# The equivariant coarse Baum-Connes conjecture for actions by a-T-menable
groups

Benyin Fu and Jiawen Zhang  College of Statistics and Mathematics, Shanghai
Lixin University of Accounting and Finance, Shanghai, 201209, China.
<EMAIL_ADDRESS>School of Mathematical Sciences, Fudan University, 220
Handan Road, Shanghai, 200433, China<EMAIL_ADDRESS>
###### Abstract.

The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe
[29] as a unified way to approach both the Baum-Connes conjecture and its
coarse counterpart. In this paper, we prove that if an a-T-menable group
$\Gamma$ acts properly and isometrically on a bounded geometry metric space
$X$ with controlled distortion such that the quotient space $X/\Gamma$ is
coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds
for this action. This answers a question posed in [8] affirmatively.

Both of the authors are supported by NSFC11871342.

##  1\. Introduction

Over the last century, one of the most significant achievements in mathematics
is the Atiyah-Singer index theorem [2]. It provides a topological formula for
Fredholm indices of elliptic pseudodifferential operators on closed manifolds,
and has profound applications in geometry and topology (see, _e.g._ , [24]).

To obtain finer topological and geometric information of the underlying
manifold, people consider lifted operators on the universal cover. These
operators are by definition equivariant under deck transformations, and hence
can be cooked up to provide an element (called the _higher index_ [7]) in the
K-theory of the reduced group $C^{*}$-algebra of the fundamental group [13,
30]. Higher indices reveals more information of the underlying manifolds, and
hence it is important to compute $K$-theories of reduced group
$C^{*}$-algebras. The famous Baum-Connes conjecture ([4], see also [5])
provides a practical and systematic approach, which has fruitful applications
in analysis, geometry and topology (see, _e.g._ , [3, 32]). It has been
verified for a large classes of groups, including a-T-menable groups (_i.e._ ,
groups with the Haagerup property) [16] and hyperbolic groups [23].

On the other hand, Roe introduced a coarse geometric approach in his
pioneering work on higher index theory for elliptic differential operators on
open manifolds [27], and formulated a coarse version to the Baum-Connes
conjecture [28] (see also [18]). Using Higson’s descent principle [15], the
coarse Baum-Connes conjecture also found significant applications in geometry
and topology (see, _e.g._ , [29, 34]), for example the Novikov conjecture
concerning homotopy invariance of higher signatures and the Gromov-Lawson
conjecture concerning positive scalar curvature on manifolds. Over the last
three decades, a lot of results on the coarse Baum-Connes conjecture have been
achieved [6, 11, 21, 22, 33] after Yu’s groundbreaking work for coarsely
embeddable metric spaces [36].

As indicated in [29, Section 5] (see also [34, Section 7]), the Baum-Connes
conjecture and its coarse counterpart can be formulated in a unified way using
the language of group actions. To be more precise, let us assume that $\Gamma$
is a countable discrete group acting on a proper metric space $X$ properly
(_not necessarily_ cocompactly) by isometries. There is an _equivariant index
map_ (see Section 2.5 for precise definitions)

$\mathrm{Ind}^{\Gamma}:\lim_{r\rightarrow\infty}K_{*}^{\Gamma}(P_{r}(X))\longrightarrow
K_{*}(C^{*}(X)^{\Gamma})$

for $\ast=0,1$ where $K_{*}^{\Gamma}(P_{r}(X))$ is the $\Gamma$-equivariant
$K$-homology group of the Rips complex $P_{r}(X)$ of $X$ with scalar $r$, and
$K_{*}(C^{*}(X)^{\Gamma})$ is the $K$-theory of the $\Gamma$-equivariant Roe
algebra of $X$. The equivariant coarse Baum-Connes conjecture asserts that the
equivariant index map $\mathrm{Ind}^{\Gamma}$ is an isomorphism.

Noted by Roe [29], the equivariant Roe algebra $C^{*}(X)^{\Gamma}$ is Morita
equivalent to the reduced group $C^{*}$-algebra $C^{*}_{r}(\Gamma)$ when the
action is cocompact, and hence the equivariant index map
$\mathrm{Ind}^{\Gamma}$ being an isomorphism is just a reformulation of the
classic Baum-Connes conjecture. A typical example is the fundamental group
$\Gamma$ of a closed manifold $M$ acting on its universal cover
$\widetilde{M}$ via deck transformations. As mentioned above, a lifted
differential operator (from $M$ to $\widetilde{M}$) is $\Gamma$-equivariant
and hence its higher index sits naturally in the $K$-theory of the equivariant
Roe algebra $C^{*}(\widetilde{M})^{\Gamma}$, which coincides with the
$K$-theory of $C^{*}_{r}(\Gamma)$. On the other hand when $\Gamma$ is trivial,
$\mathrm{Ind}^{\Gamma}$ being an isomorphism is nothing but the coarse Baum-
Connes conjecture. Therefore, the equivariant coarse Baum-Connes conjecture
can be regarded as an “interpolation” between the Baum-Connes conjecture for
groups and the coarse Baum-Connes conjecture for metric spaces.

The equivariant coarse Baum-Connes conjecture was studied by couples of
mathematicians, and recently has attracted more and more attention. In [31],
Shan proved the equivariant index map is injective for torsion free groups
acting on simply connected complete Riemannian manifolds with non-positive
scalar curvature. Later in [9], the first author together with Wang showed
that the equivariant coarse Baum-Connes conjecture holds for group actions on
metric spaces with bounded geometry which admit equivariant coarse embeddings
into Hilbert space, serving as an equivariant analogue of Yu’s celebrated work
[36].

However, a recent ingenious example due to Arzhantseva and Tessera [1]
illuminates that a metric space with a group action might not admit a coarse
embedding into Hilbert space even if both the group and the quotient space are
coarsely embeddable. Inspired by their example, the first author together with
Wang and Yu [10] managed to show that the equivariant index map remains
injective for group actions with bounded distortion when both the group and
the quotient space admit coarse embeddings into Hilbert space.

Along this way, very recently the first author together with Deng and Wang [8]
proved that the equivariant coarse Baum-Connes conjecture holds for amenable
groups acting on metric spaces such that all orbits are equivariantly
uniformly coarsely equivalent and quotient spaces are coarsely embeddable.
They also asked a question whether the result remains true when the involved
group is a-T-menable, inspired by Higson and Kasparov’s significant result
that the Baum-Connes conjecture holds for a-T-menable groups [16]. However,
the techniques in [8] are no longer applicable for a-T-menable groups due to
an obstruction in coarse $K$-amenability (see [8, Theorem 1.2]).

In this paper, we use a different approach to bypass the issue of coarse
$K$-amenability and answer the question posed in [8] affirmatively. More
precisely, we show that the equivariant coarse Baum-Connes conjecture holds
for a-T-menable groups acting on metric spaces with controlled distortion such
that quotient spaces are coarsely embeddable. To state our main result, let us
introduce some notions first (see Section 2 for more details).

For a metric space $(X,d)$ with a $\Gamma$-action, a subset
$\mathcal{D}\subseteq X$ is called a _fundamental domain_ if $X$ can be
decomposed into the disjoint union of $\Gamma$-orbits for points in
$\mathcal{D}$. An action is said to have _controlled distortion_ if there
exists a fundamental domain $\mathcal{D}\subseteq X$ such that the family of
orbit maps $\\{\mathcal{O}_{x}:\Gamma\to\Gamma x,\gamma\mapsto\gamma
x\\}_{x\in\mathcal{D}}$ are uniformly coarsely equivalent.

The following is the main result of this paper:

######  Theorem 1.1.

Let $\Gamma$ be a countable discrete group acting properly and isometrically
on a discrete metric space $X$ with bounded geometry. If the action has
controlled distortion, the quotient space $X/\Gamma$ admits a coarse embedding
into Hilbert space and $\Gamma$ is a-T-menable, then the equivariant index map

$\mathrm{Ind}^{\Gamma}:\lim_{r\rightarrow\infty}K_{*}^{\Gamma}(P_{r}(X))\longrightarrow
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism for $\ast=0,1$.

Note that when $\Gamma$ is trivial, Theorem 1.1 recovers Yu’s celebrated
result [36] on the coarse Baum-Connes conjecture for coarsely embeddable
metric spaces. On the other hand, note that when the action is cocompact then
it automatically has controlled distortion, and hence Theorem 1.1 recovers
Higson and Kasparov’s famous result [16] on the Baum-Connes conjecture for
a-T-menable groups. Therefore, Theorem 1.1 can be regarded as a combination of
[16] and [36] in virtue of the language of group actions.

Readers might already notice that the hypothesis of Theorem 1.1 is slightly
different from that of [8, Theorem 1.1] except for the requirement on the
group. More precisely, in Theorem 1.1 we assume that the action has controlled
distortion, while [8, Theorem 1.1] requires that all orbits are equivariantly
uniformly coarsely equivalent. As revealed in Section 2.2, having controlled
distortion can be deduced from equivariantly uniformly coarse equivalence of
orbits, which means that our hypothesis is (at least formally) weaker than the
one used in [8, Theorem 1.1]111In fact, we think that controlled distortion
with other assumptions are also sufficient to prove [8, Theorem 1.1].. In
Section 2.2, we also provide some analysis on the relations between our
hypothesis of Theorem 1.1 and those used in [9, 10].

The proof of Theorem 1.1 follows the machinery from [34, Chapter 12], which
was originally invented by Yu in [36]. Roughly speaking, we use an equivariant
version of the coarse Mayer-Vietoris argument to chop the space and decompose
the associated algebras preserving group actions. This leads to the reduction
of the proof for Theorem 1.1 to the case of sequences of cocompact actions
with block-diagonal operators thereon (Corollary 3.8). Then we construct
equivariant twisted Roe and localisation algebras (Definition 4.9 and 4.10),
and index maps in terms of the Bott-Dirac operators (Proposition 5.1).
Finally, we prove an equivariant version of the local isomorphism (Theorem
6.1) thanks to (a family version of) the Baum-Connes conjecture with
coefficients verified for a-T-menable groups [16], and conclude the proof via
a canonical diagram chasing argument.

We would like to highlight the difference between our approach and the one
used in [8]. Although both of them consult Yu’s machinery, the constructions
of equivariant twisted algebras and index maps are different. More precisely
in the approach of [8], index maps (usually called the _Bott_ and _Dirac_
maps) are defined on dense subalgebras in the form of asymptotic morphisms,
and therefore one needs a coarse $K$-amenability result to extend these maps.
While in the current paper, we use a coarse Mayer-Vietoris argument to chop
the space and decompose the associated algebras at first, which allows us to
construct index maps directly on the $K$-theories of the whole algebras (see
Proposition 5.7). An ingenious “ _space for speed_ ” argument from [34] is
crucial to bypass the issue of coarse $K$-amenability and accomplish the job.

The paper is organised as follows. In Section 2, we collect necessary
preliminaries and provide comparisons for different geometric hypotheses used
in [8, 9, 10]. Section 3 is devoted to the reduction of the proof for Theorem
1.1 to the case of sequences of cocompact actions. We introduce equivariant
twisted algebras in Section 4, and construct index maps in Section 5. In
Section 6, we prove that the equivariant twisted Roe and localisation algebras
have the same $K$-theory using a family version of the Baum-Connes conjecture
with coefficients for a-T-menable groups essentially from [16], and finally
conclude the proof in Section 7.

We also provide an appendix on a precise statement of the family version of
the Baum-Connes conjecture with coefficients for a-T-menable groups which we
need to conclude Theorem 6.1. This result is well-known to experts, while we
cannot find an explicit statement or proof in literature. For convenience to
readers and also for completeness, we provide a detailed proof based on ideas
from [16] and [36].

##  2\. Preliminaries

In this section, we recall some notions and definitions.

###  2.1. Notions from coarse geometry

Here we collect several basic notions.

######  Definition 2.1.

Let $(X,d)$ be a metric space and $R>0$.

  1. (1)

For $x\in X$, denote the closed _$R$ -ball_ $B(x,R):=\\{y\in X:d(x,y)\leq
R\\}$. For $A\subseteq X$, denote its _$R$ -neighbourhood_ by
$\mathcal{N}_{R}(A):=\\{y\in X:d(y,A)\leq R\\}$.

  2. (2)

A subset $A\subseteq X$ is called _bounded_ if its diameter
$\sup\\{d(x,y):x,y\in A\\}$ is finite; $A$ is called a _net_ in $X$ if there
exists a $c>0$ such that $\mathcal{N}_{c}(A)=X$.

  3. (3)

$(X,d)$ is called _proper_ if any bounded closed subset in $X$ is compact.

  4. (4)

If $(X,d)$ is discrete, we say that $(X,d)$ has _bounded geometry_ if
$\sup_{x\in X}\sharp B(x,R)$ is finite for any $R>0$. In the general case, we
say that $(X,d)$ has _bounded geometry_ if it contains a discrete net with
bounded geometry.

######  Definition 2.2.

A map $f:(X,d_{X})\to(Y,d_{Y})$ between metric spaces is called a _coarse
embedding_ if there exist two proper non-decreasing functions
$\rho_{+},\rho_{-}:[0,\infty)\to[0,\infty)$ such that

$\rho_{-}(d_{X}(x,y))\leq d_{Y}(f(x),f(y))\leq\rho_{+}(d_{X}(x,y))$

holds for any $x,y\in X$. The map $f$ is called a _coarse equivalence_ if it
is a coarse embedding and the image of $f$ is a net in $Y$. We say that $X$
and $Y$ is _coarsely equivalent_ if there exists a coarse equivalence $f:X\to
Y$.

A family of maps $\\{f_{i}:X_{i}\to Y_{i}\\}_{i\in I}$ between metric spaces
are called _uniformly coarsely equivalent_ if $f_{i}$ is a coarse equivalence
with the same control functions $\rho_{+}$ and $\rho_{-}$, and there exists a
$c>0$ such that the $c$-neighbourhood of the image of $f_{i}$ coincides with
$Y_{i}$ for each $i\in I$.

For a proper function $\rho:[0,\infty)\to[0,\infty)$ (which is not necessarily
monotonely increasing), we denote

$\rho^{-1}(R):=\sup\\{S\in[0,\infty):\rho(S)\leq
R\\},\quad\mbox{for~{}}R\in[0,\infty).$

Now we move to the case of group actions.

######  Definition 2.3.

Let $\Gamma$ be a countable discrete group acting on a proper metric space
$(X,d)$. The action is called _proper_ if for any compact set $K\subseteq X$,
the set $\\{\gamma\in\Gamma:\gamma K\cap K\neq\emptyset\\}$ is finite. The
action is called _isometric_ if $d(x,y)=d(\gamma x,\gamma y)$ for all
$\gamma\in\Gamma$ and $x,y\in X$.

To simplify the notation, we say that a proper metric space $(X,d)$ is a
_$\Gamma$ -space_ if the $\Gamma$-action is proper and isometric. For a
$\Gamma$-space $(X,d)$ and $x\in X$, the _orbit of $x$_ is defined to be
$\Gamma x$ and the _orbit map at $x$_ is defined by:

$\mathcal{O}_{x}:\Gamma\longrightarrow\Gamma x,\quad\gamma\mapsto\gamma
x\quad\mbox{for}\quad\gamma\in\Gamma.$

######  Definition 2.4.

Let $\Gamma$ be a countable discrete group. A map $f:X\to Y$ between two
$\Gamma$-spaces is called _equivariant_ if for any $x\in X$ and
$\gamma\in\Gamma$, we have $f(\gamma x)=\gamma f(x)$. The map $f$ is called an
_equivariantly coarse equivalence_ if $f$ is equivariant and coarsely
equivalent. A family of maps $\\{f_{i}:X_{i}\to Y_{i}\\}_{i\in I}$ between
$\Gamma$-spaces is called an _equivariantly uniformly coarse equivalence_ if
each $f_{i}$ is $\Gamma$-equivariant and the family $\\{f_{i}\\}_{i\in I}$ is
a uniformly coarse equivalence.

A family of $\Gamma$-spaces $X_{i}$ for $i\in I$ are called _equivariantly
uniformly coarsely equivalent_ if for any $i,j\in I$ there exists an
equivariant map $f_{i,j}:X_{i}\to X_{j}$ such that the family
$\\{f_{i,j}:X_{i}\to X_{j}\\}_{i,j\in I}$ is a uniformly coarse equivalence.

Recall that for a $\Gamma$-space $(X,d)$, a subset $\mathcal{D}\subseteq X$ is
called a _fundamental domain_ if $X$ can be decomposed into the disjoint union
of $\Gamma$-orbits of points in $\mathcal{D}$. An action is said to have
_controlled distortion_ if there exists a fundamental domain
$\mathcal{D}\subseteq X$ such that the family of orbit maps
$\\{\mathcal{O}_{x}:x\in\mathcal{D}\\}$ are uniformly coarsely equivalent. In
this case, we also say that the action has controlled distortion with respect
to $\mathcal{D}$.

Recall from [10] that for a $\Gamma$-space $(X,d)$, the action is said to have
_bounded distortion_ if there exists a fundamental domain $\mathcal{D}$ such
that for any $\gamma\in\Gamma$, we have

$\sup_{y\in\mathcal{D}}d(\gamma y,y)<+\infty.$

In this case, we also say that the action has bounded distortion with respect
to $\mathcal{D}$. It follows directly from definitions that having controlled
distortion implies bounded distortion.

###  2.2. Comparing different geometric hypotheses

In this subsection, we would like to compare the geometric hypothesis of
Theorem 1.1 with those used in the main results of [8, 9, 10]. For convenience
to readers, we record these hypotheses chronologically as follows. Assume that
$(X,d)$ is a $\Gamma$-space.

  1. (1)

_hypothesis of[9, Theorem 1.2]_: $X$ admits a $\Gamma$-equivariant coarse
embedding into Hilbert space.

  2. (2)

_hypothesis of[10, Theorem 1.2]_: both $\Gamma$ and $X/\Gamma$ are coarsely
embeddable, and the action has bounded distortion.

  3. (3)

_hypothesis of[8, Theorem 1.1]_: $\Gamma$ is amenable and $X/\Gamma$ is
coarsely embeddable, and the orbit spaces are equivariantly uniformly
equivalent.

  4. (4)

_hypothesis of Theorem 1.1 (the current paper)_: $\Gamma$ is a-T-menable and
$X/\Gamma$ is coarsely embeddable, and the action has controlled distortion.

Note that the conclusion of [10, Theorem 1.2] says that the equivariant index
map is injective, while the rest conclude isomorphisms.

The following lemma shows that our hypothesis of Theorem 1.1 is (at least
formally) weaker than the one of [8, Theorem 1.1].

######  Lemma 2.5.

For a $\Gamma$-space $(X,d)$, if all $\Gamma$-orbits are equivariantly
uniformly coarsely equivalent, then the action has controlled distortion.

###### Proof.

Fix a point $x_{0}\in X$ and a fundamental domain
$\mathcal{D}^{\prime}\subseteq X$ with $x_{0}\in\mathcal{D}^{\prime}$. Since
the action is proper, it follows from the Milnor-Švarc lemma that the orbit
map $\mathcal{O}_{x_{0}}:\Gamma\longrightarrow\Gamma x_{0}$ is a coarsely
equivalent. Now for any $y\in\mathcal{D}^{\prime}$, by assumption there exists
an equivariant map $\psi_{y}:\Gamma x_{0}\to\Gamma y$ such that
$\\{\psi_{y}:y\in\mathcal{D}^{\prime}\\}$ is a uniformly coarse equivalence.
Setting $\hat{y}:=\psi_{y}(x_{0})$, the equivariance of $\psi_{y}$ implies
that $\psi_{y}$ has the following form:

$\psi_{y}:\Gamma x_{0}\longrightarrow\Gamma\hat{y},\quad\gamma
x_{0}\mapsto\gamma\hat{y}\quad\mbox{for}~{}\gamma\in\Gamma.$

Taking $\mathcal{D}:=\\{\hat{y}:y\in\mathcal{D}^{\prime}\\}$, it is clear that
$\mathcal{D}$ is also a fundamental domain. Moreover, for
$\hat{y}\in\mathcal{D}$ the orbit map $\mathcal{O}_{\hat{y}}$ coincides with
$\psi_{y}\circ\mathcal{O}_{x_{0}}$. Hence we conclude the proof. ∎

Now we turn to the relation with [9] and [10]. The general situation is still
unclear while we try to offer some analysis in special cases.

