# A Bott periodicity theorem for $\ell^{p}$-spaces and the coarse Novikov
conjecture at infinity111Supported in part by NSFC (No. 11831006, 12171156).

Liang Guo  Zheng Luo  Qin Wang  Yazhou Zhang

###### Abstract

We formulate and prove a Bott periodicity theorem for an $\ell^{p}$-space
($1\leq p<\infty$). For a proper metric space $X$ with bounded geometry, we
introduce a version of $K$-homology at infinity, denoted by
$K_{*}^{\infty}(X)$, and the Roe algebra at infinity, denoted by
$C^{*}_{\infty}(X)$. Then the coarse assembly map descends to a map from
$\lim_{d\to\infty}K_{*}^{\infty}(P_{d}(X))$ to $K_{*}(C^{*}_{\infty}(X))$,
called the coarse assembly map at infinity. We show that to prove the coarse
Novikov conjecture, it suffices to prove the coarse assembly map at infinity
is an injection. As a result, we show that the coarse Novikov conjecture holds
for any metric space with bounded geometry which admits a fibred coarse
embedding into an $\ell^{p}$-space. These include all box spaces of a
residually finite hyperbolic group and a large class of warped cones of a
compact space with an action by a hyperbolic group.

###### Contents

  1. 1 Introduction
  2. 2 The coarse Novikov conjecture
    1. 2.1 Warped Cones and proper affine isometric group actions
  3. 3 A Bott periodicity theorem for $\ell^{p}$ spaces
    1. 3.1 A Clifford algebras for a Hilbert space
    2. 3.2 A $C^{*}$-algebra associated with a real $\ell^{p}$ space
    3. 3.3 The $K$-theory of $\mathcal{A}(B)$
  4. 4 $K$-homology at infinity and Roe algebra at infinity
  5. 5 Reduction to the coarse Novikov conjecture at infinity
    1. 5.1 The case of coarse disjoint unions
    2. 5.2 The case of spaces which admit an infinite coarse component
  6. 6 Twisted algebras at infinity and their $K$-theories
    1. 6.1 The twisted algebras at infinity
    2. 6.2 The twisted coarse Baum-Connes conjecture at infinity
  7. 7 The geometric Bott map and proof of the main theorem

##  1 Introduction

The Novikov conjecture claims that the higher signatures of closed oriented
smooth manifolds are invariant under orientation preserving homotopy
equivalences. It is implied by the injectivity of the Baum-Connes assembly map
[2]. The coarse Novikov conjecture, as a geometric analogue of the Novikov
conjecture, is the injectivity part of the coarse Baum–Connes conjecture [12,
18]. It offers a technique to determine when an elliptic operators higher
index on a noncompact complete Riemannian manifold is nonzero. A detailed
survey on the coarse Novikov conjecture can be found in [31].

In [4], X. Chen, Q. Wang and G. Yu generalized Gromov’s notion of coarse
embedding into Hilbert space and introduced the notion of fibred coarse
embedding (see Definition 2.4) and showed that if a discrete metric space $X$
with bounded geometry admits a fibred coarse embedding into Hilbert space,
then the maximal coarse Baum–Connes conjecture holds for $X$. Later in [8], M.
Finn-Sell showed that the coarse Novikov conjecture also holds for $X$ in such
a case.

Our main theorem in this paper states as follows:

######  Theorem 1.1.

_[Theorem 2.5]_ Let $X$ be a metric space with bounded geometry. If $X$ admits
a fibred coarse embedding into an $\ell^{p}$-space for some $p\in[1,\infty)$,
then the coarse Novikov conjecture holds for $X$, i.e. the coarse assembly map

$\mu:\lim_{d\to\infty}K_{*}(P_{d}(X))\to K_{*}(C^{*}(X))$

is injective.

Warped cones over actions of hyperbolic groups and box spaces of residually
finite hyperbolic groups are typical examples to which our main result appies.
A sequence of nested normal subgroups $\\{\Gamma_{i}\\}$ of a residually
finite group $\Gamma$ is said to be a filtration if the sequence has trivial
intersection. The box space of $\Gamma$ associated to a filtration
$\\{\Gamma_{i}\\}$, denoted by ${\bf Box}_{\\{\Gamma_{i}\\}}\Gamma$, is the
coarse disjoint union $\bigsqcup\Gamma/\Gamma_{i}$, where each quotient is
equipped with the word length metric. Warped cones are metric spaces
introduced by J. Roe [20] from discrete group actions on compact metric
spaces. Typically, let $\Gamma$ be a finite generated group, $M$ a compact
Riemannian manifold. The warped cone for $(\Gamma,M)$, denoted by
$\mathcal{O}_{\Gamma}(M)$, is constructed from the action $\Gamma$ on $M$ by
diffeomorphisms. It is a metric space whose underlying topological space is
the open cone $\mathcal{O}M$ on $M$, but whose coarse geometry produces large
groups of translations. The second section will have a comprehensive
introduction for warped cones.

In [22], D. Sawicki and J. Wu found a relationship between fibred coarse
embeddability of warped cones and existence of proper affine isometric actions
of discrete groups on Banach spaces. They proved that if $(\Gamma,M)$ is free
and admits a linearization by unitary operators in a Hilbert space
$\mathcal{H}$ and $\Gamma$ admits a proper affine isometric action on
$\mathcal{H}$ (i.e., $\Gamma$ is a-T-menable), then $\mathcal{O}_{\Gamma}(M)$
admits a fibred coarse embedding into Hilbert space. The result still holds if
we replace the Hilbert space with an $\ell^{p}$-space. The same question is
also considered by Q. Wang and Z. Wang in [24]. They provided a different
proof for the case of $\ell^{p}$ spaces and replace the condition
$\Gamma\curvearrowright M$ is free by that $M$ is required to contain a dense
and free orbit of $\Gamma$ in their paper.

As corollaries, we have the following results:

######  Corollary 1.2.

Let $\Gamma$ be a finite generated discrete group and let $M$ be a compact
manifold. Assume that the action of $\Gamma$ on $M$ is linearisable in an
$\ell^{p}$-space and $\Gamma\curvearrowright M$ is free (or $M$ contains a
dense free orbit). If $\Gamma$ admits a proper affine isometric action on an
$\ell^{p}$-space, then the coarse Novikov conjecture holds for the warped cone
$\mathcal{O}_{\Gamma}(M)$.

Combining the result in [30, 21, 3], we also have

######  Corollary 1.3.

Let $\Gamma$ be a finitely generated residually finite hyperbolic group, and
$G=\widehat{\Gamma}((\Gamma_{n}))$ the corresponding profinite completion.
Then the coarse Novikov conjecture hold for $\mathcal{O}_{\Gamma}(G)$ and any
box apace ${\bf Box}_{\\{\Gamma_{n}\\}}\Gamma$.

We now give a short explaination about serveral key ingredients in our proof.
We briefly recall the proof of the maximal coarse Baum-Connes conjecture for a
metric space which admits a fibred coarse embedding into a Hilbert space in
[4]. The authors firstly reduced the issue to proving the maximal coarse Baum-
Connes conjecture at infinity for the coarse disjoint unions of finite metric
spaces. By using a geometric Dirac-dual-Dirac argument, they further reduced
the problem to a twisted version of coarse Baum-Connes conjecture at infinity
by using a $C^{*}$-algebra constructed in [11]. Then the theorem follows by
using a cutting and pasting argument of G. Yu introduced in the Section 6 of
[29]. We try to follow the outline of their proof, however there are servarl
problems.

For the first step, we need to use a Mayer-Vietoris argument and Five Lemma to
reduce the problem to ’infinity’. First of all, a coarse Mayer-Vietoris
argument in [13] enables us to just take into consideration of the coarse
Baum-Connes conjecture for coarse disjoint unions of finite subsets. For the
case when $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$ where each $X_{n}$ is finite,
one can use the following diagram, see [4]:

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The vertical sequence on the right side is exact only when we take the maximal
norm and the top horizantal map is an isomorphism. Then it suffices to prove
the assembly map at infinity $\mu_{\infty}$ is an isomorphism.

However, when we consider the coarse Novikov conjecture for certain spaces,
the coarse Mayer-Vietories argument above does not work anymore as the Five
Lemma fails if we only have the injectivity part of the assembly map $\mu$.
The argument holds only for a coarse disjoint union of a sequence of finite
metric spaces $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$. One can still reduce the
coarse Novikov conjecture for $X$ to ’infinity’ by using the following
diagram, see [25, 26].

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Notice that $i_{*}:K_{*}(\mathcal{K})\to K_{*}(C^{*}(X))$ is injective only
when $X$ is a coarse disjoint union of finite spaces, see [16]. By using a
diagram chasing argument, to prove $\mu$ is injective, it suffices to prove
$\mu_{\infty}$ is injective.

For the general case (_especially the case when $X$ can not be seen as a
coarse disjoint union of finite metric spaces_), in order to make the whole
progress work well, we introduce a notion of $K$-homology at infinity for a
locally compact Hausdorff space and a notion of Roe algebra at infinity for a
proper metric space in Section 4 and construct the following diagram:

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0.0pt\hbox{$\textstyle{\lim\limits_{d\to\infty}K_{*}^{\infty}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
102.56076pt\raise-20.29373pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mu_{\infty}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
102.56076pt\raise-29.45137pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(P_{d}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.75261pt\raise-34.73051pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.92639pt\hbox{$\scriptstyle{\Phi_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
66.66206pt\raise-40.58746pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.66206pt\raise-40.58746pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}_{\infty}(P_{d}(X)))}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

A metric space with bounded geometry is said to admit an _infinite coarse
component_ if there exists $R>0$ such that the Rips complex $P_{R}(X)$ has an
unbounded component. It is an opposite statement to the coarse disjoint union
(see Proposition 5.5). We prove that $\pi_{*}$ is an injection if $X$ admits
an infinite coarse component. Thus we can still reduce the Novikov conjecture
for $X$ to proving $\mu_{\infty}$ is injective in this case. Moreover, we show
in Section 5 that our construction coincides with the construction in [4] for
the coarse disjoint union case. This means that we can always reduce the
coarse Novikov conjecture to ’infinity’ by using our construction (see Theorem
5.7).

For the second step, we need to construct a twisted version of the coarse
Baum-Connes conjecture by using an algebra associated with
$B=\ell^{p}(\mathbb{N},\mathbb{R})$ for $p\in[1,\infty)$. As $B$ is of
property (H) introduced by G. Kasparov and G. Yu in [14], we wanted to use the
$C^{*}$-algebra
$\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$
introduced in [14, 5] to solve this problem. For any
$a\in\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$,
we shall need a $C^{*}$-isomorphism
$t^{*}:\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})\to\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$
associated to an affine isometry $t$ on $B$ to translate the support of $a$.
However,
$\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$
is built by using a dense subspace of $B$ which may not be invarinat under
$t$. To solve this problem, we construct a new algebra $\mathcal{A}(B)$ for an
$\ell^{p}$-space $B=\ell^{p}(\mathbb{N},\mathbb{R})$ in Section 3. Our
construction is inspired from the paper of S. Gong, J. Wu and G. Yu [10] for
the case when $p=2$. We construct a Bott homomorphism
$\beta_{x_{0}}:\mathcal{S}\to\mathcal{A}(B)$ associated with a base point
$x_{0}\in B$ by using the $p/2$-Hölder extension of the Mazur map introduced
by E. Odell and T. Schlumprecht in [15] and extended by Q. Cheng in [6]. We
show that the Bott homomorphism induces an isomorphism on $K$-theory which
does not depend on the choice of the base point, i.e.,
$(\beta_{x_{0}})_{*}:K_{*}(\mathcal{S})\to K_{*}(\mathcal{A}(B))$ is an
isomorphism for any $x_{0}\in B$ (see Theorem 3.15).

Actually, it has been pointed out in G. Kasparov and G. Yu’s paper [14] that
the Bott periodicity theorem holds for $\ell^{p}$-spaces. Indeed, the Mazur
map can be extended to a homeomorphism between
$\ell^{p}(\mathbb{N},\mathbb{R})$ and $\ell^{2}(\mathbb{N},\mathbb{R})$ and
the Bott periodicity for $\ell^{2}(\mathbb{N},\mathbb{R})$ has been proved in
[11]. However, the constructions in [11] and [14] also rely on a dense
subspace of $\ell^{p}(\mathbb{N},\mathbb{R})$ which is sightly different from
ours.

This paper is organized as follows. In Section 2, we briefly recall the Roe
algebra and Yu’s localization algebra, and states the coarse Novikov
conjecture. After that we go over the concept of warped metric and warped
cones introduced by John Roe [20]. In Section 3, we introduce an algebra
$\mathcal{A}(B)$ associated with $B=\ell^{p}(\mathbb{N},\mathbb{R})$ and
calculate its $K$-theory. It will play an important role in the following
proof. In Section 4, we introduce $K$-homology at infinity, Roe algebra at
infinity and the coarse Novikov conjecture at infinity for a proper metric
space. In Section 5. we compare the case of coarse disjoint union and the case
of spaces which admit an infinite coarse component, and reduce both cases to
the coarse Novikov conjecture at infinity. In Section 6, we define twisted
algebras at infinity and compute their $K$-theories by using a cutting and
pasting technique introduced by G. Yu in [29]. In Section 7, we define the
Bott map and complete the proof.

##  2 The coarse Novikov conjecture

First of all, let us recall the definition of the Roe algebra and the coarse
Baum-Connes assembly map for a proper metric space (cf. [2, 18, 19]) in this
section.

Assume that $(X,d)$ is a proper metric space. For $\delta>0$, a $\delta$-net
of $X$ is a discrete subset of $X_{\delta}\subset X$ such that there exists
$r>0$ such that $d(x,x^{\prime})>r$ for all distinct $x,x^{\prime}\in
X_{\delta}$ and $\bigcup_{x\in X_{\delta}}B(x,\delta)=X$. Moreover, $X$ is
said to have bounded geometry if $X$ contains a net $X_{\delta}$ with bounded
geometry for some $\delta>0$, i.e., for any $r>0$ there exists $N>0$ such that
any ball of radius $r$ in $X_{\delta}$ contains at most $N$ elements.

We denote $C_{0}(X)$ to be the $C^{*}$-algebra of all continuous functions on
$X$ which vanish at infinity. A separable infinite-dimensional Hilbert space
$H_{X}$ is said to be an $X$-module if $H_{X}$ is equipped with a non-
degenerate $*$-representation $\pi:C_{0}(X)\to\mathcal{B}(H_{X})$. Moreover,
$H_{X}$ is said to be ample if no non-zero element of $C_{0}(X)$ acts as a
compact operator.

######  Definition 2.1 ([18]).

Let $H_{X}$ be an ample $X$-module and $T\in\mathcal{B}(H_{X})$.

