# The equivariant coarse Baum–Connes conjecture for metric spaces with proper
group actions

Jintao Deng and Benyin Fu and Qin Wang

###### Abstract.

The equivariant coarse Baum–Connes conjecture interpolates between the
Baum–Connes conjecture for a discrete group and the coarse Baum–Connes
conjecture for a proper metric space. In this paper, we study this conjecture
under certain assumptions. More precisely, assume that a countable discrete
group $\Gamma$ acts properly and isometrically on a discrete metric space $X$
with bounded geometry, not necessarily cocompact. We show that if the quotient
space $X/\Gamma$ admits a coarse embedding into Hilbert space and $\Gamma$ is
amenable, and that the $\Gamma$-orbits in $X$ are uniformly equivariantly
coarsely equivalent to each other, then the equivariant coarse Baum–Connes
conjecture holds for $(X,\Gamma)$. Along the way, we prove a $K$-theoretic
amenability statement for the $\Gamma$-space $X$ under the same assumptions as
above, namely, the canonical quotient map from the maximal equivariant Roe
algebra of $X$ to the reduced equivariant Roe algebra of $X$ induces an
isomorphism on $K$-theory.

Supported in part by NSFC (No. 11871342, 11771143, 11831006, 12171156).

##  1\. Introduction

The Baum–Connes conjecture [2, 3] provides an algorithm to compute the
$K$-theory of reduced group $C^{*}$-algebras, which has important applications
in geometry, topology and analysis (see [2] for a survey). It has been
verified for a large class of groups including a-T-menable groups [18] and
hyperbolic groups [24].

The coarse Baum–Connes conjecture [27, 28] is a geometric analogue of the
Baum–Connes conjecture, which also has significant applications in geometry
and topology, such as the Novikov conjecture and Gromov’s conjecture about
Riemannian metric of positive scalar curvature ([10, 20, 38, 39]). Many
results (c.f. [5, 7, 13, 14, 15, 22, 23, 26, 31, 32, 33]) have been achieved
in recent years after Yu’s breakthrough to the coarse Baum–Connes conjecture
for metric spaces which are coarsely embeddable into Hilbert space [37].

Let $X$ be a proper metric space with bounded geometry, and let $\Gamma$ be a
countable discrete group. Assume that $\Gamma$ acts on $X$ properly and
isometrically, not necessarily cocompact. In this case, we call $X$ a
$\Gamma$-space. There is an equivariant higher index map ([11, 29])

${\rm
Ind}^{\Gamma}:\lim\limits_{d\rightarrow\infty}K_{*}^{\Gamma}(P_{d}(X))\to
K_{*}(C^{*}(X)^{\Gamma}),$

where $K_{*}^{\Gamma}(P_{d}(X))$ is the $\Gamma$-equivariant $K$-homology
group of the Rips complex $P_{d}(X)$ of $X$ on scale $d>0$, and
$K_{*}(C^{*}(X)^{\Gamma})$ is the $K$-theory group of the equivariant Roe
algebra of the $\Gamma$-space $X$. The equivariant coarse Baum–Connes
conjecture states that the equivariant higher index map ${\rm Ind}^{\Gamma}$
above is an isomorphism provided that $X$ has bounded geometry. When the
$\Gamma$-action is cocompact, $C^{*}(X)^{\Gamma}$ is Morita equivalent to the
reduced group $C^{*}$-algebra of $\Gamma$, so that the equivariant coarse
Baum–Connes conjecture is a reformulation of the Baum–Connes conjecture (cf.
[34]). When the group is trivial, the equivariant coarse Baum–Connes
conjecture is plainly the coarse Baum–Connes conjecture.

In [29], Shan proved that the equivariant higher index map is injective when
$X$ is a simply connected complete Riemannian manifold with non-positive
sectional curvature and $\Gamma$ is a torsion free group acting on $X$
properly and isometrically. In [11], Fu and Wang showed that the equivariant
coarse Baum–Connes conjecture holds for a $\Gamma$-space $X$ with bounded
geometry which admits an equivariant coarse embedding into Hilbert space. In
[12], Fu, Wang and Yu proved that if a discrete group $\Gamma$ acts properly
and isometrically on a space $X$ with bounded geometry, such that both
$X/\Gamma$ and $\Gamma$ admit a coarse embedding into Hilbert space, and that
the action has bounded distortion, then the equivariant higher index map is
injective for the $\Gamma$-space $X$.

In a recent work [1], Arzhantseva and Tessera answer in the negative the
following well-known question [9, 17]: does coarse embeddability into Hilbert
space is preserved under group extensions of finitely generated groups? Their
constructions also provide the first example of a finitely generated group
which does not coarsely embed into Hilbert space yet does not contain any
weakly embedded expander, answering in the affirmative another open problem
[1, 25]. Their examples are a certain restricted permutational wreath products

$\mathbb{Z}_{2}\wr_{G}H:=\left(\bigoplus_{G}\mathbb{Z}_{2}\right)\rtimes H$

and

$\mathbb{Z}_{2}\wr_{G}(H\times\mathbb{F}_{n}):=\left(\bigoplus_{G}\mathbb{Z}_{2}\right)\rtimes\Big{(}H\times\mathbb{F}_{n}\Big{)},$

where $G$ is a Gromov monster group, i.e., a finitely generated group which
contains an isometrically embedded expander in its Cayley group, $H$ is a
Haagerup monster group, i.e., a finitely generated group with the Haagerup
property but without Yu’s property A, and $\mathbb{F}_{n}$ is a finitely
generated free group. Note that both examples are “abelian-by-Haagerup” group
extensions. The reason why they do not coarsely embed into Hilbert relies on
the fact that both groups contain a certain relative expanders [1]. If we take
$X=\mathbb{Z}_{2}\wr_{G}H$ or $\mathbb{Z}_{2}\wr_{G}(H\times\mathbb{F}_{n})$,
and $\Gamma=\bigoplus_{G}\mathbb{Z}_{2}$, then we come across with a situation
that $\Gamma$ is a torsion group, the $\Gamma$-action on $X$ does not have
bounded distortion, and that the space $X$ does not even coarsely embed into
Hilbert space, let alone admits a $\Gamma$-equivariant coarse embedding into
Hilbert space, namely, a situation which does not satisfy the assumptions in
each of the main results in [29, 11, 12] mentioned above.

In this paper, as motivated by these examples by Arzhantseva and Tessera, we
shall further investigate the equivariant coarse Baum–Connes conjecture along
this line. The following is our main result.

######  Theorem 1.1.

Let $\Gamma$ be a countable discrete group acting properly and isometrically
on a discrete metric space $X$ with bounded geometry, not necessarily co-
compact. Assume that all $\Gamma$-orbits in $X$ are uniformly equivariantly
coarsely equivalent. If the quotient space $X/\Gamma$ admits a coarse
embedding into Hilbert space and $\Gamma$ is amenable, then the equivariant
higher index map

${\rm
Ind}^{\Gamma}:\lim\limits_{d\rightarrow\infty}K_{*}^{\Gamma}(P_{d}(X))\to
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism.

When the group $\Gamma$ is trivial, Theorem 1.1 recovers Yu’s famous result in
[37]. On the other hand, when the $\Gamma$-action is cocompact, Theorem 1.1
recovers the amenable group case of the celebrated result of Higson and
Kasparov in [18], where they proved the theorem for more general a-T-menable
groups. Note that the examples $X=\mathbb{Z}_{2}\wr_{G}H$ or
$\mathbb{Z}_{2}\wr_{G}(H\times\mathbb{F}_{n})$, and
$\Gamma=\bigoplus_{G}\mathbb{Z}_{2}$ by Arzhantseva and Tessera discussed
above provide us with nontrivial examples of non-cocompact group actions
satisfying the assumptions in Theorem 1.1. At this point, it is natural to
expect that Theorem 1.1 be true for non-cocompact actions with $\Gamma$ being
a-T-menable. However, technically in our approach to the main theorem, we have
to establish an isomorphism between the maximal equivariant twisted Roe
algebra and the reduced equivariant twisted Roe algebra, see Proposition 3.12,
in which the amenability of $\Gamma$ plays an essential role. Nevertheless,
once we get this isomorphism on the $C^{*}$-algebra level, we are able to
further get a $K$-theoretic amenability statement for the $\Gamma$-space $X$,
in a way as the proof of Theorem 1.1 by using the idea of an infinite
dimensional geometric Bott periodicity, see also [37]. More precisely, we have
the following result:

######  Theorem 1.2.

With the same assumptions as Theorem 1.1, the map

$\lambda_{*}:K_{*}(C^{*}_{max}(X)^{\Gamma})\rightarrow
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism, where $\lambda_{*}$ is the homomorphism induced by the
canonical quotient map $\lambda$ from the maximal equivariant Roe algebra
$C^{*}_{max}(X)^{\Gamma}$ to the reduced equivariant Roe algebra
$C^{*}(X)^{\Gamma}$.

When the group $\Gamma$ is trivial, Theorem 1.2 recovers the main result in
[31] by Špakula and Willett. When the group action is co-compact, the
statement of Theorem 1.2 actually holds for a more general class of groups
(see [6]), including all a-T-menable groups, as proved by Higson and Kasparov
in the same paper [18]. Unfortunately, due to limitation of our approach, the
amenability of the group $\Gamma$ is necessary in our proof to Theorem 1.2.
For the general case where the group action is non-cocompact, we do not know
whether the conclusion of Theorem 1.2 is true or not for $\Gamma$ being a-T-
menable.

The paper is organized as follows. In Section 2, we recall the reduced and
maximal equivariant Roe algebras, and formulate the equivariant coarse
Baum–Connes conjecture, for a discrete metric space with bounded geometry
which admits a proper and isometric action, not necessarily co-compact, by a
countable discrete group. In Section 3, we use the coarse embedding of the
quotient space $X/\Gamma$ into a Hilbert space to define the reduced and
maximal equivariant twisted Roe algebras, and their localization counterparts.
We prove that the $K$-theory of the equivariant twisted localization algebra
is isomorphic to the $K$-theory of the equivariant twisted Roe algebra.
Moreover, we also prove that the maximal equivariant twisted Roe algebra and
the reduced equivariant twisted Roe algebra are isomorphic. In Section 4, we
complete the proof of the main theorems by using a geometric analogue of
Higson–Kasparov–Trout’s infinite dimensional Bott periodicity.

##  2\. The equivariant coarse Baum–Connes conjecture

In this section, we shall first recall several notions from coarse geometry,
and then define the reduced and maximal equivariant Roe algebras, so as to
formulate the equivariant coarse Baum–Connes conjecture for a discrete metric
space with bounded geometry which admits a proper and isometric action, not
necessarily co-compact, by a countable discrete group.

###  2.1. Roe algebras

In this subsection, we shall recall the notions of the (maximal) equivariant
Roe algebras.

Suppose that $Z$ is a proper metric space in the sense that every closed ball
of $Z$ is compact. Let $\Gamma$ be a countable discrete group acting on $Z$
properly and isometrically. Recall that a $\Gamma$-action on $Z$ is $proper$
if the map

$Z\times\Gamma\rightarrow Z\times Z,\qquad(x,\gamma)\mapsto(\gamma x,x)$

is proper, i.e., the preimage of a compact set is compact. The group $\Gamma$
acts on $Z$ isometrically if $d(x,y)=d(\gamma x,\gamma y)$ for all
$\gamma\in\Gamma$ and $x,y\in Z$. In this case, we call $Z$ a $\Gamma$-space.

Recall that a $Z$-$module$ is a separable Hilbert space $H$ equipped with a
faithful and non-degenerate $*$-representation $\phi:C_{0}(Z)\rightarrow B(H)$
whose range contains no nonzero compact operators, where $C_{0}(Z)$ is the
algebra of all complex-valued continuous functions on $Z$ vanishing at
infinity.

Let $Z$ be a $\Gamma$-space. we define a $\Gamma$-action on $C_{0}(Z)$ by

$\gamma(f)(x)=f(\gamma^{-1}x)$

for all $\gamma\in\Gamma$ and $f\in C_{0}(Z)$.

######  Definition 2.1.

Let $H$ be an $Z$-module. We say that $H$ is a covariant $Z$-module if it is
equipped with a unitary action $\rho$ of $\Gamma$, i.e.,
$\rho:\Gamma\rightarrow\mathcal{U}(H)$ is a group homomorphism from $\Gamma$
to the set of unitary elements in $B(H)$, compatible with the action of
$\Gamma$ on $Z$ in the sense that for $v\in H$, $T\in B(H)$ and
$\gamma\in\Gamma$,

$\gamma(T)(v)=\rho(\gamma)T\rho(\gamma)^{*}(v).$

We call such a triple $(C_{0}(Z),\Gamma,\phi)$ a $covariant\ system$.