To compare with [10], note that having controlled distortion implies bounded
distortion. We show in the following that for certain special case, these two
notions are “almost” the same.

######  Example 2.6.

Let $N,Q$ be countable discrete groups with an action
$\alpha:Q\to\mathrm{Aut}(N)$. Equip the associated semi-direct product
$N\rtimes Q$ with a proper length function $\ell$, which derives a left-
invariant proper metric $d$. Consider the action of $N$ on $N\rtimes Q$ by
left multiplication, which is clearly isometric and proper.

Assume that the action has bounded distortion with respect to the fundamental
domain $Q$. Then obviously there exists a proper function
$\rho_{+}:[0,\infty)\to[0,\infty)$ such that $d(hq,q)\leq\rho_{+}(\ell(h))$
for any $h\in N$ and $q\in Q$. Hence for any $h\in N$ and $q\in Q$, we also
have:

$\ell(h)=d(q^{-1}hq\cdot
q^{-1},q^{-1})\leq\rho_{+}(\ell(q^{-1}hq))=\rho_{+}(d(hq,q)),$

which implies that the action has controlled distortion. Finally we remark
that it is unclear to the authors that whether the same result holds if the
action has bounded distortion with respect to an arbitrary fundamental domain.

To compare with [9], recall that a recent ingenious example due to Arzhantseva
and Tessera [1] illuminates that a $\Gamma$-space might not admit a coarse
embedding into Hilbert space even if both the group and the quotient space are
coarsely embeddable. However as we show below, the situation often gets better
under the hypothesis of controlled distortion.

######  Lemma 2.7.

Let $(X,d)$ be a $\Gamma$-space with the quotient map $\pi:X\to X/\Gamma$.
Equip $\Gamma$ with a proper left-invariant metric $d_{\Gamma}$ and $X/\Gamma$
with the quotient metric $d_{q}$. Assume that there exists a fundamental
domain $\mathcal{D}\subseteq X$ such that
$\pi|_{\mathcal{D}}:\mathcal{D}\to\pi(\mathcal{D})$ is a coarse equivalence
and the action has controlled distortion with respect to $\mathcal{D}$. Then
$X$ is coarsely equivalent to the product metric $\Gamma\times(X/\Gamma)$
equipped with the product metric $d_{\Gamma}\times d_{q}$.

Consequently, if additionally both $\Gamma$ and $X/\Gamma$ are coarsely
embeddable, then so is $X$.

###### Proof.

Assume that $\mathcal{D}=\\{x_{\lambda}:\lambda\in\Lambda\\}$, and
$\pi|_{\mathcal{D}}$ is a coarse equivalence with controlled functions
$\rho_{+}$ and $\rho_{-}$. Also assume that the orbits maps
$\\{\mathcal{O}_{x_{\lambda}}:\lambda\in\Lambda\\}$ are uniformly coarsely
equivalent with the same controlled functions. For any two points
$\gamma_{1}x_{\lambda_{1}}$ and $\gamma_{2}x_{\lambda_{2}}$ in $X$, note that

$\displaystyle d(\gamma_{1}x_{\lambda_{1}},\gamma_{2}x_{\lambda_{2}})$
$\displaystyle=d(x_{\lambda_{1}},\gamma_{1}^{-1}\gamma_{2}x_{\lambda_{2}})\leq
d(x_{\lambda_{1}},x_{\lambda_{2}})+d(\gamma_{1}^{-1}\gamma_{2}x_{\lambda_{2}},x_{\lambda_{2}})$
$\displaystyle\leq\rho_{-}^{-1}(d_{q}(\pi(x_{\lambda_{1}}),\pi(x_{\lambda_{2}})))+\rho_{+}(d_{\Gamma}(\gamma_{1},\gamma_{2})).$

On the other hand, given $R>0$ and assume that
$d(\gamma_{1}x_{\lambda_{1}},\gamma_{2}x_{\lambda_{2}})\leq R$. Then

$d_{q}(\pi(x_{\lambda_{1}}),\pi(x_{\lambda_{2}}))=d_{q}(\pi(\gamma_{1}x_{\lambda_{1}}),\pi(\gamma_{2}x_{\lambda_{2}}))\leq
R,$

which implies that $d(x_{\lambda_{1}},x_{\lambda_{2}})\leq\rho_{+}(R)$.
Moreover, we have

$d(x_{\lambda_{2}},\gamma_{1}^{-1}\gamma_{2}x_{\lambda_{2}})\leq
d(x_{\lambda_{1}},x_{\lambda_{2}})+d(x_{\lambda_{1}},\gamma_{1}^{-1}\gamma_{2}x_{\lambda_{2}})\leq\rho_{+}(R)+R,$

which implies that
$d_{\Gamma}(\gamma_{1},\gamma_{2})\leq\rho_{-}^{-1}(\rho_{+}(R)+R)$.

Combining the above two paragraphs, we conclude the proof. ∎

######  Remark 2.8.

It is unclear to the authors whether Lemma 2.7 holds or not without the
assumption that $\pi|_{\mathcal{D}}$ is a coarse equivalence. On the other
hand, note that this condition holds for many examples, including the semi-
direct product case studied in Example 2.6.

More precisely, it is obvious that the quotient map $N\rtimes Q\to Q$
restricted to the fundamental domain $Q$ is a coarse equivalence. As a
consequence to Lemma 2.7, we obtain that $N\rtimes Q$ is coarsely embeddable
if both $N$ and $Q$ are coarsely embeddable and the action of $N$ on $N\rtimes
Q$ by left multiplication has bounded distortion with respect to $Q$. Finally,
note that the example constructed in [1] has the form of semi-direct products.
It follows by straightforward calculations that their example does not have
bounded distortion, alternatively by combining the fact that it is not
coarsely embeddable together with the analysis above.

###  2.3. (Equivariant) Roe algebras

Now we introduce the notion of Roe algebras and their equivariant
counterparts.

For a proper metric space $(Z,d)$, recall that a _$Z$ -module_ is a non-
degenerate $\ast$-representation
$\phi:C_{0}(Z)\to\mathfrak{B}(\mathcal{H}_{Z})$ where $\mathcal{H}_{Z}$ is
some infinite-dimensional separable Hilbert space. We also say that
$\mathcal{H}_{Z}$ is a $Z$-module if the representation is clear from the
context, and simply write $f$ as a bounded linear operator on
$\mathcal{H}_{Z}$ instead of $\phi(f)$ for $f\in C_{0}(Z)$. A $Z$-module is
called _ample_ if no non-zero element of $C_{0}(Z)$ acts as a compact operator
on $\mathcal{H}_{Z}$.

######  Definition 2.9.

Let $\mathcal{H}_{Z}$ be an ample module of a proper metric space $(Z,d)$.

  1. (1)

For $T\in B(\mathcal{H}_{Z})$, its _support_ $\mathrm{supp}(T)$ is defined to
be the complement of the set of points $(x,y)\in Z\times Z$ for which there
exist $f,g\in C_{0}(Z)$ satisfying:

$gTf=0,\quad f(x)\neq 0\quad\mbox{and}\quad g(y)\neq 0.$

  2. (2)

The propagation of $T\in B(\mathcal{H}_{Z})$ is defined to be

$\mathrm{prop}(T):=\sup\\{d(x,y):(x,y)\in\mathrm{supp}(T)\\}.$

We say that $T$ has finite propagation if the number $\mathrm{prop}(T)$ is
finite.

  3. (3)

An operator $T\in B(\mathcal{H}_{Z})$ is called _locally compact_ if $fT$ and
$Tf$ are compact for any $f\in C_{0}(Z)$.

######  Definition 2.10.

For a proper metric space $Z$ and an ample $Z$-module $\mathcal{H}_{Z}$, the
_algebraic Roe algebra_ $\mathbb{C}[\mathcal{H}_{Z}]$ of $\mathcal{H}_{Z}$ is
defined to be the $*$-algebra of locally compact finite propagation operators
on $\mathcal{H}_{Z}$, and the _Roe algebra $C^{*}(\mathcal{H}_{Z})$_ of
$\mathcal{H}_{Z}$ is defined to be the norm-closure of
$\mathbb{C}[\mathcal{H}_{Z}]$ in $\mathfrak{B}(\mathcal{H}_{Z})$.

It is a standard result that the Roe algebra $C^{*}(\mathcal{H}_{Z})$ does not
depend on the chosen ample module $\mathcal{H}_{Z}$ up to $*$-isomorphisms,
hence denoted by $C^{*}(Z)$ and called the _Roe algebra of $Z$_. Furthermore,
$C^{*}(Z)$ is a coarse invariant of the metric space $Z$ (up to non-canonical
$*$-isomorphisms), and their $K$-theories are coarse invariants up to
canonical isomorphisms (see, _e.g._ , [28]).

Now we move to the equivariant case. Here we follow the setting of [34,
Section 4.5 and 5.2], which is in fact equivalent to those in [8, 9, 29].

For a proper $\Gamma$-space $(Z,d)$, we define a $\Gamma$-action on $C_{0}(Z)$
by

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

for all $\gamma\in\Gamma$ and $f\in C_{0}(Z)$. For a unitary representation
$U:\Gamma\rightarrow\mathcal{U}(\mathcal{H})$ on some Hilbert space
$\mathcal{H}$, we denote the adjoint

$\mathrm{Ad}_{U_{\gamma}}(T)=U_{\gamma}TU_{\gamma}^{*}=:\gamma\cdot T$

for $\gamma\in\Gamma$ and $T\in\mathfrak{B}(\mathcal{H})$. An operator
$T\in\mathfrak{B}(\mathcal{H})$ is called ($\Gamma$-)_invariant_ if
$\gamma\cdot T=T$ for any $\gamma\in\Gamma$. Denote the set of all
$\Gamma$-invariant operators by $\mathfrak{B}(\mathcal{H})^{\Gamma}$.

The following notion essentially comes from [34, Section 4.5].

######  Definition 2.11.

Let $(Z,d)$ be a proper $\Gamma$-space, and $(\mathcal{H}_{Z},\phi)$ be an
ample $Z$-module.

  1. (1)

$\mathcal{H}_{Z}$ is called _covariant_ if it is equipped with a unitary
representation $U:\Gamma\rightarrow\mathcal{U}(\mathcal{H}_{Z})$ such that for
any $\gamma\in\Gamma$, we have:

$\phi(\gamma\cdot
f)=\gamma\cdot\phi(f)\left(=U_{\gamma}\phi(f)U_{\gamma}^{*}\right).$

  2. (2)

$\mathcal{H}_{Z}$ is called _locally free_ if for any finite subgroup $F$ of
$\Gamma$ and any $F$-invariant Borel subset $E$ of $Z$, there exists a Hilbert
space $\mathcal{H}_{E}$ equipped with the trivial representation of $F$ such
that $\chi_{E}\mathcal{H}_{Z}$ and $\ell^{2}(F)\otimes\mathcal{H}_{E}$ are
isomorphic as $F$-representations.

To simplify the notation, we call an ample covariant locally free $Z$-module
an _admissible_ $Z$-module.

It follows from [34, Lemma 4.5.5] that admissible modules always exist. Now we
introduce the notion of equivariant Roe algebras (see, _e.g._ , [34,
Definition 5.2.1]).

######  Definition 2.12.

Let $(Z,d)$ be a proper $\Gamma$-space, and $(\mathcal{H}_{Z},\phi)$ be an
admissible $Z$-module. The _algebraic equivariant Roe algebra_
$\mathbb{C}[\mathcal{H}_{Z}]^{\Gamma}$ of $\mathcal{H}_{Z}$ is defined to be
the $*$-algebra of locally compact finite propagation $\Gamma$-invariant
operators on $\mathcal{H}_{Z}$, and the _equivariant Roe algebra
$C^{*}(\mathcal{H}_{Z})^{\Gamma}$_ of $\mathcal{H}_{Z}$ is defined to be the
norm-closure of $\mathbb{C}[\mathcal{H}_{Z}]^{\Gamma}$ in
$\mathfrak{B}(\mathcal{H}_{Z})$.

Analogous to the case of Roe algebras, it follows from [34, Theorem 5.2.6]
that the equivariant Roe algebra $C^{*}(\mathcal{H}_{Z})^{\Gamma}$ does not
depend on the chosen admissible module $\mathcal{H}_{Z}$ up to
$*$-isomorphisms, hence denoted by $C^{*}(Z)^{\Gamma}$ and called the
_equivariant Roe algebra of $Z$_. Moreover, equivariant Roe algebras are
invariant under equivariant coarse equivalences (up to non-canonical
$*$-isomorphisms), and their $K$-theories are equivariant coarse invariants up
to canonical isomorphisms.

We would also like to point out an important observation from [29, Lemma 5.14]
that the equivariant Roe algebra $C^{*}(Z)^{\Gamma}$ is Morita equivalent to
the reduced group $C^{*}$-algebra $C_{r}^{*}(\Gamma)$ provided that the action
is cocompact in the sense that $Z/\Gamma$ is compact (see also [34, Theorem
5.3.2]). This allows us to include the setting of the classical Baum-Connes
conjecture to the scope of equivariant coarse Baum-Connes conjecture which
will be introduced below.

###  2.4. Equivariant $K$-homology groups and index maps

Let us briefly recall the notion of equivariant $K$-homology groups introduced
by Kasparov [20].

######  Definition 2.13.

Let $(Z,d)$ be a proper $\Gamma$-space. For $i=0$ and $1$, the _equivariant
$K$-homology group_ $K_{i}^{\Gamma}(Z)=KK_{i}^{\Gamma}(C_{0}(Z),\mathbb{C})$
is generated by certain cycles modulo a certain equivalent relation:

  1. (1)

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

  2. (2)

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

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

Now we define the _equivariant index map_ for a $\Gamma$-space $Z$:

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

Note that every class in $K^{\Gamma}_{*}(Z)$ can be represented by a cycle
$(\mathcal{H},\phi,F)$ such that $(\mathcal{H},\phi)$ is an admissible
$Z$-module (see, _e.g._ , [22, Section 3]).

Consider such a cycle $(\mathcal{H},\phi,F)$. Take a locally finite,
$\Gamma$-equivariant and uniformly bounded open cover $\\{U_{i}\\}_{i\in I}$
of $Z$, 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 sum converges in strong topology. It is clear that the cycle
$(\mathcal{H},\phi,F^{\prime})$ is equivalent to $(\mathcal{H},\phi,F)$ in
$K^{\Gamma}_{0}(Z)$ and $F^{\prime}$ has finite propagation. Hence
$F^{\prime}$ is a multiplier of $C^{*}(Z)^{\Gamma}$ and invertible modulo
$C^{*}(Z)^{\Gamma}$. Applying the boundary map, $F^{\prime}$ gives rise to an
element in $K_{0}(C^{*}(Z)^{\Gamma})$, denoted by $\partial([F^{\prime}])$. We
define the equivariant index map

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

by

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

with the notation as above. Similarly, we can define the equivariant index map
for the $K_{1}$-case:

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

###  2.5. 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.

Let $(X,d)$ be a discrete metric space with bounded geometry. For $r>0$, the
_Rips complex $P_{r}(X)$ with scale $r$_ is the simplicial complex with
vertices $X$ such that a finite subset
$Y=\\{x_{0},x_{1},\cdots,x_{n}\\}\subseteq X$ spans a simplex if and only if
$\mathrm{diam}(Y)\leq r$.

The Rips complex $P_{r}(X)$ is endowed with the following _spherical metric_.
On each connected component of $P_{r}(X)$, the spherical metric is the maximal
metric whose restriction on each simplex
$\Delta:=\\{\sum_{i=0}^{n}t_{i}x_{i}:t_{i}\geq 0,\sum_{i=0}^{n}t_{i}=1\\}$ is
the metric obtained by identifying $\Delta$ with $S_{+}^{n}$ via the map

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

where $S_{+}^{n}:=\\{(s_{0},s_{1},\dots,s_{n})\in\mathbb{R}^{n+1}:s_{i}\geq
0,\sum_{i=0}^{n}s_{i}^{2}=1\\}$ is endowed with the standard Riemannian
metric. If $y_{0}$ and $y_{1}$ belong to two different connected components
$Y_{0}$ and $Y_{1}$ of $P_{r}(X)$, respectively, we define

$d(y_{0},y_{1})=\min\big{\\{}d(y_{0},x_{0})+d_{X}(x_{0},x_{1})+d(x_{1},y_{1}):x_{0}\in
X\cap Y_{0},x_{1}\in X\cap Y_{1}\big{\\}}.$

Assume further that $X$ is a $\Gamma$-space. Then each Rips complex $P_{r}(X)$
also admits a $\Gamma$-action defined by

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

for all $\sum\limits_{i=1}^{k}c_{i}x_{i}\in P_{r}(X)$ and $\gamma\in\Gamma$.
It is clear that the action is proper and isometric, and hence $P_{r}(X)$ is a
proper $\Gamma$-space as well.

The following is the main conjecture we study in this paper:

######  Conjecture 2.14 (The equivariant coarse Baum–Connes conjecture).

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

$\mathrm{Ind}^{\Gamma}:\lim_{r\rightarrow\infty}K_{*}^{\Gamma}(P_{r}(X))\longrightarrow\lim_{r\rightarrow\infty}K_{*}(C^{*}(P_{r}(X))^{\Gamma})\cong
K_{*}(C^{*}(X)^{\Gamma})$

is an isomorphism for $\ast=0,1$.

As noticed in Introduction, the equivariant coarse Baum-Connes conjecture
includes both the classical Baum-Connes conjecture for groups and the coarse
Baum-Connes conjecture for metric spaces as special cases. More precisely, it
coincides with the Baum-Connes conjecture when the action is cocompact, and
when the action is trivial it is nothing but the coarse Baum-Conness
conjecture (see, _e.g._ , [34, Section 7]).

We reformulate the following significant result due to Higson and Kasparov
[16]. Recall that a discrete group $\Gamma$ is said to be _a-T-menable_ [12]
(or has the _Haagerup property_) if it admits a metrically proper action on
some Hilbert space by affine isometries.

######  Proposition 2.15 ([16]).

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

$\mathrm{Ind}^{\Gamma}:\lim_{r\to\infty}K^{\Gamma}_{\ast}(P_{r}(X))\longrightarrow
K_{\ast}(C^{*}(X)^{\Gamma})$

is an isomorphism for $\ast=0,1$.

###  2.6. Equivariant localisation algebras

Here we introduce another version on the $K$-homology group, which was
originally introduced by Yu [35].

######  Definition 2.16.

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

$f:[1,+\infty)\rightarrow C^{*}(Z)^{\Gamma}$

such that

$\mathrm{prop}(f(t))\rightarrow 0\quad\mbox{and}\quad t\rightarrow\infty.$

We recall that for a proper $\Gamma$-space $Z$, there exists an equivariant
local index map which originally comes from [35] (with an equivariant
formulation from [10]):

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

Moreover, we have the following result which is an equivariant analogue of
[35, Theorem 3.2] and [26, Theorem 3.4].

######  Proposition 2.17.

Let $Z$ be a proper $\Gamma$-space. Then the equivariant local index map

$\mathrm{Ind}^{\Gamma}_{L}:K_{*}^{\Gamma}(Z)\rightarrow
K_{*}(C_{L}^{*}(Z)^{\Gamma})$

is an isomorphism for $\ast=0,1$.

