra $C_{0}(X)$. The support
and the propagation of operators between two $X$-$G$-Hilbert $B$-modules are
defined in an obvious way. We let $\tilde{\mathcal{E}}$ be an $X$-$G$-Hilbert
$B$-module $\mathcal{E}\otimes L^{2}(G)$ equipped with a unitary
representation

$G\ni g\mapsto u_{g}\otimes\lambda_{g}$

where $u_{g}$ is the unitary on $\mathcal{E}$ corresponding to $g\in G$ and
$\lambda_{g}$ is the left-translation by $g$ and with the structure map

$C_{0}(X)\ni\phi\mapsto\phi\otimes 1.$

For any cut-off function $c$ on $X$, we have a $G$-equivariant isometry
$V_{c}\in\mathfrak{L}(\mathcal{E},\tilde{\mathcal{E}})$ defined by sending
$v\in\mathcal{E}$ to

$(V_{c}(v)\colon h\mapsto h(c)v)\in\tilde{\mathcal{E}}=L^{2}(G,\mathcal{E}).$

The isometry $V_{c}$ is a strict cover of the identity, i.e. it commutes with
$C_{0}(X)$.

######  Lemma 12.1.

Let $G$ be a locally compact group and $X$ be a $G$-compact, proper $G$-space.
Let $H_{X}$ be an $X$-$G$-module which is ample as an $X$-module and let
$\mathcal{E}$ be the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. For
any $X$-$G$-Hilbert $B$-module $\mathcal{E}_{0}$, there is a sequence $V_{n}$
of $G$-equivariant isometries in $\mathfrak{L}(\mathcal{E}_{0},\mathcal{E})$
such that for any $\phi\in C_{0}(X)$,

  * •

$[V_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},\mathcal{E})$,

  * •

$\lim_{n\to\infty}\lVert[V_{n},\phi]\rVert=0$.

###### Proof.

Let $c\in C_{c}(X)$ be a cut-off function and $X_{0}$ be the support of $c$.

Let $W_{n}\in\mathfrak{L}(\mathcal{E}_{0},H_{X}\otimes B)$ be a sequence of
isometries satisfying the following three conditions: for any $\phi$ in
$C_{0}(X)$,

  * •

$[W_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B)$,

  * •

$\lim_{n\to\infty}\lVert[W_{n},\phi]\rVert=0$,

  * •

$W_{n}c=\chi W_{n}c$ for some $\chi\in C_{c}(X)$.

We explain how we can find such $W_{n}$. Fix a metric $d$ on $X^{+}$ and let
$\delta>0$ be small enough so that the $\delta$-neighborhood of $X_{0}$ is
relatively compact in $X$. Let $(\phi_{i})_{i\in S}$ be a finite partition of
unity in $C(X^{+})$ so that the support of each $\phi_{i}$ is contained in a
ball of radius $\delta/2$. Let $X_{i}$ be the support of $\phi_{i}$ and let
$\mathcal{E}_{0}^{(i)}=C(X^{+})\phi_{i}\mathcal{E}_{0}$ which is naturally an
$X_{i}$-module. Then, we have an isometry

$\Phi\colon\mathcal{E}_{0}\to\bigoplus_{i\in
S}\mathcal{E}_{0}^{(i)},\,\,\,v\mapsto(\phi^{1/2}_{i}v).$

Let $H_{X_{i}}=C(X^{+})\phi_{i}H_{X}$ which is naturally an ample
$X_{i}$-module. We take a sequence
$W_{n}^{(i)}\in\mathfrak{L}(\mathcal{E}^{(i)}_{0},H_{X_{i}}\otimes B)$ of
isometries satisfying,

  * •

For each $n$, $(W_{n}^{(i)})_{i\in S}$ have mutually orthogonal ranges in
$H_{X}\otimes B$,

  * •

$[W^{(i)}_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},H_{X_{i}}\otimes B)$ for
any $\phi$ in $C(X_{i})$,

  * •

$\lim_{n\to\infty}\lVert[W^{(i)}_{n},\phi]\rVert=0$ for any $\phi$ in
$C(X_{i})$.

Such isometries exist because $H_{X_{i}}$ is an ample $C(X_{i})$-module (see
[16, Theorem 2.3]). Let $W_{n}$ be the composition of $\Phi$ and the
(orthogonal) sum of $W_{n}^{(i)}$. We can see that $W_{n}$ are isometries
satisfying the three conditions above.

We now define a sequence $V_{n}$ of $G$-equivariant isometries in
$\mathfrak{L}(\mathcal{E}_{0},\mathcal{E})$ as the composition

$V_{n}=\tilde{W}_{n}V_{c}\colon\mathcal{E}_{0}\to\tilde{\mathcal{E}}_{0}\to\tilde{H}_{X}\otimes
B$

where a $G$-equivariant isometry
$\tilde{W}_{n}\colon\tilde{\mathcal{E}}_{0}\to\tilde{H}_{X}\otimes B$ is of
the form

$(\tilde{W}_{n}\colon g\mapsto g(W_{n}))\in
C_{b,\mathrm{SOT}^{*}}(G,\mathfrak{L}(\mathcal{E}_{0},H_{X}\otimes B)).$

Here, $g(W_{n})=u^{\prime}_{g}W_{n}u^{-1}_{g}$ where $u^{\prime}_{g}$ (resp.
$u_{g}$) is the unitary on $H_{X}\otimes B$, (resp. on $\mathcal{E}_{0}$)
corresponding to $g$ in $G$. Note that $g(W_{n})$ is adjointable although
$u^{\prime}_{g}$ and $u_{g}$ may not. We also note that $g\mapsto g(W_{n})$ is
not necessarily norm-continuous ($W_{n}$ may not be $G$-continuous).