  * (1)

$T$ is said to be locally compact if $fT$ and $Tf$ are compact operators for
any $f\in C_{0}(X)$;

  * (2)

The support of $T$, denoted by $\textup{supp}(T)$, is defined to be the set of
all points $(x,y)\in X\times X$ such that for all $f,g\in C_{0}(X)$ with
$f(x)\neq 0$ and $g(y)\neq 0$, we have that $fTg\neq 0$;

  * (3)

The propagation of $T$ is defined to be

$\textup{Prop}(T)=\sup\\{d(x,y)\mid(x,y)\in\textup{supp}(T)\\}\in[0,\infty].$

Moreover, $T$ is said to have finite propagation if $\textup{Prop}(T)<\infty$.

  * (4)

The algebraic Roe algebra of $X$, denote by $\mathbb{C}[X,H_{X}]$ (or simply
$\mathbb{C}[X]$), is defined to be the $*$-algebra of all finite propagation
locally compact operators on $H_{X}$.

  * (5)

The Roe algebra of $X$, denoted by $C^{*}(X,H_{X})$ (or simply $C^{*}(X)$), is
the norm closure of $\mathbb{C}[X]$ in $\mathcal{B}(H_{X})$.

Note that $C^{*}(X,H_{X})$ does not depend on the choice of $X$-module $H_{X}$
up to a non-canonical $*$-isomorphism [13]. Hence, it is convenient to discuss
the following specific $X$-module. Let $Z\subseteq X$ be a countable dense
subset of $X$ and $H$ be a fixed infinite dimensional Hilbert space. We denote
$\mathcal{K}(H)$ to be the set of all compact operators on $H$. We define
$H_{X}=\ell^{2}(Z)\otimes H$ to be an ample $X$-module equipped with the
pointwise multiplication representation of $C_{0}(X)$ on $\ell^{2}(Z)$. It is
easy to check that $H_{X}$ is ample.

######  Definition 2.2.

Define $\mathbb{C}_{f}[X]$ to be the set of all bounded functions $T:Z\times
Z\to\mathcal{K}(H)$ satisfying the following conditions:

  * (1)

for any bounded subset $B\subset X$, the set

$\\#\\{(x,y)\in(B\times B)\cap(X\times X)\mid T(x,y)\neq 0\\}<\infty;$

  * (2)

there exists $L>0$ such that

$\\#\\{y\in Z|T(x,y)\neq 0\\}<L,\qquad\\#\\{y\in Z\mid T(y,x)\neq 0\\}<L$

for all $x\in Z$;

  * (3)

there exists $R>0$ such that $T(x,y)=0$ whenever $d(x,y)>R$ for $x,y\in Z$.

Notice that $\mathbb{C}_{f}[X]$ can be viewed as a dense $*$-subalgebra of
$\mathbb{C}[X,H_{X}]$ in $C^{*}(X)$, where the algebraic operation of
$\mathbb{C}_{f}[X]$ is given by viewing $T\in\mathbb{C}_{f}[X]$ as $Z$-by-$Z$
matrix. In this sequel, we will use $\mathbb{C}_{f}[X]$ to replace
$\mathbb{C}[X]$ to define the Roe algebra of $X$.

We next recall the assembly map $\mu$ for the Roe algebras. Let $X$ be a
proper metric space. Recall that the $K$-homology groups
$K_{i}(X)=KK_{i}(C_{0}(X),\mathbb{C})$ $(i=0,1)$ are generated by certain
cycles modulo certain equivalence relations

  * (1)

a cycle for $K_{0}(X)$ is a pair $(H_{X},F)$, where $H_{X}$ is an $X$-module
and $F$ is a bounded linear operator acting on $H_{X}$ such that $F^{*}F-I$
and $FF^{*}-I$ are locally compact, and $\varphi F-F\varphi$ is compact for
all $\varphi\in C_{0}(X)$;

  * (2)

a cycle for $K_{1}(X)$ is a pair $(H_{X},F)$, where $H_{X}$ is an $X$-module
and $F$ is a self-adjoint operator acting on $H_{X}$ such that $F^{2}-I$ is
locally compact, and $\varphi F-F\varphi$ is compact for all $\varphi\in
C_{0}(X)$.

Let $(H_{X},F)$ represent a cycle in $K_{0}(X)$. For any $R>0$, one can always
take a locally finite, uniformly bounded open cover $\\{U_{i}\\}_{i\in I}$ of
$X$ such that the diameter of each $U_{i}$ is no more than $R$. Let
$\\{\phi_{i}\\}_{i\in I}$ be a continuous partition of unity subordinate to
the open cover $\\{U_{i}\\}_{i\in I}$. Define

$F^{\prime}=\sum_{i}\phi^{\frac{1}{2}}_{i}F\phi^{\frac{1}{2}}_{i},$

where the sum converges in the strong operator topology. It is not hard to see
that $(H_{X},F)$ and $(H_{X},F^{\prime})$ are equivalent via
$(H_{X},(1-t)F+tF^{\prime})$, where $t\in[0,1]$. Note that both $F^{\prime}$
and $F^{\prime 2}-1$ have finite propagation, so $F^{\prime}$ is a multiplier
of $C^{*}(X)$ and $F^{\prime}$ is invertible modulo $C^{*}(X)$. Hence
$F^{\prime}$ gives rise to an element, denoted by $\partial([F^{\prime}])$ in
$K_{0}(C^{*}(X))$, where

$\partial:K_{1}(M(C^{*}(X))/C^{*}(X))\to K_{0}(C^{*}(X))$

is the boundary map of $K$-theory, and $M(C^{*}(X))$ is the multiplier algebra
of $C^{*}(X)$. We define the index of $(H_{X},F)$ to be
$\partial([F^{\prime}])$. Similarly, we can define the index map from
$K_{1}(X)$ to $K_{1}(C^{*}(X))$.

Now we are ready to introduce the coarse Baum–Connes assembly map for a metric
space with bounded geometry.

######  Definition 2.3.

Let $X$ be a discrete metric space with bounded geometry. For each $d\geq 0$,
the Rips complex of $X$ at scale $d$, denoted by $P_{d}(X)$, consists as a set
of all formal sums

$z=\sum_{x\in X}t_{x}x$

such that each $t_{x}$ is in $[0,1]$, such that $\sum_{x\in X}t_{x}=1$, and
such that the support of $z$ defined by $supp(z):=\\{x\in X\,|\,t_{x}\neq
0\\}$ has diameter at most $d$.

We will next discuss the semi-simplicial metric of the Rips complex of $X$.
One can find more details of Rips complex in the section 7.2 of [27]. We shall
first define the spherical metric $d_{S}$ on $P_{d}(X)$. On each path
connected component of $P_{d}(X)$, the spherical metric is the maximal metric
whose restriction to each simplex $\\{\sum_{i=0}^{n}t_{i}x_{i}\mid t_{i}\geq
0,\sum_{i}t_{i}=1\\}$ is the metric obtained by identifying the simplex with
$S^{n}_{+}$ via the map

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

where $S^{n}_{+}=\\{(x_{0},\cdots,x_{n})\in\mathbb{R}^{n+1}\mid x_{i}\geq
0,\sum_{i=0}^{n}x_{i}^{2}=1\\}$ endowed with the standard Riemannian metric on
the unit $n$-sphere.

For any $x,y\in P_{d}(Z)$, a semi-simplicial path $\delta$ between $x$ and $y$
is a finite sequence of points

$x=x_{0},y_{0},x_{1},y_{1},\cdots,x_{n},y_{n}=y$

where $x_{i}$ and $y_{i}$ are in $X$ for each $i=\\{1,\cdots,n\\}$. The length
of $\delta$ is defined to be

$l(\delta)=\sum_{i=0}^{n}d(x_{i},y_{i})+\sum_{i=0}^{n-1}d_{S}(y_{i},x_{i+1})$

We define the semi-spherical metric $d_{P_{d}}$ on $P_{d}(X)$ by

$d_{P_{d}}(x,y)=\inf\\{l(\delta)\mid\text{ $\delta$ is a semi-simplicial path
between $x$ and $y$}\\}.$

One can check that $(P_{0}(X),d_{P_{0}})$ identifies isometrically with
$(X,d)$. Moreover, it has been proved in [27, Proposition 7.2.11] that the
canonical inclusion $i_{d}:X\to P_{d}(X)$ is a coarse equivalence for each
$d\geq 0$.

If $d<d^{\prime}$, then $P_{d}(X)$ is included in $P_{d^{\prime}}(X)$ as a
subcomplex via a simplicial map. Passing to inductive limit, we obtain the
assembly map

$\mu:\lim_{d\to\infty}K_{*}(P_{d}(X))\to\lim_{d\to\infty}K_{*}(C^{*}(P_{d}(X)))\cong
K_{*}(C^{*}(X)).$

######  The coarse Novikov conjecture.

If $X$ is a discrete metric space with bounded geometry, then the coarse
assembly map

$\mu:\lim_{d\to\infty}K_{*}(P_{d}(X))\to K_{*}(C^{*}(X))$

is injective.

To state our main theorem, we will also need some preparations of coarse
geometry and group actions. Recall that a metric space $(X,d)$ is said to
admits a coarse embedding into a Hilbert space if there exists a map
$f:X\to\mathcal{H}$ and two non-decreasing unbounded functions
$\rho_{-},\rho_{+}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that

$\rho_{-}(d_{X}(x,x^{\prime}))\leq\|f(x)-f(x^{\prime})\|_{\mathcal{H}}\leq\rho_{+}(d_{X}(x,x^{\prime})).$

The concept of fibred coarse embedding is introduced by X. Chen, Q. Wang and
G. Yu in [4] as a generalization of coarse embedding. In this paper, we will
only focus on the case when a space admits a fibred coarse embedding into a
real $\ell^{p}$-space as following:

######  Definition 2.4 ([4]).

Let $p\geq 1$ and let $B$ denote the real Banach space
$\ell^{p}(\mathbb{N},\mathbb{R})$. A metric space $(X,d)$ is said to admit a
fibred coarse embedding into $B$ if there exists

  * •

a field of Banach space $(B_{x})_{x\in X}$ over $X$ such that each $B_{x}$ is
isometric to $B$;

  * •

a section $s:X\to\bigsqcup_{x\in X}B_{x}$, i.e. $s(x)\in B_{x}$ for each $x\in
X$;

  * •

two non-decreasing functions $\rho_{-}$ and $\rho_{+}$ from $\mathbb{R}_{+}$
to $\mathbb{R}_{+}$ with $\lim_{r\to\infty}\rho_{\pm}(r)=\infty$

such that, for any $R>0$, there exists a bounded subset $K\subset X$ for which
there exists a trivialization

$t_{x,R}:(B_{z})_{z\in B(x,R)}\to B(x,R)\times B$

for each $x\in X\backslash K$, that is, a map from $(B_{z})_{z\in B(x,R)}$ to
the constant field $B(x,R)\times B$ such that $t_{x,R}$ restrict to the fibre
$V_{z}$ is an affine-isometry $t_{x,R}(z):V_{z}\to V$, satisfying the
following conditions:

  * (1)

for any $z_{1},z_{2}\in B(x,R)$,

$\rho_{-}(d(z_{1},z_{2}))\leq\|t_{x,R}(z_{1})(s(z_{1}))-t_{x,R}(z_{2})(s(z_{2}))\|_{p}\leq\rho_{+}(d(z_{1},z_{2}));$

  * (2)

for any $x,y\subset X\backslash K$ with $B(x,R)\cap B(y,R)\neq\emptyset$,
there exists an affine-isometry $t_{xy,R}:V\to V$ such that $t_{x,R}(z)\circ
t^{-1}_{y,R}(z)=t_{xy,R}$ for all $z\in B(x,R)\cap B(y,R)$.∎

Notice that for $x\in X$ and $R^{\prime}>R>0$, if there exists two
trivilizations $t_{x,R}$ and $t_{x,R^{\prime}}$ for $B(x,R)$ and
$B(x,R^{\prime})$, respectively. Without loss of generality, we may assume
that

$t_{x,R^{\prime}}|_{B(x,R)}=t_{x,R}.$

For the notational convenience, we denote $t_{x}$ and $t_{xy}$ for all
$t_{x,R}$ and $t_{xy,R}$, respectively.

In the rest of this paper, we will prove the following theorem:

######  Theorem 2.5.

Let $X$ be a discrete metric space with bounded geometry. If $X$ admits a
fibred coarse embedding into an $\ell^{p}$ space, then the coarse Novikov
conjecture holds for $X$.

###  2.1 Warped Cones and proper affine isometric group actions

In this subsection, we shall recall some basic notions of warped cones and
proper affine group actions (cf. [20, 21]). These are prominent examples of
spaces which admit a fibred coarse embedding into an $\ell^{p}$-space.

Let $(Y,d_{Y})$ be a compact metric space. The open cone of $Y$, denoted by
$\mathcal{O}Y$, is the set $Y\times[1,\infty)$ with the metric
$d_{\mathcal{O}Y}$ defined by

$d_{\mathcal{O}Y}((y_{1},t_{1}),(y_{2},t_{2}))=|t_{1}-t_{2}|+\min\\{t_{1},t_{2}\\}\cdot\frac{d_{Y}(y_{1},y_{2})}{\textup{diam}(Y)}.$

Let $(X,d_{X})$ be a proper metric space, $\Gamma$ a finite generated
countable group acting by homeomorphisms on $X$ and $S\subseteq\Gamma$ a
finite generating set. The warped metric $\delta_{\Gamma}$ on $X$ is defined
to be the greatest metric satisfying that

$\delta_{\Gamma}(x,x^{\prime})\leq
d_{X}(x,x^{\prime})\quad\text{and}\quad\delta_{\Gamma}(sx,x)\leq 1$

for any $x,x^{\prime}\in X$ and $s\in S$. The warped metric on $X$ always
exists and the coarse structure induced by the warped metric does not depend
on the choice of generating set $S$ for $\Gamma$ (see [20, Proposition 1.7]).
For any $\gamma\in\Gamma$, let $|\gamma|$ denote the word length of $\gamma$
relative to the generating set $S$. Let $x,x^{\prime}\in X$, the warped
distance between from $x$ to $y$ is [20]

$\delta_{\Gamma}(x,x^{\prime})=\inf\left\\{\sum_{i=0}^{N-1}d(\gamma_{i}x_{i},x_{i+1})+|\gamma_{i}|\,\Big{|}\,x=x_{0},x_{1},\cdots,x_{N}=x^{\prime}\in
X,\gamma_{i}\in\Gamma\right\\}.$

######  Definition 2.6 ([21]).

Let $(Y,d_{Y})$ be a compact metric space and let $\Gamma$ be a finitely
generated group acting on $Y$ by homeomorphisms. The _warped cone_ of $Y$,
denoted by $\mathcal{O}_{\Gamma}(Y)$, is the open cone $\mathcal{O}Y$ with the
warped metric, where the warping group action is defined by
$\gamma(y,t)=(\gamma y,t)$.