On the other hand, note that for a proper $\Gamma$-space $Z$ there is a
natural evaluation-at-one map

$\mathrm{ev}:C_{L}^{*}(Z)^{\Gamma}\to C^{*}(Z)^{\Gamma}$

given by

$\mathrm{ev}(f)=f(1)$

for all $f\in C_{L}^{*}(Z)^{\Gamma}$. This is clearly a $\ast$-homomorphism,
and thus induces a homomorphism

$\mathrm{ev}_{\ast}:K_{*}(C_{L}^{*}(Z)^{\Gamma})\to K_{*}(C^{*}(Z)^{\Gamma})$

on $K$-theories. Furthermore, we have the following commutative diagram:

$\textstyle{K_{*}(C_{L}^{*}(Z)^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\ast}}$$\textstyle{K_{*}^{\Gamma}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{Ind}_{L}^{\Gamma}}$$\scriptstyle{\mathrm{Ind}^{\Gamma}}$$\textstyle{K_{*}(C^{*}(Z)^{\Gamma}).}$

Consequently, for a discrete $\Gamma$-space $X$ with bounded geometry, in
order to prove that equivariant coarse Baum-Connes conjecture holds for $X$ it
suffices to show that the map

$\mathrm{ev}_{\ast}:\lim_{r\rightarrow\infty}K_{*}(C_{L}^{*}(P_{r}(X))^{\Gamma})\longrightarrow\lim_{r\rightarrow\infty}K_{*}(C^{*}(P_{r}(X))^{\Gamma})\cong
K_{*}(C^{*}(X)^{\Gamma})$

induced by the evaluation-at-one map is an isomorphism for $\ast=0,1$.

##  3\. Reduction

The aim of this section is to reduce the proof of Theorem 1.1 to the case of
sequences of metric spaces with proper and cocompact group actions. Let us
recall the following notion:

######  Definition 3.1 ([19]).

Let $Z$ be a proper metric space and $A,B$ be closed subsets in $Z$ such that
$A\cup B=Z$. We say that $(A,B)$ is _$\omega$ -excisive_ if for any $r>0$
there exists $s>0$ such that

$\mathcal{N}_{r}(A)\cap\mathcal{N}_{r}(B)\subseteq\mathcal{N}_{s}(A\cap B).$

Throughout the rest of this section, let us assume that $(X,d)$ is a proper
$\Gamma$-space with bounded geometry. Denote by $\pi:X\to X/\Gamma$ the
quotient map and equip $X/\Gamma$ with the quotient metric $\bar{d}$.

######  Lemma 3.2.

For any $Z\subseteq X/\Gamma$ and $r>0$, we have
$\mathcal{N}_{r}(\pi^{-1}(Z))=\pi^{-1}(\mathcal{N}_{r}(Z))$.

###### Proof.

For any $x\in\mathcal{N}_{r}(\pi^{-1}(Z))$, there exists $y\in\pi^{-1}(Z)$
such that $d(x,y)\leq r$. Hence we have $\bar{d}(\pi(x),\pi(y))\leq d(x,y)\leq
r$, which implies that $x\in\pi^{-1}(\mathcal{N}_{r}(Z))$.

On the other hand, given $x\in\pi^{-1}(\mathcal{N}_{r}(Z))$ we have
$\pi(x)\in\mathcal{N}_{r}(Z)$. Hence there exists $z\in Z$ such that
$\bar{d}(\pi(x),z)\leq r$. Assume that $z=\pi(y)$ for some $y\in X$. Hence,

$r\geq\bar{d}(\pi(x),z)=\inf_{\gamma,\gamma^{\prime}\in\Gamma}d(\gamma
x,\gamma^{\prime}y)=\inf_{\gamma}d(x,\gamma y)$

where we use that the action is by isometries in the third equality. Since $X$
has bounded geometry and $\gamma y\in\pi^{-1}(Z)$ for each $\gamma\in\Gamma$,
we obtain that $x\in\mathcal{N}_{r}(\pi^{-1}(Z))$. ∎

######  Lemma 3.3.

Suppose $(A,B)$ is an $\omega$-excisive closed cover of $X/\Gamma$. Then
$(\pi^{-1}(A),\pi^{-1}(B))$ is an $\omega$-excisive closed cover of $X$.

###### Proof.

Given $r>0$ there exists $s>0$ such that
$\mathcal{N}_{r}(A)\cap\mathcal{N}_{r}(B)\subseteq\mathcal{N}_{s}(A\cap B)$.
Hence by Lemma 3.2, we have:

$\displaystyle\mathcal{N}_{r}(\pi^{-1}(A))\cap\mathcal{N}_{r}(\pi^{-1}(B))$
$\displaystyle=\pi^{-1}(\mathcal{N}_{r}(A))\cap\pi^{-1}(\mathcal{N}_{r}(B))=\pi^{-1}(\mathcal{N}_{r}(A)\cap\mathcal{N}_{r}(B))$
$\displaystyle\subseteq\pi^{-1}(\mathcal{N}_{s}(A\cap
B))=\mathcal{N}_{s}(\pi^{-1}(A\cap
B))=\mathcal{N}_{s}(\pi^{-1}(A)\cap\pi^{-1}(B)),$

which concludes the proof. ∎

Now for the quotient space $X/\Gamma$, we fix a basepoint $w_{0}\in X/\Gamma$.
For each $n\in\mathbb{N}\cup\\{0\\}$, we set

$W_{n}:=\\{w\in X/\Gamma:n^{3}-n\leq\bar{d}(w,w_{0})\leq(n+1)^{3}+(n+1)\\}.$

Let $A:=\bigsqcup_{n:even}W_{n}$ and $B:=\bigsqcup_{n:odd}W_{n}$. It is
obvious that $(A,B)$ is an $\omega$-excisive cover of $X/\Gamma$. Hence Lemma
3.3 implies that $(\pi^{-1}(A),\pi^{-1}(B))$ is an $\omega$-excisive closed
cover of $X$ such that both $\pi^{-1}(A)$ and $\pi^{-1}(B)$ are
$\Gamma$-invariant. Denote $A^{\prime}=\pi^{-1}(A)$ and
$B^{\prime}=\pi^{-1}(B)$.

Applying the Mayer-Vietoris sequence arguments to the $\Gamma$-invariant
$\omega$-excisive cover $(A^{\prime},B^{\prime})$ for the $K$-theory of
equivariant Roe algebras and the equivariant coarse $K$-homology (which is
similar to [31] and [34, Section 7.5]), we obtain the following commutative
diagram of long-exact sequences:

$\begin{CD}\cdots
@>{}>{}>\lim\limits_{r\to\infty}K_{*}^{\Gamma}(P_{r}(A^{\prime}\cap
B^{\prime}))@>{}>{}>\lim\limits_{r\to\infty}K_{*}^{\Gamma}(P_{r}(A^{\prime}))\oplus
K_{*}^{\Gamma}(P_{r}(B^{\prime}))@>{}>{}>\lim\limits_{r\to\infty}K_{*}^{\Gamma}(P_{r}(X))@>{}>{}>\cdots\\\
&&@V{}V{}V@V{}V{}V@V{}V{}V\\\ \cdots @>{}>{}>K_{*}(C^{*}(A^{\prime}\cap
B^{\prime})^{\Gamma})@>{}>{}>K_{*}(C^{*}(A^{\prime})^{\Gamma})\oplus
K_{*}(C^{*}(B^{\prime})^{\Gamma}<EMAIL_ADDRESS>
Hence it suffices to prove Theorem 1.1 for the spaces $A^{\prime},B^{\prime}$
and $A^{\prime}\cap B^{\prime}$. Therefore, we obtain the following:

######  Lemma 3.4.

To prove Theorem 1.1, it suffices to prove the result for a discrete metric
space $(X,d)$ of bounded geometry and having the form
$X=\bigsqcup_{n=1}^{\infty}X_{n}$ with $d(X_{n},X_{m})\to\infty$ as
$n+m\to\infty~{}(n\neq m)$ such that $\Gamma$ acts on each $X_{n}$ properly
and cocompactly by isometries.

Our next aim is to further reduce the proof of Theorem 1.1 to the case of
block-diagonal operators. First note that the operations of taking direct
limit and direct sum commute. Hence as a consequence of Proposition 2.15, we
obtain the following:

######  Corollary 3.5.

With the same notation as above and assuming that $\Gamma$ is a-T-menable, the
following map induced by the equivariant index map

$\lim\limits_{r\to\infty}\bigoplus_{n}K^{\Gamma}_{\ast}(P_{r}(X_{n}))\longrightarrow\bigoplus_{n}K_{\ast}(C^{*}(X_{n})^{\Gamma})$

is an isomorphism for $\ast=0,1$.

Now for each $r>0$, we consider the following $C^{*}$-subalgebras in
$C^{*}(P_{r}(X))^{\Gamma}$ and $C^{*}_{L}(P_{r}(X))^{\Gamma}$, respectively:

$\mathfrak{A}_{r}:=\lim\limits_{n\to\infty}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}=\overline{\bigcup_{n=1}^{\infty}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}}$

and

$\mathfrak{A}_{L,r}:=\lim\limits_{n\to\infty}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}=\overline{\bigcup_{n=1}^{\infty}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}}.$

The following lemma is straightforward, hence we omit the proof.

######  Lemma 3.6.

With the same notation as above and for $r>0$, we have the following:

$\displaystyle C^{*}(P_{r}(X))^{\Gamma}$ $\displaystyle=$
$\displaystyle\mathfrak{A}_{r}+\big{(}C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}\big{)},$
$\displaystyle\bigoplus_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}$
$\displaystyle=$
$\displaystyle\mathfrak{A}_{r}\cap\big{(}C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}\big{)}.$

The same holds for the case of equivariant localisation algebras.

Now we are able to reduce the proof of Theorem 1.1 to the case of block
diagonal operators.

######  Proposition 3.7.

Let $(X,d)$ be a discrete metric space of bounded geometry and having the form
$X=\bigsqcup_{n=1}^{\infty}X_{n}$ with $d(X_{n},X_{m})\to\infty$ as
$n+m\to\infty~{}(n\neq m)$ such that $\Gamma$ acts on each $X_{n}$ properly
and cocompactly by isometries. To prove that

$\mathrm{Ind}^{\Gamma}:\lim_{r\to\infty}K_{\ast}^{\Gamma}(P_{r}(X))\longrightarrow
K_{\ast}(C^{*}(X)^{\Gamma})$

is an isomorphism for $\ast=0,1$, it suffices to prove the map

$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}\big{(}C^{*}_{L}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}\big{)}\longrightarrow
K_{\ast}\big{(}C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}\big{)}$

induced by the evaluation-at-one map is an isomorphism for $\ast=0,1$.

###### Proof.

First note that for each $r>0$, we have the following commutative diagram:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus\limits_{n=1}^{\infty}K_{\ast}(C^{*}_{L}(P_{r}(X_{n}))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}C^{*}_{L}(P_{r}(X))^{\Gamma}\cap\prod\limits_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}\frac{C^{*}_{L}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}}{\bigoplus_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus\limits_{n=1}^{\infty}K_{\ast}(C^{*}(P_{r}(X_{n}))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}C^{*}(P_{r}(X))^{\Gamma}\cap\prod\limits_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}\frac{C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}}{\bigoplus_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots,}$

where all vertical lines are induced by evaluation-at-one maps and horizontal
lines are exact sequences. When $r\to\infty$, it follows from Corollary 3.5
and the assumption that the left and middle vertical lines are isomorphisms.
Hence we obtain that the right vertical line is also an isomorphism when
$r\to\infty$.

From Lemma 3.6, we obtain the following for each $r>0$:

$\frac{C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}}{\bigoplus_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}}\cong\frac{C^{*}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{r}},$

and

$\frac{C^{*}_{L}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}}{\bigoplus_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}}\cong\frac{C^{*}_{L}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{L,r}}.$

Hence the above implies that

$\lim\limits_{r\to\infty}K_{\ast}\big{(}\frac{C^{*}_{L}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{L,r}}\big{)}\longrightarrow\lim\limits_{r\to\infty}K_{\ast}\big{(}\frac{C^{*}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{r}}\big{)}$

is an isomorphism for $\ast=0,1$.

On the other hand, we have the following commutative diagram for each $r>0$:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(\mathfrak{A}_{L,r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}_{L}(P_{r}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}\frac{C^{*}_{L}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{L,r}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(\mathfrak{A}_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}(P_{r}(X))^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}\frac{C^{*}(P_{r}(X))^{\Gamma}}{\mathfrak{A}_{r}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots,}$

where all vertical lines are induced by evaluation-at-one maps and horizontal
lines are exact sequences. The analysis above implies that the third vertical
line is an isomorphism when $r\to\infty$. Hence to finish the proof, it
suffices to show that

$\lim_{r\to\infty}K_{\ast}(\mathfrak{A}_{L,r})\longrightarrow\lim_{r\to\infty}K_{\ast}(\mathfrak{A}_{r})$

is an isomorphism for $\ast=0,1$. Consider the following commutative diagram:

$\textstyle{\lim\limits_{r\to\infty}K_{\ast}(\mathfrak{A}_{L,r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle\cong}$$\textstyle{\lim\limits_{r\to\infty}\lim\limits_{n\to\infty}K_{\ast}\big{(}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle\cong}$$\textstyle{\lim\limits_{n\to\infty}\lim\limits_{r\to\infty}K_{\ast}\big{(}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lim\limits_{r\to\infty}K_{\ast}(\mathfrak{A}_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle\cong}$$\textstyle{\lim\limits_{r\to\infty}\lim\limits_{n\to\infty}K_{\ast}\big{(}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle\cong}$$\textstyle{\lim\limits_{n\to\infty}\lim\limits_{r\to\infty}K_{\ast}\big{(}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)},}$

where all vertical lines are induced by evaluation-at-one maps. Note that the
right two horizontal lines are isomorphisms due to the continuity of
$K$-theory and the following isomorphisms

$\lim\limits_{r\to\infty}\lim\limits_{n\to\infty}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\cong\lim\limits_{n\to\infty}\lim\limits_{r\to\infty}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}$

and

$\lim\limits_{r\to\infty}\lim\limits_{n\to\infty}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\cong\lim\limits_{n\to\infty}\lim\limits_{r\to\infty}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}$

(note that both limits are just increasing unions of subalgebras). For each
$n\in\mathbb{N}$, it follows from Proposition 2.15 that

$\mathrm{ev}_{\ast}:\lim\limits_{r\to\infty}K_{\ast}\big{(}C^{*}_{L}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)}\longrightarrow\lim\limits_{r\to\infty}K_{\ast}\big{(}C^{*}\big{(}P_{r}\big{(}\bigsqcup_{k\leq
n}X_{k}\big{)}\big{)}^{\Gamma}\big{)}$

is an isomorphism. Hence we conclude the proof. ∎

Consequently, we obtain the following:

######  Corollary 3.8.

To prove Theorem 1.1, it suffices to prove the following: Let $\Gamma$ be a
countable discrete group and $(X,d)$ be a discrete metric space of bounded
geometry and having the form $X=\bigsqcup_{n=1}^{\infty}X_{n}$ with
$d(X_{n},X_{m})\to\infty$ as $n+m\to\infty~{}(n\neq m)$ such that $\Gamma$
acts on each $X_{n}$ properly and cocompactly by isometries. Assume that the
action of $\Gamma$ on $X$ has controlled distortion, the quotient space
$X/\Gamma$ admits a coarse embedding into Hilbert space and $\Gamma$ is a-T-
menable, then the following map

(3.1)
$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}\big{(}C^{*}_{L}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(P_{r}(X_{n}))^{\Gamma}\big{)}\longrightarrow
K_{\ast}\big{(}C^{*}(P_{r}(X))^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(P_{r}(X_{n}))^{\Gamma}\big{)}$

induced by the evaluation-at-one map is an isomorphism for $\ast=0,1$.

##  4\. Equivariant twisted algebras

In this section, we define equivariant twisted Roe and localisation algebras,
which follows the constructions in [34, Section 12.6] and originally comes
from [36, Section 5].

###  4.1. The Bott-Dirac operators on Euclidean spaces

Let us start by recalling the Bott-Dirac operators. Here we only list
necessary notions and facts, and guide readers to [34, Section 12.1] for
details.

Let $E$ be a real Hilbert space (also called a _Euclidean space_) with even
dimension $d\in\mathbb{N}$. The _complexified Clifford algebra_ of $E$,
denoted by $\mathrm{Cliff}(E)$, is the universal unital complex algebra
containing $E$ as a real subspace and subject to the multiplicative relations
$x\cdot x=\|x\|_{E}^{2}$ for all $x\in E$. It is natural to treat
$\mathrm{Cliff}(E)$ as a graded Hilbert space (see for example [34, Example
E.2.12]), and in this case we denote it by $\mathcal{H}_{E}$.

Denote $\mathcal{L}_{E}^{2}$ the graded Hilbert space of square integrable
functions from $E$ to $\mathcal{H}_{E}$ where the grading is inherited from
$\mathcal{H}_{E}$, and $\SS_{E}$ the dense subspace consisting of Schwartz
class functions from $E$ to $\mathcal{H}_{E}$. Fix an orthonormal basis
$\\{e_{1},\ldots,e_{d}\\}$ of $E$ and let $x_{1},\ldots,x_{d}:E\to\mathbb{R}$
be the corresponding coordinates. Recall that the _Bott operator_ $C$ and the
_Dirac operator_ $D$ are unbounded operators on $\mathcal{L}_{E}^{2}$ with
domain $\SS_{E}$ defined as:

$(Cu)(x)=x\cdot
u(x),\quad\mbox{and}\quad(Du)(x)=\sum_{i=1}^{d}\hat{e_{i}}\cdot\frac{\partial
u}{\partial x_{i}}(x)$

for $u\in\SS_{E}$ and $x\in E$, where
$\hat{e_{i}}:\mathrm{Cliff}(E)\to\mathrm{Cliff}(E)$ is the operator determined
by $\hat{e_{i}}(w)=(-1)^{\partial w}w\cdot e_{i}$ for any homogeneous element
$w\in\mathrm{Cliff}(E)$.

######  Definition 4.1.

The _Bott-Dirac operator_ is the unbounded operator $B:=D+C$ on
$\mathcal{L}_{E}^{2}$ with domain $\SS_{E}$.

Given $x\in E$, the _left Clifford multiplication operator associated to $x$_
is the bounded operator $c_{x}$ on $\mathcal{L}_{E}^{2}$ defined as the left
Clifford multiplication by the fixed vector $x$, and the _translation operator
associated to $x$_ is the unitary operator $V_{x}$ on $\mathcal{L}_{E}^{2}$
defined by $(V_{x}u)(y):=u(y-x)$. Given $s\in[1,\infty)$, the _shrinking
operator associated to $s$_ is the unitary operator $S_{s}$ on
$\mathcal{L}_{E}^{2}$ defined by $(S_{s}u)(y):=s^{{-d}/{2}}u(sy)$.

######  Definition 4.2.

For $s\in[1,\infty)$ and $x\in E$, the _Bott-Dirac operator associated to
$(x,s)$_ is the unbounded operator $B_{s,x}:=s^{-1}D+C-c_{x}$ on
$\mathcal{L}_{E}^{2}$ with domain $\SS_{E}$.

Note that $B_{1,0}=B$ and
$B_{s,x}=s^{{-1}/{2}}~{}V_{x}S_{\sqrt{s}}BS^{*}_{\sqrt{s}}V_{x}^{*}$. It is
also known that for each $s\in[1,\infty)$ and $x\in E$, the operator $B_{s,x}$
is unbounded, odd, essentially self-adjoint and maps $\SS_{E}$ to itself (see,
_e.g._ , [34, Corollary 12.1.4]).

######  Definition 4.3.

Let $s\in[1,\infty)$, $x\in E$ and $B_{s,x}$ be the Bott-Dirac operator
associated to $(x,s)$. Define a bounded operator on $\mathcal{L}_{E}^{2}$ by:

$F_{s,x}:=B_{s,x}(1+B^{2}_{s,x})^{-1/2}.$

We list several important properties of the operator $F_{s,x}$. For
simplicity, denote $\chi_{x,R}:=\chi_{B(x,R)}$ for $x\in E$ and $R\geq 0$.