We show that $V_{n}$ satisfies the two desired conditions. Concretely, $V_{n}$
maps $v\in\mathcal{E}_{0}$ to

$(h\to h(\chi)h(W_{n})h(c)v)\in L^{2}(G,H_{X}\otimes B).$

Let $\phi\in C_{c}(X)$. Then, $[V_{n},\phi]$ maps $v\in\mathcal{E}_{0}$ to

$(h\to h(\chi)[h(W_{n}),\phi]h(c)v)\in L^{2}(G,H_{X}\otimes B)$

and we note that $h(\chi)[h(W_{n}),\phi]h(c)=0$ for $h\in G\backslash K$ for
some compact subset $K$ of $G$ which only depends on the support of $\phi$
(and $c$ and $\chi$) but not on $n$. Now, we have

$[V_{n},\phi]^{\ast}[V_{n},\phi]=\int_{h\in
K}h(c)[h(W_{n}),\phi]^{\ast}h(\chi^{2})[h(W_{n}),\phi]h(c)d\mu_{G}(h).$

We note $(h\mapsto[h(W_{n}),\phi])\in
C_{b}(G,\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B))$ since we have
$(h\mapsto[W_{n},h^{-1}(\phi)])\in
C_{b}(G,\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B))$. We also have
$\lim_{n\to\infty}\lVert[W_{n},h^{-1}(\phi)]\rVert=0$ uniformly in $h\in K$.
From these, we see that

  * •

$[V_{n},\phi]^{\ast}[V_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0})$,

  * •

$\lim_{n\to\infty}\lVert[V_{n},\phi]^{\ast}[V_{n},\phi]\rVert=0$.

We are done. ∎

Let $H_{X}$ be an $X$-$G$-module which is ample as an $X$-module. Let

$\pi\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$

be the structure map for the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes
B$. Note that we may view $\pi$ as a non-degenerate representation

$\pi\colon C_{0}(X)\to M(\mathfrak{K}(\tilde{H}_{X})\otimes B)$

and $B^{\prime}=\mathfrak{K}(\tilde{H}_{X})\otimes B$ is $G$-stable in a sense
that $B^{\prime}\otimes\mathfrak{K}(L^{2}(G))\cong B^{\prime}$ as
$G$-$C^{*}$-algebras.

In general, for any separable $G$-$C^{*}$-algebra $A$ and for any separable
$G$-stable $G$-$C^{*}$-algebra $B^{\prime}$, let us say that a non-degenerate,
$G$-equivariant representation

$\pi\colon A\to M(B^{\prime})$

is non-degenerately $G$-equivariantly absorbing (c.f. [43, Section 2]) if for
any non-degenerate $G$-equivariant representation

$\pi_{0}\colon A\to M(B^{\prime}),$

there is a sequence $u_{n}$ of $G$-equivariant unitaries in
$\mathfrak{L}(B^{\prime}\oplus B^{\prime},B^{\prime})$ such that for any $a\in
A$,

  * •

$u_{n}(\pi_{0}(a)\oplus\pi(a))-\pi(a)u_{n}\in\mathfrak{K}(B^{\prime}\oplus
B^{\prime},B^{\prime})$,

  * •

$\lim_{n\to\infty}\lVert u_{n}(\pi_{0}(a)\oplus\pi(a))-\pi(a)u_{n}\rVert=0$.

It is routine to deduce from Lemma 12.1 that the structure map $\pi$ for
$\tilde{H}_{X}\otimes B$ is non-degenerately $G$-equivariantly absorbing.

######  Proposition 12.2.

For any $X$-$G$-module $H_{X}$ which is ample as an $X$-module, the structure
map

$\pi\colon C_{0}(X)\to M(B^{\prime})$

for the $X$-$G$-module $\tilde{H}_{X}\otimes B$ where
$B^{\prime}=\mathfrak{K}(\tilde{H}_{X})\otimes B$, is non-degenerately
$G$-equivariantly absorbing.

###### Proof.

We use a standard trick (see the proof of [2, Corollary 2, p341]). Let
$\pi_{0}\colon C_{0}(X)\to M(B^{\prime})$ be a non-degenerate, $G$-equivariant
representation and let $\pi_{0}^{\infty}\colon C_{0}(X)\to M(B^{\prime})$ be
its amplification so that $\pi_{0}\oplus\pi_{0}^{\infty}$ is unitarily
equivalent to $\pi_{0}^{\infty}$. Lemma 12.1 gives a sequence $v_{n}$ of
$G$-equivariant isometries in $\mathfrak{L}(B^{\prime},B^{\prime})$ such that
for any $\phi\in C_{0}(X)$,

  * •

$v_{n}\pi^{\infty}_{0}(\phi)-\pi(\phi)v_{n}\in\mathfrak{K}(B^{\prime},B^{\prime})$,

  * •

$\lim_{n\to\infty}\lVert v_{n}\pi^{\infty}_{0}(\phi)-\pi(\phi)v_{n}\rVert=0$.