######  Definition 2.7 (P. C. Baayen and J. De. Groot [1]).

A group action $\Gamma\curvearrowright Y$ is said to admit a _linearization_
in Banach space $B$ if and there exists an isometric representation of
$\Gamma$ on $B$ and $Y$ admits a bi-Lipschtz equivariant embedding into $B$

One is referred to the section 3.1 in [22] for more informations about
linearization.

For a residually finite group, it is well-known that the coarse geometric
properties of the box sapces ${\bf Box}(\Gamma)$ are closely related to the
analytic properties of $\Gamma$. A summary of the relationship can be found in
[3, 7]. Actually, similar results also appear in the relationship between the
coarse geometric properties $\mathcal{O}_{\Gamma}Y$ and the dynamical and
analytic properties of $\Gamma$:

######  Theorem 2.8.

Let $\Gamma$ be a finte gnerated group and $(Y,d)$ a compact metric apace.
Assume that there is a free group action $\Gamma\curvearrowright Y$.

  1. (1)

The warped cone $\mathcal{O}_{\Gamma}Y$ has property A if and only if the
action $\Gamma\curvearrowright Y$ is amenable.

  2. (2)

Assume moreover the action $\Gamma\curvearrowright Y$ admit a linearization in
the real Banach space $\ell^{p}(\mathbb{N},\mathbb{R})$ for some $p\geq 1$. If
$\Gamma$ admits a proper affine isometric action on
$\ell^{p}(\mathbb{N},\mathbb{R})$, then $\mathcal{O}_{\Gamma}Y$ admits fibred
coarse embedding into $\ell^{p}(\mathbb{N},\mathbb{R})$.

The first term of the theorem above is proved by combining [20, Theorem 3.4]
and [22, Theorem 4.2]. The second term is proved in [22, Theorem 3.2] for the
case of $p=2$. Actually their prove also holds for any $p\geq 1$. One can also
find another proof in [24, Theorem 3.2] for the case of $p\neq 2$.

In the last of this section, we introduce a typical example constructed by
profinite completion:

######  Examples 2.9.

Let $\Gamma$ be a residually finite group and
$\\{\Gamma_{n}\\}_{n=1}^{\infty}$ a decreasing sequence of finite index normal
subgroups with $\bigcap_{n=1}^{\infty}\Gamma_{n}=\\{e\\}$. We denote the
quotient group by $G_{n}=\Gamma/\Gamma_{n}$. The identity map of $\Gamma$
induces an group homomorphism $G_{n}\to G_{n-1}$ for each $n\in\mathbb{N}$.
Then we have the following sequence:

$\cdots\to G_{n}\to\cdots\to G_{2}\to G_{1}\to 1.$

We denote $\widehat{\Gamma}((\Gamma_{n}))$ the inverse limit of this sequence
and call it the profinite completion of $\Gamma$ with respect to
$(\Gamma_{n})$. It is well-known that $\widehat{\Gamma}((\Gamma_{n}))$ is a
compact metrizable group containing $\Gamma$ as a dense subgroup and $\Gamma$
acts freely on $\widehat{\Gamma}((\Gamma_{n}))$ by left multiplication and
admit a linearization in $\ell^{p}(\mathbb{N},\mathbb{R})$ for any $1\leq
p<\infty$ ([22, Lemma 3.19]).

If $\Gamma$ admits a proper affine isometric action on
$\ell^{p}(\mathbb{N},\mathbb{R})$ for some $p\geq 1$, then the warped cone
$\mathcal{O}_{\Gamma}(\widehat{\Gamma}((\Gamma_{n})))$ admits a fibred coarse
embedding into $\ell^{p}(\mathbb{N},\mathbb{R})$.

Combining with Theorem 2.5, we have the following corollaries:

######  Corollary 2.10.

Let $\Gamma$ be a countable discrete group, $Y$ be a compact metric space with
a free $\Gamma$-action. If this action admits a linearization in the
$\ell^{p}(\mathbb{N},\mathbb{R})$ and $\Gamma$ admits a proper affine
isometric action on $\ell^{p}(\mathbb{N},\mathbb{R})$ for some $p\geq 1$, then
the coarse Novikov conjecture holds for $\mathcal{O}_{\Gamma}Y$.

G. Yu proved hyperbolic groups always admits a proper affine isometric action
on $\ell^{p}(\mathbb{N},\mathbb{R})$ for a sufficiently large $p\geq 1$ in
[30, Theorem 1.1]. In particular, we have the following:

######  Corollary 2.11.

Let $\Gamma$ be a finitely generated residually finite hyperbolic group, and
$G=\widehat{\Gamma}((\Gamma_{n}))$ the corresponding profinite completion.
Then the coarse Novikov conjecture holds for $\mathcal{O}_{\Gamma}(G)$.

##  3 A Bott periodicity theorem for $\ell^{p}$ spaces

In this section, we will introduce a $C^{*}$-algebra $\mathcal{A}(B)$
associated to a given real Banach space $B=\ell^{p}(\mathbb{N},\mathbb{R})$
and calculate its $K$-theory.

In [14], G. Kasparov and G. Yu introduced a $C^{*}$-algebra for Banach space
with Property (H). They pointed out that its $K$-theory can be calculated if
the base space is an $\ell^{p}$-space. We provide a detailed proof in this
section. We would like to mention that our construction is slightly different
from theirs. The $C^{*}$-algebra $\mathcal{A}(X)$ in [14] is generated by some
certain functions on a dense subspace $V$ of $B$. To build the twisted
algebras in Section 6, for any affine isometry in Definition 2.4, we shall
need a $C^{*}$-isomorphism associated to the affine isometry to translate the
support of a function in $\mathcal{A}(X)$. However, the affine isometries in
Definition 2.4 may not preserve the dense subspace in [14]. Our construction
$\mathcal{A}(B)$ will circumvent this problem.

###  3.1 A Clifford algebras for a Hilbert space

In this subsection, we will recall the definition of Clifford algebras for
Hilbert spaces.

Let $\mathcal{H}$ be an infinite dimensional separable real Hilbert space. For
each $n\geq 0$, denote

$\mathcal{H}^{\odot
n}_{\mathbb{C}}=\mathbb{C}\odot\underbrace{\mathcal{H}\odot\mathcal{H}\odot\cdots\odot\mathcal{H}}_{\text{n
times}},$

where $\odot$ means the algebraic tensor product over $\mathbb{R}$ and
$\mathcal{H}^{\otimes 0}_{\mathbb{C}}=\mathbb{C}$. Then $\mathcal{H}^{\odot
n}_{\mathbb{C}}$ is equipped with a natural structure of complex vector space.
We define the complex inner product on $\mathcal{H}_{\mathbb{C}}^{\odot n}$ by
the formula:

$\langle z_{1}\otimes v_{1}\otimes\cdots\otimes v_{n},z_{2}\otimes
w_{1}\otimes\cdots\otimes w_{n}\rangle=z_{1}\cdot\overline{z_{2}}\cdot\langle
v_{1},w_{1}\rangle\cdot\ldots\cdot\langle v_{n},w_{n}\rangle,$

and denote $\mathcal{H}_{\mathbb{C}}^{\otimes n}$ the completion of
$\mathcal{H}^{\odot n}_{\mathbb{C}}$ under the inner product about. We denote
$1\otimes v_{1}\otimes\cdots\otimes v_{n}$ simply by
$v_{1}\otimes\cdots\otimes v_{n}$.

Denote $K^{n}$ the closed subspace of $\mathcal{H}^{\otimes n}_{\mathbb{C}}$
which is spanned by

$\left\\{v_{1}\otimes\cdots\otimes
v_{k}-\text{sgn}(\sigma)v_{\sigma(1)}\otimes\cdots\otimes
v_{\sigma(n)}\,\big{|}\,v_{1},\cdots,v_{n}\in\mathcal{H},\text{$\sigma$ is a
$k$-permutation}\right\\}.$

Define the $k$-th complex exterior power of $\mathcal{H}$ to be the quotient
Hilbert space

$\wedge^{n}\mathcal{H}_{\mathbb{C}}=\mathcal{H}^{\otimes
n}_{\mathbb{C}}/K^{n},$

where the equivalence class in $\wedge^{n}\mathcal{H}_{\mathbb{C}}$ is always
denoted by $v_{1}\wedge\cdots\wedge v_{n}$. Let
$\\{e_{1},\cdots,e_{n},\cdots\\}$ be an orthonormal basis of $\mathcal{H}$.
Then the set $\\{e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{n}}\mid
i_{1}<i_{2}<\cdots<i_{n}\\}$ gives an orthonormal basis of
$\wedge^{k}\mathcal{H}_{\mathbb{C}}$. Define the antisymmetric Fock space:

$\wedge^{*}\mathcal{H}_{\mathbb{C}}=\bigoplus_{k\in\mathbb{N}}\wedge^{k}\mathcal{H}_{\mathbb{C}}.$

For any $v\in\mathcal{H}$, we define the creation operator $C(v)$ by

$C(v)(v_{1}\wedge\cdots\wedge v_{k})=v\wedge v_{1}\wedge\cdots\wedge v_{k}.$

It is not hard to check that the adjoint operator defined by

$C^{*}(v)(v_{1}\wedge\cdots\wedge v_{k})=\sum_{i=1}^{k}(-1)^{i+1}\langle
v,v_{i}\rangle v_{1}\wedge\cdots\wedge\widehat{v_{i}}\wedge\cdots\wedge
v_{k}.$

We define a self-adjoint operator

$\hat{v}:=C(v)+C^{*}(v)\in\mathcal{B}(\wedge^{*}\mathcal{H}_{\mathbb{C}})$

for each $v\in\mathcal{H}$, then we have that

$\hat{w}\hat{v}+\hat{v}\hat{w}=2\langle w,v\rangle.$

The complex Cliiford algebra $\textup{Cliff}_{\mathbb{C}}(\mathcal{H})$ to be
subalgebra of $\mathcal{B}(\wedge^{*}\mathcal{H}_{\mathbb{C}})$ generated by
$\\{\hat{v}\mid v\in\mathcal{H}\\}$.

If $V$ is a linear subspace of $\mathcal{H}$, then $\wedge^{*}V_{\mathbb{C}}$
is clearly a subspace of $\wedge^{*}\mathcal{H}_{\mathbb{C}}$. Thus
$\textup{Cliff}_{\mathbb{C}}(V)$ can be naturally viewed as a subalgebra of
$\textup{Cliff}_{\mathbb{C}}(\mathcal{H})$.

######  Proposition 3.1.

Let $\\{V_{n}\\}_{n\in\mathbb{N}}$ be an increasing sequence of finite-
dimensional Hilbert space such that $\bigcup_{n\in\mathbb{N}}V_{n}$ is dense
in $\mathcal{H}$. Then

$\lim_{n\to\infty}\textup{Cliff}_{\mathbb{C}}(V_{n})\cong\textup{Cliff}_{\mathbb{C}}(\mathcal{H}).$

###### Proof.

For each $n\in\mathbb{N}$, we can view $\textup{Cliff}_{\mathbb{C}}(V_{n})$ as
a subalgebra of $\textup{Cliff}_{\mathbb{C}}(\mathcal{H})$. It suffices to
prove

$\overline{\bigcup_{n\in\mathbb{N}}\textup{Cliff}_{\mathbb{C}}(V_{n})}=\textup{Cliff}_{\mathbb{C}}(\mathcal{H}),$

which follows directly from the fact that
$\|\hat{v}\|_{\textup{Cliff}_{\mathbb{C}}(\mathcal{H})}=\|v\|_{\mathcal{H}}$
and $\overline{\bigcup_{n\in\mathbb{N}}V_{n}}=\mathcal{H}$. ∎

######  Remark 3.2.

Actually, there is another way to construct the Clifford algebra for
$\mathcal{H}$. Let
$\mathcal{T}(\mathcal{H})=\oplus_{n=0}^{\infty}\mathcal{H}^{\odot
n}_{\mathbb{C}}$ be tha complex algebraic tensor algebra of $\mathcal{H}$. Let
$I$ be the ideal of $\mathcal{T}(\mathcal{H})$ which is generated by
$\left\\{v\otimes v-\|v\|_{\mathcal{H}}^{2}\cdot 1\mid
v\in\mathcal{H}\right\\}$. Denote $\mathbb{C}l(\mathcal{H})$ the quotient
algebra $\mathcal{T}(\mathcal{H})/I$. As a linear space,
$\mathcal{T}(\mathcal{H})$ is also a dense subspace of the Hilbert space
$\oplus_{n=0}^{\infty}\mathcal{H}^{\otimes n}_{\mathbb{C}}$. Denote by
$H(\mathbb{C}l(\mathcal{H}))$ the completion of $\mathbb{C}l(\mathcal{H})$
under the inner product induced by $\oplus_{n=0}^{\infty}\mathcal{H}^{\otimes
n}_{\mathbb{C}}$. Then $\mathbb{C}l(\mathcal{H})$ has a canonical faithful
representation by the left multiplication on $H(\mathbb{C}l(\mathcal{H}))$.
Then the Clifford algebra

$\textup{Cliff}_{\mathbb{C}}(\mathcal{H})=\overline{\mathbb{C}l(\mathcal{H})^{\|\cdot\|}}$

where the norm $\|\cdot\|$ is given by the canonical representation on
$H(\mathbb{C}l(\mathcal{H}))$.

###  3.2 A $C^{*}$-algebra associated with a real $\ell^{p}$ space

Fix $1\leq p<\infty$ and denote $B=\ell^{p}(\mathbb{N},\mathbb{R})$ to be the
real Banach space. Specially, we denote
$\mathcal{H}=\ell^{2}(\mathbb{N},\mathbb{R})$ the real Hilbert space. We
define the sign function $\text{sgn}:\mathbb{R}\to\mathbb{R}$ by the formula

$\text{sgn}(a)=\left\\{\begin{aligned} &0&&,a=0;\\\ &a|a|^{-1}&&,a\neq
0.\end{aligned}\right.$

Denote by $S(B)$ the closed unit sphere of $B$. Recall the Mazur map
$\psi:S(B)\to S(\mathcal{H})$, defined by

$\psi(a_{1},\cdots,a_{n},\cdots)=(sgn(a_{1})|a_{1}|^{p/2},\cdots,\text{sgn}(a_{n})|a_{n}|^{p/2},\cdots),$

is a uniform homeomorphism, i.e., the $r$-oscillation of $\psi$ and
$\psi^{-1}$ tend to $0$ as $r$ tends to $0$, where the $r$-oscillation of
$\psi$ is defined to be

$\omega_{r}(\psi)=\sup_{x,x^{\prime}\in S(B),\|x-x^{\prime}\|_{p}\leq
r}\|\psi(x)-\psi(x^{\prime})\|.$

We consider the _$p/2$ -Hölder extension_ of the Mazur map $\psi$, which is a
map $\phi:B\to\mathcal{H}$ defined by

$\phi(x)=\left\\{\begin{aligned} &0&&,x=0;\\\
&\|x\|^{p/2}_{p}\psi\left(\|x\|_{p}^{-1}x\right)&&,x\neq
0.\end{aligned}\right.$

Actually, we have the following uniform homeomorphism extension theorem due to
E. Odell and T. Schlumprecht [15, Proposition 2.9] and Q. Cheng [6, Theorem
2.4]:

######  Proposition 3.3 ([15, 6]).

For any $R>0$, the extended Mazur map restricted on the ball
$\phi|_{Ball_{B}(0,R)}:Ball_{B}(0,R)\to Ball_{\mathcal{H}}(0,R^{p/2})$ is a
uniform homeomorphism, i.e.,

$\omega_{r}\left(\phi|_{Ball_{B}(0,R)}\right)\to 0\text{ as }r\to 0.$

As a corollary, $\phi$ forms a homeomorphism between $B$ and $\mathcal{H}$.
When $p=2$, the extended Mazur map is the identity map on $\mathcal{H}$.