######  Definition 2.2 ([23]).

A covariant system $(C_{0}(Z),\Gamma,\phi)$ is $admissible$ if there exists a
$\Gamma$-Hilbert space $H_{Z}$ and a separable and infinite-dimensional
$\Gamma$-Hilbert space $H^{\prime}$ such that

  1. (1)

$H$ is isomorphic to $H_{Z}\otimes H^{\prime}$;

  2. (2)

$\varphi=\varphi_{0}\otimes I$ for some $\Gamma$-equivariant $*$-homomorphism
$\varphi_{0}$ from $C_{0}(Z)$ to $B(H_{Z})$ such that $\varphi_{0}(f)$ is not
in $K(H_{Z})$ for any nonzero function $f\in C_{0}(Z)$ and $\varphi_{0}$ is
non-degenerate in the sense that $\\{\varphi_{0}(f)H_{Z}:f\in C_{0}(Z)\\}$ is
dense in $H_{Z}$;

  3. (3)

for each finite subgroup $F\subset\Gamma$ and every $F$-invariant Borel subset
$E$ of $Z$, there exists a trivial $F$-representation $H_{E}$ such that

$\chi_{E}H^{\prime}\cong l^{2}(F)\otimes H_{E},$

where $\ell^{2}(F)$ is endowed with the left regular representation of $F$.

######  Definition 2.3.

Let $(C_{0}(Z),\Gamma,\phi)$ be an admissible covariant system.

  1. (1)

The $support$ $\text{Supp}(T)$ of a bounded linear operator $T:H\to H$ is the
complement of the set of points $(x,y)\in Z\times Z$ for which there exist $f$
and $f^{\prime}$ in $C_{0}(Z)$ such that

$\phi(f^{\prime})T\phi(f)=0,\quad f(x)\neq 0,\quad f^{\prime}(y)\neq 0.$

  2. (2)

A bounded operator $T:H\rightarrow H$ has finite propagation if

$\mbox{propagation}(T):=\text{sup}\\{d(x,y):(x,y)\in\text{Supp}(T)\\}<\infty.$

This number is called the propagation of $T$.

  3. (3)

A bounded operator $T:H\rightarrow H$ is locally compact if the operators
$\phi(f)T$ and $T\phi(f)$ are compact for all $f\in C_{0}(Z)$.

  4. (4)

A bounded operator $T:H\rightarrow H$ is $\Gamma$-invariant if $\gamma(T)=T$
for all $\gamma\in\Gamma$.

Now we are ready to define the equivariant Roe algebra.

######  Definition 2.4.

Let $(C_{0}(Z),\Gamma,\phi)$ be an admissible covariant system. The algebraic
equivariant Roe algebra $\mathbb{C}[Z]^{\Gamma}$ is the $*$-subalgebra of
$B(H)$ consisting of all locally compact and $\Gamma$-invariant operators with
finite propagation, where $B(H)$ is the algebra of all bounded operators on
$H$. The equivariant Roe algebra $C^{*}(Z)^{\Gamma}$ is the closure of the
$*$-algebra $\mathbb{C}[Z]^{\Gamma}$ under the operator norm in $B(H)$.

A proper $\Gamma$-space $Z$ is said to have bounded geometry if there exists a
$\Gamma$-invariant subset $X\subset Z$ such that

  1. (1)

$X$ is discrete and $\\#B_{R}(x)<\infty$ for all $r>0$, where $\\#B_{R}(x)$ is
the number of the elements in $\\{z\in X:d(x,z)\leq R\\}$;

  2. (2)

$X$ is a $c$-net in $Z$ for some $c>0$ in the sense that $Z=N_{c}(X)=\\{z\in
Z:d(z,X)\leq c\\}$.

The $*$-algebra $\mathbb{C}[Z]^{\Gamma}$ can be equipped with a $C^{*}$-norm
associated with any $*$-representation $\phi:\mathbb{C}[Z]^{\Gamma}\to
H_{\phi}$. To define the maximal norm $\mathbb{C}[Z]^{\Gamma}$, we need he
following result which is essentially proved in [14].

######  Lemma 2.5.

Let $Z$ be a proper $\Gamma$-space with bounded geometry. For any
$T\in\mathbb{C}[Z]^{\Gamma}$, there exists a constant $C_{T}>0$ such that for
any $*$-representation $\phi$ of $\mathbb{C}[Z]^{\Gamma}$ on a Hilbert space
$\mathcal{H}_{\phi}$, we have

$\|\phi(T)\|_{B(\mathcal{H}_{\phi})}\leq C_{T}.$

It follows from the above lemma that the maximal norm on the $*$-algebra
$\mathbb{C}[Z]^{\Gamma}$ is well-defined.

######  Definition 2.6 ([14]).

Let $Z$ be a proper $\Gamma$-space. The maximal equivariant Roe algebra,
denoted by $C^{*}_{max}(Z)^{\Gamma}$, is the completion of
$\mathbb{C}[Z]^{\Gamma}$ with respect to the $C^{*}$-norm

$\|T\|_{max}:=\sup\\{\|\phi(T)\|_{B(\mathcal{H}_{\phi})},\
\phi:\mathbb{C}[Z]^{\Gamma}\rightarrow B(\mathcal{H}_{\phi})\ \text{is a}\
\text{$\ast$-representation}\\}.$

We remark that the definition of the (maximal) equivariant Roe algebra does
not depends on the choice of admissible covariant systems. The (maximal)
equivariant Roe algebra $C^{*}(Z)^{\Gamma}$ is Morita equivalent to the
(maximal) reduced group $C^{*}$-algebra $C_{r}^{*}(\Gamma)$ when the quotient
space $Z/\Gamma$ is compact, i.e., $Z$ admits a proper and cocompact
$\Gamma$-action(c.f. [28, Lemma 5.14]). Moreover, Roe algebras is invariant
under the coarse equivalence in the sense of the following.

######  Definition 2.7.

Let $Z$ and $Z^{\prime}$ be proper $\Gamma$-spaces. A Borel map
$f:Z\rightarrow Z^{\prime}$ is said to be an _equivariantly coarse embedding_
if there exist non-decreasing functions
$\rho_{+},\rho_{-}:\mathbb{R}_{+}:=[0,+\infty)\to\mathbb{R}_{+}$ such that

  1. (1)

$\lim\limits_{r\rightarrow+\infty}\rho_{\pm}(r)\rightarrow+\infty$;

  2. (2)

$\rho_{-}(d(x,y))\leq d(f(x),f(y))\leq\rho_{+}(d(x,y))$, for all $x,y\in Z$;

  3. (3)

$f(\gamma x)=\gamma f(x)$ for all $x\in Z$ and $\gamma\in\Gamma$.

Moreover, the $\Gamma$-spaces $Z$ and $Z^{\prime}$ are called equivariantly
coarsely equivalent if there is a equivariant coarse embedding $f:Z\to
Z^{\prime}$ such that the image $f(Z)$ is _coarsely dense_ in $Z^{\prime}$ in
the sense that $Z^{\prime}=N_{S}(f(Z))$ for some $S>0$, where
$N_{S}(f(Z))=\\{y\in Y:d(y,f(Z))<S\\}$. We say that a family of metric spaces
$\\{Z_{i}\\}_{i\in I}$ are said to be _uniformly equivariantly coarsely
equivalent_ if any two metric spaces $Z_{i}$ and $Z_{j}$ are equivariantly
coarsely equivalent with the same functions $\rho_{+},\rho_{-}$ and constant
$S>0$ in the above definitions.

Let $H$ be a separable infinite-dimensional Hilbert space. A map $f:Z\to H$ is
called a coarse embedding into Hilbert space if if there exist non-decreasing
functions $\rho_{+},\rho_{-}:\mathbb{R}_{+}:=[0,+\infty)\to\mathbb{R}_{+}$
such that

  1. (1)

$\lim\limits_{r\rightarrow+\infty}\rho_{\pm}(r)\rightarrow+\infty$;

  2. (2)

$\rho_{-}(d(x,y))\leq d(f(x),f(y))\leq\rho_{+}(d(x,y))$, for all $x,y\in Z$.

The equivariant Roe algebras are invariant under equivariantly coarse
equivalence.

######  Proposition 2.8.

Let $Z$ and $Z^{\prime}$ be proper $\Gamma$-spaces. If there exists an
equivariantly coarse equivalence $f:Z\to Z^{\prime}$, then

$C^{*}(Z)^{\Gamma}\cong C^{*}(Z^{\prime})^{\Gamma},$

and

$C_{max}^{*}(Z)^{\Gamma}\cong C_{max}^{*}(Z^{\prime})^{\Gamma}.$

By the universality of the maximal norm, the identity map on
$\mathbb{C}[Z]^{\Gamma}$ extends to a canonical quotient map

$\lambda:C^{*}_{max}(Z)^{\Gamma}\rightarrow C^{*}(Z)^{\Gamma}.$

We then obtain a homomorphism

$\lambda_{*}:K_{*}(C^{*}_{max}(Z)^{\Gamma})\rightarrow
K_{*}(C^{*}(Z)^{\Gamma})$

induced by the map $\lambda$ at the $K$-theory level.

###  2.2. The assembly map

Let us briefly recall the definition of equivariant $K$-homology introduced by
Kasparov [21, 35] for a covariant system $(C_{0}(Z),\Gamma,\phi)$.

######  Definition 2.9.

For $i=0,1$, the $K$-homology groups
$K_{i}^{\Gamma}(Z)=KK_{i}^{\Gamma}(C_{0}(Z),\mathbb{C})$ ($i=0,1$) are
generated by certain cycles module a certain equivalent relation:

  1. (1)

each cycle for $K_{0}^{\Gamma}(Z)$ is a triple $(H,\phi,F)$, where
$\phi:C_{0}(Z)\to B(H)$ is a covariant $*$-representation and $F$ is a
$\Gamma$-invariant bounded operator on $H$ such that $\phi(f)F-F\phi(f)$,
$\phi(f)(FF^{*}-I)$ and $\phi(f)(F^{*}F-I)$ are compact operators for all
$f\in C_{0}(Z),\gamma\in\Gamma$;

  2. (2)

each cycle for $K_{1}^{\Gamma}(Z)$ is a triple $(H,\phi,F)$,
where$\phi:C_{0}(Z)\to B(H)$ is a covariant $*$-representation and $F$ is a
$\Gamma$-invariant bounded operator on $H$, and $F$ is a $\Gamma$-invariant
and self-adjoint operator on $H$ such that $\phi(f)(F^{2}-I)$, and
$\phi(f)F-F\phi(f)$ are compact for all $f\in C_{0}(Z),\gamma\in\Gamma$.

In both cases, the equivalence relation on cycles is given by homotopy of the
operator $F$.

Let us now define the equivariant index map

${\rm Ind}^{\Gamma}:K^{\Gamma}_{*}(Z)\to K_{*}(C^{*}(Z)^{\Gamma}).$

Note that every class in $K^{\Gamma}_{*}(Z)$ can be represented by a cycle
$(H,\phi,F)$ such that $(C_{0}(Z),\Gamma,\phi)$ is an admissible covariant
system [23]. Let $(H,\phi,F)$ be a cycle in $K_{0}^{\Gamma}(Z)$ such that
$(C_{0}(Z),\Gamma,\phi)$ is an admissible covariant system.

Let $\\{U_{i}\\}_{i\in I}$ be a locally finite, $\Gamma$-equivariant and
uniformly bounded open cover of $X$, and let $\\{\psi_{i}\\}_{i\in I}$ be a
$\Gamma$-equivariant partition of unity subordinate to $\\{U_{i}\\}_{i\in I}$.
Define

$F^{\prime}=\sum\limits_{i\in I}\phi(\sqrt{\psi_{i}})F\phi(\sqrt{\psi_{i}}),$

where the infinite sum converges in strong topology. Note that
$(H,\phi,F^{\prime})$ is equivalent to $(H,\phi,F)$ in $K^{\Gamma}_{0}(Z)$.
Moreover, the operator $F^{\prime}$ has finite propagation, therefore
$F^{\prime}$ is a multiplier of $C^{*}(Z)^{\Gamma}$ and is invertible modulo
$C^{*}(Z)^{\Gamma}$. Thus, $F^{\prime}$ gives rise to an element in
$K_{0}(C^{*}(Z)^{\Gamma})$, denoted by $\partial([F^{\prime}])$. We define the
equivariant index map

${\rm Ind}^{\Gamma}:K^{\Gamma}_{0}(Z)\to K_{0}(C^{*}(Z)^{\Gamma})$

by

${\rm Ind}^{\Gamma}([H,\phi,F])=\partial([F^{\prime}])$

for all $[H,\phi,F]\in K^{\Gamma}_{0}(Z)$, where $F^{\prime}$ is defined as
above. Similarly, we can define the equivariant index map

${\rm Ind}^{\Gamma}:K^{\Gamma}_{1}(Z)\to K_{1}(C^{*}(Z)^{\Gamma}).$

###  2.3. The equivariant coarse Baum–Connes conjecture

In this subsection, we shall recall the equivariant coarse Baum–Connes
conjecture for a discrete metric space with bounded geometry.

A metric space $X$ is said to have bounded geometry if

$\sup_{x\in X}\\#B_{R}(x)<\infty$

for each $R>0$, where $B_{R}(x)=\\{y\in X:d(x,y)\leq R\\}$.