Let $p_{n}=1-v_{n}v_{n}^{\ast}$ and $\sigma_{n}$ be a $G$-equivariant c.c.p.
map from $C_{0}(X)$ to $\mathfrak{L}(p_{n}B^{\prime})$ defined by
$\sigma_{n}(\phi)=p_{n}\pi(\phi)p_{n}$. We set $w_{n}$ to be a unitary in
$\mathfrak{L}(p_{n}B^{\prime}\oplus B^{\prime},B^{\prime})$ given by the
direct sum of the identity map on $p_{n}B^{\prime}$ and $v_{n}$. Then, $w_{n}$
is a $G$-equivariant unitary in $\mathfrak{L}(p_{n}B^{\prime}\oplus
B^{\prime},B^{\prime})$ such that for any $\phi\in C_{0}(X)$,

  * •

$w_{n}(\sigma_{n}(\phi)\oplus\pi^{\infty}_{0}(\phi))-\pi(\phi)w_{n}\in\mathfrak{K}(p_{n}B^{\prime}\oplus
B^{\prime},B^{\prime})$,

  * •

$\lim_{n\to\infty}\lVert
w_{n}(\sigma_{n}(\phi)\oplus\pi^{\infty}_{0}(\phi))-\pi(\phi)w_{n}\rVert=0$.

Define unitaries $u_{n}$ in $\mathfrak{L}(B^{\prime}\oplus
B^{\prime},B^{\prime})$ by the composition

$\textstyle{B^{\prime}\oplus
B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{B^{\prime}}\oplus
w_{n}^{\ast}}$$\textstyle{B^{\prime}\oplus(p_{n}B^{\prime}\oplus
B^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{p_{n}B^{\prime}\oplus(B^{\prime}\oplus
B^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{p_{n}B^{\prime}}\oplus
w}$$\textstyle{p_{n}B^{\prime}\oplus
B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w_{n}}$$\textstyle{B^{\prime}}$

Where $w\in\mathfrak{L}(B^{\prime}\oplus B^{\prime},B^{\prime})$ is a unitary
equivalence from $\pi_{0}\oplus\pi_{0}^{\infty}$ to $\pi_{0}^{\infty}$. Then,
we have for any $\phi\in C_{0}(X)$,

  * •

$u_{n}(\pi_{0}(\phi)\oplus\pi(\phi))-\pi(\phi)u_{n}\in\mathfrak{K}(B^{\prime}\oplus
B^{\prime},B^{\prime})$,

  * •

$\lim_{n\to\infty}\lVert
u_{n}(\pi_{0}(\phi)\oplus\pi(\phi))-\pi(\phi)u_{n}\rVert=0$.

∎

For any $G$-equivariant representation

$\pi\colon A\to M(B^{\prime}),$

we define $\mathcal{D}(\pi)^{G}$ to be the $C^{*}$-algebra of $G$-equivariant
elements $x$ in $M(B^{\prime})$ such that $[x,\pi(a)]\in B^{\prime}$ for any
$a\in A$ and $\mathcal{C}(\pi)^{G}$ to be the $C^{*}$-algebra of
$G$-equivariant elements $x$ in $M(B^{\prime})$ such that $x\pi(a),\pi(a)x\in
B^{\prime}$ for any $a\in A$.

We have a group homomorphism

$\Theta\colon K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\to
KK_{0}^{G}(A,B^{\prime})$

which sends a unitary $u$ in
$M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$ to the following even
Kasparov triple for $KK_{0}^{G}(A,B^{\prime})$,

$(B^{\prime\oplus n}\oplus B^{\prime\oplus n},\pi^{\oplus n}\oplus\pi^{\oplus
n},\,\begin{bmatrix}0&v\\\ v^{\ast}&0\end{bmatrix})$

where $v$ is any lift of $u$ in $M_{n}(\mathcal{D}(\pi)^{G})$. Similarly, we
have a group homomorphism

$\Theta\colon K_{0}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\to
KK_{1}^{G}(A,B^{\prime})$

which sends a projection $p$ in
$M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$ to the following odd
Kasparov triple for $KK_{1}^{G}(A,B^{\prime})$,

$(B^{\prime\oplus n},\pi^{\oplus n},2P-1)$

where $P$ is any (self-adjoint) lift of $p$ in
$M_{n}(\mathcal{D}({\pi})^{G})$.

We recall some terminologies from Section 3 [44]. An even Kasparov triple
$\mathcal{E}=(E,\phi,F)$ for $KK_{0}^{G}(A,B^{\prime})$ is called elementary
if $E=B^{\prime}\oplus B^{\prime}$ and it it called essential if $\phi(A)E=E$.
Similarly, an odd Kasparov triple $\mathcal{E}=(E,\phi,P)$ for
$KK_{1}^{G}(A,B^{\prime})$ is called elementary if $E=B^{\prime}$ and it it
called essential if $\phi(A)E=E$.

######  Proposition 12.3.

([44, Theorem 3.9, Theorem 4.2]) Let $A$ be a separable proper
$G$-$C^{*}$-algebra and $B^{\prime}$ be a separable, $G$-stable
$G$-$C^{*}$-algebra.

  1. (1a)

Every element of $KK^{G}_{0}(A,B^{\prime})$ is represented by an even Kasparov
triple which is both elementary and essential.

  2. (1b)

Two elementary and essential even Kasparov triples, $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ define the same element of $KK^{G}_{0}(A,B^{\prime})$ if and
only if there are degenerate even Kasparov triples, $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ for $KK^{G}_{0}(A,B^{\prime})$ which are both elementary and
essential, such that $\mathcal{E}_{1}\oplus\mathcal{D}_{1}$ is operator
homotopic to $\mathcal{E}_{2}\oplus\mathcal{D}_{2}$.

  3. (2a)

Every element of $KK^{G}_{1}(A,B^{\prime})$ is represented by an odd Kasparov
triple which is both elementary and essential.