######  Proposition 4.4 ([34, Proposition 12.1.10]).

For each $\varepsilon>0$ there exists an odd function
$\Psi:\mathbb{R}\rightarrow[-1,1]$ with $\Psi(t)\rightarrow 1$ as
$t\rightarrow+\infty$, satisfying the following:

  1. (1)

For all $s\in[1,\infty)$ and $x\in E$, we have
$\|F_{s,x}-\Psi(B_{s,x})\|<\varepsilon$.

  2. (2)

There exists $R_{0}>0$ such that for all $s\in[1,\infty)$ and $x\in E$, we
have $\mathrm{prop}_{E}(\Psi(B_{s,x}))\leq s^{-1}R_{0}$.

  3. (3)

For all $s\in[1,\infty)$ and $x\in E$, $\Psi(B_{s,x})^{2}-1$ is compact.

  4. (4)

For all $s\in[1,\infty)$ and $x,y\in E$, $\Psi(B_{s,x})-\Psi(B_{s,y})$ is
compact.

  5. (5)

For all $s\in[1,\infty)$ and $x,y\in E$, $\|F_{s,x}-F_{s,y}\|\leq
3\|x-y\|_{E}$. And there exists $c>0$ such that for all $s\in[1,\infty)$ and
$x,y\in E$, we have

$\|\Psi(B_{s,x})-\Psi(B_{s,y})\|\leq c\|x-y\|_{E}.$

  6. (6)

For all $x\in E$, the function

$[1,\infty)\rightarrow\mathfrak{B}(\mathcal{L}_{E}^{2}),\
s\mapsto\Psi(B_{s,x})$

is strong-$*$ continuous.

  7. (7)

The family of functions

$[1,\infty)\to\mathfrak{B}(\mathcal{L}_{E}^{2}),\ s\mapsto\Psi(B_{s,x})^{2}-1$

is norm equi-continuous as $x$ varies over $E$ and $s$ varies over any fixed
compact subset of $[1,\infty)$.

  8. (8)

For any $r\geq 0$, the family of functions

$[1,\infty)\to\mathfrak{B}(\mathcal{L}_{E}^{2}),\
s\mapsto\Psi(B_{s,x})-\Psi(B_{s,y})$

is norm equi-continuous as $(x,y)$ varies over the elements of $E\times E$
with $|x-y|\leq r$, and $s$ varies over any fixed compact subset of
$[1,\infty)$.

  9. (9)

For any $\varepsilon_{1}>0$, there exists $R_{1}>0$ such that for all $R\geq
R_{1}$, $s\geq 2d$ and $x\in E$, we have

$\|(\Psi(B_{s,x})^{2}-1)(1-\chi_{x,R})\|<\varepsilon_{1}.$

  10. (10)

For any $\varepsilon_{2}>0,r>0$ there exists $R_{2}>0$ such that for all
$R\geq R_{2}$, $s\geq 2d$ and $x,y\in E$ with $\|x-y\|_{E}\leq r$, we have

$\|(\Psi(B_{s,x})-\Psi(B_{s,y}))(1-\chi_{x,R})\|<\varepsilon_{2}.$

Moreover, we can require that the function $\Psi$, constants $R_{0}$ in (2),
$c$ in (5), $R_{1}$ in (9) and $R_{2}$ in (10) are independent of the
dimension $d$ of the Euclidean space $E$.

The strong-$\ast$ topology used in (6) above is defined as follows: A net
$(T_{i})$ of bounded operators converges to a bounded operator $T$ in the
_strong- $\ast$ topology_ if all $v$ in the underlying Hilbert space,
$T_{i}v\to Tv$ and $T_{i}^{*}v\to T^{*}v$. We need the following elementary
result:

######  Lemma 4.5 ([34, Lemma 12.3.5]).

Let $\mathcal{S}$ and $\mathcal{T}$ be norm bounded sets of operators on a
Hilbert space $\mathcal{H}$ such that $\mathcal{T}$ consists only of compact
operators. Equip $\mathcal{S}$ with the strong-$\ast$ topology and
$\mathcal{T}$ with the norm topology. Then the product maps

$\mathcal{S}\times\mathcal{T}\to\mathfrak{K}\quad\mbox{and}\quad\mathcal{T}\times\mathcal{S}\to\mathfrak{K}$

are jointly continuous where $\mathfrak{K}$ denotes the compact operators on
$\mathcal{H}$.

###  4.2. Equivariant twisted algebras

Throughout the rest of this paper, thanks to Section 3, we always assume that
$(X,d)$ is a discrete metric space of bounded geometry and having the form
$X=\bigsqcup_{n=1}^{\infty}X_{n}$ with $d(X_{n},X_{m})\to\infty$ as
$n+m\to\infty~{}(n\neq m)$ such that the a-T-menable group $\Gamma$ acts on
each $X_{n}$ properly and cocompactly by isometries. Furthermore, assume that
all $\Gamma$-orbit maps are uniformly coarsely equivalent and the quotient
space admits a coarse embedding $\xi:X/\Gamma\to\mathcal{H}$ into some Hilbert
space $\mathcal{H}$.

Denoting by $E$ the underlying Euclidean space of $H$, it is clear that the
map $\xi:X/\Gamma\to E$ is also a coarse embedding. Note that each
$X_{n}/\Gamma$ is finite, hence the image of $\xi$ restricted to
$X_{n}/\Gamma$ sits inside a finite-dimensional Euclidean space, denoted by
$E_{n}$. Without loss of generality, we assume that the dimension $d_{n}$ of
$E_{n}$ is even. For $n\in\mathbb{N}$, denote
$f_{n}:=\xi\circ\pi|_{X_{n}}:X_{n}\to E_{n}$, where $\pi:X\to X/\Gamma$ is the
quotient map.

For $n\in\mathbb{N}$ and $r>0$, denote the Rips complex of $X_{n}$ with scale
$r$ by $P_{r,n}:=P_{r}(X_{n})$. The $\Gamma$-action on $X_{n}$ induces a
proper and cocompact $\Gamma$-action on $P_{r,n}$ by isometries as explained
in Section 2.5.

For $x\in X_{n}$, we consider the open star neighbourhood
$B_{x,r,n}:=St_{n,r}(x)$ of $x$ in the barycentric subdivision of $P_{r,n}$.
Note that $B_{x,r,n}\cap B_{y,r,n}=\emptyset$ for $x\neq y\in X_{n}$,
$\gamma\cdot B_{x,r,n}=B_{\gamma x,r,n}$ for $\gamma\in\Gamma$, and
$B_{x,r,n}\subseteq B_{x,s,n}$ for each $r\leq s$. Taking
$Y_{r,n}:=\bigcup_{x\in X_{n}}B_{x,r,n}$, it is clear that $Y_{r,n}$ is
$\Gamma$-invariant (since the action is by isometries) and dense in $P_{r,n}$.
Moreover, let $Z_{r,n}$ be the collection of points $\sum{c_{i}x_{i}}$ in
$Y_{r,n}$ such that all the coefficients $c_{i}$ take rational values. It is
clear that $Z_{r,n}$ is a $\Gamma$-invariant countable subset which is also
dense in $P_{r,n}$.

For $n\in\mathbb{N}$ and $r>0$, we extend the above $f_{n}:X_{n}\to E_{n}$ to
$f_{r,n}:Y_{r,n}\to E_{n}$ by sending all points in $B_{x,r,n}$ to $f_{n}(x)$
for $x\in X_{n}$. It is clear that $f_{r,n}(\gamma\cdot y)=f_{r,n}(y)$ for any
$\gamma\in\Gamma$ and $y\in Y_{r,n}$.

Let us fix an infinite-dimensional separable Hilbert space $\mathscr{H}$. For
$r>0$ and $n\in\mathbb{N}$, consider the following Hilbert spaces:

$\mathcal{H}_{r,n}:=\ell^{2}(Z_{r,n})\otimes\mathscr{H}\otimes\ell^{2}(\Gamma)\quad\mbox{and}\quad\mathcal{H}_{r,n,E}:=\ell^{2}(Z_{r,n})\otimes\mathscr{H}\otimes\ell^{2}(\Gamma)\otimes\mathcal{L}_{E_{n}}^{2}.$

The group $\Gamma$ acts on $\mathcal{H}_{r,n}$ (respectively,
$\mathcal{H}_{r,n,E}$) as follows: for any $\gamma\in\Gamma$,

$\displaystyle U_{\gamma}:$
$\displaystyle\delta_{z}\otimes\xi\otimes\delta_{g}$
$\displaystyle\mapsto~{}~{}\delta_{\gamma z}\otimes\xi\otimes\delta_{\gamma
g}$ $\displaystyle(\mbox{respectively,}~{}U_{\gamma}:$
$\displaystyle\delta_{z}\otimes\xi\otimes\delta_{g}\otimes u$
$\displaystyle\mapsto~{}~{}\delta_{\gamma z}\otimes\xi\otimes\delta_{\gamma
g}\otimes u).$

It is clear that $\mathcal{H}_{r,n}$ is an admissible $P_{r,n}$-module under
the amplified multiplication representation, and similarly
$\mathcal{H}_{r,n,E}$ is both an admissible $P_{r,n}$-module and an ample
$E_{n}$-module. We use these modules to the build equivariant Roe algebras
$C^{*}(\mathcal{H}_{r,n})^{\Gamma}$ and $C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$
of $P_{r,n}$, and the equivariant localisation algebras
$C^{*}_{L}(\mathcal{H}_{r,n})^{\Gamma}$ and
$C^{*}_{L}(\mathcal{H}_{r,n,E})^{\Gamma}$ of $P_{r,n}$. Moreover for
$T\in\mathfrak{B}(\mathcal{H}_{r,n,E})$, we write $\mathrm{prop}_{P}(T)$ and
$\mathrm{prop}_{E}(T)$ for the propagation of $T$ with respect to the
$P_{r,n}$-module structure and the $E_{n}$-module structure, respectively.
Also denote the Hilbert spaces

$\mathcal{H}_{r}:=\bigoplus_{n=1}^{\infty}\mathcal{H}_{r,n}\quad\mbox{and}\quad\mathcal{H}_{r,E}:=\bigoplus_{n=1}^{\infty}\mathcal{H}_{r,n,E},$

and use these modules to build the equivariant Roe algebras
$C^{*}(\mathcal{H}_{r})^{\Gamma}$ and $C^{*}(\mathcal{H}_{r,E})^{\Gamma}$ of
$P_{r}:=\bigsqcup_{n=1}^{\infty}P_{r,n}$, and the equivariant localisation
algebras $C^{*}_{L}(\mathcal{H}_{r})^{\Gamma}$ and
$C^{*}_{L}(\mathcal{H}_{r,E})^{\Gamma}$ of $P_{r}$.

For $r\leq s$, the canonical inclusion $Z_{r}\to Z_{s}$ induces isometric
inclusions of Hilbert spaces

$\mathcal{H}_{r,n}\longrightarrow\mathcal{H}_{s,n}\quad\mbox{and}\quad\mathcal{H}_{r,n,E}\longrightarrow\mathcal{H}_{s,n,E},$

and further implies inclusions of $C^{*}$-algebras

$C^{*}(\mathcal{H}_{r,n})\longrightarrow C^{*}(\mathcal{H}_{s,n}),\quad
C^{*}(\mathcal{H}_{r,n,E})\longrightarrow C^{*}(\mathcal{H}_{s,n,E})$

and

$C^{*}_{L}(\mathcal{H}_{r,n})\longrightarrow
C^{*}_{L}(\mathcal{H}_{s,n}),\quad
C^{*}_{L}(\mathcal{H}_{r,n,E})\longrightarrow C^{*}_{L}(\mathcal{H}_{s,n,E}).$

For $r>0$ and $n\in\mathbb{N}$, also note that although $\\{B_{x,r,n}:x\in
X_{n}\\}$ does not cover $P_{r,n}$ we still have the following decomposition
of Hilbert spaces:

$\mathcal{H}_{r,n}=\bigoplus_{x\in
X_{n}}\chi_{B_{x,r,n}}\mathcal{H}_{r,n}\quad\mbox{and}\quad\mathcal{H}_{r,n,E}=\bigoplus_{x\in
X_{n}}\chi_{B_{x,r,n}}\mathcal{H}_{r,n,E}$

since $\\{B_{x,r,n}:x\in X_{n}\\}$ covers $Z_{r,n}$. Write

$\mathcal{H}_{x,r,n}:=\chi_{B_{x,r,n}}\mathcal{H}_{r,n}\quad\mbox{and}\quad\mathcal{H}_{x,r,n,E}=\chi_{B_{x,r,n}}\mathcal{H}_{r,n,E}.$

We can represent a bounded linear operator $T$ on $\mathcal{H}_{r,n}$
(respectively, $\mathcal{H}_{r,n,E}$) as an $X_{n}$-by-$X_{n}$ matrix
$(T_{x,y})_{x,y\in X_{n}}$, where each $T_{x,y}$ is a bounded linear operator
$\mathcal{H}_{y,r,n}\to\mathcal{H}_{x,r,n}$ (respectively,
$\mathcal{H}_{y,r,n,E}\to\mathcal{H}_{x,r,n,E}$). Moreover, $T$ is
$\Gamma$-invariant if and only if the following diagram commutes:

$\textstyle{\mathcal{H}_{y,r,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{x,y}}$$\scriptstyle{U_{\gamma}}$$\textstyle{\mathcal{H}_{x,r,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{\gamma}}$$\textstyle{\mathcal{H}_{\gamma
y,r,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{\gamma
x,\gamma y}}$$\textstyle{\mathcal{H}_{\gamma x,r,n}}$

for any $\gamma\in\Gamma$ and $x,y\in X_{n}$, _i.e._ , $T_{\gamma x,\gamma
y}=U_{\gamma}T_{x,y}U_{\gamma}^{\ast}=:\gamma\cdot T_{x,y}$.

It was pointed out in [34, Inequality (12.11)] that for any $r>0$, there
exists a proper non-decreasing function $\varphi_{r}:[0,\infty)\to[0,\infty)$
such that for any $T\in\mathbb{C}[\mathcal{H}_{r,n,E}]^{\Gamma}$ we have:

(4.1) $\mathrm{prop}_{P}(T)-2\leq\sup\\{d(x,y):T_{x,y}\neq
0\mbox{~{}where~{}}x,y\in X_{n}\\}\leq\varphi_{r}(\mathrm{prop}_{P}(T)).$

We also record the following elementary result (see, _e.g._ , [34, Lemma
12.2.4]) for later use:

######  Lemma 4.6.

For $s,r\geq 0$, there exists an $N\in\mathbb{N}$ such that for any
$n\in\mathbb{N}$ and any bounded operator $T=(T_{x,y})_{x,y\in X}$ on
$\mathcal{H}_{r,n,E}$ with $P$-propagation at most $s$, we have

$\|T\|\leq N\cdot\sup_{x,y\in X}\|T_{x,y}\|.$

To introduce the twisted algebras, we recall the following construction from
[34, Definition 12.3.1]:

######  Definition 4.7.

Given $r>0$, $n\in\mathbb{N}$ and $T\in\mathfrak{B}(\mathcal{L}_{E_{n}}^{2})$,
define a bounded linear operator $T^{V}$ on
$\mathcal{H}_{r,n,E_{n}}=\ell^{2}(Z_{r,n})\otimes\mathscr{H}\otimes\ell^{2}(\Gamma)\otimes\mathcal{L}^{2}_{E_{n}}$
by the formula

$T^{V}:\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
u\mapsto\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
V_{f_{r,n}(z)}TV_{f_{r,n}(z)}^{*}u$

for $z\in Z_{r,n}$, $\xi\in\mathscr{H}$, $g\in\Gamma$ and
$u\in\mathcal{L}^{2}_{E_{n}}$, where $V_{f_{r,n}(z)}$ is the translation
operator defined in Section 4.1.

Writing in the matrix representation, we have

$T^{V}_{x,y}=\begin{cases}~{}\mathrm{Id}_{\mathcal{H}_{x,r,n}}\otimes
V_{f_{r,n}(x)}TV_{f_{r,n}(x)}^{*},&y=x;\\\ ~{}0,&\mbox{otherwise}.\end{cases}$

Since $f_{r,n}(\gamma z)=f_{r,n}(z)$ for any $z\in Z_{r,n}$ and
$\gamma\in\Gamma$, it is easy to see that $\gamma\cdot T_{x,y}=T_{\gamma
x,\gamma y}$. In other words, $T^{V}$ is invariant. Hence we obtain:

######  Lemma 4.8.

For any $T\in\mathfrak{B}(\mathcal{L}_{E_{n}}^{2})$, we have
$T^{V}\in\mathbb{C}[\mathcal{H}_{r,n,E}]^{\Gamma}$.

Now we introduce the notion of twisted algebras, which is slightly different
from [34, Section 12.6].

######  Definition 4.9.

Fix an $r>0$. Denote
$\prod_{n\in\mathbb{N}}C_{b}([1,\infty),C^{*}(\mathcal{H}_{r,n,E})^{\Gamma})$
the product $C^{*}$-algebra of all bounded continuous functions from
$[1,\infty)$ to $C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$ with supremum norm.
Write elements of this $C^{*}$-algebra as a collection
$(T_{n,s})_{n\in\mathbb{N},s\in[1,\infty)}$ for $T_{n,s}=(T_{n,s,x,y})_{x,y\in
X_{n}}\in C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$, whose norm is

$\|(T_{n,s})\|=\sup_{n\in\mathbb{N},s\in[1,\infty)}\|T_{n,s}\|_{\mathfrak{B}(\mathcal{H}_{r,n,E})}.$

Let $\mathbb{A}^{r}(X;E)^{\Gamma}$ denote the $*$-subalgebra of
$\prod_{n\in\mathbb{N}}C_{b}([1,\infty),C^{*}(\mathcal{H}_{r,n,E})^{\Gamma})$
consisting of elements satisfying the following conditions:

  1. (1)

$\sup\limits_{s\in[1,\infty),n\in\mathbb{N}}\mathrm{prop}_{P}(T_{n,s})<\infty$;

  2. (2)

for each $n\in\mathbb{N}$,
$\lim\limits_{s\rightarrow\infty}\mathrm{prop}_{E}(T_{n,s})=0$;

  3. (3)

$\lim\limits_{R\rightarrow\infty}\sup\limits_{s\in[1,\infty),n\in\mathbb{N}}\|\chi_{0,R}^{V}T_{n,s}-T_{n,s}\|=\lim\limits_{R\rightarrow\infty}\sup\limits_{s\in[1,\infty),n\in\mathbb{N}}\|T_{n,s}\chi_{0,R}^{V}-T_{n,s}\|=0$;

  4. (4)

for each $n\in\mathbb{N}$ and $x,y\in X_{n}$, the map $s\mapsto T_{n,s,x,y}$
belongs to the subalgebra
$\mathfrak{K}(\mathcal{H}_{y,r,n},\mathcal{H}_{x,r,n})\otimes
C_{b}([1,\infty),\mathfrak{K}(\mathcal{L}^{2}_{E_{n}}))$ of
$C_{b}([1,\infty),\mathfrak{K}(\mathcal{H}_{y,r,n},\mathcal{H}_{x,r,n})\otimes\mathfrak{K}(\mathcal{L}^{2}_{E_{n}}))$.

The _equivariant twisted Roe algebra_ $A^{r}(X;E)^{\Gamma}$ of $X$ is defined
to be the norm-closure of $\mathbb{A}^{r}(X;E)^{\Gamma}$ in
$\prod_{n\in\mathbb{N}}C_{b}([1,\infty),C^{*}(\mathcal{H}_{r,n,E})^{\Gamma})$.