######  Definition 2.10.

Let $d>0$. The Rips complex $P_{d}(X)$ at scale $d$ is the simplicial
polyhedron whose vertex set is $X$ and where a finite subset $Y\subseteq X$
spans a simplex if and only if $d(x,y)\leq d$ for all $x,y\in Y$.

Let $X$ be a discrete $\Gamma$-space. Each Rips complex $P_{d}(X)$ ie quipped
with the spherical metric. Recall that on each path connected component of
$P_{d}(X)$, the spherical metric is the maximal metric whose restriction to
the simplex spanned by a finite subset
$\left\\{x_{0},x_{1},\cdots,x_{n}\right\\}$ is the metric obtained by
identifying the simplex with

$S_{+}^{n}:=\Big{\\{}(s_{0},s_{1},\cdots,s_{n})\in\mathbb{R}^{n+1}:s_{i}\geq
0,\sum\limits_{i=0}^{n}s_{i}^{2}=1\Big{\\}}$

via the map

$\sum\limits_{i=0}^{n}t_{i}x_{i}\mapsto\left(\frac{t_{0}}{\sqrt{\sum_{i=0}^{n}t_{i}^{2}}},\frac{t_{1}}{\sqrt{\sum_{i=0}^{n}t_{i}^{2}}},\cdots,\frac{t_{n}}{\sqrt{\sum_{i=0}^{n}t_{i}^{2}}}\right),$

where $S_{+}^{n}$ is endowed with the standard Riemannian metric. The distance
of a pair of points in different connected components of $P_{d}(X)$ is defined
to be infinity.

Each Rips complex $P_{d}(X)$ admits a proper $\Gamma$-action by isometries
defined by

$\gamma\cdot\sum\limits_{i=1}^{k}c_{i}x_{i}=\sum\limits_{i=1}^{k}c_{i}(\gamma\cdot
x_{i})$

for all $\sum\limits_{i=1}^{k}c_{i}x_{i}\in P_{d}(X)$ and $\gamma\in\Gamma$.

The following conjecture is called _the equivariant coarse Baum–Connes
conjecture_ :

######  Conjecture 2.11 (The equivariant coarse Baum–Connes conjecture).

Let $X$ be a discrete metric space with bounded geometry and let $\Gamma$ be a
countable discrete group acting on $X$ properly by isometries. Then the
equivariant index map

$\rm{Ind}^{\Gamma}:\lim\limits_{d\rightarrow\infty}K_{*}^{\Gamma}(P_{d}(X))\rightarrow\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X))^{\Gamma})\cong
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism.

###  2.4. Localization algebras and local index map

In this subsection, we shall recall the notions of the localization algebras
and the local index map.

######  Definition 2.12.

Let $Z$ be a proper $\Gamma$-space. The equivariant localization algebra
$C_{L}^{*}(Z)^{\Gamma}$ is the norm closure of the algebra of all bounded and
uniformly norm-continuous functions

$f:[0,+\infty)\rightarrow C^{*}(Z)^{\Gamma},$

such that

$\text{propagation}(f(t))\rightarrow 0\ \text{as}\ t\rightarrow\infty,$

with respect to the norm

$\|f\|=\sup\limits_{t\in[0,+\infty)}\|f(t)\|.$

Now, let us recall the local $\Gamma$-index map

${\rm Ind}:K_{*}^{\Gamma}(Z)\to K_{*}(C^{*}_{L}(Z)^{\Gamma}).$

For each positive integer $n$, let $\\{U_{n,i}\\}_{i\in I}$ be a locally
finite and $\Gamma$-invariant open cover for $Z$ such that

${\rm diam}(U_{n,i})<\frac{1}{n}$

for all $i\in I$. Let $\\{\phi_{n,i}\\}_{I}$ be a continuous partition of
unity subordinate to $\\{U_{n,i}\\}_{I}$. Let $[(H,\phi,F)]\in
K_{0}^{\Gamma}(Z)$. Define a continuous path of operators
$F(t)_{t\in[0,\infty)}$ on $H$ by

$F(t)=\sum_{i}\bigl{(}(1-(t-n))\phi_{n,i}^{\frac{1}{2}}F\phi_{n+1,i}^{\frac{1}{2}}+(t-n)\phi_{n+2,i}^{\frac{1}{2}}F\phi_{n+1,i}^{\frac{1}{2}}\bigr{)}$

for all $t\in[n,n+1)$, where the infinite sum converges in the strong operator
topology. Notice that ${\rm propagation}(F(t))\rightarrow 0$ as
$t\rightarrow\infty.$ Therefore, the path $(F(t))_{t\in[0,\infty)}$ is a
multiplier of $C^{*}_{L}(X)^{\Gamma}$ and $F(t)$ is a unitary modulo
$C^{*}_{L}(Z)^{\Gamma}$. Hence $F(t)$ gives rise to an element
$\partial[F(t)]$ in $K_{0}(C^{*}_{L}(Z)^{\Gamma})$. We define the local index
map

$Ind^{\Gamma}_{L}:K_{0}^{\Gamma}(Z)\to K_{0}(C^{*}_{L}(Z)^{\Gamma}).$

Similarly, we can define

${\rm Ind}:K_{1}^{\Gamma}(Z)\to K_{1}(C^{*}_{L}(Z)^{\Gamma}).$

Now we have the following result which is an equivariant analogue of Theorem
3.2 in [36]. It can be proved by the

######  Theorem 2.13.

Let $X$ be a discrete metric space with bounded geometry and let $\Gamma$ be a
countable discrete group acting on $X$ properly and isometrically. Then the
local index map

$\rm{Ind}^{\Gamma}_{L}:K_{*}^{\Gamma}(P_{d}(X))\rightarrow
K_{*}(C_{L}^{*}(P_{d}(X))^{\Gamma})$

is an isomorphism.

For any $d>0$, there is a natural evaluation map

$e:C_{L}^{*}(P_{d}(X))^{\Gamma}\to C^{*}(P_{d}(X))^{\Gamma}$

by

$e(f)=f(0)$

for all $f\in C_{L}^{*}(P_{d}(X))^{\Gamma}$. This is a $*$-homomorphism, thus
induces a homomorphism

$e_{*}:K_{*}(C_{L}^{*}(P_{d}(X))^{\Gamma})\to K_{*}(C^{*}(P_{d}(X))^{\Gamma})$

on $K$-theory.

We have the following commutative diagram:

$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}^{\Gamma}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rm{Ind}_{L}^{\Gamma}}$$\scriptstyle{\rm{Ind}^{\Gamma}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X))^{\Gamma}).}$

As a result, in order to prove that the map $\rm{Ind}^{\Gamma}$ is an
isomorphism, it suffices to show that the map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X))^{\Gamma})\rightarrow\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X))^{\Gamma})$

induced by the evalaution map on $K$-theory is an isomorphism.

##  3\. Equivariant twisted Roe algebras and localization algebras

In this section, we shall follow the constructions in [37] to define
$C^{*}$-algebras encoding the geometry of the quotient space $X/\Gamma$. These
algebras include maximal and reduced equivariant twisted Roe algebras, and
equivariant twisted localization algebras. We prove that the K-theory of
equivariant twisted localization algebras is isomorphic to the K-theory of
equivariant twisted Roe algebras under the assumptions on the geometry of
$X/\Gamma$ and the orbits. We also prove that the canonical quotient map
between the maximal equivariant twisted Roe algebras and the reduced
equivariant twisted Roe algebras is an isomorphism under the same assumptions.

###  3.1. The equivariant twisted Roe algebras along the quotient space

Let $\Gamma$ be a countable discrete group, and let $X$ be a $\Gamma$-space
with bounded geometry. Assume that $\xi:X/\Gamma\to H$ is a coarse embedding
of $X$ into Hilbert space $H$. Let $\pi:X\rightarrow X/\Gamma$ be the quotient
map. We shall first recall the $C^{*}$-algebra associated to an infinite-
dimensional Euclidean space introduced by Higson–Kasparov–Trout [19].

Let $H$ be a real (countably infinite-dimensional) Hilbert space. Denote by
$V_{a}$, $V_{b}$, etc., the finite-dimensional affine subspaces of $H$. Let
$V_{a}^{0}$ be the finite-dimensional linear subspaces of $H$ consisting of
differences of elements of $V_{a}$. Let ${\rm Cliff}(V_{a}^{0})$ be the
complexified Clifford algebra of $V_{a}^{0}$, and let $\mathcal{C}(V_{a})$ be
the graded $C^{*}$-algebra of continuous functions from $V_{a}$ to ${\rm
Cliff}(V_{a}^{0})$ vanishing at infinity. Let $\mathcal{S}$ be the
$C^{*}$-algebra $C_{0}(\mathbb{R})$, graded according to the odd and even
functions. Define the graded tensor product

$\mathcal{A}(V_{a})=\mathcal{S}\widehat{\otimes}\mathcal{C}(V_{a}).$

If $V_{a}\subseteq V_{b}$, we have a decomposition $V_{b}=V_{ba}^{0}+V_{a}$,
where $V_{ba}^{0}$ is the orthogonal complement of $V_{a}^{0}$ in $V_{b}^{0}$.
For each $v_{b}\in V_{b}$, there is a unique decomposition
$v_{b}=v_{ba}+v_{a}$, where $v_{ba}\in V_{ba}^{0}$ and $v_{a}\in V_{a}$. Every
function $h$ on $V_{a}$ can be extended to a function $\tilde{h}$ on $V_{b}$
by the formula

$\tilde{h}(v_{ba}+v_{a})=h(v_{a}).$

If $V_{a}\subseteq V_{b}$, we use $C_{ba}$ to denote the function
$V_{b}\rightarrow{\rm Cliff}(V_{b}^{0})$, $v_{b}\mapsto v_{ba}$, where
$v_{ba}$ is considered as an element of ${\rm Cliff}(V_{b}^{0})$ via the
inclusion $V_{ba}^{0}\subset\textrm{Cliff}(V_{b}^{0})$. Let $X$ be the
unbounded multiplier of $\mathcal{S}$ with degree one given by the point-wise
multiplication with the function $x\mapsto x$. Note that
$X\widehat{\otimes}1+1\widehat{\otimes}C_{ba}$ is an unbouded, essentially
self-adjoint operator with degree one [19]. So we can use functional calculus
to define a $*$-homomorphism

$\beta_{ba}:\mathcal{A}(V_{a})\rightarrow\mathcal{A}(V_{b})$

by the formula

$\beta_{ba}(g\widehat{\otimes}h)=g(X\widehat{\otimes}1+1\widehat{\otimes}C_{ba})(1\widehat{\otimes}\tilde{h}),$

for $g\in\mathcal{S}$ and $h\in\mathcal{C}(V_{a})$, where
$g(X\widehat{\otimes}1+1\widehat{\otimes}C_{ba}).$

######  Remark 3.1.

Let $g_{0}(x)=e^{-x^{2}}$ and $g_{1}(x)=xe^{-x^{2}}$. It is not difficult to
check that

$g_{0}(X\widehat{\otimes}1+1\widehat{\otimes}C_{ba})=g_{0}(X)\widehat{\otimes}g_{0}(C_{ba})$

for $g_{0}=e^{-x^{2}}$ and

$g_{1}(X\widehat{\otimes}1+1\widehat{\otimes}C_{ba})=g_{0}(X)\widehat{\otimes}g_{1}(C_{ba})+g_{1}(X)\widehat{\otimes}g_{0}(C_{ba})$

for $g_{1}(x)=xe^{-x^{2}}$. Moreover $g_{0}(X)=g_{0}(x)$, $g_{1}(X)=g_{1}(x)$
as multiplication operators, and
$(g_{0}(C_{ba})\tilde{h})(v_{b})=g_{0}(\|v_{ba}\|)h(v_{a})$,
$(g_{1}(C_{ba})\tilde{h})(v_{b})=v_{ba}g_{0}(\|v_{ba}\|)h(v_{a})$ when we view
$C_{ba}$ as an unbounded multiplier on $\mathcal{C}(V_{b})$.

The maps $(\beta_{ba})$ make the collection $(\mathcal{A}(V_{a}))$ into a
directed system as $V_{a}$ ranges over finite-dimensional affine subspaces of
$H$. Define the $C^{*}$-algebra $\mathcal{A}(H)$ by

$\mathcal{A}(H)=\lim\limits_{\rightarrow}\mathcal{A}(V_{a}),$

where the inductive limit is taken over all finite-dimensional affine
subspaces of $H$.