  4. (2b)

Two elementary and essential odd Kasparov triples, $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ define the same element of $KK^{G}_{1}(A,B^{\prime})$ if and
only if there are degenerate odd Kasparov triples, $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ for $KK^{G}_{1}(A,B^{\prime})$ which are both elementary and
essential, such that $\mathcal{E}_{1}\oplus\mathcal{D}_{1}$ is operator
homotopic to $\mathcal{E}_{2}\oplus\mathcal{D}_{2}$.

###### Proof.

In [44], the proof was given for $A$ which is $G$-stable but the only property
of $A$ which is used for the proof is that for any $G$-equivariant
representation $\phi\colon A\to\mathfrak{L}(E)$ on a $G$-Hilbert
$B^{\prime}$-module $E$ with $\phi(A)E=E$, we have $E\oplus B^{\prime}\cong
B^{\prime}$ (see [44, Theorem 2.8, Corollary 2.9]). Any proper
$G$-$C^{*}$-algebra has this property (see [35, Proposition 8.6]). ∎

######  Theorem 12.4.

(c.f. [43, Theorem 3.2]) Let $A$ be a separable proper $G$-$C^{*}$-algebra and
$B^{\prime}$ be a separable, $G$-stable $G$-$C^{*}$-algebra. Suppose a non-
degenerate $G$-equivariant representation $\pi\colon A\to M(B^{\prime})$ is
non-degenerately $G$-equivariantly absorbing. Then, the group homomorphisms
$\Theta$ induce isomorphisms

$K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\cong
KK_{0}^{G}(A,B^{\prime})\,\,\,\text{and}\,\,\,K_{0}(\mathcal{D}(\pi)/\mathcal{C}(\pi)^{G})\cong
KK_{1}^{G}(A,B^{\prime}).$

###### Proof.

The proof of [43, Theorem 3.2] works with minor changes so we shall be brief.
We give a proof for the isomorphism
$K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\cong
KK_{0}^{G}(A,B^{\prime})$. The other one is parallel. Surjectivity: Take any
even elementary, essential, Kasparov triple $\mathcal{E}=(B^{\prime}\oplus
B^{\prime},\pi_{0}\oplus\pi_{1},\begin{bmatrix}0&x\\\
x^{\ast}&0\end{bmatrix})$ for $KK^{G}_{0}(A,B^{\prime})$. By adding an
essential, degenerate Kasparov triple

$\mathcal{D}=(B^{\prime\infty}\oplus
B^{\prime\infty},(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1})\oplus(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1}),\begin{bmatrix}0&1\\\
1&0\end{bmatrix})\oplus(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$

to $\mathcal{E}$, where $(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1})$ is the
infinite direct sum of $(\pi_{0}\oplus\pi_{1})$, we see that the triple
$\mathcal{E}\oplus\mathcal{D}$ is isomorphic to the essential triple
$\mathcal{E}_{1}$ of the form $(B^{\prime}\oplus
B^{\prime},\pi^{\prime}\oplus\pi^{\prime},\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})$ where $\pi^{\prime}(a)-\pi(a)\in B^{\prime}$ for any
$a\in A$, i.e. $\mathcal{E}_{1}$ is a compact perturbation of the triple
$(B^{\prime}\oplus B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})$. Using a cut-off function (as usual), the latter
triple is a compact perturbation of the triple of the same form where $v$ is
$G$-equivariant. Such a triple is in the image of $\Theta$.

Injectivity: We may assume $\pi$ is of infinite multiplicity. We take a
unitary $u\in\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G}$ and let
$v\in\mathcal{D}(\pi)^{G}$ be any lift of $u$ (it is enough to consider
$1\times 1$-matrix since $\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G}$ is
properly infinite.). Suppose $\Theta(u)=0$ in $KK^{G}_{0}(A,B^{\prime})$.
Then, there are essential, degenerate triples
$\mathcal{D}_{1},\mathcal{D}_{2}$ such that $(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})\oplus\mathcal{D}_{1}$ is operator homotopic to
$(B^{\prime}\oplus B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\
1&0\end{bmatrix})\oplus\mathcal{D}_{2}$. By adding infinite copies of
$\mathcal{D}_{1}\oplus\mathcal{D}_{2}$ to both, we may assume that
$\mathcal{D}_{1}=\mathcal{D}_{2}$. We can further arrange that
$\mathcal{D}_{1}=\mathcal{D}_{2}$ are of the form $(B^{\prime}\oplus
B^{\prime},\lambda\oplus\lambda,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$ with
$\lambda$ of infinite multiplicity. By adding $(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$, we can
assume that there is a $G$-equivariant unitary $w\in M(B^{\prime})$ such that
$w\lambda(a)w^{\ast}-\pi(a)\in B^{\prime}$ for any $a\in A$. From here, using
the same reasoning of the proof [43, Theorem 3.2], we get a $G$-equivariant
norm-continuous path $G_{t}$ in the matrix algebra $M_{n}(M(B^{\prime}))$ such
that $G_{0}=1_{n}$ and $G_{1}=v\oplus 1_{n-1}$ and