######  Definition 4.10.

Define $\mathbb{A}^{r}_{L}(X;E)^{\Gamma}$ to be the collection of uniformly
continuous bounded functions $T:[1,\infty)\to\mathbb{A}^{r}(X;E)^{\Gamma}$
satisfying the following: writing
$T=(T_{t})_{t\in[1,\infty)}=(T_{t,n,s})_{t,s\in[1,\infty),n\in\mathbb{N}}$
then

  1. (1)

$\lim\limits_{t\to\infty}\sup\limits_{s\in[1,\infty),n\in\mathbb{N}}\mathrm{prop}_{P}(T_{t,n,s})=0$;

  2. (2)

$\lim\limits_{R\rightarrow\infty}\sup\limits_{t,s\in[1,\infty),n\in\mathbb{N}}\|\chi_{0,R}^{V}T_{t,n,s}-T_{t,n,s}\|=\lim\limits_{R\rightarrow\infty}\sup\limits_{t,s\in[1,\infty),n\in\mathbb{N}}\|T_{t,n,s}\chi_{0,R}^{V}-T_{t,n,s}\|=0$.

The _equivariant twisted localisation algebra_ $A^{r}_{L}(X;E)^{\Gamma}$ of
$X$ is defined to be the completion of $\mathbb{A}^{r}_{L}(X;E)^{\Gamma}$ for
the norm $\|(T_{t})\|:=\sup_{t}\|T_{t}\|_{A^{r}(X;E)^{\Gamma}}$.

######  Remark 4.11.

Readers might already notice that the above definition is slightly different
from [34, Definition 12.6.2]. More precisely, we weaken condition (2) to
level-wise convergence and restrict the living space of each matrix entry (see
condition (4) above) for later use. On the other hand, the following lemma
shows that the original condition (4) in [34, Definition 12.6.2] can be
recovered.

######  Lemma 4.12.

Given $n\in\mathbb{N}$, $r>0$ and $T\in C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$,
we have

$\lim_{i\in I}\|p_{i}^{V}T-T\|=\lim_{i\in I}\|Tp_{i}^{V}-T\|=0$

where $\\{p_{i}\\}_{i\in I}$ is the net of finite rank projections on
$\mathcal{L}_{E_{n}}^{2}$.

###### Proof.

We fix an operator $T\in\mathbb{C}[\mathcal{H}_{r,n,E}]^{\Gamma}$ with
$P$-propagation at most $R^{\prime}$, and a point $x_{0}\in X_{n}$. For
$x,y\in X_{n}$ with $T_{x,y}\neq 0$, we have
$d(x,y)\leq\varphi_{r}(R^{\prime})=:R$ where the function $\varphi_{r}$ comes
from Inequality (4.1). Let $N$ be the number from Lemma 4.6 for the parameters
$R^{\prime}$ and $r$, _i.e._ , for any bounded operator
$\tilde{T}=(\tilde{T}_{x,y})_{x,y\in X}$ on $\mathcal{H}_{r,n,E}$ with
$P$-propagation at most $R^{\prime}$, then

$\|\tilde{T}\|\leq N\cdot\sup_{x,y\in X}\|\tilde{T}_{x,y}\|.$

Recall that the $\Gamma$-action on $X_{n}$ is cocompact, hence there exists an
$S>0$ such that $\Gamma\cdot B(x_{0},S)=X_{n}$.

We consider a finite set of operators:

$\mathfrak{F}:=\\{T_{x,y}:x\in B(x_{0},S)\mbox{~{}and~{}}y\in B(x,R)\\}.$

Since $T$ is locally compact, each $T_{x,y}$ is a compact operator from
$\mathcal{H}_{y,r,n,E}$ to $\mathcal{H}_{x,r,n,E}$. Note that for any
projection $p\in\mathfrak{B}(\mathcal{L}^{2}_{E_{n}})$ and $x,y\in X_{n}$, we
have:

$(p^{V}T-T)_{x,y}=\big{(}\mathrm{Id}_{\mathcal{H}_{x,r,n}}\otimes
V_{f_{r,n}(x)}pV_{f_{r,n}(x)}^{*}-\mathrm{Id}_{\mathcal{H}_{x,r,n,E}}\big{)}\cdot
T_{x,y},$

and the net

$\big{\\{}\mathrm{Id}_{\mathcal{H}_{x,r,n}}\otimes
V_{f_{r,n}(x)}pV_{f_{r,n}(x)}^{*}-\mathrm{Id}_{\mathcal{H}_{x,r,n,E}}:p\mbox{~{}is~{}a~{}finite~{}rank~{}projection~{}in~{}}\mathfrak{B}(\mathcal{L}^{2}_{E_{n}})\big{\\}}$

converges to $0$ in the strong operator topology. Hence for any
$\varepsilon>0$ there exists a finite rank projection
$p_{\varepsilon}\in\mathfrak{B}(\mathcal{L}^{2}_{E_{n}})$ such that for any
finite rank projection $p\geq p_{\varepsilon}$ and $T_{x,y}\in\mathfrak{F}$,
we have

$\|(p^{V}T-T)_{x,y}\|<\frac{\varepsilon}{N}.$

Now for any $x^{\prime},y^{\prime}\in X_{n}$ with
$d(x^{\prime},y^{\prime})\leq R$, there exists $\gamma\in\Gamma$ such that
$\gamma x^{\prime}\in B(x_{0},S)$. Letting $x=\gamma x^{\prime}$ and $y=\gamma
y^{\prime}$, we have $d(x,y)\leq R$ and hence $T_{x,y}\in\mathfrak{F}$.
Moreover, we obtain:

$\|(p^{V}T-T)_{x^{\prime},y^{\prime}}\|=\|(p^{V}T-T)_{\gamma^{-1}x,\gamma^{-1}y}\|=\|\gamma^{-1}\cdot(p^{V}T-T)_{x,y}\|=\|(p^{V}T-T)_{x,y}\|<\frac{\varepsilon}{N}.$

Note that the $P$-propagation of $p^{V}T-T$ is again at most $R^{\prime}$.
Hence from Lemma 4.6 we obtain:

$\|p^{V}T-T\|\leq N\cdot\sup_{x,y\in X_{n}:d(x,y)\leq
R}\|(p^{V}T-T)_{x,y}\|<N\cdot\frac{\varepsilon}{N}=\varepsilon.$

Therefore, we obtain that $\lim_{i\in I}\|p_{i}^{V}T-T\|=0$. Finally taking
adjoints, we obtain $\lim_{i\in I}\|Tp_{i}^{V}-T\|=0$ as well and conclude the
proof. ∎

Finally, we introduce the following operators:

######  Definition 4.13 ([34, Section 12.3 and 12.6]).

Fix an $r>0$. For each $n\in\mathbb{N}$, $s\in[1,\infty)$ and $x\in E_{n}$,
Definition 4.3 provides a bounded linear operator
$F_{s,x}\in\mathfrak{B}(\mathcal{L}_{E_{n}}^{2})$, also denoted by
$F_{n,s,x}$. Applying Definition 4.7, we obtain an operator

$F_{n,s}:=F^{V}_{n,s+2d_{n},0}\in\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}$

where $d_{n}$ is the dimension of $E_{n}$. Let
$F_{s}:=(F_{n,s})_{n\in\mathbb{N}}$ be the block diagonal operator in
$\prod_{n}\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}\subseteq\mathfrak{B}(\mathcal{H}_{r,E})^{\Gamma}$.
Finally, we define $F$ to be an element in
$\prod_{n}\mathfrak{B}(L^{2}([1,\infty);\mathcal{H}_{r,n,E}))\subseteq\mathfrak{B}(L^{2}([1,\infty);\mathcal{H}_{r,E}))$
defined by $(F(u))(s):=F_{s}u(s)$.

Similarly given $\varepsilon>0$, let $\Psi$ be a function as in Proposition
4.4 and set $F^{\Psi}_{n,s,x}:=\Psi(B_{n,s,x})$. Let $F^{\Psi}_{s}$ be the
bounded diagonal operator on $\mathcal{H}_{r,E}$ defined by:

$F^{\Psi}_{s}:=(F^{\Psi}_{n,s})_{n\in\mathbb{N}}\quad\mbox{where}\quad
F^{\Psi}_{n,s}:=(F^{\Psi}_{n,s+2d_{n},0})^{V}=\Psi(B_{n,s+2d_{n},0})^{V}\in\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}.$

We also define $F^{\Psi}$ to be the element in
$\prod_{n}\mathfrak{B}(L^{2}([1,\infty);\mathcal{H}_{r,n,E}))\subseteq\mathfrak{B}(L^{2}([1,\infty);\mathcal{H}_{r,E}))$
by $(F^{\Psi}(u))(s):=F^{\Psi}_{s}u(s)$.

##  5\. The index maps

In this section, we construct equivariant index maps (with the same notation
as in Section 4.2):

(5.1)
$\mathrm{Ind}_{F}:K_{*}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\rightarrow
K_{*}(A^{r}(X;E)^{\Gamma})$

and

(5.2)
$\mathrm{Ind}_{F_{L}}:K_{*}\big{(}C^{*}_{L}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\rightarrow
K_{*}(A^{r}_{L}(X;E)^{\Gamma}),$

where $F$ is the operator from Definition 4.13. We use these maps to transfer
$K$-theoretic information from equivariant Roe and localisation algebras to
their twisted counterparts, which allow us to prove Theorem 1.1 via local
isomorphisms. This approach is mainly based on [34, Secition 12.3 and 12.6] in
the case of coarse embedding, while several changes are needed to involve
group actions.

The main result of this section is the following:

######  Proposition 5.1.

Fix an $r>0$. With notation as in Section 4.2, for each $s\in[1,\infty)$ the
composition

$K_{*}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\stackrel{{\scriptstyle\mathrm{Ind}_{F}}}{{\longrightarrow}}K_{*}(A^{r}(X;E)^{\Gamma})\stackrel{{\scriptstyle\iota^{s}_{*}}}{{\longrightarrow}}K_{*}\big{(}C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\big{)}$

is an isomorphism, where $\iota^{s}:A^{r}(X;E)^{\Gamma}\rightarrow
C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$
is the evaluation map at $s$. The analogous statement holds for the
equivariant localisation algebras. Moreover, we have the following commutative
diagram:

$\textstyle{K_{*}\big{(}C^{*}_{L}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ind}_{F_{L}}}$$\textstyle{K_{\ast}(A^{r}_{L}(X;E)^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{s}_{*}}$$\textstyle{K_{\ast}\big{(}C^{*}_{L}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(\mathcal{H}_{r,n,E})^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ind}_{F}}$$\textstyle{K_{\ast}(A^{r}(X;E)^{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{s}_{*}}$$\textstyle{K_{\ast}\big{(}C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\big{)}}$

where all vertical lines are induced by the evaluation-at-one maps.

We follow the same notation as in Section 4.2. Recall that $(X,d)$ is a
discrete metric space of bounded geometry and having the form
$X=\bigsqcup_{n=1}^{\infty}X_{n}$ with $d(X_{n},X_{m})\to\infty$ as
$n+m\to\infty~{}(n\neq m)$ such that $\Gamma$ acts on each $X_{n}$ properly
and cocompactly by isometries. For each $n\in\mathbb{N}$, we have a map
$f_{n}:X_{n}\to E_{n}$ coming from the uniformly coarse embedding of
$X_{n}/\Gamma$ into some Euclidean space $E_{n}$ of even dimension $d_{n}$.

To construct the index map $\mathrm{Ind}_{F}$, we need a series of lemmas. Let
us fix an $r>0$ throughout the rest of this section.

######  Lemma 5.2.

The operator $F$ is a self-adjoint, norm one, odd operator in the multiplier
algebra of $A^{r}(X;E)^{\Gamma}$.

###### Proof.

The operator $F$ is self-adjoint, norm one and odd since each $F_{n,s,x}$ is.
Given $\varepsilon>0$, let $\Psi:\mathbb{R}\to[-1,1]$ be a function as in
Proposition 4.4 for this $\varepsilon$. Then Proposition 4.4(1) implies:

$\|F-F^{\Psi}\|\leq\sup_{n\in\mathbb{N},s\in[1,\infty)}\|F^{V}_{n,s,0}-\Psi(B_{n,s,0})^{V}\|\leq\sup_{n\in\mathbb{N},s\in[1,\infty)}\sup_{x\in
X_{n}}\|F_{n,s,f_{n}(x)}-\Psi(B_{n,s,f_{n}(x)})\|\leq\varepsilon.$

Hence it suffices to show that $(T_{n,s})F^{\Psi}=(T_{n,s}F^{\Psi}_{n,s})$
belongs to $\mathbb{A}^{r}(X;E)^{\Gamma}$ for any
$(T_{n,s})\in\mathbb{A}^{r}(X;E)^{\Gamma}$.

We first claim that for each $n\in\mathbb{N}$ the map $s\mapsto
T_{n,s}F^{\Psi}_{n,s}$ is norm continuous. In fact, this can be proved using
the same argument as in the proof of [34, Lemma 12.3.6] thanks to Lemma 4.12.
Here we provide a direct proof.

Fix $n\in\mathbb{N}$ and $x_{0}\in X_{n}$. Assume that each $T_{n,s}$ has
$P$-propagation at most $R^{\prime}$, and set $R:=\varphi_{r}(R^{\prime})$
where the function $\varphi_{r}$ comes from Inequality (4.1). Since the
$\Gamma$-action on $X_{n}$ is cocompact, there exists an $S>0$ such that
$\Gamma\cdot B(x_{0},S)=X_{n}$. For each $x,y\in X_{n}$, it follows from
Proposition 4.4(6) and Lemma 4.5 that the map $s\to T_{n,s,x,y}\cdot
F^{\Psi}_{n,s+2d_{n},f_{n}(y)}=(T_{n,s}F^{\Psi}_{n,s})_{x,y}$ is norm
continuous. On the other hand, for any $x,y\in X_{n}$ with $d(x,y)\leq R$,
there exists $\gamma\in\Gamma$ such that $\bar{x}:=\gamma x\in B(x_{0},S)$ and
hence $\bar{y}:=\gamma y\in B(\bar{x},R)$. Moreover, we have

$(T_{n,s}F^{\Psi}_{n,s})_{x,y}=(T_{n,s}F^{\Psi}_{n,s})_{\gamma^{-1}\bar{x},\gamma^{-1}\bar{y}}=\gamma^{-1}\cdot(T_{n,s}F^{\Psi}_{n,s})_{\bar{x},\bar{y}}.$

Note that the set $\\{(\bar{x},\bar{y}):\bar{x}\in
B(x_{0},S)\mbox{~{}and~{}}\bar{y}\in B(\bar{x},R)\\}$ is finite, hence the
family $\\{s\mapsto(T_{n,s}F^{\Psi}_{n,s})_{x,y}:x,y\in X_{n}\\}$ is uniformly
norm continuous. This concludes the claim due to Lemma 4.6.

For conditions (1)-(3) in Definition 4.9, it suffices to note that the
$P$-propagation of $F^{\Psi}_{n,s}$ is $0$ while the $E$-propagation of
$F^{\Psi}_{n,s}$ is uniformly bounded (in both $s$ and $n$) and (uniformly)
tends to $0$ (in $n$) as $s\to\infty$ by Proposition 4.4(2). Finally for
condition (4), note that for each $n\in\mathbb{N}$ and $x,y\in X_{n}$ we have

$s\mapsto(T_{n,s}F^{\Psi}_{n,s})_{x,y}=T_{n,s,x,y}\cdot\big{(}\mathrm{Id}_{\mathcal{H}_{y,r,n}}\otimes
F^{\Psi}_{n,s+2d_{n},f_{n}(y)}\big{)}$

Since the map $s\mapsto T_{n,s,x,y}$ belongs to
$\mathfrak{K}(\mathcal{H}_{y,r,n},\mathcal{H}_{x,r,n})\otimes
C_{b}([1,\infty),\mathfrak{K}(\mathcal{L}^{2}_{E_{n}}))$, it follows from
Proposition 4.4(6) and Lemma 4.5 again that the map
$s\mapsto(T_{n,s}F^{\Psi}_{n,s})_{x,y}$ belongs to
$\mathfrak{K}(\mathcal{H}_{y,r,n},\mathcal{H}_{x,r,n})\otimes
C_{b}([1,\infty),\mathfrak{K}(\mathcal{L}^{2}_{E_{n}}))$ as well. Hence we
conclude the proof. ∎

######  Lemma 5.3.

Considered as represented on $L^{2}([1,\infty))\otimes\mathcal{H}_{r,E}$ via
the amplification of identity,
$C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$
is a subalgebra of the multiplier algebra of $A^{r}(X;E)^{\Gamma}$.

###### Proof.

It suffices to show that $(S_{n}T_{n,s})\in\mathbb{A}^{r}(X;E)^{\Gamma}$ for
any $(T_{n,s})\in\mathbb{A}^{r}(X;E)^{\Gamma}$ and
$(S_{n})\in\mathbb{C}[\mathcal{H}_{r}]^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$.
It is clear that the map $s\mapsto S_{n}T_{n,s}$ is norm-continuous and
bounded for each $n\in\mathbb{N}$.

For conditions (1) and (2) in Definition 4.9, it suffices to note that $S_{n}$
has uniformly finite $P$-propagation (in $n$) and $E$-propagation $0$.
Condition (4) follows from the fact that $S_{n}$ is constant in $s$. Finally
for condition (3), it is clear that

$\lim\limits_{R\rightarrow\infty}\sup\limits_{s\in[1,\infty),n\in\mathbb{N}}\|S_{n}T_{n,s}\chi_{0,R}^{V}-S_{n}T_{n,s}\|=0.$

On the other hand, set

$R_{0}:=\sup_{n}\big{\\{}|f_{n}(x)-f_{n}(y)|:x,y\in
X_{n}\mbox{~{}with~{}}d(x,y)\leq\mathrm{prop}_{P}(S_{n})\big{\\}}.$

This is clear that $R_{0}$ is finite. For any $n\in\mathbb{N}$, $x,y\in X_{n}$
and $R\geq R_{0}$, we have:

$\displaystyle(\chi_{0,R}^{V}\cdot S_{n}\cdot\chi_{0,R-R_{0}}^{V})_{x,y}$
$\displaystyle=$ $\displaystyle\chi_{f_{n}(x),R}\cdot
S_{n,x,y}\cdot\chi_{f_{n}(y),R-R_{0}}=S_{n,x,y}\cdot\chi_{f_{n}(x),R}\cdot\chi_{f_{n}(y),R-R_{0}}$
$\displaystyle=$ $\displaystyle
S_{n,x,y}\cdot\chi_{f_{n}(y),R-R_{0}}=(S_{n}\cdot\chi_{0,R-R_{0}}^{V})_{x,y}.$

In other words, we obtain

$\chi_{0,R}^{V}\cdot
S_{n}\cdot\chi_{0,R-R_{0}}^{V}=S_{n}\cdot\chi_{0,R-R_{0}}^{V},$

which implies that

$\|\chi_{0,R}^{V}S_{n}T_{n,s}-S_{n}T_{n,s}\|\leq\|\chi_{0,R}^{V}\cdot
S_{n}\cdot(\chi_{0,R-R_{0}}^{V}T_{n,s}-T_{n,s})\|+\|S_{n}\cdot(\chi_{0,R-R_{0}}^{V}T_{n,s}-T_{n,s})\|.$

This tends to $0$ as $R\to\infty$ uniformly (in $s$ and $n$) by assumption. ∎

Regarding
$C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$
as a subalgebra in $\mathfrak{B}(L^{2}([1,\infty))\otimes\mathcal{H}_{r,E})$
as in Lemma 5.3, we have the following:

######  Lemma 5.4.

For any $(S_{n})\in
C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$,
we have $[(S_{n}),F]\in A^{r}(X;E)^{\Gamma}$.