Let $\mathbb{R}_{+}\times H$ be the topological space endowed with the weakest
topology for which the projection onto $H$ is weakly continuous and the
function $(t,w)\mapsto t^{2}+\|w\|^{2}$ is continuous. This topology makes
$\mathbb{R}_{+}\times H$ a locally compact Hausdorff space. Note that for
$v\in H$ and $r>0$, the subset

$B(v,r)=\\{(t,w)\in\mathbb{R}_{+}\times H:t^{2}+\|v-w\|^{2}<r^{2}\\}$

is an open subset of $\mathbb{R}_{+}\times H$. For each finite-dimensional
subspace $V_{a}\subseteq H$, $C_{0}(\mathbb{R}_{+}\times V_{a})$ is included
in $\mathcal{A}(V_{a})$ as its center. If $V_{a}\subseteq V_{b}$, then
$\beta_{ba}$ takes $C_{0}(\mathbb{R}_{+}\times V_{a})$ into
$C_{0}(\mathbb{R}_{+}\times V_{b})$. Then $C^{*}$-algebra
$\lim\limits_{\rightarrow}C_{0}(\mathbb{R}_{+}\times V_{a})$ is isomorphic to
$C_{0}(\mathbb{R}_{+}\times H)$, where the direct limit is over the directed
set of all finite-dimensional affine subspaces $V_{a}$ of $H$.

######  Definition 3.2.

The support of an element $a\in\mathcal{A}(H)$ is the complement of all
$(t,v)\in\mathbb{R}_{+}\times H$ such that there exists $g\in
C_{0}(\mathbb{R}_{+}\times H)$ with $g(t,v)\neq 0$ and $g\cdot a=0$.

For each $d>0$, the $\Gamma$-action on $X$ induces a proper and isometric
$\Gamma$-action on $P_{d}(X)$ by

$\gamma\cdot\sum\limits_{i=1}^{k}c_{i}x_{i}=\sum\limits_{i=1}^{k}c_{i}(\gamma\cdot
x_{i})$

for $\sum\limits_{i=1}^{k}c_{i}x_{i}\in P_{d}(X)$ and $\gamma\in\Gamma$.

Note that the coarse embedding $\xi:X/\Gamma\rightarrow H$ induces a coarse
embedding

$\xi:P_{d}(X)/\Gamma\rightarrow H$

by

$\xi\Big{(}\pi(\sum\limits_{i=1}^{k}c_{i}x_{i})\Big{)}=\sum\limits_{i=1}^{k}c_{i}\xi(\pi(x_{i})),$

for all $\sum\limits_{i=1}^{k}c_{i}x_{i}\in P_{d}(X)$ where $\pi:P_{d}(X)\to
P_{d}(X)/\Gamma$ is the quotient map induced by the $\Gamma$-action on
$P_{d}(X)$.

For any element $x\in P_{d}(X)$, we define

$W_{k}(\pi(x))=\xi(\pi(x))+\,\mathrm{span}\\{\xi(\pi(y))-\xi(\pi(x)):y\in
X/\Gamma,d(\pi(x),\pi(y))\leq k^{2}\\}.$

The subspace $W_{k}(\pi(x))$ is a finite-dimensional affine subspace of $H$ by
the bounded geometry of $X$. We can assume that $\bigcup\limits_{k\geq
1}W_{k}(\pi(x))$ is dense in $H$, otherwise we can replace $H$ with the norm
closure of $\bigcup\limits_{k\geq 1}W_{k}(\pi(x))$. For $k<k^{\prime}$, let

$\beta_{k^{\prime},k}(\pi(x)):\mathcal{A}(W_{k}(\pi(x)))\rightarrow\mathcal{A}(W_{k^{\prime}}(\pi(x)))$

be the map defined by the inclusion of $W_{k}(\pi(x))$ into
$W_{k^{\prime}}(\pi(x))$.

Let

$\beta_{k}(\pi(x)):\mathcal{A}(W_{k}(\pi(x))\rightarrow\mathcal{A}(H)$

the map defined by the definition of $\mathcal{A}(H)$. We write
$\beta(\pi(x))$ for

$\beta_{0}(\pi(x)):\mathcal{S}\cong\mathcal{A}(W_{0}(\pi(x)))\rightarrow\mathcal{A}(H).$

For each $d>0$, fix a countable and $\Gamma$-invariant dense subset $X_{d}$ of
$P_{d}(X)$ such that $X_{d_{1}}\subseteq X_{d_{2}}$ if $d_{1}\leq d_{2}$.

######  Definition 3.3.

The _algebraic twisted equivariant Roe algebra_
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ is defined to be the set of
all functions $T:X_{d}\times X_{d}\to\mathcal{A}(H)\widehat{\otimes}K$ such
that

  1. (1)

There exists $M\geq 0$ such that $\|T(x,y)\|\leq M$ for all $x,y\in X$;

  2. (2)

There exists an integer $N$ such that

$T(x,y)\in\beta_{N}(\pi(x))(\mathcal{A}(W_{N}(\pi(x))))\widehat{\otimes}K\subseteq\mathcal{A}(H)\widehat{\otimes}K$

for all $x,y\in X$, where
$\beta_{N}(\pi(x)):\mathcal{A}(W_{N}(\pi(x)))\rightarrow\mathcal{A}(H)$ is the
$*$-homomorphism associated to the inclusion of $W_{N}(\pi(x))$ into $H$, and
$K$ is the algebra of compact operators on the Hilbert space
$H_{0}\widehat{\otimes}l^{2}(\Gamma)$;

  3. (3)

There exists $r_{1}>0$ such that if $d(x,y)>r_{1}$, then $T(x,y)=0$;

  4. (4)

There exists $L>0$ such that for each $y\in X_{d}$,

$\sharp\\{x:T(x,y)\neq 0\\}\leq L,\qquad\sharp\\{x:T(y,x)\neq 0\\}\leq L;$

  5. (5)

For any bounded set $B\subseteq P_{d}(X)$, the set

$\\{(x,y)\in B\times B\cap X_{d}\times X_{d}|T(x,y)\neq 0\\}$

is finite;

  6. (6)

There exists $r_{2}>0$ such that

$\mbox{Supp}(T(x,y))\subseteq\left\\{(s,h)\in\mathbb{R}_{+}\times
H:s^{2}+\|h-\xi(\pi(x))\|^{2}<r_{2}^{2}\right\\}$

for $x,y\in X$;

  7. (7)

There exists $c>0$ such that if $Y=(s,h)\in\mathbb{R}_{+}\times W_{N}(\pi(x))$
with $\|Y\|=\sqrt{s^{2}+\|h\|^{2}}\leq 1$ and if $T_{1}(x,y)$ satisfies that
$(\beta_{N}(\pi(x))\widehat{\otimes}1)(T_{1}(x,y))=T(x,y)$, then the
derivative of the function $T_{1}(x,y):\mathbb{R}\times
W_{N}(\pi(x))\rightarrow\text{Cliff}(W_{N}(\pi(x)))\widehat{\otimes}K$ in the
direction of $Y$ exists, denoted by $D_{Y}(T_{1}(x,y))$, and
$\|D_{Y}(T_{1}(x,y))\|\leq c$ for all $x,y\in X$;

  8. (8)

$T(\gamma x,\gamma y)=T(x,y)$ for all $x,y\in X$ and $\gamma\in\Gamma$.

We define a multiplication on $C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$
by

$(T_{1}T_{2})(x,y)=\sum\limits_{z\in X_{d}}T_{1}(x,z)T_{2}(z,y)$

and an involution by $T^{*}(x,y)=\big{(}T(y,x)\big{)}^{*}$. Then
$C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ is a $*$-algebra. Let

$E=\left\\{\sum\limits_{x\in
X_{d}}a_{x}[x]:a_{x}\in\mathcal{A}(H)\widehat{\otimes}K,\sum\limits_{x\in
X_{d}}a_{x}^{*}a_{x}\ \text{converge in norm}\right\\}.$

Then $E$ is a Hilbert module over $\mathcal{A}(H)\widehat{\otimes}K$ with

$\left\langle\sum\limits_{x\in X_{d}}a_{x}[x],\sum\limits_{x\in
X_{d}}b_{x}[x]\right\rangle=\sum\limits_{x\in X_{d}}a_{x}^{*}b_{x},$
$\left(\sum\limits_{x\in X_{d}}a_{x}[x]\right)a=\sum\limits_{x\in
X_{d}}a_{x}a[x]$

for all $a\in\mathcal{A}(H)\widehat{\otimes}K$ and $\sum\limits_{x\in
X_{d}}a_{x}[x]\in E$.

Now we can define a $*$-representation of
$C^{*}_{alg}(X,\mathcal{A}(H))^{\Gamma}$ on $E$ by

$T\left(\sum\limits_{x\in X_{d}}a_{x}[x]\right)=\sum\limits_{y\in
X_{d}}\left(\sum\limits_{x\in X_{d}}T(y,x)a_{x}\right)[y]$

for all $T\in C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ and
$\sum\limits_{x\in X_{d}}a_{x}[x]\in E$.

######  Definition 3.4.

The reduced equivariant twisted Roe algebra
$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ is the operator norm closure of
$C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ in $B(E)$, where $B(E)$ is the
algebra of all bounded adjointable homomorphisms from $E$ to $E$.

By similar arguments as in Lemma 2.5, the maximal norm on the algebraic
equivariant twisted Roe algebra is also well-defined. Thus, we can define the
maximal equivariant twisted Roe algebra.

######  Definition 3.5.

The maximal equivariant twisted Roe algebra
$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ is defined to be the
completion of the $*$-algebra $C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$
with respect to the maximal norm

$\|T\|_{max}:=\sup\\{\|\phi(T)\|:\phi:C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))^{\Gamma}\rightarrow
B(E_{\phi}),\ \text{a}\ \text{$\ast$-representation}\\}.$

Let $C_{L,alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ be the set of all
bounded, uniformly norm continuous functions

$g:\mathbb{R}_{+}\rightarrow C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$

such that:

  1. (1)

there exists $N$ such that

$g(t)(x,y)\in(\beta_{N}(\pi(x))\hat{\otimes}1)(\mathcal{A}(W_{N}(\pi(x)))\hat{\otimes}K)\subseteq\mathcal{A}(H)\hat{\otimes}K$

for all $x,y\in X_{d}$ and $t\in\mathbb{R}_{+}$;

  2. (2)

$\lim\limits_{t\to\infty}\mbox{propagation}(f(t))\to 0\ \text{as}\
t\rightarrow\infty$;

  3. (3)

there exists $R>0$ such that $\text{Supp}(g(t)(x,y))\subseteq
B_{\mathbb{R}_{+}\times H}(\xi(\pi(x)),R)$ for all $x,y\in X_{d}$ and
$t\in\mathbb{R}_{+}$;

  4. (4)

there exists $L>0$ such that for any $y\in P_{d}(X)$,

$\sharp\\{x:g(t)(x,y)\neq 0\\}<L$, $\sharp\\{x:g(t)(y,x)\neq 0\\}<L$

for any $t\in\mathbb{R}_{+}$.

  5. (5)

There exists $c>0$ such that if $Y=(s,h)\in\mathbb{R}_{+}\times W_{N}(\pi(x))$
with $\|Y\|=\sqrt{s^{2}+\|h\|^{2}}\leq 1$ and if $g_{1}(t)(x,y)$ satisfies
that

$(\beta_{N}(\pi(x))\widehat{\otimes}1)(g(t)_{1}(x,y))=T(x,y),$

then the derivative of the function

$g_{1}(t)(x,y):\mathbb{R}\times
W_{N}(\pi(x))\rightarrow\mbox{Cliff}(W_{N}(\pi(x)))\widehat{\otimes}K$

in the direction of $Y$ exists, denoted by $D_{Y}(g(t)_{1}(x,y))$, and

$\|D_{Y}(g(t)_{1}(x,y))\|\leq c$

for all $x,y\in X$ and $t\in\mathbb{R}_{+}$;

######  Definition 3.6.

The twisted equivariant localization algebra
$C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ is the completion of
$C_{L,alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ with respect to the norm

$\|g\|=\sup\limits_{t\in\mathbb{R}_{+}}\|g(t)\|_{C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}}.$

There is an evaluation map

$e:C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma},$

by

$e(g)=g(0),$

for all $g\in C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$. We have a
homomorphism

$e_{*}:K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\rightarrow
K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}),$

induced by the evaluation map on $K$-theory.

In the rest of this subsection, we shall prove the following result.

######  Proposition 3.7.

Let $\Gamma$ be a countable discrete group acting properly and isometrically
on a discrete metric space $X$ with bounded geometry. Assume that all
$\Gamma$-orbits in $X$ are uniformly equivariantly coarsely equivalent. If the
quotient space $X/\Gamma$ admits a coarse embedding into Hilbert space and
$\Gamma$ satisfies the Baum–Connes conjecture with coefficients, then the map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\rightarrow\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})$

is an isomorphism.

To prove the above proposition, we need to analyze the ideals of the twisted
Roe algebras and localization algebras associated to the open subsets of
$\mathbb{R}_{+}\times H$. Let $O$ be an open subset of $\mathbb{R}_{+}\times
H$. Define $C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ to be a
$*$-subalgebra of $C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ by

$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}=\\{T\in
C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}:\text{Supp}(T(x,y))\subseteq
O,\forall x,y\in X\\}.$

We can define $C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ to be the norm closure of
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ in
$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$, respectively. We would like to
point out that the norm on $C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$
is not the maximal norm of the $*$-algebra
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$.