$(G_{t}^{\ast}G_{t}-1_{n})\pi(a),(G_{t}G_{t}^{\ast}-1_{n})\pi(a),[\pi(a),G_{t}]\in
M_{n}(B^{\prime})$

for all $t$ and for all $a\in A$. The $G$-equivariance can be arranged by
perturbing the operator homotopy into $G$-equivariant one using a cut-off
function $c$ for a proper $G$-space $X$ for which $A$ is a
$G$-$C_{0}(X)$-algebra. More precisely, $G_{t}$ would be a homotopy between
$G_{0}=1_{n}$ and $G_{1}=v^{\prime}\oplus 1_{n-1}$ where
$v^{\prime}=\int_{g\in G}g(c)g(v)g(c)d\mu_{G}(g)=\int_{g\in
G}g(c)vg(c)d\mu_{G}(g)$ but $v^{\prime}$ is still a lift of $u$. This gives a
path of unitaries in $M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$
connecting $u\oplus 1_{n-1}$ to $1_{n}$. ∎

######  Corollary 12.5.

Let $G$ be a locally compact group, $X$ be a $G$-compact proper $G$-space and
$B$ be a separable $G$-$C^{*}$-algebra. Let $H_{X}$ be an $X$-$G$-module which
is ample as an $X$-module. Let $\pi\colon
C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$ be the structure map for the
$X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. Then, there are canonical
isomorphisms

$\Theta\colon K_{1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\cong
KK_{0}^{G}(C_{0}(X),B),$ $\Theta\colon
K_{0}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\cong KK_{1}^{G}(C_{0}(X),B).$

######  Corollary 12.6.

Let $G$ be a locally compact group, $X$ be a $G$-compact proper $G$-space and
$B$ be a separable $G$-$C^{*}$-algebra. Let $H_{X}$ be a universal
$X$-$G$-module. Assume that $H_{X}$ is of infinite multiplicity. Let
$\pi\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$ be the structure
map for the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. We have the
following sequence of isomorphisms.

$\textstyle{KK_{\ast}^{G}(C_{0}(X),B)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast+1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Theta}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}^{-1}}$$\textstyle{K_{\ast+1}(D^{*}(\tilde{H}_{X}\otimes
B)^{G}/C^{*}(\tilde{H}_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$$\textstyle{K_{\ast+1}(\mathcal{D}_{T}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(\mathcal{D}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$$\textstyle{K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho_{\ast}}$$\textstyle{K_{\ast}(\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})}$

In particular, for any universal $X$-$G$-module $H_{X}$, there is a canonical
isomorphism

$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
KK_{\ast}^{G}(C_{0}(X),B).$

###### Proof.

Combine Theorem 10.12, Proposition 11.7 and Corollary 12.5. ∎

##  13\. $\rho_{X}$ is an isomorphism, part IV

Now we are ready to show that the group homomorphism

$\rho_{X}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$

is an isomorphism for any $G$-compact proper $G$-space $X$ where $H_{X}$ is a
universal $X$-$G$-module.

For any $X$-$G$-module $H_{X}$, we have a group homomorphism

$\iota_{X}\colon K_{\ast}(C_{L,u}^{*}(H_{X}\otimes B)^{G})\to
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$

defined in the same way as $\rho_{X}$ using a canonical asymptotic morphism

$\pi_{X}\otimes\iota\colon C_{0}(X)\otimes C_{L,u}^{*}(H_{X}\otimes
B)^{G}\to\mathfrak{A}(\mathfrak{K}(H_{X}\otimes B))$

such that the image of $\phi\otimes T$ is represented by

$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)).$

It is clear that we have $\rho_{X}=\iota_{X}\circ\rho_{\ast}$, that is
$\rho_{X}$ factors through the map

$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G})$

induced by the right-regular representation which is an isomorphism whenever
$H_{X}$ is universal.

######  Lemma 13.1.

(c.f. [16, Theorem 5.1]) Let $G$ be a locally compact group, $X$ be a
$G$-compact proper $G$-space and $B$ be a separable $G$-$C^{*}$-algebra. Let
$H_{X}$ be any $X$-$G$-module. Let $\pi\colon
C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$ be the structure map for the
$X$-$G$-Hilbert $B$-module $H_{X}\otimes B$. Then, the following diagram
commutes.

$\textstyle{K_{\ast+1}(D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\iota}$$\textstyle{K_{\ast+1}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{K_{\ast+1}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{KK_{\ast}^{G}(C_{0}(X),B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{\ast}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast}(C^{\ast}_{L,u}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}).}$$\scriptstyle{\iota_{X}}$

###### Proof.

The argument of the proof of [16, Theorem 5.1] works with minor changes. We
give a proof for the case $\ast=0$. The case of $\ast=1$ is simpler and this
is the one of the two cases which is explained carefully in [16]. Take a
unitary $\dot{u}$ in $D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G}$
(for simplicity we are taking a unitary in the $1\times 1$-matrix algebra). We
first compute the image of $\dot{u}$ in $E_{0}^{G}(C_{0}(X),B)$ by the clock-
wise composition. The functional calculus for $\dot{u}$ gives a
$\ast$-homomorphism

$\Sigma\ni f\mapsto f(\dot{u})\in D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G}$

where $\Sigma\cong C_{0}(0,1)$ is identified as $C_{0}(S^{1}-\\{1\\})$. We let

$\Sigma\ni f\mapsto f(u)\in D^{*}(H_{X}\otimes B)^{G}$

be its (not necessarily linear) continuous, bounded lift to
$D^{*}(H_{X}\otimes B)^{G}$ (which exists by Bartle–Graves Theorem [4, Theorem
4]). This is an abuse of notations and $f(u)$ is not the functional calculus
applied to some element $u$. We may compose this map $f\mapsto f(u)$ with a
c.c.p. map