Denote by $\mathbb{R}_{+}=[0,\infty)$. Let
$C(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$
be the algebra of all continuous functions from $B\times\mathbb{R}_{+}$ to
$\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R})$. We view
$\textup{Cliff}_{\mathbb{C}}(\mathcal{H})$ as a subalgebra of
$\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R})$ in the following
paper.

######  Definition 3.4.

Define
$C_{b,0}(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$
to be the subalgebra of
$C(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$
consisting of all bounded functions $f$ such that

$f(x,0)\in\textup{Cliff}_{\mathbb{C}}(\mathcal{H})\subseteq\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R})$

for all $x\in B$.

######  Definition 3.5.

For any $x_{0}\in B$, we define the Clifford operator $C_{x_{0}}\in
C(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$
by

$C_{x_{0}}(x,t)=(\phi(x-x_{0}),t)\in\mathcal{H}\oplus\mathbb{R}\subseteq\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R})$

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

We denote $C_{0}(\mathbb{R})_{ev}$ (or $C_{0}(\mathbb{R})_{odd}$,
respectively) the subset of $C_{0}(\mathbb{R})$ of all even (or odd,
respectively) functions. For any Hilbert space $H$, we next define the
functional calculus of $\textup{Cliff}_{\mathbb{C}}(H)$: for any $f\in
C_{0}(\mathbb{R})_{ev}$ and
$v\in\mathcal{H}\subset\textup{Cliff}_{\mathbb{C}}(H)$, we define
$f(v)=f(\|v\|)\in\mathbb{C}\subset\textup{Cliff}_{\mathbb{C}}(H)$ and for
$g\in C_{0}(\mathbb{R})_{odd}$, we define the functional calculus by

$g(v)=\left\\{\begin{aligned} &0&&,v=0\\\ &g(\|v\|)\frac{v}{\|v\|}&&,v\neq
0,\end{aligned}\right.$

where $g(v)$ is an element of $\textup{Cliff}_{\mathbb{C}}(H)$.

######  Lemma 3.6.

For any $x_{0}\in B$ and $f\in C_{0}(\mathbb{R})$, the Bott map
$\beta_{x_{0}}$ is defined by functional calculus such that

$(\beta_{x_{0}}(f))(x,t)=f(C_{x_{0}}(x,t)),$

for each $x\in B$ and $t\in\mathbb{R}_{+}$. Then $\beta_{x_{0}}$ gives a
graded homomorphism from $C_{0}(\mathbb{R})$ to
$C_{b,0}(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$.

###### Proof.

Combining the definition of the functional calculus and the fact that $\phi$
is continuous, it is obvious that $\beta_{x_{0}}(f)$ is continuous. ∎

######  Definition 3.7.

The algebra $\mathcal{A}(B)$ is the $C^{*}$-subalgebra
$C_{b,0}(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$
generated by

$\\{\beta_{x}(f)\mid x\in B,f\in C_{0}(\mathbb{R})\\}.$

Let $W$ be a subspace of $B$, the algebra $\mathcal{A}(B,W)$ is defined to be
the $C^{*}$-subalgebra of $\mathcal{A}(B)$ generated by

$\\{\beta_{x}(f)\mid x\in W\subseteq B,f\in C_{0}(\mathbb{R})\\}.$

######  Lemma 3.8.

Fix $f\in\mathcal{S}$.

  1. (1)

For any $\varepsilon>0$, there exists $\delta>0$ such that

$\|\beta_{x}(f)-\beta_{y}(f)\|\leq\varepsilon$

whenever $x,y\in B$ satisfies $\|x-y\|_{p}\leq\delta$.

  2. (2)

As a corollary, let $\\{x_{n}\\}\in B$ be a sequence which converges to
$x_{0}\in B$. Then

$\lim_{n\to\infty}\|\beta_{x_{n}}(f)-\beta_{x_{0}}(f)\|=0$

for each $f\in C_{0}(\mathbb{R})$.

###### Proof.

Let $f_{0}$ and $f_{1}$ be the even part and odd part of $f$. There exists
$R>0$ such that $|f(t)|\leq\varepsilon/2$ for all $|t|>R$.

Case 1. For the even part, by definition, for each $x,y,z\in B$, we have that

$\|(\beta_{x}(f_{0})-\beta_{y}(f_{0}))(z,t)\|=|f_{0}(\|C_{x}(z,t)\|)-f_{0}(\|C_{y}(z,t)\|)|.$

If $(z,t)$ satisfies that both $\|C_{x}(z,t)\|$ and $\|C_{y}(z,t)\|$ are
greater than $R$, then

$|f_{0}(\|C_{x}(z,t)\|)-f_{0}(\|C_{y}(z,t)\|)|\leq|f_{0}(\|C_{x}(z,t)\|)|+|f_{0}(\|C_{y}(z,t)\|)|\leq\varepsilon.$

Notice that
$\|C_{x}(z,t)\|=(\|\phi(z-x)\|^{2}+t^{2})^{1/2}=(\|z-x\|^{p}_{p}+t^{2})^{1/2}$.
If one of $\|C_{x}(z,t)\|$ or $\|C_{y}(z,t)\|$ is less than $R$, without loss
of generality, assume that $\|C_{x}(z,t)\|<R$. Then $\|z-x\|^{p}_{p}<R^{2}$.
Notice that

$\begin{split}\left|\|C_{x}(z,t)\|-\|C_{y}(z,t)\|\right|&=\left(\|z-x\|^{p}_{p}+t^{2}\right)^{1/2}-\left(\|z-y\|^{p}_{p}+t^{2}\right)^{1/2}\\\
&\leq\left(\|z-x\|^{p}_{p}-\|z-y\|^{p}_{p}\right)^{1/2},\end{split}$

and

$\left|\|z-x\|_{p}-\|z-y\|_{p}\right|\leq\|x-y\|_{p}.$

Set $\delta_{0}>0$ satisfying that
$|f_{0}(t_{1})-f_{0}(t_{2})|<\frac{\varepsilon}{2}$ for all $t_{1},t_{2}$ with
$|t_{1}-t_{2}|<\delta_{0}$. As $y(t)=t^{p}$ is uniform continuous in
$[0,R^{p/2}+1]$, then there exists $\delta_{1}>0$ such that
$|t_{1}^{p}-t_{2}^{p}|<\delta_{0}^{2}$ for all $t_{1},t_{2}\in[0,R^{p/2}+1]$
with $|t_{1}-t_{2}|<\delta_{1}$. Set $\delta_{ev}=\min\\{\delta_{1},1\\}$. If
$\|x-y\|_{p}\leq\delta_{ev}$, then both $\|z-x\|_{p}$ and $\|z-y\|_{p}$ are in
$[0,R^{p/2}+1]$ and we have that

$|f_{0}(\|C_{x}(z,t)\|)-f_{0}(\|C_{y}(z,t)\|)|\leq\frac{\varepsilon}{2}.$

Case 2. For the odd part, assume that there exists $g\in
C_{0}(\mathbb{R})_{ev}$ such that $f_{1}(t)=tg(t)$. Such elements are dense in
$C_{0}(\mathbb{R})_{odd}$. Then, by definition,

$\begin{split}\|(\beta_{x}(f_{1})-\beta_{y}(f_{1}))(z,t)\|=&\left\|g(\|C_{x}(z,t)\|)C_{x}(z,t)-g(\|C_{y}(z,t)\|)C_{y}(z,t)\right\|\\\
\leq&\left\|g(\|C_{x}(z,t)\|)-g(\|C_{y}(z,t)\|)C_{x}(z,t)\right\|\\\
&+|g(C_{y}(z,t))|\cdot\|C_{x}(z,t)-C_{y}(z,t)\|.\end{split}$

Similarly, if both $\|C_{x}(z,t)\|$ and $\|C_{y}(z,t)\|$ are greater than $R$,
then

$\left\|g(\|C_{x}(z,t)\|)C_{x}(z,t)-g(\|C_{y}(z,t)\|)C_{y}(z,t)\right\|\leq|f_{1}(\|C_{x}(z,t)\|)|+|f_{1}(\|C_{y}(z,t)\|)|\leq\varepsilon.$

If one of $\|C_{x}(z,t)\|$ or $\|C_{y}(z,t)\|$ is less than $R$, with a
similar argument as above, we can find a $\delta_{3}>0$ such that
$|g(\|C_{x}(z,t)\|)-g(\|C_{y}(z,t)\|)|\leq\frac{\varepsilon}{4R}$. For the
second term, we have that

$\|C_{x}(z,t)-C_{y}(z,t)\|=\|\phi(z-x)-\phi(z-y)\|.$

By Proposition 3.3, there exists $0<\delta_{4}<1$ associated with
$\phi|_{Ball_{B}(0,R^{2/p}+1)}$ such that if $\|x-y\|\leq\delta_{4}$, then
$\|\phi(z-x)-\phi(z-y)\|\leq\frac{\varepsilon}{4\|g\|}$. Set
$\delta_{odd}=\min\\{\delta_{3},\delta_{4}\\}$, we conclude that if
$\|x-y\|_{p}\leq\delta_{odd}$, then

$\|(\beta_{x}(f_{1})-\beta_{y}(f_{1}))(z,t)\|\leq\frac{\varepsilon}{2}$

for all $(z,t)\in B\times\mathbb{R}$.

Set $\delta=\min\\{\delta_{ev},\delta_{odd}\\}$, then

$\|\beta_{x}(f)-\beta_{y}(f)\|\leq\|\beta_{x}(f_{0})-\beta_{y}(f_{0})\|+\|\beta_{x}(f_{1})-\beta_{y}(f_{1})\|\leq\varepsilon.$

This completes the proof. ∎

######  Corollary 3.9.

Let $\\{W_{n}\\}_{n\in\mathbb{N}}$ be an increasing sequence of finite
dimensional linear subspaces of $B$ with $\bigcup_{n\in\mathbb{N}}W_{n}$ dense
in $B$. Then

  1. (1)

$\mathcal{A}(B,B)=\mathcal{A}(B)$ and
$\mathcal{A}(B,W_{n})\subseteq\mathcal{A}(B,W_{n+1})$;

  2. (2)

$\lim_{n\to\infty}\mathcal{A}(B,W_{n})=\mathcal{A}(B,\bigcup_{n\in\mathbb{N}}W_{n})=\mathcal{A}(B,\overline{\bigcup_{n\in\mathbb{N}}W_{n}})$.

###### Proof.

The first term and the first equality in (2) are immediate from the
definition. The second equality in (2) comes from Lemma 3.8. ∎

###  3.3 The $K$-theory of $\mathcal{A}(B)$

In this subsection, we shall calculate the $K$-theory of $\mathcal{A}(B)$. We
will show that the Bott map $(\beta_{x_{0}})_{*}:K_{*}(\mathcal{S})\to
K_{*}(\mathcal{A}(B))$ is an isomorphism for any $x_{0}\in B$. One can find a
proof in [10, Remark 7.7] for the case when $p=2$. We remark that the Bott
periodicity theorem holds for any Banach space which is spherical equivalent
to $\ell^{2}$ by using a similar argument.

Let $N_{n}=\\{0,1,\cdots,n\\}\subseteq\mathbb{N}$. We denote
$B_{n}=\ell^{p}(N_{n},\mathbb{R})$ and $H_{n}=\ell^{2}(N_{n},\mathbb{R})$ to
be the finite dimensional subspaces of $B$ and $\mathcal{H}$, respectively. It
is clear that $\overline{\bigcup_{n=0}^{\infty}B_{n}}=B$ and
$\overline{\bigcup_{n=0}^{\infty}H_{n}}=\mathcal{H}$. Denote
$\mathcal{S}=C_{0}(\mathbb{R})$ equipped with the grading according to even
and odd functions.

Define
$C_{b,0}(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
to be the subalgebra of
$C_{b}(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
consisting of all bounded function $f$ such that

$f(x,0)\in\textup{Cliff}_{\mathbb{C}}(H_{n})\subseteq\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}).$

Since the Mazur map restricts to a uniform homeomorphism
$\psi_{n}=\psi|_{B_{n}}:S(B_{n})\to S(H_{n})$, for any $x_{0}\in B_{n}$, we
can still define the Clifford operator $C^{n}_{x_{0}}\in
C(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
on $B_{n}$ by

$C^{n}_{x_{0}}(x,t)=(\phi_{n}(x-x_{0}),t)\in\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}),$

where $\phi_{n}:B_{n}\to H_{n}$ is the extended Mazur map of $\psi_{n}$ and
$x\in B_{n}$, $t\in\mathbb{R}_{+}$. The Bott map
$\beta_{x_{0}}^{n}:\mathcal{S}\to
C_{b,0}(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
is defined to be

$\left(\beta^{n}_{x_{0}}(f)\right)(x,t)=f(C^{n}_{x_{0}}(x,t)).$

######  Definition 3.10.

Define $\mathcal{A}(B_{n})$ to be the subalgebra of
$C_{b,0}(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
generated by

$\\{\beta^{n}_{x}(f)\mid f\in\mathcal{S},x\in B_{n}\\}.$

Similarly, one can also define the Clifford operator on $H_{n}$ and
$\mathcal{A}(H_{n})$ by taking $p=2$. To clearfy the notation, we write the
Bott map for $H_{n}$ associated to $v\in H_{n}$ by $\beta^{H_{n}}_{v}$.

######  Lemma 3.11.

$\mathcal{A}(B_{n})$ is a $(B_{n}\times\mathbb{R}_{+})$-$C^{*}$-algebra.