Similarly, let

$C_{L,alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O}:=\\{g\in
C_{L,alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}:\text{Supp}(g(t))\subseteq
X_{d}\times X_{d}\times O\\}.$

Define $C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O}$ to be the norm closure
of $C_{L,alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O}$ in
$C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$.

Let

$B(\xi(\pi(x)),r)=\\{(s,v)\in\mathbb{R}_{+}\times
H,s^{2}+\|v-\xi(\pi(x))<r^{2}\|\\},$

where $\pi:X\to X/\Gamma$ is the quotient map and $\xi:X/\Gamma\to H$ is the
coarse embedding. Denote $O_{r}=\bigcup\limits_{x\in X}B(\xi(\pi(x)),r)$. We
have

$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}$

and

$C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}.$

Recall that $\pi:X\to X/\Gamma$ is the natural quotient map associated to the
$\Gamma$-action on $X$.

For any $r>0$, since $X/\Gamma$ has bounded geometry, there exists finitely
many mutually disjoint subsets of $X/\Gamma$, say $\widetilde{X}_{k}$ for
$1\leq k\leq k_{0}$, such that

  1. (1)

$X/\Gamma=\bigsqcup\limits_{k=1}^{k_{0}}\widetilde{X_{k}};$

  2. (2)

for each $\widetilde{X}_{k}$ and for all $z,z^{\prime}\in\widetilde{X}_{k}$,
if $z\neq z^{\prime}$, then

$\|\xi(z)-\xi(z^{\prime})\|\geq 2r,$

where $\xi:X/\Gamma\to H$ is the coarse embedding.

Let $Y\subset X$ be a $\Gamma$-domain of $X$ in the sense that each orbit
intersects with $Y$ at exactly one point. For simplicity, we denote

$Y_{k}:=(\pi|_{Y})^{-1}(\widetilde{X}_{k})\subset Y$

for $k=1,2,\cdots,k_{0}$, where $\pi|_{Y}:Y\to X/\Gamma$ is the restriction of
$\pi:X\to X/\Gamma$ to the $\Gamma$-domain $Y\subset X$. For each
$k=1,2,\cdots,k_{0}$ and $r>0$, denote

$O_{r,k}=\bigcup\limits_{x\in Y_{k}}B(\xi(\pi(x)),r).$

######  Lemma 3.8.

With the same assumptions as Proposition 3.7 and $\\{O_{r,k}\\}_{r,k}$ defined
above, the map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma})\rightarrow\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma})$

induced by the evaluation map is an isomorphism for all $r>0$ and $1\leq k\leq
k_{0}$.

###### Proof.

For each $x\in X$, set $O_{x}=B(\xi(\pi(x)),r)$ for brevity. Define

$\Pi^{u}_{x\in Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$

to be the $*$-subalgebra of the product algebra $\Pi_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma},$ such that for
each element

$\Pi_{x\in Y_{k}}T_{x}\in\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma},$

the conditions in Definition 3.3 are satisfied for all $T_{x}$ with $x\in
X_{k}$ uniformly.

For each element $a\in\mathcal{A}(H)$ with support contained in $O_{r,k}$, it
can be decomposed as a sum

$a=\sum_{x\in Y_{k}}a_{x},$

where each $a_{x}$ is a restriction of $a$ to $O_{x}$ and can be viewed as a
function supported in $O_{x}$ for all $x\in X$. It is obvious that

$C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma}=\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}.$

We define $\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$ to be the norm closure
of

$\left\\{\Pi_{x\in Y_{k}}T_{x}~{}\Big{|}~{}T_{x}\in
C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma},\sup_{x\in
X_{k}}\|T_{x}\|<\infty\right\\}$

under the supremum norm.

Similarly, we define $\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$ to be the
$C^{*}$-subalgebra of

$\left\\{\Pi_{x\in Y_{k}}b_{x}~{}\Big{|}\Pi_{x\in
Y_{k}}b_{x}(t)\in\Pi^{u}_{x\in
Y_{k}}C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}\ \mbox{and}\
\sup_{x\in Y_{k}}\|b_{x}\|<\infty\right\\}$

generated by elements $\Pi_{x\in Y_{k}}b_{x}$, such that

  1. (1)

the function

$\Pi_{x\in Y_{k}}b_{x}:\mathbb{R}_{+}\rightarrow\Pi^{u}_{x\in
Y_{k}}C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$

is uniformly norm-continuous in $t\in\mathbb{R}_{+}$;

  2. (2)

there exists a bounded function $c(t)$ on $\mathbb{R}_{+}$ with
$\lim\limits_{t\rightarrow\infty}c(t)=0$ such that $b_{x}(t)(y,y^{\prime})=0$
whenever $d(y,y^{\prime})>c(t)$ for all $x\in Y_{k},y,y^{\prime}\in X_{d}$ and
$t\in\mathbb{R}_{+}$.

By the definition of the algebraic equivariant twisted Roe algebra, we can
prove that

$C^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O{r,k}}\cong\lim\limits_{S\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}$

and

$C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O_{r,k}}\cong\lim\limits_{S\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma},$

where $N_{S}(\Gamma\cdot x)$ is the $S$-neighbourhood of $\Gamma\cdot x$ in
$X$. So

$\lim\limits_{d\rightarrow\infty}C^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O{r,k}}\cong\lim\limits_{d\rightarrow\infty}\lim\limits_{S\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}$ $\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cong\lim\limits_{S\rightarrow\infty}\lim\limits_{d\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}$

and

$\lim\limits_{d\rightarrow\infty}C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}_{O{r,k}}\cong\lim\limits_{d\rightarrow\infty}\lim\limits_{S\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cong\lim\limits_{S\rightarrow\infty}\lim\limits_{d\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}.$

It suffices to show that

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}\left(\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\to\lim\limits_{d\rightarrow\infty}K_{*}(\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(N_{S}(\Gamma\cdot x)),\mathcal{A}(H))_{O_{x}}^{\Gamma})$

induced by the evaluation map on $K$-theory is an isomorphism for each $S>0$.

For each $S>0$ and $x\in Y_{k}$, we define a map:

$f_{x}:P_{d}(\Gamma)\to P_{d}(N_{S}(\Gamma\cdot x))$

by

$f_{x}\left(\sum c_{g}g\right)=\sum_{g}c_{g}gx$

for each $\sum_{g\in\Gamma}c_{g}g\in P_{d}(\Gamma)$. When $d$ is suitably
large, we obtain a family of continuous maps $\\{f_{j}\\}_{j\in J_{k}}$
satisfying the following conditions:

  1. (1)

each $f_{x}$ is $\Gamma$-equivariant;

  2. (2)

there exists constants $c_{1},c_{2}>0$ such that

$c_{1}\cdot d(y,y^{\prime})\leq d(f_{x}(y),f_{x}(y^{\prime}))\leq c_{2}\cdot
d(x,y),$

for all $y,y^{\prime}\in P_{d}(\Gamma)$ and $x\in Y_{k}$;

  3. (3)

there exists a constant $c>0$ such that $f_{x}(P_{d}(\Gamma))$ is a $c$-net in
$P_{d}(N_{S}(\Gamma\cdot x))$ for all $x\in Y_{k}$, i.e.
$P_{d}(N_{S}(\Gamma\cdot x))=N_{c}(f_{x}(P_{d}(\Gamma)))$, where
$N_{c}(f_{x}(P_{d}(\Gamma)))$ is the $c$-neighborhood of
$f_{x}(P_{d}(\Gamma))$ in $P_{d}(N_{S}(\Gamma\cdot x))$;

  4. (4)

$\\{f_{x}\\}_{x\in Y_{k}}$ is a $\Gamma$-equivariantly strongly Lipschitz
homotopy equivalence.

Following from conditions (1) and (3) above, we have that

$\displaystyle\Pi^{u}_{x\in Y_{k}}C^{*}(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}\cong$ $\displaystyle\Pi^{u}_{x\in
Y_{k}}C^{*}(P_{d}(\Gamma),\mathcal{A}(H))_{O_{x}}^{\Gamma}$
$\displaystyle\cong$ $\displaystyle C^{*}(P_{d}(\Gamma),\Pi^{u}_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}.$

For the localization algebras, following the proof of the Proposition 3.7 in
[36] by using the storngly $\Gamma$-equivalently Lipschitz homotopy
equivalence, we also have that

$\displaystyle\Pi^{u}_{x\in Y_{k}}C_{L}^{*}(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H))_{O_{x}}^{\Gamma}\cong$ $\displaystyle\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}(P_{d}(\Gamma),\mathcal{A}(H))_{O_{x}}^{\Gamma}$
$\displaystyle\cong$ $\displaystyle C_{L}^{*}(P_{d}(\Gamma),\Pi^{u}_{x\in
X_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}.$

Since the Baum–Connes conjecture with coefficients holds for the group
$\Gamma$, the assembly map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}\left(C_{L}^{*}\left(P_{d}(\Gamma),\Pi^{u}_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}}\right)^{\Gamma}\right)\to\lim\limits_{d\rightarrow\infty}K_{*}\left(C^{*}\left(P_{d}(\Gamma),\Pi^{u}_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}}\right)^{\Gamma}\right)$

is an isomorphism.

When $d>0$ is large enough, from the above discussion, we have the following
commutative diagram:

${K_{*}\left(\Pi^{u}_{x\in Y_{k}}C_{L}^{*}\left(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H)\right)_{O_{x}}^{\Gamma}\right)}$${K_{*}\left(\Pi^{u}_{x\in
Y_{k}}C^{*}\left(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H)\right)_{O_{x}}^{\Gamma}\right)}$${K_{*}\left(C_{L}^{*}\left(P_{d}(\Gamma),\Pi^{u}_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}}\right)^{\Gamma}\right)}$${K_{*}\left(C^{*}\left(P_{d}(\Gamma),\Pi^{u}_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}}\right)^{\Gamma}\right).}$$\scriptstyle{e_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{e_{*}}$

It follows that the assembly map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}\left(\Pi^{u}_{x\in
Y_{k}}C_{L}^{*}\left(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H)\right)_{O_{x}}^{\Gamma}\right)\to\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\lim\limits_{d\rightarrow\infty}K_{*}\left(\Pi^{u}_{x\in
Y_{k}}C^{*}\left(P_{d}(N_{S}(\Gamma\cdot
x)),\mathcal{A}(H)\right)_{O_{x}}^{\Gamma}\right)$

is an isomorphism. ∎

To prove proposition 3.7, we also need the following lemma, which can be
proved by the same arguments in Lemma 6.3 in [37].

######  Lemma 3.9.

Let $X_{1}$ and $X_{2}$ be $\Gamma$-invariant subsets of $X$, and let

$O_{r}^{1}=\cup_{x\in X_{1}}B_{\mathbb{R}_{+}\times H}(\xi(\pi(x)),r),\quad
O_{r}^{2}=\cup_{x\in X_{2}}B_{\mathbb{R}_{+}\times H}(\xi(\pi(x)),r).$

Then we have that

$\begin{split}\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}}^{\Gamma}+\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{2}_{r}}^{\Gamma}\\\
=\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}\cup
O^{2}_{r}}^{\Gamma};\end{split}$
$\begin{split}\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}}^{\Gamma}+\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{2}_{r}}^{\Gamma}\\\
=\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}\cup
O^{2}_{r}}^{\Gamma};\end{split}$
$\begin{split}\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}}^{\Gamma}\cap\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{2}_{r}}^{\Gamma}\\\
=\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}\cap
O^{2}_{r}}^{\Gamma};\end{split}$
$\begin{split}\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}}^{\Gamma}\cap\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{2}_{r}}^{\Gamma}\\\
=\lim_{\begin{subarray}{c}r<r_{0},r\rightarrow
r_{0}\end{subarray}}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O^{1}_{r}\cap
O^{2}_{r}}^{\Gamma}.\end{split}$

###### Proof of Proposition 3.7.

Since

$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}$

and

$C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma},$

where $O_{r}=\bigcup\limits_{1\leq k\leq k_{0}}O_{r,k}$, following the similar
arguments in [30] we obtain that

$\lim\limits_{d\rightarrow\infty}\lim\limits_{r\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma})\cong\lim\limits_{r\rightarrow\infty}\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma},$

and

$\lim\limits_{d\rightarrow\infty}\lim\limits_{r\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}\cong\lim\limits_{r\rightarrow\infty}\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}.$

We get the following commuting diagram:

$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}_{L}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\lim\limits_{d\rightarrow\infty}\lim\limits_{r\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}\lim\limits_{r\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\lim\limits_{r\rightarrow\infty}\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{r\rightarrow\infty}\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}).}$

The conclusion follows that the bottom horizon map is an isomorphism by using
Lemma 3.8, Lemma 3.9 and the Mayer–Vietoris sequence argument. ∎

###  3.2. The maximal and reduced equivariant twisted Roe algebras

In this subsection, we shall prove that the canonical quotient map

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}\to
C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$

is an isomorphism.