$x\to\int_{g\in G}g(c)xg(c)d\mu_{G}(g)$

on $D^{*}(H_{X}\otimes B)^{G}$ to assume that the image of $\Sigma$ is
properly supported as a set. Note that such a composition is still a lift of
$f\mapsto f(\dot{u})$. Let $x\in\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ be as given by Lemma 11.2 for

$D=\\{f(u)\in D^{*}(H_{X}\otimes B)^{G}\subset\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\mid f\in\Sigma\\}.$

Then,

$\Sigma\ni f\mapsto(1-x)f(u)\in\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$

is a lift of the $\ast$-homomorphism from $\Sigma$ to
$\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ associated to the unitary $\dot{u}$, or more
precisely the unitary $\iota(\dot{u})\in\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. The boundary map $\partial$ sends the element
$[\dot{u}]$ in $K_{1}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})$ to the element in
$K_{0}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})$ represented by
an asymptotic morphism

$h\otimes f\mapsto h(v_{t})(1-x)f(u)$

from $\Sigma^{2}$ to $\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.
Here, we let $M$ be a separable $C^{*}$-subalgebra of
$\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ generated by
$(1-x)f(u)$ and $v_{t}$ is a continuous approximate unit for
$M\cap\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, quasi-central
with respect to $M$. The morphism $\iota_{X}$ sends this element in
$K_{0}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})$ to the element
in $E_{0}^{G}(C_{0}(X),B)$ represented by an asymptotic morphism

$h\otimes f\otimes\phi\mapsto h(v_{t}(s(t)))(1-x(s(t)))f(u)\phi$

from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)$ where
$t\mapsto s(t)$ is a continuous, increasing map on $[1,\infty)$ with
$s(t)\to\infty$ sufficiently fast as $t\to\infty$.

On the other hand, the downward composition sends the element $[\dot{u}]$ in
$K_{1}(D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G})$ to the element in
$E_{0}^{G}(C_{0}(X),B)$ represented by an asymptotic morphism

$h\otimes f\otimes\phi\mapsto h(u_{t})f(u)\phi$

from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)$ where
$u_{t}$ is an asymptotically $G$-equivariant, continuous approximate (see [26,
Definition 6.2]) in $\mathfrak{K}(H_{X}\otimes B)$ which is quasi-central with
respect to $C_{0}(X)$ and $f(u)$ for $f\in\Sigma$. We can take $u_{t}$ so that
we have

$\lVert(1-u_{t})x(s(t))\phi\rVert\to 0$

as $t\to\infty$ for any $\phi\in C_{0}(X)$. Then, the above asymptotic
morphism is asymptotic to the one

$h\otimes f\otimes\phi\mapsto h(u_{t})(1-x(s(t)))f(u)\phi.$

The two asymptotic morphisms

$h\otimes f\otimes\phi\mapsto h(u_{t})(1-x(s(t)))f(u)\phi,\,\,\,h\otimes
f\otimes\phi\mapsto h(v_{t}(s(t)))(1-x(s(t)))f(u)\phi$

are homotopic through an asymptotic morphism

$h\otimes f\otimes\phi\mapsto h(w^{r}_{t})(1-x(s(t)))f(u)\phi,$

from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)\otimes
C[0,1]$ where

$w^{r}_{t}=(1-r)v_{t}(s(t))+ru_{t}$

for $0\leq r\leq 1$. That this is well-defined can be seen from that

$f\mapsto(1-x(s(t)))f(u)$

is asymptotically a $\ast$-homomorphism modulo the product of $\phi\in
C_{0}(X)$ and $h(w^{r}_{t})$ and from that

$\lVert[h(w^{r}_{t})(1-x(s(t)))f(u),\phi]\rVert\to
0,\,\,\,\lVert[h(w^{r}_{t}),(1-x(s(t)))f(u)]\rVert\to 0$

as $t\to\infty$. ∎

######  Theorem 13.2.

Let $G$ be a locally compact group. The natural transformation

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$

is an isomorphism for any proper $G$-space $X$ and for any separable
$G$-$C^{*}$-algebra $B$.

###### Proof.

The case when $X$ is $G$-compact follows from Proposition 7.5, Corollary 12.6
and Lemma 13.1. The general case follows since the transformation is natural
with respect to $X$ and both sides are representable by $G$-compact subsets. ∎

######  Theorem 13.3.

Let $G$ be a locally compact group, $X$ be a proper $G$-space and $B$ be a
separable $G$-$C^{*}$-algebra. The forget-control map

$\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to K_{\ast}(B\rtimes_{r}G)$

is naturally equivalent to the Baum–Connes assembly map

$\mu_{X}^{B,G}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(X),B)\to K_{\ast}(B\rtimes_{r}G).$

That is, there is a natural isomorphism

$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$

of the functors from $\mathcal{PR}^{G}$ to $\mathcal{GA}$ and the following
diagram commutes

$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\cong}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{B,G}_{X}}$

###### Proof.

Combine Theorem 9.13 and Theorem 13.2. ∎

As before, let $RL^{0}_{c}(H_{X}\otimes B)$ be the kernel of the evaluation
map $\mathrm{ev}_{1}$ on $RL_{c}^{\ast}(H_{X}\otimes B)$ (see the end of
Section 9).

######  Corollary 13.4.

Let $G$ be a locally compact group and $B$ be a separable $G$-$C^{*}$-algebra.
The Baum–Connes assembly map $\mu^{B,G}_{r}$ is an isomorphism if and only if

$K_{\ast}(RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G)=0$

for a universal $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.