###### Proof.

Let $\mathcal{A}_{ev}(B_{n})$ be the $C^{*}$-subalgebra of
$\mathcal{A}(B_{n})$ generated by

$\left\\{\beta^{n}_{x}(f)\mid x\in B_{n},\ f\in
C_{0}(\mathbb{R})_{ev}\right\\}.$

Then $\mathcal{A}_{ev}(B_{n})$ is a subalgebra of the center
$\mathcal{Z}(\mathcal{A}(B_{n}))$. It is clear that $\beta^{n}_{x}(f)\in
C_{0}(B_{n}\times\mathbb{R}_{+})$ for any $x\in B_{n}$ and $f\in
C_{0}(\mathbb{R})_{ev}$. For any different points
$(x_{1},t_{1}),(x_{2},t_{2})\in B_{n}\times\mathbb{R}_{+}$ with $t_{1}\leq
t_{2}$, we have that

$\|C_{x_{1}}(x_{1},t_{1})\|=|t_{1}|<\sqrt{\|x_{1}-x_{2}\|_{p}^{p}+t_{2}^{2}}=\|C_{x_{1}}(x_{2},t_{2})\|.$

Choose a suitable $f\in C_{0}(\mathbb{R})_{ev}$ such that
$f(\|C_{x_{1}}(x_{1},t_{1})\|)\neq f(\|C_{x_{1}}(x_{2},t_{2})\|)$, i.e.,

$\left(\beta^{n}_{x_{1}}(f)\right)(x_{1},t_{1})\neq\left(\beta^{n}_{x_{1}}(f)\right)(x_{2},t_{2})$

By Stone-Weierstrass theorem, we conclude that $\mathcal{A}_{ev}(B_{n})\cong
C_{0}(B_{n}\times\mathbb{R}_{+})$.

For any $f\in C_{c}(\mathbb{R})$, let $f_{0}\in C_{0}(\mathbb{R})$ be an even
function such that $f_{0}(t)=1$ for all $t\in\textup{supp}(f)$, thus
$\beta^{n}_{x}(f)\beta^{n}_{x}(f_{0})=\beta^{n}_{x}(f)$ for any $x\in B_{n}$.
This means that $C_{0}(B_{n}\times\mathbb{R}_{+})\cdot\mathcal{A}(B_{n})$ is
dense in $\mathcal{A}(B_{n})$. Then $\mathcal{A}(B_{n})$ is a
$(B_{n}\times\mathbb{R}_{+})$-$C^{*}$-algebra. ∎

In [14], G. Kasparov and G. Yu introduced a $C^{*}$-algebra
$\mathcal{S}\mathcal{C}(B_{n})$ associated with $B_{n}$. We briefly recall the
definition here. Denote
$\mathcal{C}(B_{n})=C_{0}(B_{n},\textup{Cliff}_{\mathbb{C}}(H_{n}))$ equipped
with the grading induced from $\textup{Cliff}_{\mathbb{C}}(H_{n})$. Denote
$\mathcal{S}\mathcal{C}(H_{n})=\mathcal{S}\widehat{\otimes}\mathcal{C}(H_{n})$
to be the graded tensor product.

######  Theorem 3.12.

With notations as above, $\mathcal{A}(B_{n})$ is isomorphic to
$\mathcal{S}\mathcal{C}(B_{n})$ for any $n\in\mathbb{N}$.

###### Proof.

By definition, $\mathcal{S}\mathcal{C}(B_{n})=\\{f\widehat{\otimes}h\mid f\in
C_{0}(\mathbb{R}),h\in C_{0}(B_{n},\textup{Cliff}_{\mathbb{C}}(H_{n}))\\}$.
Define

$\mathcal{B}=\\{f\in
C_{0}(\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathbb{R}))\mid
f(0)\in\mathbb{C}\\}.$

$\mathcal{B}$ is induced with the grading of
$\textup{Cliff}_{\mathbb{C}}(\mathbb{R})\cong\mathbb{C}\oplus\mathbb{C}$, the
scalar part and the vector part. We define a homomorphism
$\varphi:\mathcal{S}\to\mathcal{B}$ by the following formula:

$(\varphi(f))(t)=f_{ev}(t)1+f_{odd}(t)e,$

where $f_{ev}$ and $f_{odd}$ are respectively the even and odd parts of $f$
and $e$ is a unit vector of $\mathbb{R}$. It is not hard to see that $\varphi$
is an isomorphism preserving the grading.

As a corollary, we have the identification:

$\mathcal{S}\mathcal{C}(B_{n})=\\{f\in
C_{0}(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))\mid
f(x,0)\in\textup{Cliff}_{\mathbb{C}}(V_{n})\\},$

where we view $\textup{Cliff}_{\mathbb{C}}(H_{n})$ as a subalgebra of
$\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R})$. Notice that
$C_{0}(B_{n}\times\mathbb{R}_{+})\cong C_{0}(\mathbb{R})_{ev}\otimes
C_{0}(B_{n})$ is a subalgebra of $\mathcal{Z}(\mathcal{S}\mathcal{C}(B_{n}))$,
the center of $\mathcal{S}\mathcal{C}(B_{n})$, which makes
$\mathcal{S}\mathcal{C}(B_{n})$ a
$(B_{n}\times\mathbb{R}_{+})$-$C^{*}$-algebra.

Notice that $\mathcal{A}(B_{n})$ is a subalgebra of
$\mathcal{S}\mathcal{C}(B_{n})$. Indeed,

$\lim_{(x,t)\to\infty}(\beta^{n}_{x_{0}}(f))(x,t)=0\quad\text{and}\quad
f(C(v,0))=f(v)\in\textup{Cliff}_{\mathbb{C}}(V).$

It suffices to prove $\mathcal{A}(B_{n})$ is dense in
$\mathcal{S}\mathcal{C}(B_{n})$. As both algebras are
$(B_{n}\times\mathbb{R}_{+})$-$C^{*}$-algebras, it suffices to prove the fiber
of $\mathcal{A}(B_{n})$ at $(x,t)$ is the same as
$\mathcal{S}\mathcal{C}(B_{n})$ for any $(x,t)\in B_{n}\times\mathbb{R}_{+}$.
The fiber of $\mathcal{A}(B_{n})$ at $(x,t)$ is given by

$\mathcal{A}(B_{n})/C_{0}(B_{n}\times\mathbb{R}\backslash\\{(x,t)\\})\cdot\mathcal{A}(B_{n})\cong\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus
t\mathbb{R}),$

where

$t\mathbb{R}=\left\\{\begin{aligned} &0&&,t=0;\\\ &\mathbb{R}&&,t\neq
0,\end{aligned}\right.$

which is actually same as $\mathcal{S}\mathcal{C}(B_{n})$. We then finish the
proof that $\mathcal{A}(B_{n})\cong\mathcal{S}\mathcal{C}(B_{n})$. ∎

######  Corollary 3.13.

The Bott map induces an isomorphism on $K$-theory:

$(\beta_{x}^{n})_{*}:K_{*}(\mathcal{S})\xrightarrow{\cong}K_{*}(\mathcal{A}(B_{n}))$

for any $x\in B_{n}$.

###### Proof.

For any $x_{0},x_{1}\in B_{n}$, we take $x_{s}=(1-s)x_{0}+sx_{1}$ for each
$s\in[0,1]$. Then $s\mapsto x_{s}$ is a continuous path connecting $x_{0}$ and
$x_{1}$. By Lemma 3.8, we have that $\beta_{x_{s}}$ forms a homotopy between
$\beta_{x_{0}}$ and $\beta_{x_{1}}$. Thus it suffices to prove the theorem for
$0\in B_{n}$.

Define $\phi^{*}:\mathcal{A}(H_{n})\to\mathcal{A}(B_{n})$ by

$(\phi^{*}(\sigma))(x,t)=\sigma(\phi(x),t),\quad\text{$\sigma\in\mathcal{A}(H_{n})$
and $x\in B_{n}$}.$

As $\phi$ is a homeomorphism, it is clearly to see that $\phi^{*}$ a
$*$-isomorphism. For any $f\in\mathcal{S}$ and $(x,t)\in
B_{n}\times\mathbb{R}_{+}$, one can check that

$\begin{split}\left(\phi^{*}\left(\beta_{0}^{H_{n}}(f)\right)\right)(x,t)&=\left(\beta_{0}^{H_{n}}(f)\right)(\phi(x),t)\\\
&=f(\phi(x),t)\\\ &=\left(\beta_{0}^{n}(f)\right)(x,t).\end{split}$

This shows that $\phi^{*}\circ\beta^{H_{n}}_{0}(f)=\beta^{n}_{0}(f)$. Thus, we
have the following commuting diagram:

${K_{*}(\mathcal{S})}$${K_{*}(\mathcal{A}(H_{n}))}$${K_{*}(\mathcal{A}(B_{n})).}$$\scriptstyle{(\beta^{B_{n}}_{0})_{*}}$$\scriptstyle{(\beta^{H_{n}}_{0})_{*}}$$\scriptstyle{\phi^{*}}$

It is proved that $\left(\beta^{H_{n}}_{0}\right)_{*}$ is an isomorphism in
[11] and [10]. As a result, $(\beta^{n}_{0})_{*}$ is an isomorphism. ∎

For any $n\in\mathbb{N}$, we denote
$B^{\prime}_{n}=\ell^{p}(\mathbb{N}\backslash N_{n},\mathbb{R})$. Then one can
easily see that $B=B_{n}\oplus_{p}B^{\prime}_{n}$, where $\oplus_{p}$ means
that $B_{n}\oplus B^{\prime}_{n}$ is equipped with the norm

$\|(x,y)\|_{p}=(\|x\|^{p}_{p}+\|y\|^{p}_{p})^{1/p}$

for any $(x,y)\in B_{n}\oplus_{p}B^{\prime}_{n}$. Denote by $P_{n}:B\to B_{n}$
and $Q_{n}:B\to B^{\prime}_{n}$ the direct sum projection, i.e., for any
$x=(x_{1},x_{2})\in B=B_{n}\oplus_{p}B^{\prime}_{n}$ we have that
$P_{n}(x)=x_{1}$ and $Q_{n}(x)=x_{2}$. The following fact which will be very
useful in the next lemma can be checked with a easy calculation:

Fact. For any $(x,y)\in B_{n}\oplus_{p}B^{\prime}_{n}=B$, we have that

$\phi(x,y)=(\phi(x),\phi(y))\in H_{n}\oplus H_{n}^{\prime}=\mathcal{H},$

i.e.,
$\phi=\phi|_{B_{n}}\oplus\phi_{B^{\prime}_{n}}:B_{n}\oplus_{p}B_{n}^{\prime}\to
H_{n}\oplus H_{n}^{\bot}$.

For any $\sigma\in\mathcal{A}(B,B_{n})$, we define the restriction
homomorphism $\pi_{n}:\mathcal{A}(B,B_{n})\to\mathcal{A}(B_{n})$ by

$(\pi_{n}(\sigma))(x,t)=\sigma(x,t)$

for all $x\in W_{n}$. To see $\pi_{n}$ is well-defined, one can check that

$(\pi_{n}\circ\beta_{x_{0}}(f))(x,t)=\pi_{n}(f(C_{x_{0}}(x,t)))=f(C^{n}_{x_{0}}(x,t)),$

for any $x_{0},x\in B_{n}$ and $t\geq 0$.

######  Lemma 3.14.

The restriction homomorphism $\pi_{n}:\mathcal{A}(B,W_{n})\to\mathcal{A}(B)$
is an isomorphism.

###### Proof.

For each $(w,t)\in B_{n}^{\prime}\oplus\mathbb{R}$, we define

$\tau_{w,t}^{n}:H_{n}\oplus\mathbb{R}\to\mathcal{H}\oplus\mathbb{R}$

to be the inclusion which is identity on $H_{n}$ and maps
$(0,(\|w\|_{p}^{2}+t^{2})^{1/2})\in V_{n}\oplus\mathbb{R}$ to $(0,\phi(w),t)$,
i.e., $\tau^{n}_{w,t}(0\oplus\mathbb{R})$ is the one-dimensional subspace of
$H_{n}^{\bot}\oplus\mathbb{R}$ spanned by $(\phi(w),t)$ and the inclusion is
an isometry. This map clearly induces an inclusion of Clifford algebras
$\textup{Cliff}_{\mathbb{C}}(V_{n}\oplus\mathbb{R})\to\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R})$.

Define

$\Phi:C(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))\to
C(B\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}))$

by the following formula:

$(\Phi(\sigma))(x,t)=\tau^{n}_{Q_{n}(x),t}\left(\sigma(P_{n}(x),(\|Q_{n}(x)\|_{p}^{2}+t^{2})^{1/2})\right)\in\textup{Cliff}_{\mathbb{C}}(\mathcal{H}\oplus\mathbb{R}),$

where $(P_{n}(x),(\|Q_{n}(x)\|_{p}^{2}+t^{2})^{1/2})\in
B_{n}\times\mathbb{R}_{+}$. It is clear that $\Phi$ is a $*$-homomorphism.

Let $x_{0}\in B_{n}$, consider the Clifford operator $C^{n}_{x_{0}}\in
C(B_{n}\times\mathbb{R}_{+},\textup{Cliff}_{\mathbb{C}}(H_{n}\oplus\mathbb{R}))$
on $B_{n}$, i.e., $C_{x_{0}}^{n}(x,t)=(x-x_{0},t)$ for each $(x,t)\in
B_{n}\oplus\mathbb{R}$. Then for any $y\in B$, we have

$\begin{split}(\Phi(C^{n}_{x_{0}}))(y,t)&=\tau^{n}_{Q_{n}(y),t}(C^{n}_{x_{0}}(P_{n}(y),(\|Q_{n}(y)\|_{p}^{2}+t^{2})^{1/2}))\\\
&=\tau^{n}_{Q_{n}(y),t}(\phi(P_{n}(y)-x_{0}),(\|Q_{n}(y)\|_{p}^{2}+t^{2})^{1/2})\\\
&=(\phi(P_{n}(y)-x_{0})\oplus\phi(Q_{n}(y)),t).\end{split}$ (1)

As
$\phi=\phi|_{B_{n}}\oplus\phi|_{B_{n}^{\prime}}:B_{n}\oplus_{p}B_{n}^{\prime}\to
H_{n}\oplus H_{n}^{\bot}$, then $\phi(x)=\phi(x_{1})+\phi(x_{2})$ for any
$x=(x_{1},x_{2})\in B_{n}\oplus B^{\prime}_{n}=B$. Thus
$\phi(P_{n}(y)-x_{0})+\phi(Q_{n}(y))=\phi(P_{n}(y)+Q_{n}(y)-x_{0})=\phi(y-x_{0})$.
Combining the equation (1), we have that $\Phi(C_{x_{0}}^{n})=C_{x_{0}}$ the
Clifford operator of $B$. Write $t_{y}=\sqrt{\|Q_{n}(y)\|_{p}^{2}+t^{2}}$.
Then for any $f\in C_{0}(\mathbb{R})$, we have that

$\begin{split}(\Phi(\beta_{x_{0}}^{n}(f)))(y,t)=&\tau_{Q_{n}(y),t}^{n}(f(C_{x_{0}}^{n}(P_{n}(y),t_{y})))\\\
=&\tau_{Q_{n}(y),t}^{n}\left(f_{ev}\left(\|C_{x_{0}}^{n}\left(P_{n}(y),t_{y}\right)\|\right)\right)+\\\
&\tau_{Q_{n}(y),t}^{n}\left(f_{odd}\left(\|C_{x_{0}}^{n}\left(P_{n}(y),t_{y}\right)\|\right)\frac{C_{x_{0}}^{n}(P_{n}(y),t_{y})}{\|C_{x_{0}}^{n}(P_{n}(y),t_{y})\|}\right)\\\
=&f_{ev}\left(\|C_{x_{0}}\left(y,t\right)\|\right)+f_{odd}\left(\|C_{x_{0}}\left(y,t\right)\|\right)\frac{\tau_{Q_{n}(y),t}^{n}(C_{x_{0}}^{n}(P_{n}(y),t_{y}))}{\|C_{x_{0}}(y,t)\|}\\\
=&f_{ev}\left(\|C_{x_{0}}\left(y,t\right)\|\right)+f_{odd}\left(\|C_{x_{0}}\left(y,t\right)\|\right)\frac{C_{x_{0}}(y,t)}{\|C_{x_{0}}(y,t)\|}=\beta_{x_{0}}(f)(y,t)\end{split}$
(2)

This shows that $\Phi$ maps $\mathcal{A}(W_{n})$ to $\mathcal{A}(B)$ and
$\Phi(\mathcal{A}(W_{n}))$ is actually equal to $\mathcal{A}(B,W_{n})$. This
also shows that $\Phi$ is the inverse of $\pi_{n}$. ∎

######  Theorem 3.15.

For any $x_{0}\in B$, the Bott homomorphism

$\beta_{x_{0}}:C_{0}(\mathbb{R})\to\mathcal{A}(B)$

induces an isomorphism on $K$-theory, i.e.,

$(\beta_{x_{0}})_{*}:K_{*}(\mathcal{S})\to K_{*}(\mathcal{A}(B))$

is an isomorphism.