Recall that in Section 3.1, we defined a $2$-sided $*$-ideal
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ of
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ consisting of the operators
with entries supported in $O$ for each open subset $O$ in
$\mathbb{R}_{+}\times H$, and the $C^{*}$-subalgebras
$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ are the operator norm closure of
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}$ in
$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$, respectively. Recall that

$O_{r,k}=\bigcup\limits_{x\in
Y_{k}}B(\xi(\pi(x)),r)\subset\mathbb{R}_{+}\times H.$

######  Lemma 3.10.

Let $X$ be a discrete metric space with bounded geometry, and let $\Gamma$ be
a countable discrete amenable group. Assume that $\Gamma$ acts on $X$ properly
and isometrically such that the $\Gamma$-orbits are uniformly equivariantly
coarsely equivalent. If the quotient space $X/\Gamma$ admits a coarse
embedding into Hilbert space, then the canonical quotient map

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A})_{O_{r,k}}^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma}$

extended from the identity map on
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma}$ is an isomorphism.

###### Proof.

For brevity, we denote by $O$ the open subset $O_{r,k}$ and denote
$O_{x}=B(\xi(\pi(x)),r)$. Let $\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$ be the
$*$-algebra consisting of the operators

$\Pi_{x\in Y_{k}}T_{x}\in\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma},$

such that the family of operators $\\{T_{x}\\}_{x\in Y_{k}}$ satisfies
conditions in Definition 3.3 with uniform constants. Since $O=\bigsqcup_{x\in
Y_{k}}O_{x}$, we have that

$C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}=\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}.$

By the definition of each
$C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}$, we have

$\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(P_{d}(X),\mathcal{A}(H))_{O_{x}}^{\Gamma}=\lim\limits_{R\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(N_{R}(\Gamma\cdot x),\mathcal{A}(H))_{O_{x}}^{\Gamma},$

where the inductive limit can be viewed as the union of these $*$-algebras.

Denote by $|\Gamma|$ the metric space of $\Gamma$ endowed with a word-length
metric. Let $\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(|\Gamma|,\mathcal{A}(H))_{O_{x}}^{\Gamma}$ be the
$C^{*}$-algebra of all the operators

$\Pi_{x\in Y_{k}}T_{x}\in\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(|\Gamma|,\mathcal{A}(H))_{O_{x}}^{\Gamma}$

satisfying that

  1. (1)

each $T_{x}$ is a $\Gamma\times\Gamma$-matrix and $T_{x}(g,h)=T_{x}(\gamma
g,\gamma h)$ for $g,h,\gamma\in\Gamma$;

  2. (2)

there exists a constant $M>0$, such that $\sup_{x\in
Y_{k},g,h\in\Gamma}\|T_{x}(g,h)\|\leq M$;

  3. (3)

$\sup_{x\in Y}\mbox{propagation}(T_{x})<\infty$;

  4. (4)

there exists $c>0$ such that if $Y=(s,v)\in\mathbb{R}_{+}\times W_{N}(\pi(x))$
with $\|Y\|=\sqrt{s^{2}+\|v\|^{2}}\leq 1$ and if $T^{\prime}_{x}(g,h)$ is an
element satisfying
$(\beta_{N}(\pi(x_{x}))\widehat{\otimes}1)(T^{\prime}_{x}(g,h))=T(g,h)$, then
the derivative of the function $T^{\prime}_{x}(g,h):\mathbb{R}\times
W_{N}(\pi(x))\rightarrow\text{Cliff}(W_{N}(\pi(x)))\widehat{\otimes}K$ in the
direction of $Y$ exists, denoted by $D_{Y}(T^{\prime}_{x}(g,h))$, and
$\|D_{Y}(T^{\prime}_{x}(g,h))\|\leq c$ for all $g,h\in\Gamma$.

Since the $\Gamma$-orbits are uniformly coarsely equivalent, all the metric
spaces $N_{R}(\Gamma\cdot x)$ with $x\in Y_{k}$ are uniformly coarsely
equivalent to $|\Gamma|$. It follows that

$\lim\limits_{R\rightarrow\infty}\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(N_{R}(\Gamma\cdot
x),\mathcal{A}(H))_{O_{x}}^{\Gamma}\cong\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(|\Gamma|,\mathcal{A}(H))_{O_{x}}^{\Gamma}.$

Moreover, since

$\Pi^{u}_{x\in
Y_{k}}C^{*}_{alg}(|\Gamma|,\mathcal{A}(H))_{O_{x}}^{\Gamma}=C^{*}_{alg}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma},$

we have that

$C^{*}_{max}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}\cong
C^{*}_{max}(|\Gamma|,\Pi_{j\in J_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$

and

$C^{*}(P_{d}(X),\mathcal{A}(H))_{O}^{\Gamma}\cong C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}.$

Since $\Gamma$ is amenable, there exists a sequence of finitely supported
positive type functions $\phi_{n}:\Gamma\rightarrow\mathbb{C}$ such that
$\phi_{n}$ converge pointwise to $1$(c.f. [4, Theorem 2.6.8]). Denote

$k_{n}(g,h)=\phi_{n}(g^{-1}h)$

for all $g,h\in\Gamma$. It is obvious that each $k_{n}$ is a positive definite
kernel $k_{n}:\Gamma\times\Gamma\to[0,1]$ such that

$k_{n}(\gamma g,\gamma h)=k_{n}(g,h),$

for all $\gamma,g,h\in\Gamma$. Therefore, one can define a sequence of unital
and completely positive Schur multipliers

$M^{alg}_{k_{n}}:C^{*}_{alg}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\rightarrow
C^{*}_{alg}(|\Gamma|,\Pi_{x\in Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$

by

$M^{alg}_{k_{n}}(T)(g,h)=k_{n}(g,h)T(g,h)$

for all $T\in C^{*}_{alg}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$ and $g,h\in\Gamma$. Each operator
$M^{alg}_{k_{n}}$ continuously extends to completely positive maps

$M^{max}_{k_{n}}:C^{*}_{max}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\rightarrow
C^{*}_{max}(|\Gamma|,\Pi_{x\in Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$

and

$M_{k_{n}}:C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\rightarrow C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$

By the definition of $k_{n}$, we know that $M^{max}_{k_{n}}$ and $M_{k_{n}}$
converge in point-norm to identity maps on $C_{max}^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$ and $C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}$, respectively.

Now we have the following commutative diagram:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
49.99475pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C^{*}_{max}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\lambda}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.02444pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78334pt\hbox{$\scriptstyle{M^{max}_{k_{n}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.99475pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.99475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
123.9895pt\raise-21.21889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\lambda}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
123.9895pt\raise-30.02444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.15314pt\raise-36.00722pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
73.99475pt\raise-42.43777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.99475pt\raise-42.43777pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{$\textstyle{C^{*}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H)_{O_{x}})^{\Gamma}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

For each operator $T\in\text{Ker}(\lambda)$, we have

$M_{k_{n}}(\lambda(T))=\lambda(M^{max}_{k_{n}}(T))=0.$

Since $M^{max}_{k_{n}}(T)\in C^{*}_{alg}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H))_{O_{x}}^{\Gamma}$ and the restriction of $\lambda$ on
the $*$-subalgebra $C^{*}_{alg}(|\Gamma|,\Pi_{x\in
Y_{k}}\mathcal{A}(H))_{O_{x}}^{\Gamma}$ is the identity map, we have that
$M^{max}_{k_{n}}(T)=0$. It follows that

$0=T=\lim\limits_{n\to\infty}M^{max}_{k_{n}}(T).$

Therefore, $\lambda$ is injective. As a result, the map

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r,k}}^{\Gamma}$

is an isomorphism.

∎

We shall glue all the isomorphisms between ideals together using the following
result.

######  Lemma 3.11.

Let $A,A^{\prime}$ be $C^{*}$-algebras, $\varphi:A\rightarrow A^{\prime}$ a
$*$-homomorphism. Assume $A=I+J$, $A^{\prime}=I^{\prime}+J^{\prime}$, where
$I,J$ are 2-sided ideals of $A$, and $I^{\prime},J^{\prime}$ are 2-sided
ideals of $A^{\prime}$. If the restriction maps $\varphi_{I}:I\rightarrow
I^{\prime}$, $\varphi_{J}:J\rightarrow J^{\prime}$ and $\varphi_{I\cap
J}:I\cap J\rightarrow I^{\prime}\cap J^{\prime}$ are all isomorphisms, then
$\varphi:A\rightarrow A^{\prime}$ is an isomorphism.

###### Proof.

Since $A/I=I+J/I=J/I\cap J$, so the quotient map $\varphi_{A/I}:A/I\rightarrow
A^{\prime}/I^{\prime}$ is an isomorphism. We have the following commutative
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{I}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{A/I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A/I}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}/I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

As a consequence of diagram-chasing, we obtain that $\varphi$ is an
isomorphism. ∎

######  Proposition 3.12.

Let $X$ be a discrete metric space with bounded geometry, and $\Gamma$ a
countable discrete group. Assume $\Gamma$ acts on $X$ properly and
isometrically with the $\Gamma$-orbits uniformly equivariantly coarsely
equivalent. If the quotient space $X/\Gamma$ is coarsely embeddable into
Hilbert space and $\Gamma$ is amenable, then the homomorphism

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$

is an isomorphism.

###### Proof.

Denote

$O_{r}=\bigcup\limits_{\pi(x)\in X/\Gamma}B(\pi(x),r).$

Note that $C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}$ are 2-sided ideals of
$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$ and
$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$, respectively. Moreover, we have
that

$C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma},$

and

$C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}=\lim\limits_{r\rightarrow+\infty}C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}.$

For each $r>0$, $O_{r}=\bigcup\limits_{k=1}^{k_{0}}O_{r,k}$. By Lemma 3.9,
3.10, 3.11, we have that the canonical quotient

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}\to
C^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma},$

extended from the identity map on
$C_{alg}^{*}(P_{d}(X),\mathcal{A}(H))_{O_{r}}^{\Gamma}$ is an isomorphism. As
a result, the homomorphism

$\lambda:C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma}$

is an isomorphism. ∎

##  4\. Proof of the main results

In this section, we shall show that the maps

$e_{*}:\lim\limits_{d\to\infty}K_{*}(C_{L}(P_{d}(X))^{\Gamma})\to
K_{*}(C^{*}(X)^{\Gamma})$

and

$\lambda_{*}:K_{*}(C_{max}^{*}(X)^{\Gamma})\to K_{*}(C^{*}(X)^{\Gamma})$

are isomorphisms under the assumptions in Theorem 1.1. In order to prove the
main result, we shall introduce a geometric analogue of Bott periodicity
following the constructions of Yu [37].

Let $H$ be a separable infinite-dimensional real Hilbert space. Let
$V\subseteq H$ be a finite dimensional affine subspace of $H$. Denote by
$V^{0}$ the finite-dimensional linear subspace consisting of differences of
elements in $V$. Let $\mathcal{L}^{2}(V)=L^{2}(V,{\rm Cliff}(V^{0}))$ be the
graded infinite dimensional Hilbert space of $L^{2}$-maps from $V$ to the
complexified Clifford algebra of $V^{0}$.

Let $V_{a}\subseteq V_{b}$ be finite-dimensional affine subspaces of $H$. Then
we have an algebraic decomposition

$V_{b}=V_{ba}^{0}\oplus V_{a},$

where $V_{ba}^{0}$ is the orthogonal complement of $V_{a}^{0}$ in $V_{b}^{0}$.
Consider a unit vector $\xi_{0}\in\mathcal{L}^{2}(V_{ba}^{0})$ defined by

$\xi_{0}(w)=\pi^{-\frac{\text{dim}(V_{ba})}{4}}\exp(-\frac{1}{2}\|w\|^{2}),$

for all $w\in V_{ba}^{0}$. Then we can regard $\mathcal{L}^{2}(V_{a})$ as a
subspace of $\mathcal{L}^{2}(V_{b})$ via an isometric inclusion given by

(4.1)
$i_{ba}:\mathcal{L}^{2}(V_{a})\rightarrow\mathcal{L}^{2}(V_{ba}^{0})\widehat{\otimes}\mathcal{L}^{2}(V_{a})\cong\mathcal{L}^{2}(V_{b}),\quad\xi\mapsto\xi_{0}\widehat{\otimes}\xi.$

For any finite-dimensional affine subspaces $V_{a}\subseteq V_{b}\subseteq
V_{c}$, we have

$i_{ca}=i_{cb}\circ i_{ba}.$

Then we define

$\mathcal{L}^{2}(H)=\lim\limits_{\rightarrow}\mathcal{L}^{2}(V),$

where the limit is taken over all the finite-dimensional affine subspaces
$V\subseteq H$.