##  14\. $X$-$G$-localized element in $KK^{G}(\mathbb{C},\mathbb{C})$

We recall some materials on Kasparov’s $G$-equivariant $KK$-theory from [30]
(see also [7]). The graded (minimal) tensor product is denoted by
$\hat{\otimes}$.

Let $G$ be a second countable, locally compact group. For any (not necessarily
separable) graded $G$-$C^{*}$-algebras $A$ and $B$, Kasparov defines the
abelian group $KK^{G}(A,B)=KK_{0}^{G}(A,B)$ ([30, Definition 2.2] ) as the
group of homotopy classes (up to isomorphisms) of even Kasparov triples
$(E,\pi,T)$ where $E$ is a countably generated, graded $G$-Hilbert $B$-module
$E$ which is equipped with a (graded, $G$-equivariant) $\ast$-homomorphism
$\pi\colon A\to\mathfrak{L}(E)$ and where $T$ is an odd, $G$-continuous
operator in $\mathfrak{L}(E)$ such that for any $a\in A$ and $g\in G$, (we
write $a=\pi(a)$)

$a(1-T^{2}),\,\,[a,T],\,\,a(T-T^{\ast}),\,\,a(g(T)-T)\in\mathfrak{K}(\mathcal{E}).$

We often assume that $T$ is self-adjoint without loss of generality. The
homotopy is defined by the even Kasparov triple for $KK^{G}(A,B[0,1])$ for
$B[0,1]=B\otimes C[0,1]$ and the group structure is given by the direct sum
operation. For any $\ast$-homomorphisms $\phi\colon A_{1}\to A_{2}$,
$\psi\colon B_{1}\to B_{2}$, we have canonically defined homomorphisms ([30,
Definition 2.5]

$\phi^{\ast}\colon KK^{G}(A_{2},B)\to KK^{G}(A_{1},B),\,\,\,\psi_{\ast}\colon
KK^{G}(A,B_{1})\to KK^{G}(A,B_{2})$

and the group $KK^{G}(A,B)$ is homotopy invariant in both variables. If $D$ is
$\sigma$-unital, we have a canonically defined homomorphism

$\sigma_{D}\colon KK^{G}(A,B)\to KK^{G}(A\hat{\otimes}D,B\hat{\otimes}D)$

which sends $(E,\pi,T)$ to
$(E\hat{\otimes}D,\pi\hat{\otimes}\mathrm{id}_{D},T\hat{\otimes}1)$.

For $A$, separable, Kasparov defines the well-defined, bilinear, product law
(Kasparov product) (see [30, Definition 2.10, Theorem 2.11])

$KK^{G}(A,B_{1})\otimes KK^{G}(B_{1},B_{2})\to
KK^{G}(A,B_{2}),\,\,\,(x_{1},x_{2})\mapsto x_{1}\otimes_{B_{1}}x_{2}$

for any $B_{1},B_{2}$. The Kasparov product satisfies several functorial
properties, assuming separability or $\sigma$-unital for the relevant slots
(see [30, Theorem 2.14]).

The descent homomorphism [30, Section 3.11]

$j^{G}_{r}\colon KK^{G}(A,B)\to KK(A\rtimes_{r}G,B\rtimes_{r}G)$

is defined for any $A,B$ and it satisfies functorial properties, assuming
separability or $\sigma$-unital for the relevant slots (see [30, Theorem
3.11]).

The abelian group $KK^{G}_{1}(A,B)$ is defined to be

$KK^{G}_{1}(A,B)=KK^{G}(A\hat{\otimes}\mathbb{C}_{1},B)=KK^{G}(A,B\hat{\otimes}\mathbb{C}_{1})$

where $\mathbb{C}_{1}$ is the first Clifford algebra. We define

$K_{\ast}(A)=KK_{\ast}(\mathbb{C},A)$

for any graded $C^{*}$-algebra $A$ and when $A$ is ungraded, this group is
canonically isomorphic to the topological K-theory of $A$.

In particular, a cycle for the commutative ring
$R(G)=KK^{G}(\mathbb{C},\mathbb{C})$ is a pair $(H,T)$ of an odd, self-
adjoint, $G$-continuous operator $T$ on a (separable) graded $G$-Hilbert space
$H$ satisfying for any $g\in G$,

$1-T^{2},g(T)-T\in\mathfrak{K}(H).$

We call such a pair, a Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. A
cycle $(\mathbb{C},0)$ defines the unit of the ring $R(G)$, denoted by
$1_{G}$,

For a proper $G$-space $X$, a graded $X$-$G$-module $H_{X}$ is a graded
$G$-Hilbert space equipped with a non-degenerate (graded, $G$-equivariant)
representation of $C_{0}(X)$. It is just a pair $H_{X}=H_{X}^{(0)}\oplus
H_{X}^{(1)}$ of $X$-$G$-modules. For any graded $X$-$G$-module $H_{X}$, the
representable localization algebra $RL^{*}_{c}(H_{X})$ is naturally a graded
$G$-$C^{*}$-algebra.

######  Definition 14.1.

An $X$-$G$-localized Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$ is a
pair $(H_{X},T)$ of a graded $X$-$G$-module $H_{X}$ and an odd, self-adjoint,
$G$-continuous element $T$ in the multiplier algebra $M(RL_{c}^{*}(H_{X}))$
satisfying for any $g\in G$,

$1-T^{2},\,\,\,g(T)-T\in RL^{*}_{c}(H_{X}).$

######  Remark 14.2.

Although $RL^{*}_{c}(H_{X})$ is not $\sigma$-unital, one may think
$(RL_{c}^{*}(H_{X}),T)$ as a cycle for $KK^{G}(\mathbb{C},RL^{*}_{c}(H_{X}))$
(see [42, Section 3]).