###### Proof.

Fix an
$(S_{n})\in\mathbb{C}[\mathcal{H}_{r}]^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$.
From Proposition 4.4(1), it suffices to show that

$\left(s\mapsto[S_{n},F_{n,s}^{\Psi}]\right)_{n}$

belongs to $\mathbb{A}^{r}(X;E)^{\Gamma}$ for any $\Psi$ as in Proposition
4.4.

First note that for any $n\in\mathbb{N}$ and $x,y\in X_{n}$, we have

$([S_{n},F_{n,s}^{\Psi}])_{x,y}=S_{n,x,y}\otimes\big{(}F^{\Psi}_{n,s+2d_{n},f_{n}(y)}-F^{\Psi}_{n,s+2d_{n},f_{n}(x)}\big{)}.$

Hence for each $n\in\mathbb{N}$, the function $s\mapsto[S_{n},F_{n,s}^{\Psi}]$
is bounded and norm continuous by Proposition 4.4(8) and Lemma 4.6. Also
condition (4) in Definition 4.9 holds by Proposition 4.4(4) and (8). Moreover,
condition (1) and (2) holds since $F_{n,s}^{\Psi}$ has $P$-propagation $0$,
and $S_{n}$ has $E$-propagation $0$ together with Proposition 4.4(2).

Finally for condition (3), note that

$(\chi_{0,R}^{V}\cdot[S_{n},F_{n,s}^{\Psi}])_{x,y}=S_{n,x,y}\otimes\chi_{f_{n}(x),R}\big{(}F^{\Psi}_{n,s+2d_{n},f_{n}(y)}-F^{\Psi}_{n,s+2d_{n},f_{n}(x)}\big{)}.$

Hence condition (3) follows from Proposition 4.4(10), finite $P$-propagation
of $(S_{n})$ and Lemma 4.6. ∎

######  Lemma 5.5.

For any projection $(p_{n})\in
C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$,
the element

$\left(s\mapsto(p_{n}F_{n,s}p_{n})^{2}-p_{n}\right)_{n\in\mathbb{N}}$

is in $(p_{n})A^{r}(X;E)^{\Gamma}(p_{n})$.

###### Proof.

From Lemma 5.4, it suffices to show that $\left(s\mapsto
p_{n}(F_{n,s})^{2}-p_{n}\right)_{n}$ is in $A^{r}(X;E)^{\Gamma}$. Moreover, we
only need to show that

$\left(s\mapsto q_{n}(F_{n,s}^{\Psi})^{2}-q_{n}\right)_{n}$

is in $\mathbb{A}^{r}(X;E)^{\Gamma}$ for any $\Psi$ as in Proposition 4.4,
where $(q_{n})$ is a finite propagation approximation to $(p_{n})$. For each
$n\in\mathbb{N}$, it follows from Proposition 4.4(7) and Lemma 4.6 that the
function $s\mapsto q_{n}(F_{n,s}^{\Psi})^{2}-q_{n}$ is bounded and continuous.
It is routine to check condition (1)-(4) in Definition 4.9 for this map using
Proposition 4.4(3) and (9), the finite propagation of $(q_{n})$ together with
Lemma 4.6. ∎

Now we are in the position to construct the index map $\mathrm{Ind}_{F}$ in
(5.1). It follows from a standard construction in $K$-theories (see, _e.g._ ,
[34, Definitoin 2.8.5]):

######  Definition 5.6.

Let $\mathcal{H}=\mathcal{H}^{+}\oplus\mathcal{H}^{-}$ be a graded Hilbert
space with grading operator $U$ (_i.e._ , $U$ is a self-adjoint unitary
operator in $\mathfrak{B}(\mathcal{H})$ such that $\mathcal{H}^{\pm}$
coincides with the $(\pm 1)$-eigenspace of $U$), and $A$ be a
$C^{*}$-subalgebra of $\mathfrak{B}(\mathcal{H})$ such that $U$ is in the
multiplier algebra of $A$. Let $F\in\mathfrak{B}(\mathcal{H})$ be an odd
operator of the form

$F=\begin{pmatrix}0&V\\\ W&0\end{pmatrix}$

for some operators $V:\mathcal{H}^{-}\rightarrow\mathcal{H}^{+}$ and
$W:\mathcal{H}^{+}\rightarrow\mathcal{H}^{-}$. Suppose $F$ satisfies:

  * •

$F$ is in the multiplier algebra of $A$;

  * •

$F^{2}-1$ is in $A$.

Then we define the _index class_ $\mathrm{Ind}[F]\in K_{0}(A)$ of $F$ to be

$\mathrm{Ind}[F]:=\begin{bmatrix}(1-VW)^{2}&V(1-WV)\\\
W(2-VW)(1-VW)&WV(2-WV)\end{bmatrix}-\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}.$

Combining Lemma 5.2$\sim$Lemma 5.5, we obtain that for each projection
$(p_{n})\in
C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$
the operator $((p_{n})F(p_{n}))$ is an odd self-adjoint operator on the graded
Hilbert space $\bigoplus_{n}p_{n}(L^{2}([1,\infty),\mathcal{H}_{r,n,E}))$
satisfying:

  * •

$((p_{n})F(p_{n}))$ is in the multiplier algebra of
$(p_{n})A^{r}(X;E)^{\Gamma}(p_{n})$;

  * •

$((p_{n})F(p_{n}))^{2}-(p_{n})$ is in $(p_{n})A^{r}(X;E)^{\Gamma}(p_{n})$.

Hence Definition 5.6 produces an index class in
$K_{0}((p_{n})A^{r}(X;E)^{\Gamma}(p_{n}))$. Composing with the $K_{0}$-map
induced by the inclusion $(p_{n})A^{r}(X;E)^{\Gamma}(p_{n})\hookrightarrow
A^{r}(X;E)^{\Gamma}$, we get an element in $K_{0}(A^{r}(X;E)^{\Gamma})$,
denoted by $\mathrm{Ind}_{F}[(p_{n})]$. An argument analogous to [34, Lemma
12.3.11] provides a well-defined homomorphism

$\mathrm{Ind}_{F}:K_{0}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\longrightarrow
K_{0}(A^{r}(X;E)^{\Gamma}).$

Passing to suspension and applying the above construction pointwise, we can
also define $\mathrm{Ind}_{F}$ on the level of $K_{1}$-groups. Similarly, we
can also deal with the localisation case by applying the above construction
pointwise in the parameter $t$. Consequently, we obtain the following:

######  Proposition 5.7.

The process above provides well-defined homomorphisms:

$\mathrm{Ind}_{F}:K_{\ast}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\rightarrow
K_{\ast}(A^{r}(X;E)^{\Gamma})$

and

$\mathrm{Ind}_{F_{L}}:K_{\ast}\big{(}C^{*}_{L}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}_{L}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\rightarrow
K_{\ast}(A^{r}_{L}(X;E)^{\Gamma})$

for $\ast=0,1$, which are called the _equivariant index maps_ associated to
$F$.

Finally, we prove Proposition 5.1.

###### Proof of Proposition 5.1.

The proof follows the outline of [34, Proposition 12.3.13 and Proposition
12.6.3]. We will only focus on the case of equivariant (twisted) Roe algebras,
while the localisation case follows from the same argument applied pointwise.
Throughout the proof, we fix an $s\in[1,\infty)$.

For each $n\in\mathbb{N}$, we define a map $\kappa_{n}:E_{n}\rightarrow E_{n}$
by

$\kappa_{n}(x)=\begin{cases}~{}\frac{x}{|x|}(|x|-1),&\mbox{if }|x|\geq 1;\\\
~{}0,&\mbox{otherwise}\end{cases}$

and a sequence of maps:

$F_{n}^{(k)}:\mathcal{H}_{r,n,E}\rightarrow\mathcal{H}_{r,n,E},\quad\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
u\mapsto\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
F_{n,s+2d_{n},\kappa_{n}^{k}(f_{r,n}(z))}u$

for $k\in\mathbb{N}\cup\\{\infty\\}$ and $z\in Z_{r,n}$, where
$\kappa_{n}^{\infty}(v):=0$ for all $v\in E_{n}$. Denote

$F^{(k)}:=(F_{n}^{(k)})_{n}\in\prod_{n}\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}\subseteq\mathfrak{B}(\mathcal{H}_{r,E})^{\Gamma}.$

We note that for the fixed $s$ at the beginning, we have that $F^{(0)}=F_{s}$
from Definition 4.13 and $F^{(\infty)}=(\mathrm{Id}_{\mathcal{H}_{r,n}}\otimes
F_{n,s+2d_{n},0})_{n}$. Quite analogous to the construction in Proposition
5.7, we obtain:

$\mathrm{Ind}_{F^{(k)}}:K_{*}\big{(}C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}\big{)}\rightarrow
K_{*}\big{(}C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\big{)}$

for each $k\in\mathbb{N}\cup\\{\infty\\}$. It is clear that
$\mathrm{Ind}_{F^{(0)}}=\iota^{s}_{*}\circ\mathrm{Ind}_{F}$. In the following,
we will show that $\mathrm{Ind}_{F^{(0)}}=\mathrm{Ind}_{F^{(\infty)}}$ and
that $\mathrm{Ind}_{F^{(\infty)}}$ is an isomorphism to conclude the proof. We
will only focus on the case of $K_{0}$, and the case of $K_{1}$ can be handled
using a standard suspension argument.

First we show that $\mathrm{Ind}_{F^{(0)}}=\mathrm{Ind}_{F^{(\infty)}}$. Let

$\mathcal{H}_{r,n,E,\infty}:=(\mathcal{H}_{r,n,E})^{\oplus\infty}=\mathcal{H}_{r,n}\otimes(\mathcal{L}^{2}_{E_{n}})^{\oplus\infty}~{}\mbox{for~{}all~{}}n\in\mathbb{N}\quad\mbox{and}\quad\mathcal{H}_{r,E,\infty}:=\bigoplus_{n}\mathcal{H}_{r,n,E,\infty}.$

Denote $C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}$ the corresponding
equivariant Roe algebra. It is known that the “top-left corner inclusion”:

$\iota:C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\rightarrow
C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E,\infty})^{\Gamma},\quad
T\mapsto\begin{pmatrix}T&0&\cdots\\\ 0&0&\\\ \vdots&&\ddots\end{pmatrix}$

induces an isomorphism on $K$-theory. To show that
$\mathrm{Ind}_{F^{(0)}}=\mathrm{Ind}_{F^{(\infty)}}$, it suffices to show that
$\iota_{\ast}\circ\mathrm{Ind}_{F^{(0)}}=\iota_{\ast}\circ\mathrm{Ind}_{F^{(\infty)}}$.

Without loss of generality, it suffices to show that
$\iota_{\ast}\circ\mathrm{Ind}_{F^{(0)}}[(p_{n})]=\iota_{\ast}\circ\mathrm{Ind}_{F^{(\infty)}}[(p_{n})]$
for any projection $(p_{n})\in
C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$.
For each $k\in\mathbb{N}\cup\\{\infty\\}$, it follows from Definition 5.6 that
the class $\mathrm{Ind}_{F^{(k)}}[(p_{n})]$ can be represented by a concrete
difference of projections, say

$[(p_{n}^{(k)})]-[(q_{n})],$

where $(q_{n})$ is independent of $k$ and we have

$(p_{n}^{(k)})-(q_{n})\in
M_{2}\left((p_{n})\cdot\big{(}C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\big{)}\cdot(p_{n})\right).$

Now we consider the projections

$\begin{pmatrix}(p_{n}^{(0)})&0&0&\cdots\\\ 0&(p_{n}^{(1)})&0&\\\
0&0&(p_{n}^{(2)})&\\\
\vdots&&&\ddots\end{pmatrix}\quad\mbox{and}\quad\begin{pmatrix}(p_{n}^{(\infty)})&0&0&\cdots\\\
0&(p_{n}^{(\infty)})&0&\\\ 0&0&(p_{n}^{(\infty)})&\\\
\vdots&&&\ddots\end{pmatrix}$

in the multiplier algebra of
$M_{2}\big{(}C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E,\infty})^{\Gamma}\big{)}$.
For any compact subset $K\subseteq P_{r}$ there exists $k\in\mathbb{N}$ such
that $\kappa^{k}(f_{r,n}(K\cap P_{r,n}))=0$ for any $n\in\mathbb{N}$ (in fact
there are only finitely many $n\in\mathbb{N}$ satisfying $K\cap
P_{r,n}\neq\emptyset$). Hence it is easy to see that the difference of these
projections is in
$M_{2}\big{(}C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E,\infty})^{\Gamma}\big{)}$.
Therefore the formal difference

$\begin{bmatrix}\begin{pmatrix}(p_{n}^{(0)})&0&0&\cdots\\\
0&(p_{n}^{(1)})&0&\\\ 0&0&(p_{n}^{(2)})&\\\
\vdots&&&\ddots\end{pmatrix}\end{bmatrix}-\begin{bmatrix}\begin{pmatrix}(p_{n}^{(\infty)})&0&0&\cdots\\\
0&(p_{n}^{(\infty)})&0&\\\ 0&0&(p_{n}^{(\infty)})&\\\
\vdots&&&\ddots\end{pmatrix}\end{bmatrix}$

defines a class in
$K_{0}\big{(}C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E,\infty})^{\Gamma}\big{)}$,
denoted by $a$.

On the other hand, there is a class $b\in
K_{0}\big{(}C^{*}(\mathcal{H}_{r,E,\infty})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E,\infty})^{\Gamma}\big{)}$
defined by the formal difference

$\begin{bmatrix}\begin{pmatrix}(p_{n}^{(0)})&0&0&\cdots\\\ 0&0&0&\\\ 0&0&0&\\\
\vdots&&&\ddots\end{pmatrix}\end{bmatrix}-\begin{bmatrix}\begin{pmatrix}(p_{n}^{(\infty)})&0&0&\cdots\\\
0&0&0&\\\ 0&0&0&\\\ \vdots&&&\ddots\end{pmatrix}\end{bmatrix}.$

We claim that $a+b=a$, whence $b=0$. This implies that
$\mathrm{Ind}_{F^{(0)}}[(p_{n})]=\mathrm{Ind}_{F^{(\infty)}}[(p_{n})]$. In
fact, for each $k\in\mathbb{N}$ we consider the path
$(F^{(k),r})_{r\in[0,1]}=\big{(}(F_{n}^{(k),r})\big{)}_{r\in[0,1]}$ in
$\mathfrak{B}(\mathcal{H}_{r,E})^{\Gamma}$ defined by

$F_{n}^{(k),r}:\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
u\mapsto\delta_{z}\otimes\xi\otimes\delta_{g}\otimes
F_{n,s+2d_{n},(1-r)\kappa_{n}^{k}(f_{r,n}(z))+r\kappa_{n}^{k+1}(f_{r,n}(z))}u\quad\mbox{on
each }\mathcal{H}_{r,n,E}.$

It follows from Proposition 4.4(5) that $\|F^{(k),r}-F^{(k),r^{\prime}}\|\leq
3|r-r^{\prime}|$ for each $n\in\mathbb{N}$. Hence $a$ is homotopic to

$\begin{bmatrix}\begin{pmatrix}(p_{n}^{(1)})&0&0&\cdots\\\
0&(p_{n}^{(2)})&0&\\\ 0&0&(p_{n}^{(3)})&\\\
\vdots&&&\ddots\end{pmatrix}\end{bmatrix}-\begin{bmatrix}\begin{pmatrix}(p_{n}^{(\infty)})&0&0&\cdots\\\
0&(p_{n}^{(\infty)})&0&\\\ 0&0&(p_{n}^{(\infty)})&\\\
\vdots&&&\ddots\end{pmatrix}\end{bmatrix}.$

which implies that $a+b=a$.

Finally we explain that $\mathrm{Ind}_{F^{(\infty)}}$ is an isomorphism. For
each $n\in\mathbb{N}$, note that
$F^{(\infty)}_{n}=\mathrm{Id}_{\mathcal{H}_{r,n}}\otimes F_{n,s+2d_{n},0}$.
Let $p_{n,0}$ be the projection onto the kernel of $F_{n,s+2d_{n},0}$, which
is one-dimensional. Recall that $F_{n,s+2d_{n},0}=g((s+2d_{n})^{-1}D+C)$ where
$g(x)=x(1+x^{2})^{-1/2}$. Now consider the path
$(F_{n,s+2d_{n},0}^{t})_{t\in[1,\infty]}$ defined by

$F_{n,s+2d_{n},0}^{t}:=g\big{(}t\big{(}\frac{D}{s+2d_{n}}+C\big{)}\big{)}.$

This defines a homotopy between $F_{n,s+2d_{n},0}$ and
$F_{n,s+2d_{n},0}^{\infty}$ (for each $n\in\mathbb{N}$), which decomposes with
respect to the grading as

$F_{n,s+2d_{n},0}^{\infty}=\left(\begin{matrix}0&1\\\
1-p_{n,0}&0\end{matrix}\right).$

Hence for each $n$, this homotopy provides that

$\mathrm{Ind}_{F_{n}^{(\infty)}}[p_{n}]=[p_{n}\otimes p_{n,0}],$

for any projection $(p_{n})\in
C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$.
Since the above homotopies are _not_ uniformly continuous with respect to $n$,
we need an extra argument (which is called a “stacking argument” in the proof
of [34, Proposition 12.6.3]) to conclude that

$\mathrm{Ind}_{F^{(\infty)}}[(p_{n})]=[(p_{n}\otimes p_{n,0})].$

For the convenience to readers, here we provide more details. Without loss of
generality, for any projection $(p_{n})\in
C^{*}(\mathcal{H}_{r})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n})^{\Gamma}$
we assume that

$\mathrm{Ind}_{F^{(\infty)}}[(p_{n})]=[(q_{n})]-[(q_{n}^{(0)})],$

where $(q_{n})$ and $(q_{n}^{(0)})$ are projections in
$C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}$.
For each $n\in\mathbb{N}$, the path $F_{n,s+2d_{n},0}^{t}$ above provides a
homotopy of projections $\\{H_{n,t}\\}_{t\in[0,1]}$ such that $H_{n,0}=q_{n}$.
Hence for each $n\in\mathbb{N}$, there exists $\delta_{n}>0$ such that for any
$t,s\in[0,1]$ with $|t-s|<\delta_{n}$, we have
$\|H_{n,t}-H_{n,s}\|<\frac{1}{2}$. For each $n\in\mathbb{N}$, take an
$K_{n}\in\mathbb{N}$ such that $\frac{1}{K_{n}}<\delta_{n}$ and an isometry

$V_{n}:\mathcal{H}_{r,n,E}\longrightarrow\mathcal{H}_{r,n,E}^{\oplus
K_{n}},\quad v\mapsto(v,0,\cdots,0).$

They give rise to an isometry

$V=(V_{n})_{n}:\mathcal{H}_{r,E}\longrightarrow\mathcal{H}^{\prime}_{r,E}:=\bigoplus_{n}\mathcal{H}_{r,n,E}^{\oplus
K_{n}}.$

Note that $V$ equivariantly covers the identity map on $P_{r}$, hence the
following homomorphism

$\Phi:=\mathrm{Ad}_{V}:C^{*}(\mathcal{H}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}C^{*}(\mathcal{H}_{r,n,E})^{\Gamma}\longrightarrow
C^{*}(\mathcal{H}^{\prime}_{r,E})^{\Gamma}\cap\prod_{n=1}^{\infty}\mathfrak{B}(\mathcal{H}_{r,n,E}^{\oplus
K_{n}})^{\Gamma}$

gives rise to an isomorphism on $K$-theories (see, _e.g._ , [34, Theorem
5.2.6] for more details). Therefore, it suffices to show that

$\Phi_{\ast}(\mathrm{Ind}_{F^{(\infty)}}[(p_{n})])=\Phi_{\ast}([(p_{n}\otimes
p_{n,0})]).$