###### Proof.

Similarly, it suffices to prove the theorem for some fixed $x_{0}\in B$.

Viewing $B_{n}$ as a subspace of $B_{n+1}$, one can similarly define a map

$\Phi_{n}:\mathcal{A}(B_{n})\to\mathcal{A}(B_{n+1},B_{n})\subseteq\mathcal{A}(B_{n+1}).$

By compute on the generators, it is not hard to check that we have the
following commuting diagram

$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{n}_{x_{0}}}$$\scriptstyle{\beta^{n+1}_{x_{0}}}$$\textstyle{\mathcal{A}(B_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{n}}$$\scriptstyle{\Phi}$$\textstyle{\mathcal{A}(B,B_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}(B_{n+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathcal{A}(B,B_{n+1})}$

where $\beta$ is defined as in [11]. Combining the results we prove before, we
have that

$\mathcal{A}(B)=\lim_{n\to\infty}\mathcal{A}(B,B_{n})\cong\lim_{n\to\infty}\mathcal{A}(B_{n}).$

As the $K$-functor commutes with the direct limits (see [27]), the theorem
holds directly from Theorem 3.12. ∎

##  4 $K$-homology at infinity and Roe algebra at infinity

In this section, we will introduce $K$-homology at infinity and Roe algebra at
infinity for a proper metric space.

The approach to $K$-homology via localization algebras is first introduced by
G. Yu in [28]. In this paper, we will also need to work with a version of
localization algebras introduced by R. Willett and G. Yu in [27]. For the
convenience of the readers, we shall briefly recall its definition and its
functoriality. Let $X$ be a proper metric space.

######  Definition 4.1 ([27]).

Let $H_{X}^{+}$ be an $X^{+}$-module, where $X^{+}$ is the one point
compactification. Define $\mathbb{L}[H_{X}^{+};\infty]$ to be the collection
of all bounded functions $(T_{t})$ from $[1,\infty)$ to
$\mathcal{B}(H_{X}^{+})$ such that:

  * (1)

for any compact subset $K$ of $X$, there exists $t_{K}\geq 0$ such that for
all $t\geq t_{K}$, the operators $\chi_{K}T_{t}$ and $T_{t}\chi_{K}$ are
compact and the functions

$t\mapsto\chi_{K}T_{t}\quad\mbox{and}\quad t\mapsto T_{t}\chi_{K}$

are uniformly continuous when restricted to $[t_{K},\infty)$;

  * (2)

for any open neighbourhood $U$ of diagonal in $X^{+}\times X^{+}$, there
exists $t_{U}\geq 1$ such that for all $t>t_{U}$

$\textup{supp}(T_{t})\subseteq U.$

Define $L^{*}(H_{X}^{+};\infty)$, or simply $L^{*}(X^{+};\infty)$, to be the
$C^{*}$-algebra completion of $\mathbb{L}[H_{X}^{+};\infty]$ for the norm

$\|(T_{t})\|=\sup_{t}\|T_{t}\|_{\mathcal{B}(H_{X})}.$

By [27, Proposition 6.6.2], [28, Theorem 3.2] and [17, Theorem 3.4], we have
the following result:

######  Theorem 4.2.

The $K$-homology group $K_{*}(X)$ is isomorphic to the $K$-theory group
$K_{*}(L^{*}(H_{X}^{+};\infty))$, i.e.,

$K_{*}(X)\cong K_{*}(L^{*}(H_{X}^{+};\infty))$

and $K_{*}(L^{*}(H_{X}^{+};\infty))$ does not depend on the choice of ample
modules up to the canonical equivalence.

Let $H_{X}^{+}$ be a geometric module. Let $\mathbb{L}_{0}[H_{X}^{+};\infty]$
be the collection of all $(T_{t})\in\mathbb{L}[H_{X}^{+};\infty]$ such that
for any compact subset $K\subseteq X$ there exists $t_{K}\geq 0$ such that for
all $t\geq t_{K}$,

$\chi_{K}T_{t}=T_{t}\chi_{K}=0.$

It is not difficult to see that $\mathbb{L}_{0}[H_{X}^{+};\infty]$ is a
$*$-ideal in $\mathbb{L}[H_{X}^{+};\infty]$. Let $L^{*}_{0}(H_{X}^{+};\infty)$
be the closure of $\mathbb{L}_{0}[H_{X}^{+};\infty]$ inside
$L^{*}(H_{X}^{+};\infty)$, let

$L^{*}_{Q}(H_{X}^{+};\infty)=L^{*}(H_{X}^{+};\infty)/L^{*}_{0}(H_{X}^{+};\infty)$

be the corresponding quotient $C^{*}$-algebra. Using an Eilenberg swindle, we
have the following result as in [27, Lemma 6.4.11]

######  Lemma 4.3.

The quotient map $L^{*}(H_{X}^{+};\infty)\to L^{*}_{Q}(H_{X}^{+};\infty)$
induces an isomorphism on $K$-theory. $\square$

To see the functoriality, we still need some preparations:

######  Definition 4.4.

Let $K$ be a closed subspace of $X$, and let $H_{X}^{+}$ be an $X^{+}$-module.
Define $\mathbb{L}[H^{+}_{X};K^{+}]$ to be the subset of
$\mathbb{L}[H_{X}^{+};\infty]$ consisting of $(T_{t})$ such that for any open
subset $U$ of $X^{+}\times X^{+}$ that contains $K^{+}\times K^{+}$, there
exists $t_{U}$ such that for all $t\geq t_{U}$

$\textup{supp}(T_{t})\subseteq U$

Define $L^{*}(H^{+}_{X};K^{+})$, or simply $L^{*}(X^{+};K^{+})$, to be the
closure of $\mathbb{L}[H^{+}_{X};K^{+}]$ inside
$\mathbb{L}[H_{X}^{+};\infty]$.

Similarly, we shall define
$L^{*}_{0}(H_{X}^{+};K^{+})=L^{*}(H_{X}^{+};K^{+})\cap
L^{*}_{0}(H_{X}^{+};\infty)$ and

$L^{*}_{Q}(H_{X}^{+};K^{+})=L^{*}(H_{X}^{+};K^{+})/L^{*}_{0}(H^{+}_{X};K^{+}).$

It is easy to show that the quotient map $L^{*}(H_{X}^{+};K^{+})\to
L^{*}_{Q}(H_{X}^{+};K^{+})$ induces an isomorphism on the $K$-theory level.
The following lemma has been proved in [27, Lemma 6.3.6].

######  Lemma 4.5.

The inclusion homomorphism from $L^{*}(H_{K}^{+};\infty)$ to
$L^{*}(H_{X}^{+};K^{+})$ induces an isomorphism from
$K_{*}(L^{*}(H_{K}^{+};\infty))$ to $K_{*}(L^{*}(H_{X}^{+};K^{+}))$, i.e.,

$K_{*}(K)\cong K_{*}(L^{*}(X^{+};K^{+}))\cong K_{*}(L^{*}_{Q}(X^{+};K^{+})).$

∎

Let $K\subset X$ be a close subset and $U=X\backslash K$, then we have the
short sequence of topological spaces

$K^{+}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}X^{+}\stackrel{{\scriptstyle
c}}{{\longrightarrow}}U^{+}$

where $i:K^{+}\to X^{+}$ is the inclusion and $c:X^{+}\to U^{+}$ is the
collapse map which is the identity on $U$ and sends $X^{+}\backslash U$ to the
point at infinity in $U^{+}$.

######  Proposition 4.6.

With the notation above, there is a natural sic-term exact sequence

$\textstyle{K_{0}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{K_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{K_{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$

induced by the short exact sequence of $C^{*}$-algebra

$0\to L^{*}_{Q}(X^{+};K^{+})\stackrel{{\scriptstyle
ad(V^{i}_{t})}}{{\longrightarrow}}L^{*}_{Q}(X^{+};\infty)\stackrel{{\scriptstyle
ad(V^{c}_{t})}}{{\longrightarrow}}L^{*}_{Q}(U^{+};\infty)\to 0.$

where $V^{i}_{t}$ and $V^{c}_{t}$ are the covering isometries for $i$ and $c$,
respectively.

###### Proof.

Let $d$ be a bounded metric function on $X^{+}$. Let $Z\subset X$ be a
countable dense subset of $X$ whose intersection with $K$, $U$ is also dense
in $K$, $U$ respectively for each $n\in\mathbb{N}$. Let $H$ be a separable,
infinite-dimensional Hilbert space, and define ample $X^{+}$, $K^{+}$ and
$U^{+}$-module respectively by

$H^{+}_{X}=\ell^{2}(Z,H)\oplus H,\,H^{+}_{K}=\ell^{2}(Z\cap K,H)\oplus
H,\,H^{+}_{U}=\ell^{2}(Z\cap U,H)\oplus H$

Define $v:\ell^{2}(Z\cap K,H)\to\ell^{2}(Z,H)$ to be the canonical inclusion.
Let $u:\ell^{2}(Z\cap K,H)\oplus H\to H$ be an unitary operator and define

$w=id_{\ell^{2}(Z\cap U,H)}\oplus u:H_{X}^{+}=\ell^{2}(Z\cap
U,H)\oplus(\ell^{2}(Z\cap K,H)\oplus H)\to H_{U}^{+}$

Define $V^{i}_{t}=v\oplus id_{H}$ and $V^{c}_{t}=w$. It is easy to see
$V_{t}^{i}$ and $V_{t}^{c}$ are the covering isometries for $i$ and $c$,
respectively.

Notice that $L^{*}_{Q}(X^{+},K^{+})$ is a two-side ideal of
$L^{*}_{Q}(X^{+},\infty)$ and $ad(V_{t}^{i})$ is actually the inclusion. Thus
it suffices to prove

$\frac{L^{*}_{Q}(X^{+},\infty)}{L^{*}_{Q}(X^{+},K^{+})}\cong
L^{*}_{Q}(U^{+},\infty).$

Let $(C_{n})$ be an increasing sequence of compact subsets of $U$ whose union
is all of $U$. Let $\chi:\mathbb{R}_{+}\to B(U)$ be a continuous map from
$\mathbb{R}_{+}$ to the set of all bounded Borel functions on $U$ defined by

$\chi(t)=(n+1-t)\chi_{C_{n}}+(t-n)\chi_{C_{n+1}}$

if $t\in[n,n+1)$, where $\chi_{C_{n}}$ is the characteristic function of
$C_{n}$. Provisionally define maps

$\Phi:\frac{L^{*}_{Q}(X^{+},\infty)}{L^{*}_{Q}(X^{+},K^{+})}\to
L^{*}_{Q}(U^{+},\infty),\quad[[T_{t}]]\mapsto[ad(V^{c}_{t})(T_{t})]$

and

$\Psi:L^{*}_{Q}(U^{+},\infty)\to\frac{L^{*}_{Q}(X^{+},\infty)}{L^{*}_{Q}(X^{+},K^{+})},\quad[T_{t}]\mapsto[[\chi(t)T_{t}\chi(t)]],$

where $\chi(t)T_{t}\chi(t)$ can be view as a operator on
$\mathcal{B}(H_{X}^{+})$ by viewing $\ell^{2}(Z\cap U,H)$ as a subspace of
$H^{+}_{X}$ for each $t\in\mathbb{R}_{+}$. It is easy to see
$ad(V_{t}^{c})(T_{t})$ and $\chi(t)T_{t}\chi(t)$ satisfy the conditions in
Definition 4.1.

First, we show that $\Phi$ and $\Psi$ are well-defined $*$-homomorphisms. Let
$[T_{t}]\in L^{*}_{Q}(X^{+},K^{+})$. For any compact subset $C\subset U$, we
have that $d(C,K^{+})>0$. Take $\varepsilon<d(C,K^{+})$, there exists $T>0$
such that

$\textup{supp}(T_{t})\subset\\{(x,y)\in X\times X\mid
d(x,K^{+})<\varepsilon,d(y,K^{+})<\varepsilon\\},$

for all $t>T$. Then $\chi_{C}(ad(V^{c}_{t})(T_{t}))=\chi_{C}T_{t}=0$ for all
$t>T$, which means that $ad(V_{t})(T_{t})\in L^{*}_{0}(U^{+},\infty)$.
Therefore, $\Phi$ is a well-defined map $*$-homomorphism.