The evaluation

$\mathrm{ev}_{t=1}\colon RL^{*}_{c}(H_{X})\to\mathfrak{K}(H_{X})$

extends to

$\mathrm{ev}_{t=1}\colon M(RL^{*}_{c}(H_{X}))\to\mathfrak{L}(H_{X}).$

For any $T\in M(RL^{*}_{c}(H_{X}))$, we write $T_{1}\in\mathfrak{L}(H_{X})$,
its image by $\mathrm{ev}_{t=1}$.

######  Lemma 14.3.

Let $X$ be a proper $G$-space and $(H_{X},T)$ be an $X$-$G$-localized Kasparov
cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. Then, a pair $(H_{X},T_{1})$ is a
cycle for $KK^{G}(\mathbb{C},\mathbb{C})$.

######  Definition 14.4.

Let $X$ be a proper $G$-space. We say that an element $x$ in
$KK^{G}(\mathbb{C},\mathbb{C})$ is $X$-$G$-localized if there is an
$X$-$G$-localized Kasparov cycle $(H_{X},T)$ for
$KK^{G}(\mathbb{C},\mathbb{C})$ such that

$[H_{X},T_{1}]=x\,\,\text{in}\,\,\,KK^{G}(\mathbb{C},\mathbb{C}).$

We say that $x\in KK^{G}(\mathbb{C},\mathbb{C})$ factors through a
$G$-$C^{*}$-algebra $B$ if there is $y\in KK^{G}(\mathbb{C},B)$ and $z\in
KK^{G}(B,\mathbb{C})$ such that $x=y\otimes_{B}z$. By definition, $x\in
KK^{G}(\mathbb{C},\mathbb{C})$ is the gamma element if it factors through a
(separable) proper $G$-$C^{*}$-algebra $A$ and it satisfies $x=1_{K}$ in
$R(K)$ for any compact subgroup $K$ of $G$.

######  Proposition 14.5.

Suppose that an element $x\in KK^{G}(\mathbb{C},\mathbb{C})$ factors through a
separable $G$-$C_{0}(X)$-algebra $A$ for a proper $G$-space $X$. Then $x$ is
$X$-$G$-localized.

The following is an immediate corollary.

######  Theorem 14.6.

The gamma element $\gamma$ for $G$, if exists, is $X$-$G$-localized for
$X=\underline{E}G$.

Before giving a proof of Proposition 14.5, we prove two lemmas.

Let $A$ be a graded $G$-$C_{0}(X)$ algebra and $(H,\pi,F)$ be a Kasparov
triple for $KK^{G}(\mathbb{C},A)$. If $\pi$ is non-degenerate, $\pi$ induces a
natural non-degenerate representation of $C_{0}(X)$ on $H$. We view $H$
naturally as a (graded) $X$-$G$-module through this representation and set
$H_{X}=H$. Any element $a$ in $M(A)$ commutes with $C_{0}(X)$ and hence $a$
(as a constant function) is naturally an element in
$M(RL^{*}_{c}(H_{X}))\subset\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$.

######  Lemma 14.7.

Let $X$ be a proper $G$-space and $A$ be a graded, separable
$G$-$C_{0}(X)$-algebra. Let $(H,\pi,F_{0})$ be a Kasparov triple for
$KK^{G}(A,\mathbb{C})$ with $\pi$ non-degenerate. We view $H$ naturally as a
graded $X$-$G$-module through this representation and set $H_{X}=H$. Then,
there is an odd, $G$-continuous, self-adjoint element $F$ in
$M(RL^{*}_{c}(H_{X}))\subset\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$ such
that

  1. (I)

$a(F-F_{0})\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$ for any $a\in A$, where
$F_{0}\in\mathfrak{L}(H_{X})$ is regarded as a constant function in
$\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$,

  2. (II)

for any $a\in A$ and $g\in G$,

$a(1-F^{2}),[a,F],a(g(F)-F)\in RL^{*}_{c}(H_{X}).$

###### Proof.

We assume $F_{0}$ is self-adjoint. We assume $F_{0}$ is $G$-equivariant using
the standard trick: Let $c\in C_{b}(X)$ be a cut-off function on $X$. Then,

$F_{0}^{\prime}=\int_{g\in G}g(c)g(F_{0})g(c)d\mu_{G}(g)$

is $G$-equivariant and satisfies

$a(F_{0}^{\prime}-F_{0})\in\mathfrak{K}(H_{X})$

for any $a\in A$. Here, for an isometry $V_{c}\colon H_{X}\to H_{X}\otimes
L^{2}(G)$ defined by

$V_{c}\colon v\mapsto(g\mapsto g(c)v),$

we have $F_{0}^{\prime}=V_{c}^{\ast}\tilde{F}V_{c}$ where $\tilde{F}$ on
$H_{X}\otimes L^{2}(G)$ is the diagonal operator $(g\mapsto g(F_{0}))\in
C_{b}(G,\mathfrak{L}(H_{X}))$. That
$a(F_{0}^{\prime}-F_{0})\in\mathfrak{K}(H_{X})$ can be checked by

$a(F_{0}^{\prime}-F_{0})=\int_{g\in
G}ag(c)(g(F_{0})-F_{0})g(c)+ag(c)[F_{0},g(c)]d\mu_{G}(g)$

which is norm convergent in $\mathfrak{K}(H_{X})$ for compactly supported $a$.
We remark that a cut-off function $c$ may not be $G$-continuous in general