Note that for each $n\in\mathbb{N}$, we have

$\begin{bmatrix}\begin{pmatrix}q_{n}&0&\cdots&0\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{pmatrix}\end{bmatrix}=\begin{bmatrix}\begin{pmatrix}H_{n,0}&0&\cdots&0\\\
0&H_{n,\frac{1}{K_{n}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&H_{n,1}\end{pmatrix}\end{bmatrix}-\begin{bmatrix}\begin{pmatrix}0&0&\cdots&0\\\
0&H_{n,\frac{1}{K_{n}}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&H_{n,1}\end{pmatrix}\end{bmatrix}.$

On the other hand, note that

$\left\|~{}\begin{pmatrix}0&0&\cdots&0\\\ 0&H_{n,\frac{1}{K_{n}}}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&H_{n,1}\end{pmatrix}-\begin{pmatrix}0&0&\cdots&0\\\
0&H_{n,0}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&H_{n,\frac{K_{n}-1}{K_{n}}}\end{pmatrix}~{}\right\|<\frac{1}{2}$

Hence for each $n\in\mathbb{N}$ there exists a homotopy connecting these two
matrices, which are _uniformly_ continuous with respect to $n$. Following by a
rotation homotopy between

$\begin{pmatrix}0&0&\cdots&0\\\ 0&H_{n,0}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&H_{n,\frac{K_{n}-1}{K_{n}}}\end{pmatrix}\quad\mbox{and}\quad\begin{pmatrix}H_{n,0}&0&\cdots&0&0\\\
0&H_{n,\frac{1}{K_{n}}}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&H_{n,\frac{K_{n}-1}{K_{n}}}&0\\\ 0&0&\cdots&0&0\end{pmatrix},$

we obtain that

$\begin{bmatrix}\begin{pmatrix}q_{n}&0&\cdots&0\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{pmatrix}\end{bmatrix}=\begin{bmatrix}\begin{pmatrix}0&\cdots&0&0\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\cdots&0&0\\\
0&\cdots&0&H_{n,1}\end{pmatrix}\end{bmatrix}=\begin{bmatrix}\begin{pmatrix}H_{n,1}&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{pmatrix}\end{bmatrix}.$

Note that these homotopies are _uniformly_ continuous with respect to $n$,
hence from the construction of $\Phi$ we obtain that

$\Phi_{\ast}(\mathrm{Ind}_{F^{(\infty)}}[(p_{n})])=\Phi_{\ast}([(p_{n}\otimes
p_{n,0})]),$

which concludes the proof. ∎

##  6\. Isomorphisms between $K$-theories of equivariant twisted algebras

In this section, we study the $K$-theories of equivariant twisted algebras
$A^{r}(X;E)^{\Gamma}$ and $A^{r}_{L}(X;E)^{\Gamma}$ introduced in Section 4.2.
First recall that for $r>0$, there is the evaluation-at-one homomorphism

$\mathrm{ev}:A^{r}_{L}(X;E)^{\Gamma}\longrightarrow
A^{r}(X;E)^{\Gamma},\quad(T_{t})\mapsto T_{1}.$

Also recall that for $r\leq s$, the inclusion $Z_{r}\to Z_{s}$ induces
isometric embeddings of Hilbert spaces

$\mathcal{H}_{r,n}\longrightarrow\mathcal{H}_{s,n}\quad\mbox{and}\quad\mathcal{H}_{r,n,E}\longrightarrow\mathcal{H}_{s,n,E}$

for each $n\in\mathbb{N}$. These maps give rise to the following commutative
diagram:

$\textstyle{A^{r}_{L}(X;E)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}}$$\textstyle{A^{r}(X;E)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{s}_{L}(X;E)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}}$$\textstyle{A^{s}(X;E)^{\Gamma}}$

for each $r\leq s$, which further induces a homomorphism between direct limits
when taking $r\to+\infty$.

The main result of this section is the following:

######  Theorem 6.1.

The evaluation-at-one map induces the following isomorphism in $K$-theories:

$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}(A^{r}_{L}(X;E)^{\Gamma})\longrightarrow\lim_{r\to\infty}K_{\ast}(A^{r}(X;E)^{\Gamma})$

for $\ast=0,1$.

The proof follows the outline of that in [34, Section 12.4], and the main
ingredient is to use appropriate Mayer-Vietoris arguments for twisted algebras
(Proposition 6.5). This allows us to chop the space into easily-handled
pieces, on which we apply the result from [16] on the Baum-Connes conjecture
with coefficients for a-T-menable groups.

Let us start with some more notions. By saying that $(F_{n})_{n\in\mathbb{N}}$
is a sequence of closed subsets in $(E_{n})_{n\in\mathbb{N}}$, we mean that
each $F_{n}$ is a closed subset of $E_{n}$. Firstly, we define the following
subalgebras associated to $(F_{n})$:

######  Definition 6.2.

Fix an $r>0$. For a sequence of closed subsets $(F_{n})$ in $(E_{n})$, we
define $\mathbb{A}^{r}_{(F_{n})}(X;E)^{\Gamma}$ to be the set of elements
$(T_{n,s})\in\mathbb{A}^{r}(X;E)^{\Gamma}$ satisfying: for each $n$ and
$\varepsilon>0$ there exists $s_{n,\varepsilon}>0$ such that for $s\geq
s_{n,\varepsilon}$ we have:

$\mathrm{supp}_{E}(T_{n,s})\subseteq\mathcal{N}_{\varepsilon}(F_{n})\times\mathcal{N}_{\varepsilon}(F_{n}).$

Denote by $A^{r}_{(F_{n})}(X;E)^{\Gamma}$ the $C^{*}$-algebra of the norm
closure of $\mathbb{A}^{r}_{(F_{n})}(X;E)^{\Gamma}$ in $A^{r}(X;E)^{\Gamma}$.

Denote by $\mathbb{A}^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ the set of elements
$(T_{t})$ in $\mathbb{A}^{r}_{L}(X;E)^{\Gamma}$ satisfying that
$T_{t}\in\mathbb{A}^{r}_{(F_{n})}(X;E)^{\Gamma}$ for each $t\in[1,\infty)$.
Define $A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ to be the $C^{*}$-algebra of the
completion of $\mathbb{A}^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ with respect to the
norm $\|(T_{t})\|=\sup_{t}\|T_{t}\|$. Equivalently, the algebra
$A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ consists of elements $(T_{t})$ in
$A^{r}_{L}(X;E)^{\Gamma}$ such that each $T_{t}$ belongs to
$A^{r}_{(F_{n})}(X;E)^{\Gamma}$.

The following lemma is straightforward, hence we omit the proof.

######  Lemma 6.3.

With the same notation as above, $A^{r}_{(F_{n})}(X;E)^{\Gamma}$ is an ideal
in $A^{r}(X;E)^{\Gamma}$ and $A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ is an ideal in
$A^{r}_{L}(X;E)^{\Gamma}$.

Moreover, we have the following.

######  Lemma 6.4.

Fix an $r>0$. Let $(F_{n})$ and $(G_{n})$ be two sequences of compact subsets
in $(E_{n})$. Then

$A^{r}_{(F_{n})}(X;E)^{\Gamma}\cap
A^{r}_{(G_{n})}(X;E)^{\Gamma}=A^{r}_{(F_{n}\cap G_{n})}(X;E)^{\Gamma}$

and

$A^{r}_{(F_{n})}(X;E)^{\Gamma}+A^{r}_{(G_{n})}(X;E)^{\Gamma}=A^{r}_{(F_{n}\cup
G_{n})}(X;E)^{\Gamma}.$

The same holds for the localisation case.

###### Proof.

We first prove the case of equivariant twisted Roe algebras. For the first
equation, note that $A^{r}_{(F_{n}\cap G_{n})}(X;E)^{\Gamma}\subseteq
A^{r}_{(F_{n})}(X;E)^{\Gamma}\cap A^{r}_{(G_{n})}(X;E)^{\Gamma}$ holds
trivial. The converse comes from a $C^{*}$-algebraic fact that the
intersection of two ideals coincides with their product together with a basic
fact for metric space: For a compact metric space $K$, a closed cover $(C,D)$
of $K$ and $\varepsilon>0$, there exists $\delta>0$ such that
$\mathcal{N}_{\delta}(C)\cap\mathcal{N}_{\delta}(D)\subseteq\mathcal{N}_{\varepsilon}(C\cap
D)$.

For the second equation, note that
$A^{r}_{(F_{n})}(X;E)^{\Gamma}+A^{r}_{(G_{n})}(X;E)^{\Gamma}\subseteq
A^{r}_{(F_{n}\cup G_{n})}(X;E)^{\Gamma}$ holds trivially. As for the converse,
we fix a $(T_{n,s})\in\mathbb{A}^{r}_{(F_{n}\cup G_{n})}(X;E)^{\Gamma}$. For
$n\in\mathbb{N}$ there is a strictly increasing sequence
$(s_{n,k})_{k\in\mathbb{N}}$ in $[1,\infty)$ tending to infinity such that for
$s\geq s_{n,k}$, we have

$\mathrm{supp}_{E}(T_{n,s})\subseteq\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup
G_{n})\times\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup G_{n}).$

For each $n\in\mathbb{N}$ and $s\in[1,\infty)$, we construct an operator
$W_{n,s}\in\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}$ as follows:

$W_{n,s}=\begin{cases}~{}\chi_{\mathcal{N}_{1}(F_{n})},&\mbox{if~{}}1\leq
s\leq s_{n,1};\\\\[8.5359pt]
~{}\frac{s_{n,k+1}-s}{s_{n,k+1}-s_{n,k}}\chi_{\mathcal{N}_{\frac{1}{k}}(F_{n})}+\frac{s-s_{n,k}}{s_{n,k+1}-s_{n,k}}\chi_{\mathcal{N}_{\frac{1}{k+1}}(F_{n})},&\mbox{for~{}}s_{n,k}\leq
s\leq s_{n,k+1}\mbox{~{}where~{}}k\in\mathbb{N}.\end{cases}$

Then the map $s\mapsto W_{n,s}$ is in
$C_{b}([1,\infty),\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma})$. Moreover, it
is clear that $(s\mapsto W_{n,s})_{n}$ is also in the multiplier algebra of
$A^{r}(X;E)^{\Gamma}$. Now we consider:

$\displaystyle(T_{n,s})$
$\displaystyle=(W_{n,s})(T_{n,s})+(1-W_{n,s})(T_{n,s})$
$\displaystyle=(W_{n,s})(T_{n,s})+(1-W_{n,s})(T_{n,s})(W_{n,s})+(1-W_{n,s})(T_{n,s})(1-W_{n,s}).$

It is clear that $(W_{n,s})(T_{n,s})$ and $(1-W_{n,s})(T_{n,s})(W_{n,s})$ are
in $A^{r}_{(F_{n})}(X;E)^{\Gamma}$. Also note that from the construction
above, for each $n\in\mathbb{N}$ and $s\geq s_{n,k}$ we have:

$\mathrm{supp}_{E}((1-W_{n,s})T_{n,s}(1-W_{n,s}))\subseteq\mathcal{N}_{\frac{1}{k+1}}(G_{n})\times\mathcal{N}_{\frac{1}{k+1}}(G_{n}).$

Hence we obtain that
$A^{r}_{(F_{n})}(X;E)^{\Gamma}+A^{r}_{(G_{n})}(X;E)^{\Gamma}$ is dense in
$A^{r}_{(F_{n}\cup G_{n})}(X;E)^{\Gamma}$, which concludes the proof.

For the localisation case, we apply the above argument pointwise to obtain the
first equation. Concerning the second, note that the above $(W_{n,s})$ might
not vary continuously with respect to the parameter $t$. To get around the
issue, we need an approximation argument.

Fix a $(T_{t})=(T_{t,n,s})\in\mathbb{A}^{r}_{L,(F_{n}\cup
G_{n})}(X;E)^{\Gamma}$. Note that the map $t\mapsto T_{t}$ is uniformly
continuous, hence for any $\varepsilon>0$ there is a $\delta>0$ such that for
any $t^{\prime},t^{\prime\prime}\in[1,\infty)$ with
$|t^{\prime}-t^{\prime\prime}|\leq\delta$, we have
$\|T_{t^{\prime}}-T_{t^{\prime\prime}}\|\leq\varepsilon$. For each
$p\in\mathbb{N}$ we set $t_{p}=1+p\delta$, and also set $t_{-1}=1$. We
construct another element
$(T^{\varepsilon}_{t})=(T^{\varepsilon}_{t,n,s})\in\mathbb{A}^{r}_{L,(F_{n}\cup
G_{n})}(X;E)^{\Gamma}$ by setting $T^{\varepsilon}_{t_{p}}=T_{t_{p}}$ for each
$p\in\mathbb{N}$ and do linear combination between $t_{p}$ and $t_{p+1}$. It
is clear that $\|(T_{t})-(T^{\varepsilon}_{t})\|\leq 2\varepsilon$. Since
$\varepsilon$ is arbitrarily chosen, it suffices to show that each
$(T^{\varepsilon}_{t})$ is in
$A^{r}_{L,(F_{n})}(X;E)^{\Gamma}+A^{r}_{L,(G_{n})}(X;E)^{\Gamma}$.

For each $n\in\mathbb{N}$ and $p\in\mathbb{N}\cup\\{0\\}$, there is a strictly
increasing sequence $(s_{n,k,p})_{k\in\mathbb{N}}$ in $[1,\infty)$ tending to
infinity such that for $s\geq s_{n,k,p}$ and $t=t_{p-1},t_{p},t_{p+1}$ we have

$\mathrm{supp}_{E}(T_{t,n,s})\subseteq\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup
G_{n})\times\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup G_{n}).$

For each $n\in\mathbb{N}$, $p\in\mathbb{N}\cup\\{0\\}$ and $s\in[1,\infty)$,
we construct an operator
$W_{t_{p},n,s}\in\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma}$ as follows:

$W_{t_{p},n,s}=\begin{cases}~{}\chi_{\mathcal{N}_{1}(F_{n})},&\mbox{if~{}}1\leq
s\leq s_{n,1,p};\\\\[8.5359pt]
~{}\frac{s_{n,k+1}-s}{s_{n,k+1}-s_{n,k}}\chi_{\mathcal{N}_{\frac{1}{k}}(F_{n})}+\frac{s-s_{n,k}}{s_{n,k+1}-s_{n,k}}\chi_{\mathcal{N}_{\frac{1}{k+1}}(F_{n})},&\mbox{for~{}}s_{n,k,p}\leq
s\leq s_{n,k+1,p}\mbox{~{}where~{}}k\in\mathbb{N}.\end{cases}$

Then the map $s\mapsto W_{t_{p},n,s}$ is in
$C_{b}([1,\infty),\mathfrak{B}(\mathcal{H}_{r,n,E})^{\Gamma})$. Moreover, it
is clear that for each $p\in\mathbb{N}\cup\\{0\\}$, $(s\mapsto
W_{t_{p},n,s})_{n}$ is also in the multiplier algebra of
$A^{r}(X;E)^{\Gamma}$. Now we define a path $(W_{t})_{t\in[1,\infty)}$ by
setting $W_{t_{p}}=(W_{t_{p},n,s})$ for each $p\in\mathbb{N}\cup\\{0\\}$, and
do linear combination between $t_{p}$ and $t_{p+1}$. It is clear that
$t\mapsto W_{t}=(W_{t,n,s})$ is uniformly continuous and bounded, and hence it
is in the multiplier algebra of $A^{r}_{L}(X;E)^{\Gamma}$. Now consider:

$\displaystyle(T^{\varepsilon}_{t,n,s})$
$\displaystyle=(W_{t,n,s})(T^{\varepsilon}_{t,n,s})+(1-W_{t,n,s})(T^{\varepsilon}_{t,n,s})$
$\displaystyle=(W_{t,n,s})(T^{\varepsilon}_{t,n,s})+(1-W_{t,n,s})(T^{\varepsilon}_{t,n,s})(W_{t,n,s})+(1-W_{t,n,s})(T^{\varepsilon}_{t,n,s})(1-W_{t,n,s}).$

It is clear that $t\mapsto W_{t}\cdot T^{\varepsilon}_{t}$ and
$t\mapsto(1-W_{t})\cdot T^{\varepsilon}_{t}\cdot W_{t}$ are in
$A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$. Note that for any
$p\in\mathbb{N}\cup\\{0\\}$ and $s\leq s_{n,k+1,p}$, we have

$\mathrm{supp}_{E}(1-W_{t_{p},n,s})\subseteq(\mathcal{N}_{\frac{1}{k+1}}(F_{n})^{c})\times(\mathcal{N}_{\frac{1}{k+1}}(F_{n})^{c}).$

Setting $s^{\prime}_{n,k,p}$ to be the minimal of $s_{n,k,p}$ and
$s_{n,k,p+1}$, it follows that for any $t\in[t_{p},t_{p+1}]$ and
$s\in[s^{\prime}_{n,k,p},s^{\prime}_{n,k+1,p}]$, we have

$\mathrm{supp}_{E}(1-W_{t,n,s})=\mathrm{supp}_{E}\big{(}1-(\lambda
W_{t_{p},n,s}+(1-\lambda)W_{t_{p+1},n,s})\big{)}\subseteq(\mathcal{N}_{\frac{1}{k+1}}(F_{n})^{c})\times(\mathcal{N}_{\frac{1}{k+1}}(F_{n})^{c})$

and

$\mathrm{supp}_{E}(T^{\varepsilon}_{t,n,s})=\mathrm{supp}_{E}\big{(}\lambda
T_{t_{p},n,s}+(1-\lambda)T_{t_{p+1},n,s}\big{)}\subseteq\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup
G_{n})\times\mathcal{N}_{\frac{1}{k+1}}(F_{n}\cup G_{n}),$

where $\lambda$ is the combinatorial parameter determined by
$t\in[t_{p},t_{p+1}]$. These inequalities imply that

$\mathrm{supp}_{E}\big{(}(1-W_{t,n,s})T^{\varepsilon}_{t,n,s}(1-W_{t,n,s})\big{)}\subseteq\mathcal{N}_{\frac{1}{k+1}}(G_{n})\times\mathcal{N}_{\frac{1}{k+1}}(G_{n}).$

Finally note that the sequence $(s^{\prime}_{n,k,p})_{k\in\mathbb{N}}$ tends
to infinity, hence we obtain that $(T^{\varepsilon}_{t})\in
A^{r}_{L,(F_{n})}(X;E)^{\Gamma}+A^{r}_{L,(G_{n})}(X;E)^{\Gamma}$ as required.
∎

Consequently, we obtain the following Mayer-Vietoris sequences for twisted
algebras:

######  Proposition 6.5.

Let $(F_{n})$ and $(G_{n})$ be two sequences of compact subsets in $(E_{n})$
and fix an $r>0$. Then we have the following six-term exact sequence:

$\textstyle{K_{0}\big{(}A_{(F_{n}\cap
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}\big{(}A_{(F_{n})}^{r}(X;E)^{\Gamma}\big{)}\oplus
K_{0}\big{(}A_{(G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}\big{(}A_{(F_{n}\cup
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}\big{(}A_{(F_{n}\cup
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}\big{(}A_{(F_{n})}^{r}(X;E)^{\Gamma}\big{)}\oplus
K_{1}\big{(}A_{(G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}\big{(}A_{(F_{n}\cap
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The same holds in the case of equivariant twisted localisation algebras.
Furthermore, we have the following commutative diagram:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}\big{(}A_{L,(F_{n}\cap
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}A_{L,(F_{n})}^{r}(X;E)^{\Gamma}\big{)}\oplus
K_{1}\big{(}A_{L,(G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}A_{L,(F_{n}\cup
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}\big{(}A_{(F_{n}\cap
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}A_{(F_{n})}^{r}(X;E)^{\Gamma}\big{)}\oplus
K_{\ast}\big{(}A_{(G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}\big{(}A_{(F_{n}\cup
G_{n})}^{r}(X;E)^{\Gamma}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

where the vertical maps are induced by the evaluation-at-one homomorphisms.