Similarly, one can show $\Psi$ is also well-defined. Indeed, let $T_{t}\in
L^{*}_{0}(U^{+},\infty)$. For any $\varepsilon>0$, denote
$\mathcal{N}_{\varepsilon}(K^{+})=\\{x\in X^{+}\mid
d(x,K^{+})<\varepsilon\\}$. Then
$C=X^{+}\backslash\mathcal{N}_{\varepsilon}(K^{+})\subset U$ is a compact
subset of $U$. By definition, there exists $T>0$ such that
$\chi_{C}T_{t}=T_{t}\chi_{C}=0$ for all $t>T$. Thus, for all $t>T$, we have
that

$\textup{supp}(\chi(t)T_{t}\chi(t))\subset\mathcal{N}_{\varepsilon}(K^{+})\times\mathcal{N}_{\varepsilon}(K^{+}),$

i.e., $\chi(t)T_{t}\chi(t)\in L^{*}_{Q}(X^{+},K^{+})$. By [27, Lemma 6.1.2],
the multiplier of $L^{*}_{Q}(U^{+},\infty)$ defined by $\chi(t)$ is central.
Thus $\Psi$ is a well-defined $*$-homomorphism.

At last, we show that $\Psi$ induces a mutually inverse of $\Phi$. Computing,
for $[T_{t}]\in L^{*}_{Q}(U^{+},\infty)$

$\Phi(\Psi([T_{t}]))=[ad(V_{t}^{c})(\chi(t)T_{t}\chi(t))]=[\chi(t)T_{t}\chi(t)],$

here we view $\chi(t)T_{t}\chi(t)$ as an operator on $H^{+}_{U}$ for each
$t\in\mathbb{R}_{+}$. Notice that for any compact subset $C\subset U$,

$\chi_{C}(T_{t}-\chi(t)T_{t}\chi(t))=\chi_{C}T_{t}-\chi(t)\chi_{C}T_{t}\chi(t).$

As $\lim_{t\to\infty}\textup{Prop}(T_{t})=0$ and
$U=\bigcup_{n\in\mathbb{N}}C_{n}$, there exists $T>0$ and a compact subset
$C^{\prime}\subset U$ such that $\chi_{C}\chi(t)=\chi_{C}$ and
$\chi_{C}T_{t}=\chi_{C}T_{t}\chi_{C^{\prime}}$ for all $t>T$. Thus
$\chi_{C}(T_{t}-\chi(t)T_{t}\chi(t))=0$ for all $t>T$, i.e.,
$[T_{t}]=[\chi(t)T_{t}\chi(t)]$.

On the other hand, for
$[[T_{t}]]\in\frac{L^{*}_{Q}(X^{+},\infty)}{L^{*}_{Q}(X^{+},K^{+})}$,

$\Psi(\Phi([[T_{t}]]))=[[\chi(t)ad(V^{c}_{t})(T_{t})\chi(t)]]=[\chi(t)T_{t}\chi(t)],$

here we view $\chi(t)T_{t}\chi(t)$ as an operator on $H^{+}_{X}$ for each
$t\in\mathbb{R}_{+}$. For any $\varepsilon>0$, let
$C=X^{+}\backslash\mathcal{N}_{\varepsilon}(K^{+})\subset U$ be a compact
subset of $U$. Then by a similar argument above

$\chi_{C}(T_{t}-\chi(t)T_{t}\chi(t))=0$

for sufficiently large $t$, this completes the proof. ∎

Let $X$ be a unbounded proper metric space. Fix a based point $x_{0}\in X$,
and let $K_{R}=\\{x\in X\mid d(x,x_{0})\leq R\\}$. Denoted by
$U_{R}=X\backslash K_{R}$. Thus we have the following commuting diagram by
[27, Proposition B.2.3]:

$\textstyle{K_{0}(K_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{K_{0}(U_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(U_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{K_{1}(K_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$

where $i_{*}$ is induced by the natural inclusion $i:K_{R}\to X$ and $j^{*}$
is induced by the collapse $K_{R}$ to $\infty$ map $c:X^{+}\to U_{R}^{+}$ .

If $R<R^{\prime}\in\mathbb{R}$, we denote $i_{RR^{\prime}}:K_{R}\to
K_{R^{\prime}}$ and $j_{RR^{\prime}}:U_{R^{\prime}}\to U_{R}$ the inclusion
map. Let $c_{RR^{\prime}}:U^{+}_{R}\to U^{+}_{R^{\prime}}$ be the collapse map
that is identity on $U$ and sends $U^{+}\backslash U_{R^{\prime}}$ to the
point at infinity in $U^{+}$. Then we have the following commuting diagram

$\textstyle{K_{R}\sqcup\\{\infty\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{i_{RR^{\prime}}}$$\textstyle{X^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{U_{R}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{RR^{\prime}}}$$\textstyle{K_{R^{\prime}}\sqcup\\{\infty\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{U_{R}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$
(3)

Taking the direct limit as $R$ tends to infinity, we get a commutative diagram

$\textstyle{\lim\limits_{R\to\infty}K_{0}(K_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\lim\limits_{R\to\infty}K_{0}(U_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lim\limits_{R\to\infty}K_{1}(U_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\lim\limits_{R\to\infty}K_{1}(K_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$

######  Definition 4.7.

The $K$-homology at infinity of $X$, denote by $K_{*}^{\infty}(X)$, is defined
to be the group $\lim\limits_{R\to\infty}K_{*}(U_{R})$.

For any locally compact, secondly countable, Hausdorff space $X$, we can
define a unbounded proper metric on $X$. We can still define the $K$-homology
at infinity for $X$ according to this metric. By the definition, we can see
that $K^{\infty}_{*}(X)$ does not depend on the choice of the metric.
Moreover, if $X$ is compact, then $K^{\infty}_{*}(X)$ is zero group. Recall
the definition of the representable $K$-homology:

######  Definition 4.8 ([27]).

Let $H_{X}$ be an $X$ module. Define $R\mathbb{L}[H_{X}]$ to be the collection
of all elements $(T_{t})$ of $\mathbb{L}[H^{+}_{X},\infty]$ such that there
exists a compact subset $K$ of $X$ and $t_{K}\geq 1$ such that

$T_{t}=\chi_{K}T_{t}\chi_{K}$

for all $t\geq t_{K}$. Define $RL^{*}(H_{X})$ to be the completion of
$R\mathbb{L}[H_{X}]$ for the norm

$\|(T_{t})\|=\sup_{t}\|T_{t}\|_{\mathcal{B}(H_{X})}.$

The representable $K$-homology of $X$, denoted by $RK_{*}(X)$, is defined to
be the $K$-theory group

$RK_{*}(X):=K_{*}(RL^{*}(H_{X}))$

######  Remark 4.9.

It is obvious that $RL^{*}(H_{X})$ is a closed ideal of
$L^{*}(H_{X}^{+},\infty)$. Notice that $\lim\limits_{R\to\infty}K_{*}(K_{R})$
is actually the representable $K$-homology [27, Proposition 9.4.7]. Thus the
diagram can be rewrite as following:

$\textstyle{RK_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{K_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{K^{\infty}_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\infty}_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{RK_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{i_{*}}$

Let $H_{X}$ be an ample $X$-module, $L^{*}(H_{X}^{+},\infty)$ and
$RL^{*}(H_{X})$ be as above. Define $L^{*}_{\infty}(H_{X})$ to be the quotient
algebra $L^{*}(H_{X}^{+},\infty)/RL^{*}(H_{X})$.

######  Proposition 4.10.

The $K$-homology at infinity $K_{*}^{\infty}(X)$ is isomorphic to the
$K$-theory group of $L^{*}_{\infty}(H_{X})$, i.e.,

$K_{*}^{\infty}(X)\cong K_{*}(L^{*}_{\infty}(H_{X})).$

###### Proof.

Let $Z\subset X$ be a countable dense subset of $X$ whose intersection with
$K_{n}$, $U_{n}$ is also dense in $K_{n}$, $U_{n}$ respectively for each
$n\in\mathbb{N}$. Let $H$ be a separable, infinite-dimensional Hilbert space,
and define ample $X^{+}$, $K_{n}^{+}$ and $U^{+}_{n}$-module respectively by

$H^{+}_{X}=\ell^{2}(Z,H)\oplus H,\,H^{+}_{K_{n}}=\ell^{2}(Z\cap K_{n},H)\oplus
H,\,H^{+}_{U_{n}}=\ell^{2}(Z\cap U_{n},H)\oplus H$

For each $n,n^{\prime}\in\mathbb{N}$, assume that $n<n^{\prime}$, define
$v_{n}:\ell^{2}(Z\cap K_{n},H)\to\ell^{2}(Z,H)$ and
$v_{nn^{\prime}}:\ell^{2}(Z\cap K_{n},H)\to\ell^{2}(Z\cap K_{n}^{\prime},H)$
to be the canonical inclusion. Let $u_{n}:\ell^{2}(Z\cap K_{n},H)\oplus H\to
H$ and $u_{nn^{\prime}}:\ell^{2}(U_{n}\cap K_{n^{\prime}},H)\oplus H\to H$ be
unitaries and define

$w_{n}=id_{\ell^{2}(Z\cap U_{n},H)}\oplus u_{n}:H_{X}^{+}=\ell^{2}(Z\cap
U_{n},H)\oplus(\ell^{2}(Z\cap K_{n},H)\oplus H)\to H_{U_{n}}^{+}$

and

$w_{nn^{\prime}}=id_{\ell^{2}(Z\cap U_{n^{\prime}},H)}\oplus
u_{nn^{\prime}}:H_{U_{n}}^{+}=\ell^{2}(Z\cap
U_{n^{\prime}},H)\oplus(\ell^{2}(U_{n}\cap K_{n^{\prime}},H))\to
H_{U_{n^{\prime}}}^{+}$

Define $(V_{t})_{n}=v_{n}$, $(V_{t})_{nn^{\prime}}=v_{nn^{\prime}}$,
$(W_{t})_{n}=w_{n}$ and $(W_{t})_{nn^{\prime}}=w_{nn^{\prime}}$. Then we have
the following commuting diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{*}_{Q}(X^{+},K^{+}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad((V_{t})_{n})}$$\scriptstyle{ad((V_{t})_{nn^{\prime}})}$$\textstyle{L^{*}_{Q}(X^{+},\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad((W_{t})_{n})}$$\textstyle{L^{*}_{Q}(U_{n}^{+},\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad((W_{t})_{nn^{\prime}})}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{*}_{Q}(X^{+};K^{+}_{n^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad((V_{t})_{n^{\prime}})}$$\textstyle{L^{*}_{Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ad((W_{t})_{n^{\prime}})}$$\textstyle{L^{*}_{Q}(U_{n^{\prime}}^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

Taking the direct limit as $n$ tends to infinity, by [27, Proposition 9.4.7],
we have that

$0\to RL^{*}_{Q}(X^{+};\infty)\to
L^{*}_{Q}(X^{+};\infty)\stackrel{{\scriptstyle
J}}{{\longrightarrow}}\lim\limits_{n\to\infty}L^{*}_{Q}(U^{+}_{n};\infty)\to
0.$

We denote the map

$J=\lim_{n\to\infty}ad((W_{t})_{n}):L^{*}_{Q}(X^{+};\infty)\to\lim\limits_{n\to\infty}L^{*}_{Q}(U^{+}_{n};\infty).$

For any $T_{t}\in R\mathbb{L}[X^{+};\infty]$, there exists a compact subset
$K\subset X$ and $t_{K}>1$ such that $T_{t}=\chi_{K}T_{t}\chi_{K}$ for all
$t\geq t_{K}$. Set $N\in\mathbb{N}$ such that $K\subset K_{N}$, then
$[ad((W_{t})_{N})(T_{t})]=0\in L^{*}_{Q}(U_{N}^{+},\infty)$. Thus
$J([T_{t}])=0$ for all $[T_{t}]\in RL^{*}_{Q}(X^{+},\infty)$. Then $J$
descends to a map on the quotient:

$J_{\infty}:L^{*}_{\infty,Q}(X^{+};\infty)\to\lim\limits_{n\to\infty}L^{*}_{Q}(U^{+}_{n};\infty).$

Thus we have the following commuting diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL^{*}_{Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{*}_{Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{*}_{\infty,Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{\infty}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL^{*}_{Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{*}_{Q}(X^{+};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lim\limits_{n\to\infty}L^{*}_{Q}(U^{+}_{n};\infty)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Thus $(J_{\infty})_{*}:K^{\infty}_{*}(X)\to K_{*}(L^{*}_{\infty}(H_{X}))$ is
an isomorphism by Five Lemma, as desire. ∎

For technical convenience, we will consider the following version of
localization algebra and representable localization algebra (c.f. [28, 29]).
Let $H_{0}$ be an infinite dimensional Hilbert space, and $Z\subseteq X$ be a
countable dense subset and $\mathbb{C}_{f}[X]$ be as in Definition 2.2.

######  Definition 4.11 ([28]).

The algebraic localization algebra, denoted by $\mathbb{C}_{L}[X]$, is the
$*$-algebra of all uniformly bounded and uniformly continuous functions

$g:[0,\infty)\to\mathbb{C}_{f}[X]\subseteq\mathbb{C}[X]$

such that the family $(g(t))_{t\in[0,\infty)}$ satisfy the conditions in
Definition 2.2 with uniform constants and there exists a bounded function
$R(t):[0,\infty)\to[0,\infty)$ with $\lim_{t\to\infty}R(t)=0$ such that

$(g(t))(x,y)=0\mbox{ whenever }d(x.y)>R(t)$

for all $x,y\in Z$ and $t\in[0,\infty)$.

Define $C^{*}_{L}(X)$ to be the completion of $\mathbb{C}_{L}[X]$ with respect
to the norm

$\|g\|=\sup_{t\in\mathbb{R}_{+}}\|g(t)\|.$

######  Theorem 4.12 ([28, 17]).

Suppose that $X$ is a proper metric space with bounded geometry, then there
exists a local index map $\mu_{L}:K_{*}(P_{d}(X))\to
K_{*}(C^{*}_{L}(P_{d}(X)))$ which is an isomorphism.

Consequently, if $X$ is a discrete metric space with bounded geometry, we have
the following commuting diagram:

$\textstyle{\lim\limits_{d\to\infty}K_{*}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\mu_{L}}$$\textstyle{\lim\limits_{d\to\infty}K_{*}(C^{*}_{L}(P_{d}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev_{*}}$$\textstyle{K_{*}(C^{*}(X)),}$

where the evaluation homomorphism $ev:C^{*}_{L}(X)\to C^{*}(X)$ is defined by
$ev(f(t))=f(0)$ for all $f\in C^{*}_{L}(X)$. Then, to proof the coarse Novikov
conjecture for $X$, it suffices to prove the evaluation homomorphism

$ev_{*}:\lim_{d\to\infty}K_{*}(C^{*}_{L}(P_{d}(X)))\to\lim_{d\to\infty}K_{*}(C^{*}(P_{d}(X)))$

is an injection.

######  Definition 4.13.

The representable localization algebra of $X$, denoted by $RC^{*}_{L}(X)$, is
the closure of the collection of all elements $g(t)$ of $C^{*}_{L}(X)$ such
that there exists a compact subset $K$ of $X$ such that

$g(t)=\chi_{K}g(t)\chi_{K}$

for all $t\in[0,\infty)$. It is not hard to see that $RC^{*}_{L}(X)$ is a two-
sided closed ideal of $C^{*}_{L}(X)$.