Let $\mathscr{S}(V)\subseteq\mathcal{L}^{2}(H)$ be the subspace of Schwartz
class functions from $V$ to Cliff$(V^{0})$. Choose an orthonormal basis
$\\{e_{1},e_{2},\cdots,e_{n}\\}$ for $V^{0}$, and let
$\\{x_{1},x_{2},\cdots,x_{n}\\}$ be the dual coordinates to
$\\{e_{1},e_{2},\cdots,e_{n}\\}$, we define the Dirac operator $D_{V}$, an
unbounded operator on $\mathcal{L}^{2}(H)$ with domain $\mathscr{S}(V)$, by
the formula

$D_{V}\xi=\sum\limits_{i=1}^{n}(-1)^{\text{deg}\xi}\frac{\partial\xi}{\partial
x_{i}}e_{i},$

where $e_{i}$ is the Clifford multiplication by $e_{i}\in
V^{0}\subseteq\text{Cliff}(V^{0})$. Define the Clifford operator $C_{V,v}$
with domain $\mathscr{S}(V)$, by

$(C_{V,v}\xi)(w)=(w-v)\cdot\xi(w),$

where $v\in V$ is a fixed base point, and the multiplication is the Clifford
multiplication by the vector $w-v\in V^{0}$.

Now fix $x\in P_{d}(X)$. Let $W_{k}(\pi(x))$ be the finite-dimensional
subspace as in Definition 3.2, and identify $\mathcal{L}^{2}(W_{k}(\pi(x)))$
as a subspace of $\mathcal{L}^{2}(W_{k+1}(\pi(x)))$ via the isometric
inclusion

$\mathcal{L}^{2}(W_{k}(\pi(x)))\rightarrow\mathcal{L}^{2}(W_{k+1}(\pi(x)))$

defined in (4.1). Note that these inclusions preserve the Schwartz subspaces
$\mathscr{S}(W_{k}(\pi(x)))$. Define a Schwartz subspace of
$\mathcal{L}^{2}(H)$ by taking the algebraic direct limit

$\mathscr{S}(\pi(x))=\lim\limits_{\longrightarrow}\mathscr{S}(W_{k}(\pi(x))).$

Let $V_{0}(\pi(x))=W_{1}(\pi(x))$ and $V_{k}(\pi(x))=W_{k+1}(\pi(x))\ominus
W_{k}(\pi(x))$ if $k\geq 1$, where $x\in P_{d}(X)$. We consider the Dirac
operator $D_{k}$ defined by

$D_{k}=D_{V_{k}(\pi(x))}$

and Clifford operators $C_{k,\pi(x)}$ defined by

$C_{0,\pi(x)}=C_{V_{0}(\pi(x)),f(\pi(x))};\;C_{k,\pi(x)}=C_{V_{k}(\pi(x)),0},\;k\geq
1,$

associated to each $V_{k}(\pi(x))$. For each $n\in\mathbb{N}$ and $t\geq 1$,
define an unbounded operator $B_{n,t}(\pi(x))$ on $\mathcal{L}^{2}(H)$ by

$B_{n,t}(\pi(x))=\sum\limits_{k=0}^{n-1}(1+kt^{-1})D_{k}+\sum\limits_{k=n}^{\infty}(1+kt^{-1})(D_{k}+C_{k,\pi(x)})$

associated to the decomposition

$V_{0}(\pi(x))\oplus V_{1}(\pi(x))\oplus\cdots\oplus
V_{n}(\pi(x))\oplus\cdots.$

Note that the operator $B_{n,t}(\pi(x))$ is well-defined on the Schwartz space
$\mathscr{S}(\pi(x))$.

Let $\mathcal{K}$ be the algebra of compact operators on
$\mathcal{H}=\mathcal{L}^{2}(H)$. We can define the $*$-algebra
$C_{alg}^{*}(X,\mathcal{K})^{\Gamma}$ similarly as $C^{*}(X)^{\Gamma}$ by
changing the coefficients of $T\in C_{alg}^{*}(X)^{\Gamma}$ from $K$ to
$\mathcal{K}\widehat{\otimes}K$. Let $C_{max}^{*}(X,\mathcal{K})^{\Gamma}$ and
$C_{red}^{*}(X,\mathcal{K})^{\Gamma}$ be the maximal and reduced Roe algebra
which are the completions of $C_{alg}^{*}(X,\mathcal{K})^{\Gamma}$ under the
maximal and reduced norm, respectively.

Recall that in Section 2, we choose a $\Gamma$-invariant countable dense
subset $X_{d}\subset P_{d}(X)$ for each $d>0$.

######  Definition 4.1.

Let $C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$
be the set of all functions $T$ from $X_{d}\times X_{d}$ to
$\mathcal{S}\widehat{\otimes}\mathcal{K}\widehat{\otimes}K$ such that

  1. (1)

there exists $M>0$ such that $\|T(x,y)\|\leq M$ for any $(x,y)\in X_{d}\times
X_{d}$;

  2. (2)

there exists $r_{1}>0$ such that $T(x,y)=0$ if $d(x,y)>r_{1}$;

  3. (3)

there exists $L>0$ such that for each $y\in X_{d}$,

$\sharp\\{x:T(x,y)\neq 0\\}\leq L,\qquad\sharp\\{x:T(y,x)\neq 0\\}\leq L;$

  4. (4)

for any bounded set $B\subseteq P_{d}(X)$, the set $\\{(x,y)\in B\times B\cap
X_{d}\times X_{d}|T(x,y)\neq 0\\}$ is finite;

  5. (5)

there exists $r_{2}>0$, such that if $|t|>r_{2}$, then $T(x,y)(t)=0$ for all
$(x,y)\in X\times X$, where
$T(x,y)\in{S}\widehat{\otimes}\mathcal{K}\widehat{\otimes}K$ can be viewed as
a $\mathcal{K}\widehat{\otimes}K$ valued function on $\mathbb{R}$;

  6. (6)

there exists $c>0$ such that $\|\frac{d}{dt}T(x,y)\|\leq c$ for all $(x,y)\in
X\times X$;

  7. (7)

$\gamma(T)=T$, i.e. $T(\gamma^{-1}x,\gamma^{-1}y)=T(x,y)$.

Following Definition 3.3, we can view
$C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ as a
$*$-algebra acting on the graded Hilbert module

$\mathcal{E}=\ell^{2}(X_{d})\widehat{\otimes}\ell^{2}(\Gamma)\widehat{\otimes}\mathcal{S}\widehat{\otimes}\mathcal{K}\widehat{\otimes}K$

over $\mathcal{S}\widehat{\otimes}\mathcal{K}\widehat{\otimes}K$, where the
$\mathbb{Z}_{2}$-grading on $\ell^{2}(X_{d})$, $\ell^{2}(\Gamma)$ and $K$ are
trivial and $\mathcal{S}$ is graded by the odd and even functions.

######  Definition 4.2.

  1. (1)

The $C^{*}$-algebra
$C^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ is defined
to be the operator norm closure of
$C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ in
$B(\mathcal{E})$, where $B(\mathcal{E})$ is the $C^{*}$-algebra of all
adjointable module homomorphisms from $\mathcal{E}$ to $\mathcal{E}$.

  2. (2)

The $C^{*}$-algebra
$C_{max}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ is
the completion of
$C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ with
respect to the $C^{*}$-norm

$\|T\|_{max}:=\sup\\{\|\phi(T)\|,\phi:C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\rightarrow
B(\mathcal{E}_{\phi}),\ \text{a}\ \text{$\ast$-representation}\\},$

where $\mathcal{E}_{\phi}$ is a Hilbert module over the $C^{*}$-algebra
$\mathcal{S}\widehat{\otimes}\mathcal{K}\widehat{\otimes}K$.

Naturally, we have the canonical quotient map

$\lambda:C^{*}_{max}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})\to
C^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})$

extended from the identity map on the $*$-algebra
$C^{*}_{alg}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})$.

We can similarly define the localization algebra, denoted
$C^{*}_{L,alg}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$, to
be the $*$-algebra of all uniformly bounded and uniformly continuous functions
$f:[0,\infty)\to
C^{*}_{alg}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ such
that

  1. (1)

${\rm propagation}(f(t))\to 0$ as $t\to\infty$,

  2. (2)

the constants in condition $3$, $4$, $5$ and $6$ of Definition 4.1 are
independent on $t\in[0,\infty)$.

######  Definition 4.3.

The localization algebra, denoted by
$C^{*}_{L}(P_{d}(X),\mathcal{K})^{\Gamma}$, is the completion of the
$*$-algebra $C^{*}_{alg}(P_{d}(X),\mathcal{K})^{\Gamma}$ under the norm

$\|f\|=\sup_{t\geq 0}\|f(t)\|.$

From the definition of
$C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$ and
the nuclearity of $\mathcal{S}$, we have

$C^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X),\mathcal{K})^{\Gamma},$
$C_{L}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X),\mathcal{K})^{\Gamma},$

and

$C_{max}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}.$

For every non-negative integer $n$ and $x\in X$, we define

$(\theta_{t}^{n}(x)):\mathcal{A}(W_{n}(\pi(x)))\widehat{\otimes}K\rightarrow\mathcal{S}\widehat{\otimes}K(\mathcal{L}^{2}(H))\widehat{\otimes}K$

by

$(\theta_{t}^{n}(x))(g\widehat{\otimes}h\widehat{\otimes}k)=g_{t}(X\widehat{\otimes}1+1\widehat{\otimes}B_{n,t}(\pi(x)))(1\widehat{\otimes}M_{h_{t}})\widehat{\otimes}k$

for all $g\in\mathcal{S}$, $h\in
C_{0}(W_{n}(\pi(x)),\text{Cliff}(W_{n}(\pi(x))))$, $k\in K$, $x\in X_{d}$,
where $g_{t}(s)=g(t^{-1}s)$ for $s\in\mathbb{R}$,
$h_{t}(v)=h(\xi(\pi(x))+t^{-1}(v-\xi(\pi(x))))$ for $v\in W_{n}(\pi(x))$,
$M_{h_{t}}$ is the pointwise multiplication operator on $\mathcal{L}^{2}(V)$
(where $W_{n}(\pi(x))\subset V$ is an affine subspace of $H$) via the formula

$(M_{h_{t}}\xi)(v+w)=h_{t}(v)\xi(v+w)$

for all $\xi\in\mathcal{L}^{2}(V)$ and $v\in W_{n}(\pi(x))$, $w\in V\ominus
W_{n}(\pi(x))$. For every $T\in
C_{alg}^{*}(X,\mathcal{\mathcal{A}})^{\Gamma}$, let $n$ be a non-negative
integer such that for every $(x,y)\in X_{d}\times X_{d}$, there exists
$T_{1}(x,y)\in\mathcal{A}(W_{n}(\pi(x)))\widehat{\otimes}K$ satisfying
$\beta_{n}(x)(T_{1}(x,y))=T(x,y)$. We define

$(\alpha_{t}(T))(x,y)=(\theta_{t}^{n}(x))(T_{1}(x,y))$

for every $T\in C_{alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$, by Lemma 5.8 in
[19], we know that $(\alpha_{t}(T))(x,y)\in
K(\mathcal{L}^{2}(H))\widehat{\otimes}K$.

######  Definition 4.4.

For each $t\geq 1$, we define

$\alpha_{t}:C_{alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}\rightarrow
C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}$

via the formula

$(\alpha_{t}(T))(x,y)=(\theta_{t}^{n}(x))(T_{1}(x,y))$

for every $T\in C_{alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$, where $n$ is a
non-negative integer such that for every $(x,y)\in X_{d}\times X_{d}$, there
exists $T_{1}(x,y)\in\mathcal{A}(W_{n}(\pi(x)))\widehat{\otimes}K$ satisfying
$\beta_{n}(\pi(x))(T_{1}(x,y))=T(x,y)$.

By the similar arguments as in Lemma 7.2 in [37] and Proposition 3.12, we have
the asymptotic morphisms

$\alpha_{max,t}:C_{max}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}\rightarrow
C_{max}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X),\mathcal{K})^{\Gamma},$

and

$\alpha_{t}:C^{*}(P_{d}(X),\mathcal{A})^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X),\mathcal{K})^{\Gamma}$

for all $t\geq 1$. Then we have the induced homomorphisms

$\alpha_{max,*}:K_{*}(C_{max}^{*}(P_{d}(X),\mathcal{A})^{\Gamma})\rightarrow
K_{*}(\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}),$

and

$\alpha_{*}:K_{*}(C^{*}(P_{d}(X),\mathcal{A})^{\Gamma})\rightarrow
K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X),\mathcal{K})^{\Gamma})$

on $K$-theory.

Moreovere, we can define an asymptotic morphism

$\alpha_{L,t}:C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}\rightarrow\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}$

by

$\alpha_{L,t}(T_{s})(t)=\alpha_{t}(T_{s})$

for all path $(T_{s})_{s\in[0,\infty)}\in
C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$ and all $t\geq 1$. In addition, we
have the homomorphism

$\alpha_{L,*}:K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma})\rightarrow
K_{*}(\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X),\mathcal{K})^{\Gamma})$

induced by the asymptotic morphism $(\beta_{L,t})_{t\in[0,\infty)}$.