Proposition 6.5 allows us to chop the space into small pieces, on which we
have the following “local isomorphism” result. Recall that a family
$\\{Z_{i}\\}_{i\in I}$ of subspaces in a metric space $Z$ is _mutually
$R$-separated_ for some $R>0$ if $d(Z_{i},Z_{j})>R$ for $i\neq j$.

######  Proposition 6.6.

Let $(F_{n})$ be a sequence of closed subsets in $(E_{n})$ such that
$F_{n}=\bigsqcup_{j=1}^{\infty}F_{j}^{(n)}$ for a mutually $3$-separated
family $\\{F_{j}^{(n)}\\}_{j}$, and there exist $R>0$ and $y^{(n)}_{j}\in
X_{n}/\Gamma$ such that $F^{(n)}_{j}\subseteq B(\xi(y^{(n)}_{j});R)$. Then the
homomorphism induced by the evaluation-at-one map

$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}\big{(}A_{L,(F_{n})}^{r}(X;E)^{\Gamma}\big{)}\longrightarrow\lim_{r\to\infty}K_{\ast}\big{(}A_{(F_{n})}^{r}(X;E)^{\Gamma}\big{)}$

is an isomorphism for $\ast=0,1$.

The proof of Proposition 6.6 is technical and divided into several steps.
Before we present the detailed proof, let us first use it to conclude the
proof of Theorem 6.1. To achieve that, we need an extra lemma from [34, Lemma
12.4.5]:

######  Lemma 6.7.

For any $s>0$, there exist $M\in\mathbb{N}$ and decompositions

$X_{n}/\Gamma=Y_{n,1}\sqcup Y_{n,2}\sqcup\cdots\sqcup Y_{n,M}\quad\mbox{for
all }n\in\mathbb{N},$

such that the family $\big{\\{}\overline{B(\xi(y);s)}\big{\\}}_{y\in Y_{n,i}}$
is mutually $3$-separated for each $n\in\mathbb{N}$ and $i=1,2,\ldots,M$.

###### Proof of Theorem 6.1.

Given $s>0$, let $M\in\mathbb{N}$ and $\\{Y_{n,i}\\}_{n\in\mathbb{N},1\leq
i\leq M}$ be provided by Lemma 6.7. Setting
$W^{s}_{n}:=\mathcal{N}_{s}(\xi(Y_{n}))=\mathcal{N}_{s}(f_{n}(X_{n}))$ and
$W_{n,i}^{s}:=\bigsqcup_{y\in Y_{n,i}}\overline{B(\xi(y);s)}$, we have
$W^{s}_{n}=\bigcup_{i=1}^{M}W_{n,i}^{s}$. For each $i$ applying Proposition
6.6 to the sequence of subsets $(W_{n,i}^{s})_{n}$, we obtain that the
homomorphism induced by the evaluation-at-one map

$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}\big{(}A_{L,(W_{n,i}^{s})}^{r}(X;E)^{\Gamma}\big{)}\longrightarrow\lim_{r\to\infty}K_{\ast}\big{(}A_{(W_{n,i}^{s})}^{r}(X;E)^{\Gamma}\big{)}$

is an isomorphism for each $i$. Applying the Mayer-Vietoris sequence from
Proposition 6.5 $(M-1)$-times (and Proposition 6.6 again to deal with the
intersection) together with the Five Lemma, we obtain that the homomorphism
induced by the evaluation-at-one map

$\mathrm{ev}_{\ast}:\lim_{r\to\infty}K_{\ast}\big{(}A_{L,(W^{s}_{n})}^{r}(X;E)^{\Gamma}\big{)}\longrightarrow\lim_{r\to\infty}K_{\ast}\big{(}A_{(W^{s}_{n})}^{r}(X;E)^{\Gamma}\big{)}$

is an isomorphism. Finally note that condition $(3)$ in Definition 4.9 and
condition $(2)$ in Definition 4.10 imply that for each $r>0$, we have

$A^{r}(X;E)^{\Gamma}=\lim_{s\rightarrow\infty}A^{r}_{(W^{s}_{n})}(X;E)^{\Gamma}\quad\mbox{and}\quad
A^{r}_{L}(X;E)^{\Gamma}=\lim_{s\rightarrow\infty}A^{r}_{L,(W^{s}_{n})}(X;E)^{\Gamma}.$

Hence we conclude the proof. ∎

The rest of this section is devote to the proof of Proposition 6.6. First let
us introduce some more notation. Fix an $r>0$. Let $(F_{n})$ and $(G_{n})$ be
sequences of closed subsets in $(E_{n})$. Define:

$A^{r}(X;(G_{n}))^{\Gamma}:=(\mathrm{Id}_{L^{2}([1,\infty))}\otimes\mathrm{Id}_{\mathcal{H}_{r,n}}\otimes\chi_{G_{n}})_{n}\cdot
A^{r}(X;E)^{\Gamma}\cdot(\mathrm{Id}_{L^{2}([1,\infty))}\otimes\mathrm{Id}_{\mathcal{H}_{r,n}}\otimes\chi_{G_{n}})_{n}$

and

$A^{r}_{(F_{n})}(X;(G_{n}))^{\Gamma}:=A^{r}(X;(G_{n}))^{\Gamma}\cap
A^{r}_{(F_{n})}(X;E)^{\Gamma}.$

Moreover, given a sequence of $\Gamma$-invariant subspaces $O_{n}\subseteq
X_{n}~{}(n\in\mathbb{N})$ we define:

$A^{r}((O_{n});(G_{n}))^{\Gamma}:=\big{(}\mathrm{Id}_{L^{2}([1,\infty))}\otimes\chi_{P_{r}(O_{n})}\otimes\mathrm{Id}_{\mathcal{L}^{2}_{E_{n}}}\big{)}_{n}\cdot
A^{r}(X;(G_{n}))^{\Gamma}\cdot\big{(}\mathrm{Id}_{L^{2}([1,\infty))}\otimes\chi_{P_{r}(O_{n})}\otimes\mathrm{Id}_{\mathcal{L}^{2}_{E_{n}}}\big{)}_{n}$

and

$A^{r}_{(F_{n})}((O_{n});(G_{n}))^{\Gamma}:=\big{(}\mathrm{Id}_{L^{2}([1,\infty))}\otimes\chi_{P_{r}(O_{n})}\otimes\mathrm{Id}_{\mathcal{L}^{2}_{E_{n}}}\big{)}_{n}\cdot
A^{r}_{(F_{n})}(X;(G_{n}))^{\Gamma}\cdot\big{(}\mathrm{Id}_{L^{2}([1,\infty))}\otimes\chi_{P_{r}(O_{n})}\otimes\mathrm{Id}_{\mathcal{L}^{2}_{E_{n}}}\big{)}_{n}.$

Define the $C^{*}$-algebras $A^{r}_{L}(X;(G_{n}))^{\Gamma}$ and
$A^{r}_{L}((O_{n});(G_{n}))^{\Gamma}$ to be those consisting of elements
$(T_{t})\in A^{r}_{L}(X;E)^{\Gamma}$ such that each $T_{t}$ belongs to
$A^{r}(X;(G_{n}))^{\Gamma}$ and $A^{r}((O_{n});(G_{n}))^{\Gamma}$,
respectively. Also define the $C^{*}$-algebras
$A^{r}_{L,(F_{n})}(X;(G_{n}))^{\Gamma}$ and
$A^{r}_{L,(F_{n})}((O_{n});(G_{n}))^{\Gamma}$ to be those consisting of
elements $(T_{t})\in A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$ such that each $T_{t}$
belongs to $A^{r}_{(F_{n})}(X;(G_{n}))^{\Gamma}$ and
$A^{r}_{(F_{n})}((O_{n});(G_{n}))^{\Gamma}$, respectively.

On the other hand, we define

$A^{r}_{0}(X;E)^{\Gamma}:=\big{\\{}(T_{n,s})\in
A^{r}(X;E)^{\Gamma}:\lim\limits_{s\rightarrow\infty}T_{n,s}=0\mbox{~{}for~{}each~{}}n\in\mathbb{N}\big{\\}}$

and

$A^{r}_{0}(X;(G_{n}))^{\Gamma}:=A^{r}_{0}(X;E)^{\Gamma}\cap
A^{r}(X;(G_{n}))^{\Gamma}$

It is clear that $A^{r}_{0}(X;(G_{n}))^{\Gamma}$ is an ideal in
$A^{r}(X;(G_{n}))^{\Gamma}$, and also an ideal in
$A^{r}_{(F_{n})}(X;(G_{n}))^{\Gamma}$ for any sequence of closed subsets
$(F_{n})$. Define the algebras $A^{r}_{L,0}(X;E)^{\Gamma}$ and
$A^{r}_{L,0}(X;(G_{n}))^{\Gamma}$ to be those consisting of elements
$(T_{t})\in A^{r}_{L}(X;E)^{\Gamma}$ such that each $T_{t}$ belongs to
$A^{r}_{0}(X;E)^{\Gamma}$ and $A^{r}_{0}(X;(G_{n}))^{\Gamma}$, respectively.

######  Lemma 6.8.

Fix an $r>0$. Then we have:

$K_{\ast}(A^{r}_{0}(X;E)^{\Gamma})=0\quad\mbox{and}\quad
K_{\ast}(A^{r}_{0}(X;(G_{n}))^{\Gamma})=0.$

And we also have:

$K_{\ast}(A^{r}_{L,0}(X;E)^{\Gamma})=0\quad\mbox{and}\quad
K_{\ast}(A^{r}_{L,0}(X;(G_{n}))^{\Gamma})=0.$

###### Proof.

Here we only prove $K_{\ast}(A^{r}_{0}(X;E)^{\Gamma})=0$ since the other is
similar, and the case of localisation algebras holds by using the same
argument pointwise. Let

$\mathcal{H}_{r,n,E,\infty}:=(\mathcal{H}_{r,n,E})^{\oplus\infty}\quad\mbox{and}\quad\mathcal{H}_{r,E,\infty}:=\bigoplus_{n}\mathcal{H}_{r,n,E,\infty}.$

Using these admissible modules we construct the equivariant twisted Roe
algebra, denoted by $A^{r}_{\infty}(X;E)^{\Gamma}$, and similarly construct
$A^{r}_{0,\infty}(X;E)^{\Gamma}$. For each $n\in\mathbb{N}$, define an
isometry

$V_{n}:\mathcal{H}_{r,n,E}\rightarrow\mathcal{H}_{r,n,E,\infty},\quad
v\mapsto(v,0,0,\cdots).$

Then $(V_{n})$ induces the “top-left corner inclusion”

$\mathrm{Ad}_{(V_{n})}:A^{r}_{0}(X;E)^{\Gamma}\rightarrow
A^{r}_{0,\infty}(X;E)^{\Gamma},\quad(T_{n,s})\mapsto\big{(}T_{n,s}\oplus\bigoplus_{k=1}^{\infty}0\big{)},$

which induces the identity map on the $K$-groups.

It is also easy to check that the formula

$(\alpha_{n}):(T_{n,s})_{n,s}\rightarrow\big{(}0\oplus\bigoplus_{k=1}^{\infty}T_{n,s+k}\big{)}_{n,s}\mathrm{\
where\
}\alpha_{n}:(T_{n,s})_{s}\rightarrow\big{(}0\oplus\bigoplus_{k=1}^{\infty}T_{n,s+k}\big{)}_{s}$

and

$(\beta_{n}):(T_{n,s})_{n,s}\rightarrow\big{(}\bigoplus_{k=0}^{\infty}T_{n,s+k+1}\big{)}_{n,s}\mathrm{\
where\
}\beta_{n}:(T_{n,s})_{s}\rightarrow\big{(}\bigoplus_{k=0}^{\infty}T_{n,s+k+1}\big{)}_{s}$

give two well-defined homomorphisms from $A^{r}_{0}(X;E)^{\Gamma}$ to
$A^{r}_{0,\infty}(X;E)^{\Gamma}$. Let

$S_{n}:\mathcal{H}_{r,n,E,\infty}\rightarrow\mathcal{H}_{r,n,E,\infty},\quad(v_{0},v_{1},v_{2},\dots)\mapsto(0,v_{0},v_{1},\dots)$

be the shift operator. It is clear that $(S_{n})_{n}$ is in the multiplier
algebra of $A^{r}_{0,\infty}(X;E)^{\Gamma}$, which builds a conjugation
between $(\alpha_{n})$ and $(\beta_{n})$. More precisely, for each
$n\in\mathbb{N}$ we have

$\alpha_{n}\big{(}(T_{n,s})_{s}\big{)}=S_{n}\cdot\beta_{n}\big{(}(T_{n,s})_{s}\big{)}\cdot
S_{n}^{*}.$

Applying [34, Proposition 2.7.5], we have

$(\alpha_{n})_{*}=(\beta_{n})_{*}.$

On the other hand, notice that

$\big{(}\mathrm{Ad}_{(V_{n})}+(\alpha_{n})\big{)}\big{(}(T_{n,s})_{n,s}\big{)}=\big{(}\bigoplus_{k=0}^{\infty}T_{n,s+k}\big{)}_{n,s}.$

Since $\mathrm{Ad}_{(V_{n})}$ and $(\alpha_{n})$ have orthogonal images and
$\big{(}\mathrm{Ad}_{(V_{n})}+(\alpha_{n})\big{)}\big{(}(T_{n,s})_{n,s}\big{)}$
is homotopic to $(\beta_{n})\big{(}(T_{n,s})_{n,s}\big{)}$, we have

$(\beta_{n})_{*}=(\mathrm{Ad}_{(V_{n})}+(\alpha_{n}))_{*}=(\mathrm{Ad}_{(V_{n})})_{*}+(\alpha_{n})_{*}.$

Consequently we obtain that $(\mathrm{Ad}_{(V_{n})})_{*}=0$, which concludes
the proof. ∎

Now we move back to Proposition 6.6. Fix an $r>0$. Recall that
$F_{n}=\bigsqcup_{j=1}^{\infty}F_{j}^{(n)}$ where $\\{F_{j}^{(n)}\\}_{j}$ is a
mutually $3$-separated family, and there exist $R>0$ and $y^{(n)}_{j}\in
X_{n}/\Gamma$ such that $F^{(n)}_{j}\subseteq B(\xi(y^{(n)}_{j});R)$. Also
recall that the $\Gamma$-action on $\bigsqcup_{n}X_{n}$ has controlled
distortion with certain fundamental domain $\mathcal{D}$. For each $n$ and
$j$, denote $x^{(n)}_{j}$ the unique point in $X_{n}\cap\mathcal{D}$ such that
$\pi(x^{(n)}_{j})=y^{(n)}_{j}$. Since each $X_{n}/\Gamma$ is finite, then for
each $n\in\mathbb{N}$ there are only finitely many $j$’s such that
$F_{j}^{(n)}$ is non-empty. Denote the set of such $j$ by $J_{n}$. Taking
$G_{j}^{(n)}=\mathcal{N}_{1}(F_{j}^{(n)})$ for each $n$ and $j$, we define the
“restricted product”:

$\displaystyle\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}:$
$\displaystyle=\left(\prod_{j}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\right)\cap
A^{r}_{(F_{n})}(X;E)^{\Gamma}$ $\displaystyle=\left\\{(T_{n,s})\in
A^{r}_{(F_{n})}(X;E)^{\Gamma}:\mathrm{supp}_{E}(T_{n,s})\subseteq\bigsqcup_{j\in
J_{n}}G_{j}^{(n)}\times G_{j}^{(n)}\right\\}.$

Also define the algebra
$\prod_{j}^{res}A^{r}_{L,(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}$ to be those
consisting of elements $(T_{t})\in A^{r}_{L}(X;E)^{\Gamma}$ such that each
$T_{t}$ belongs to
$\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}$. We have the
following lemma, whose proof is again from a standard Eilenberg Swindle
argument (as in Lemma 6.8), hence omitted.

######  Lemma 6.9.

With the same notations as above, the algebras

$\prod_{j}^{res}A^{r}_{0}(X;(G_{j}^{(n)}))^{\Gamma}:=\big{(}\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\big{)}\cap
A^{r}_{0}(X;E)^{\Gamma}$

and

$\prod_{j}^{res}A^{r}_{L,0}(X;(G_{j}^{(n)}))^{\Gamma}:=\big{(}\prod_{j}^{res}A^{r}_{L,(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\big{)}\cap
A^{r}_{L,0}(X;E)^{\Gamma}$

have trivial $K$-theories.

The following lemma is a key step in the proof of Proposition 6.6:

######  Lemma 6.10.

With the same notation as above, the following inclusions

$\iota:\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\hookrightarrow
A^{r}_{(F_{n})}(X;E)^{\Gamma}$

and

$\iota_{L}:\prod_{j}^{res}A^{r}_{L,(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\hookrightarrow
A^{r}_{L,(F_{n})}(X;E)^{\Gamma}$

induce isomorphisms in $K$-theory.

###### Proof.

We only prove the first, and the second can be proved using the same argument
pointwise. The proof follows the outline of [34, Theorem 6.4.20].

Consider the following quotient algebras:

$A^{r}_{(F_{n}),Q}(X;E)^{\Gamma}:=\frac{A^{r}_{(F_{n})}(X;E)^{\Gamma}}{A^{r}_{0}(X;E)^{\Gamma}}\quad\mbox{and}\quad\prod_{j}^{res,Q}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}:=\frac{\prod\limits_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}}{\prod\limits_{j}^{res}A^{r}_{0}(X;(G_{j}^{(n)}))^{\Gamma}}.$

It follows from Lemma 6.8 and 6.9 that the quotient maps

$A^{r}_{(F_{n})}(X;E)^{\Gamma}\rightarrow
A^{r}_{(F_{n}),Q}(X;E)^{\Gamma}\quad\mbox{and}\quad\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\rightarrow\prod_{j}^{res,Q}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}$

induce isomophisms in $K$-theories.

It is clear that the inclusion $\iota$ induces a $*$-homomorphism:

$\iota_{Q}:\prod_{j}^{res,Q}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\longrightarrow
A^{r}_{(F_{n}),Q}(X;E)^{\Gamma}.$

We also define a map

$\gamma:A^{r}_{(F_{n})}(X;E)^{\Gamma}\longrightarrow\prod_{j}^{res}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}\quad\mbox{by}\quad(T_{n,s})\mapsto\prod_{j}(\chi_{G_{j}^{(n)}}T_{n,s}\chi_{G_{j}^{(n)}}),$

which induces a $*$-homomorphism

$\gamma_{Q}:A^{r}_{(F_{n}),Q}(X;E)^{\Gamma}\longrightarrow\prod_{j}^{res,Q}A^{r}_{(F_{j}^{(n)})}(X;(G_{j}^{(n)}))^{\Gamma}.$

We claim that the composition $\iota_{Q}\circ\gamma_{Q}$ equals to the
identity map on the quotient algebra, _i.e._ , the map $\iota\circ\gamma$
equals to the identity map modulo $A^{r}_{0}(X;E)^{\Gamma}$. Given
$(T_{n,s})\in A^{r}_{(F_{n})}(X;E)^{\Gamma}$, then for each $n\in\mathbb{N}$
there exists $s_{n}\in[1,\infty)$ such that for any $s>s_{n}$ we have
$\mathrm{supp}_{E}(T_{n,s})\subseteq\bigsqcup_{j}\big{(}G_{j}^{(n)}\times
G_{j}^{(n)}\big{)}$. Hence for $s>s_{n}$, we obtain:

$\sum_{j}\chi_{G_{j}^{(n)}}T_{n,s}\chi_{G_{j}^{(n)}}-T_{n,s}=0,$

which implies that

$(T_{n,s})-\sum_{j}(\chi_{G_{j}^{(n)}}T_{n,s}\chi_{G_{j}^{(n)}})\in
A^{r}_{0}(X;E)^{\Gamma}.$

On the other hand, it is clear that $\gamma\circ\iota$ is the identity map and
descends to identity map $\gamma_{Q}\circ\iota_{Q}$. Hence $\iota_{Q}$ induces
an isomorphism in $K$-theory, which implies that the inclusion $i$ induces an