######  Definition 4.5.

For each $t\geq 1$, define a map

$\beta_{t}:\mathcal{S}\widehat{\otimes}C_{alg}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$

by

$(\beta_{t}(g\widehat{\otimes}T))(x,y)=(\beta(\pi(x)))(g_{t})\widehat{\otimes}T(x,y)$

for all $g\in\mathcal{S}$, $T\in C_{alg}^{*}(X)^{\Gamma}$ and $x,y\in X$,
where $g_{t}(s)=g(t^{-1}s)$ for all $s\in\mathbb{R}$, and
$\beta(\pi(x)):\mathcal{S}=\mathcal{A}(f(\pi(x)))\rightarrow\mathcal{A}(H)$ is
the $*$-homomorphism defined in Section 3.

Following the arguments in Lemma 7.6 in [37], we know that $\beta_{t}$ extends
to an asymptotic morphisms

$\beta_{max,t}:\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{max}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$

and

$\beta_{t}:\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma}\rightarrow
C^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$

for all $t\geq 0$. Then we have the induced homomorphism

$\beta_{max,*}:K_{*}(\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X))^{\Gamma})\rightarrow
K_{*}(C_{max}^{*}(P_{d}(X),\mathcal{A})^{\Gamma})$

and

$\beta_{*}:K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma})\rightarrow
K_{*}(C^{*}(P_{d}(X),\mathcal{A})^{\Gamma})$

on $K$-theory.

Applying the map
$\beta_{t}:\mathcal{S}\widehat{\otimes}C_{alg}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{alg}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$ point-wise gives rise to an
asymptotic morphism

$\beta_{L,t}:\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma}$

for all $t\geq 1$. Then we have a homomorphism

$\beta_{L,*}:K_{*}(\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X))^{\Gamma})\rightarrow
K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A})^{\Gamma})$

induced by the asymptotic morphism $(\beta_{L,t})_{t\in[0,\infty)}$ on
$K$-theory. Now we are ready to prove the geometric analogue of Bott
periodicity.

######  Proposition 4.6.

The compositions

$\alpha_{*}\circ\beta_{*}:K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma})\to
K_{*}(C^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}),$
$\alpha_{max,*}\circ\beta_{max,*}:K_{*}(\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X))^{\Gamma})\to
K_{*}(C_{max}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma})$

and

$\alpha_{L,*}\circ\beta_{L,*}:K_{*}(\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X))^{\Gamma})\to
K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma})$

are identity maps, respectively.

###### Proof.

We shall only prove the first composition $\alpha_{*}\circ\beta_{*}$ is an
identity map, and the other two can be proved similarly.

Define the asymptotic morphism

$\gamma:C_{c}(\mathbb{R})\widehat{\otimes}C_{alg}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong
C_{c}(\mathbb{R})\widehat{\otimes}C_{alg}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}$

by

$\gamma_{t}(g\widehat{\otimes}T)(x,y)=g_{t^{2}}(X\widehat{\otimes}1+1\widehat{\otimes}B_{0,t}(\pi(x)))\widehat{\otimes}T(x,y),$

where $g\in C_{c}(\mathbb{R})$ has compact support and continuously
differentiable, $T\in C_{alg}^{*}(P_{d}(X))^{\Gamma}$, $t\geq 1$.

For any $x\in X$, we define

$\eta:\mathcal{A}(W_{1}(\pi(x)))\rightarrow\mathcal{K}(\mathcal{H})\widehat{\otimes}K$

by

$(\eta_{t}(x))((g\widehat{\otimes}h)\widehat{\otimes}k)=g_{t^{2}}(X\widehat{\otimes}1+1\widehat{\otimes}B_{1,t}(\pi(x)))(1\widehat{\otimes}M_{h_{t^{2}}})\widehat{\otimes}k$

for each $g\in\mathcal{S}$, $h\in\text{Cliff}(W_{1}(\pi(x)))$, $k\in K$ and
$t\geq 0$. Similarly, we have an asymptotic morphism

$\gamma^{\prime}:C_{c}(\mathbb{R})\widehat{\otimes}C_{alg}^{*}(P_{d}(X))^{\Gamma}\rightarrow
C_{alg}^{*}(P_{d}(X),\mathcal{S}\widehat{\otimes}\mathcal{K})^{\Gamma}\cong
C_{c}(\mathbb{R})\widehat{\otimes}C_{alg}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}$

by

$(\gamma^{\prime}_{t}(g\widehat{\otimes}T))(x,y)=(\eta_{t}(x))(\beta_{W_{1}(\pi(x))}(g)\widehat{\otimes}T(x,y)),$

where $g\in C_{c}(\mathbb{R})$ has compact support and continuously
differentiable, $T\in C_{alg}^{*}(P_{d}(X))^{\Gamma}$, $t\geq 1$ and
$\beta_{W_{1}(\pi(x))}:\mathcal{S}=\mathcal{A}(\pi(x))\rightarrow\mathcal{A}(W_{1}(\pi(x)))$
is the homomorphism induced by the map $g\mapsto
g(X\widehat{\otimes}1+1\widehat{\otimes}C_{W_{1}(\pi(x)),\pi(x)})$.

With the similar proof of Lemma 7.6 in [37], we have

$\gamma_{t}(g\widehat{\otimes}T)\leq\|g\|\|T\|,\gamma^{\prime}_{t}(g\widehat{\otimes}T)\leq\|g\|\|T\|$

for $g\in C_{c}(\mathbb{(}R))$ and $T\in C_{alg}^{*}(P_{d}(X))^{\Gamma}$, so
$\gamma$ and $\gamma^{\prime}$ can be extended to the asymptotic morphism
$\gamma$ and $\gamma^{\prime}$ from
$\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma}$ to
$\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X),\mathcal{K})^{\Gamma}$
respectively. By the proof of Proposition 4.2 in [19], we know that $\gamma$
is asymptotically equivalent to $\gamma^{\prime}$. Hence we have
$\gamma_{*}=\gamma^{\prime}_{*}$ at the $K$-theory level.

Recall that
$\mathcal{H}=\mathcal{L}^{2}(H)=\lim\limits_{\rightarrow}\mathcal{L}^{2}(V)$,
where the limit is taken over all the finite-dimensional affine subspaces
$V\subseteq H$. For any $x\in X$ and $t\geq 1$, let $U_{x}(t)$ be the unitary
operator on $\mathcal{H}$ induced by the translation

$v\mapsto v-t\xi(\pi(x))$

on $V$. Let

$R(s)=\left(\begin{array}[]{cc}\cos\frac{\pi s}{2}&\sin\frac{\pi s}{2}\\\
-\sin\frac{\pi s}{2}&\cos\frac{\pi s}{2}\\\ \end{array}\right),s\in[0,1].$

For each $x\in X$, $s\in[0,1]$ and $t\geq 1$, define a unitary operator
$U_{x,s}(t)$ on
$(l^{2}(X)\widehat{\otimes}\mathcal{H}\widehat{\otimes}H)\oplus(l^{2}(X)\widehat{\otimes}\mathcal{H}\widehat{\otimes}H)$
by

$U_{x,s}(t)=R(s)\left(\begin{array}[]{cc}1\widehat{\otimes}U_{x}(t)\widehat{\otimes}1&0\\\
0&1\\\ \end{array}\right)R^{-1}(s).$

Now we define a family of asymptotic morphisms

$\gamma(s):C_{c}(\mathbb{R})\widehat{\otimes}C^{*}_{alg}(P_{d}(X))^{\Gamma}\longrightarrow
C_{c}(\mathbb{R})\widehat{\otimes}C_{alg}^{*}(P_{d}(X),\mathcal{K})^{\Gamma}\widehat{\otimes}M_{2}(\mathbb{C})$

by

$\gamma_{t}(s)(g\widehat{\otimes}T)(x,y)=U_{x,s}(t)\left(\begin{array}[]{cc}(\gamma^{\prime}_{t}(g\widehat{\otimes}T))(x,y)&0\\\
0&0\\\ \end{array}\right)U^{-1}_{x,s}(t)$

for $g\in C_{c}(\mathbb{R})$ has compact support and continuously
differentiable, $T\in C_{alg}^{*}(P_{d}(X))^{\Gamma}$, $t\geq 1$, $s\in[0,1]$
and $(x,y)\in X_{d}\times X_{d}$.

Similarly, $\gamma(s)$ can be extended to the asymptotic morphisms $\gamma(s)$
for the reduced extensions. And from the definition, it is easy to know that
$\gamma(s)$ is a homotopy of asymptotic morphisms.

Let $g\in C_{c}(\mathbb{R})$ be a continuously differentiable function with
compact support. For any $T\in C_{alg}^{*}(P_{d}(X))^{\Gamma}$, and $t\geq 1$,
we have that

$\gamma_{t}(0)(g\widehat{\otimes}T)-\left(\begin{array}[]{cc}\alpha_{t}(\beta_{t}(g\widehat{\otimes}T))&0\\\
0&0\\\ \end{array}\right)\rightarrow 0.$

As a result, we have

$\gamma(0)_{*}=\alpha_{*}\circ\beta_{*}$

at the $K$-theory level. On the other hand,

$\gamma(1)=\left(\begin{array}[]{cc}\gamma^{\prime}_{t}&0\\\ 0&0\\\
\end{array}\right),$

so $\gamma(1)_{*}=\gamma^{\prime}_{*}$. It follows that
$\alpha_{*}\circ\beta_{*}=\gamma_{*}$.

Replacing $B_{0,t}(\pi(x))$ with $s^{-1}B_{0,t}(\pi(x))$ in the definition of
$\gamma$ for all $s\in(0,1]$, we obtain a homotopy between $\gamma$ and the
homomorphism induced by $g\widehat{\otimes}T\mapsto
g\widehat{\otimes}(P\widehat{\otimes}T)$, where $P$ is a projection onto the
one-dimensional kernel of $B_{0,t}(\pi(x))$ which does not depend on $x$. It
turns out that $\gamma_{*}$ is an identity, so $\alpha_{*}\circ\beta_{*}$ is
the identity.

∎

Finally, we are ready to complete the proofs of the main results in this
paper.

###### Proof of Theorem 1.1.

We have the following commutative diagram

$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{L,*}}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{L,*}}$$\scriptstyle{\cong}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(\mathcal{S}\widehat{\otimes}C_{L}^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{*}}$$\textstyle{\lim\limits_{d\rightarrow\infty}K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma}).}$

By Proposition 4.6, we know that $\alpha_{*}\circ\beta_{*}$ and
$(\alpha_{L})_{*}\circ(\beta_{L})_{*}$ are identity maps, respectively. It
follows immediately from the diagram chasing that the map

$e_{*}:\lim\limits_{d\rightarrow\infty}K_{*}(C_{L}^{*}(P_{d}(X))^{\Gamma})\rightarrow\lim\limits_{d\rightarrow\infty}K_{*}(C^{*}(P_{d}(X))^{\Gamma})$

induced by evaluation-at-zero map between the localization algebra and Roe
algebra is an isomorphism on $K$-theory. ∎

###### Proof of Theorem 1.2.

For any $d>0$, we have the following commutative diagram

$\textstyle{K_{*}(\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{max,*}}$$\scriptstyle{\lambda_{*}}$$\textstyle{K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{*}}$$\textstyle{K_{*}(C_{max}^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{max,*}}$$\scriptstyle{\cong}$$\scriptstyle{\lambda_{*}}$$\textstyle{K_{*}(C^{*}(P_{d}(X),\mathcal{A}(H))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\textstyle{K_{*}(\mathcal{S}\widehat{\otimes}C_{max}^{*}(P_{d}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{*}}$$\textstyle{K_{*}(\mathcal{S}\widehat{\otimes}C^{*}(P_{d}(X))^{\Gamma}).}$

By Proposition 4.6, we know that $\alpha_{max,*}\circ\beta_{max,*}$ and
$\alpha_{*}\circ\beta_{*}$ are identity maps, respectively. By Theorem 3.12
and the diagram chasing, we have that the map

$\lambda_{*}:K_{*}(C_{max}^{*}(P_{d}(X))^{\Gamma})\rightarrow
K_{*}(C^{*}(P_{d}(X))^{\Gamma})$

induced by the canonical quotient is an isomorphism on $K$-theory.
Consequently, we have that

$\lambda_{*}:K_{*}(C_{max}^{*}(X)^{\Gamma})\rightarrow
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism since $P_{d}(X)$ is $\Gamma$-equivariantly coarsely
equivalent to $X$ for a suitably large $d$. ∎

Acknowledgement. The authors would like to thank Professor Guoliang Yu for
helpful conversations. The first author thanks Professor Matthew Kennedy for
his support.

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Jintao Deng
Department of Mathematics,
University of Waterloo, Waterloo N2L3G1, Canada
E-mail<EMAIL_ADDRESS>

Benyin Fu
College of Statistics and Mathematics,
Shanghai Lixin University of Accounting and Finance,
Shanghai 201209, P. R. China.
E-mail<EMAIL_ADDRESS>

Qin Wang
Research Center for Operator Algebras, and Shanghai Key Laboratory of Pure
Mathematics and Mathematical Practice, School of Mathematical Sciences, East
China Normal University,
Shanghai 200241, P. R. China.
Email<EMAIL_ADDRESS>