The proof of the following lemma is similar with [27, Proposition 9.4.2,
Proposition 6.6.2] and [28, Theorem 3.2].

######  Lemma 4.14.

Let $X$ be a proper metric space. Then we have

$K_{*}(L^{*}(H_{X}^{+},\infty))\cong K_{*}(C^{*}_{L}(X))\quad\mbox{and}\quad
K_{*}(RL^{*}(H_{X}))\cong K_{*}(RC^{*}_{L}(X)).$

As a consequence, the $K$-homoloogy at infinity $K^{\infty}_{*}(X)$ is
isomorphic to $K_{*}(C^{*}_{L}(X)/RC^{*}_{L}(X))$. $\square$

Next, we will introduce the Roe algebra at infinity for a proper metric space
$X$. Fix $x_{0}\in X$, let $K_{n}=B(x_{0},n)$ and $U_{n}=X\backslash K_{n}$.
Let $Z$ be a countable subset of $X$.

######  Definition 4.15.

The algebraic Roe algebra at infinity, denoted by $\mathbb{C}_{\infty}[X]$, is
the set of all equivalence classes $[T]$, where $T\in\mathbb{C}_{f}(X)$ and
the equivalence relation $\sim$ on $[T]$ is defined by $T\sim S$ if and only
if

$\lim_{n\to\infty}\sup_{x,y\in U_{n}}\|T(x,y)-S(x,y)\|=0.$

The algebraic operation of $\mathbb{C}_{\infty}[X]$ is given by the usual
matrix operations of $\mathbb{C}_{f}[X]$. Define the Roe algebra at infinity
$C^{*}_{\infty}(X)$ to be the completion of $\mathbb{C}_{\infty}[X]$ with
respect to the norm

$\|[T]\|=\limsup_{n\to\infty}\|\chi_{U_{n}}T\|.$

One can see that the Roe algebra at infinity is also invariant up to coarse
equivalence, so $C^{*}_{\infty}(X)$ is isomorphic to
$C^{*}_{\infty}(P_{d}(X))$ for each $d\geq 0$. If $X$ is bounded, then
$C^{*}_{\infty}(X)=0$.

######  Definition 4.16.

The localization algebra at infinity of $X$, denoted by
$\mathbb{C}_{L,\infty}[X]$, is defined to be the $*$-algebra of all bounded
and unformly norm-continuous functions

$g:[0,\infty)\to\mathbb{C}_{\infty}[X]$

such that $g$ is of the form $g(t)=[T_{t}]$ with $T_{t}\in\mathbb{C}[X]$ where
the family $(T_{t})_{t\in\mathbb{R}_{+}}$ satisfies the conditions in
Definition 2.2 with uniform constants and there exists a bounded function
$R(t):[0,\infty)\to[0,\infty)$ with $\lim_{t\to\infty}R(t)=0$ such that

$(T_{t})(x,y)=0\mbox{ whenever }d(x.y)>R(t)$

for all $x,y\in Z$ and $t\in[0,\infty)$.

Define $C^{*}_{L,\infty}(X)$ to be the completion of
$\mathbb{C}_{L,\infty}[X]$ with respect to the norm

$\|g\|=\sup_{t\in\mathbb{R}_{+}}\|g(t)\|.$

There exists a canonical quoitent map $C^{*}_{L}(X)\to C^{*}_{L,\infty}(X)$
defined by $g\mapsto g^{\prime}$ where $g^{\prime}(t)=[g(t)]$ for each
$t\in\mathbb{R}_{+}$. One can also see that $g^{\prime}=0$ for any $g\in
RC^{*}_{L}(X)\cap\mathbb{C}_{L}[X]$ as $g(t)$ is uniformly supported in some
compact subset of $X$. Then we have a homomorphism

$q:\frac{C^{*}_{L}(X)}{RC^{*}_{L}(X)}\to C^{*}_{L,\infty}(X)$

which induces a homomorphism on $K$-theory

$q_{*}:K_{*}^{\infty}(X)\to K_{*}(C^{*}_{L,\infty}(X)).$

######  Theorem 4.17.

Suppose $X$ is a discrete metric space with bounded geometry. Then

$q_{*}:K_{*}^{\infty}(P_{d}(X))\to K_{*}(C^{*}_{L,\infty}(P_{d}(X)))$

is an isomorphism for each $d\geq 0$.

###### Proof.

By using an Eilenberg swindle argument as [28, Theorem 3.4], one can show that
both groups above are invariant under strongly Lipschitz homotopies and both
groups admits a Mayer-Vietories sequence. Then it is suffices to prove the
theorem for the case when $P_{d}(X)$ is $0$-dimensional, which obviously holds
as both groups are trivial. ∎

##  5 Reduction to the coarse Novikov conjecture at infinity

In this section, we discuss the strategy to prove the main theorem. As general
metric spaces (such as warped cones) are always not of the form of coarse
disjoint unions of a sequence of bounded subsets, we can not use the
techniques in [4] directly. We will show that there exists an assmbly map at
infinity from the $K$-homology group at infinity to the $K$-theory group of
the Roe algebra at infinity, and the coarse Novikov conjecture can always be
reduced to verifying the assembly map at infinity to be injective whether the
space is of the form of coarse disjoint union or not.

###  5.1 The case of coarse disjoint unions

In this section, we assume that $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$ is a
coarse disjoint union of a sequence of uniformly bounded geometry, finite
metric spaces such that $\\#X_{n}\to\infty$ as $n\to\infty$. Actually, this
situation has been discussed in Section 4 of [4].

######  Lemma 5.1.

Let $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$ be a coarse disjoint union of a
sequence of uniformly bounded geometry, finite metric spaces such that
$\\#X_{n}\to\infty$ as $n\to\infty$. Then for sufficiently large $d$, we have
that

$K_{*}^{\infty}(P_{d}(X))=\frac{\prod_{n=0}^{\infty}K_{*}(P_{d}(X_{n}))}{\bigoplus_{n=0}^{\infty}K_{*}(P_{d}(X_{n}))}.$

###### Proof.

For any $d\geq 0$, there exists $N_{d}\in N$ large enough such that
$d(X_{n},X_{m})>d$ for $n,m\in N_{d}$. Let
$X_{N_{d}}=\bigcup^{N_{d}-1}_{n=0}X_{n}$, then we have that
$P_{d}(X)=P_{d}(X_{N_{d}})\cup\mathop{\sqcup}\limits_{n>N_{d}}P_{d}(X_{n})$.
Then the maps $RK_{*}(P_{d}(X_{n}))\to RK_{*}(P_{d}(X))$ and
$K_{*}(P_{d}(X_{n}))\to K_{*}(P_{d}(X))$ induced by the inclusions $X_{n}\to
X$ induce isomorphisms

$RK_{*}(P_{d}(X))\cong
K_{*}(P_{d}(X_{N_{d}}))\oplus\bigoplus_{n>N_{d}}K_{*}(P_{d}(X_{n}))$

and

$K_{*}(P_{d}(X))\cong
K_{*}(P_{d}(X_{N_{d}}))\oplus\prod_{n>N_{d}}K_{*}(P_{d}(X_{n})).$

The inclusion
$\bigoplus_{n>N_{d}}K_{*}(P_{d}(X_{n}))\to\prod_{n>N_{d}}K_{*}(P_{d}(X_{n}))$
is an injection, i.e., $RK_{*}(P_{d}(X))\to K_{*}(P_{d}(X))$ is an injection.
Thus, we have the short exact sequence

$0\to RK_{*}(P_{d}(X))\to K_{*}(P_{d}(X))\to K^{\infty}_{*}(P_{d}(X))\to 0$

This means that

$K_{*}^{\infty}(P_{d}(X))=\frac{K_{*}(P_{d}(X))}{RK_{*}(P_{d}(X))}=\frac{\prod_{n=0}^{\infty}K_{*}(P_{d}(X_{n}))}{\bigoplus_{n=0}^{\infty}K_{*}(P_{d}(X_{n}))},$

which completes the proof. ∎

For each $d>0$. let $Z_{d}\subset P_{d}(X)$ be a countable dense subset, and
let $Z_{d,n}=Z_{d}\cap P_{d}(X_{n})$ for any $d\geq 0$, $n\in\mathbb{N}$. We
assume that $Z_{d,n}\subset Z_{d,n^{\prime}}$, if $d^{\prime}<d$. In this
case, the Roe algebra at infinity can be also described as follow:

######  Definition 5.2 ([4]).

For each $d\geq 0$, define
$\mathbb{C}_{u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$ to be the set of all
equivalence classes $T=[(T^{(0)},\cdots,T^{(n)},\cdots)]$ of sequences
$(T^{(0)},\cdots,T^{(n)},\cdots)$ described as follows:

  * (1)

$\\{T^{(n)}\\}_{n\in\mathbb{N}}$ is a family of uniformly bounded functions,
where $T^{(n)}$ is a function from $Z_{d,n}\times Z_{d,n}$ to $\mathcal{K}$
for all $n\in\mathbb{N}$;

  * (2)

for any bounded subset $B\subset P_{d}(X_{n})$, we have that

$\\#\\{(x,y)\in B\times B\cap Z_{d,n}\times Z_{d,n}\mid T^{(n)}(x,y)\neq
0\\}<\infty;$

  * (3)

there exists $L>0$ such that

$\\#\\{y\in Z_{d,n}|T^{(n)}(x,y)\neq 0\\}<L,\qquad\\#\\{y\in Z_{d,n}\mid
T^{(n)}(y,x)\neq 0\\}<L$

for all $x\in Z_{d,n}$, $n\in\mathbb{N}$;

  * (4)

there exists $R>0$ such that $T^{(n)}(x,y)=0$ whenever $d(x,y)>R$ for $x,y\in
Z_{d,n}$, $n\in\mathbb{N}$. The least such $R$ is called the propagation of
the sequence $(T^{(0)},\cdots,T^{(n)},\cdots)$.

The equivalence relation $\sim$ on these sequences is defined by

$(T^{(0)},\cdots,T^{(n)},\cdots)\sim(S^{(0)},\cdots,S^{(n)},\cdots)$

if and only if

$\lim_{n\to\infty}\sup_{x,y\in
Z_{d,n}}\|T^{(n)}(x,y)-S^{(n)}(x,y)\|_{\mathcal{K}}=0.$

By viewing $T^{(n)}$ as $Z_{d,n}\times Z_{d,n}$ matrices, the product
structure for $\mathbb{C}_{u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$ is
defined as the usual matrix operations. Define
$C^{*}_{u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$ to be the completion of
$\mathbb{C}_{u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$ with respect to the
norm

$\|T\|=\limsup_{n\to\infty}\|T^{(n)}\|,$

where each operator $T^{(n)}$ is viewed as an element of the Roe algebra
$C^{*}(P_{d}(X_{n}))$.

The following notion of localization algebra has its origin in [28], we shall
recall its relation with $K$-homology at infinity.

######  Definition 5.3.

Let $\mathbb{C}_{L,u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$ be the set of
all bounded, uniformly norm-continuous functions

$g:\mathbb{R}_{+}\to\mathbb{C}_{u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$

such that $g(t)$ is of the form $g(t)=[g^{(0)}(t),\cdots,g^{(n)}(t),\cdots]$
and satisfies that there exists a bounded function
$R(t):\mathbb{R}_{+}\to\mathbb{R}_{+}$ with $\lim_{t\to\infty}R(t)=0$ such
that $(g^{(n)}(t))(x,y)=0$ whenever $d(x,y)>R(t)$ and $n\in\mathbb{N}$;

The localization algebra at infinity
$C^{*}_{L,u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$ is defined to be the
norm completion of $\mathbb{C}_{L,u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$,
where $C^{*}_{L,u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$ is endowed with
the norm

$\|g\|_{\infty}=\sup_{t\in\mathbb{R}_{+}}\|g(t)\|.$

One can see that the $C^{*}_{u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$
coincides with $C^{*}_{\infty}(X)$ and
$C^{*}_{L,u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$ coincides with
$C^{*}_{L,\infty}(P_{d}(X))$.

For each $d>0$, there is a $*$-homomorphism

$\Phi:\mathbb{C}_{f}[P_{d}(X)]\to\mathbb{C}_{u,\infty}[(P_{d}(X_{n}))_{n\in\mathbb{N}}]$

defined by $\Phi(T)=[(\Phi^{(0)}(T),\cdots,\Phi^{(n)}(T),\cdots)]$ for
$T\in\mathbb{C}_{f}[P_{d}(X)]$, with

$\Phi^{(n)}(T)(x,y)=\left\\{\begin{aligned} &0,&&\mbox{if }n<N_{R}\\\
&T|_{Z_{d,n}\times Z_{d,n}},&&\mbox{if }n\geq N_{R},\end{aligned}\right.$

where $R=\textup{Prop}(T)$ and $N_{R}\in\mathbb{N}$ is large enough such that

$d_{P_{d}(X)}\left(P_{d}(X_{n}),P_{d}\left(\bigsqcup_{i=0}^{n-1}X_{i}\right)\right)>R,$

for all $n\geq N_{R}$. Then $\Phi$ extends to a $C^{*}$-homomorphism
$\Phi:C^{*}(P_{d}(X))\to C^{*}_{u,\infty}((P_{d}(X_{n}))_{n\in\mathbb{N}})$,
see [4, Theorem 4.5].

For any $d\geq 0$, there exists $N_{d}\in N$ large enough such that
$d(X_{n},X_{m})>d$ for $n,m\in N_{d}$. Let
$X_{N_{d}}=\bigcup^{N_{d}-1}_{n=0}X_{n}$. As
$P_{d}(X)=P_{d}(X_{N_{d}})\bigsqcup\left(\sqcup_{n\geq
N_{d}}P_{d}(X_{n})\right)$, we have that

$K_{*}(P_{d}(X))=K_{*}(P_{d}(X_{N_{d}}))\bigoplus\prod_{n=N_{d}}^{\infty}K_{*}(P_{d}(X_{n})).$

By the definition of the assembly maps, we have the following commutative
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}(P_{d}(X_{N_{d}}))\oplus\bigoplus_{n=N_{d}}^{\infty}K_{*}(P_{d}(X_{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}(\mathcal{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev_{*}}$$\textstyle{K_{*}(C^{*}(P_{d}(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{*}}$$\textstyle{K^{\infty}_{*}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev_{\infty,*}}$$\textstyle{K_{*}(C^{*}_{u,\infty}(P_{d}(X_{n})))}$$\textstyle{0}$

where $ev_{\infty,*}$ is induced by the evaluation map

$ev_{\infty}:C^{*}_{L,\infty}(P_{d}(X))\to C^{*}_{\infty}(X),\qquad g\mapsto
g(0).$

By [16, Remark 2.12], one has that $K_{*}(\mathcal{K})\to
K_{*}(C^{*}(P_{d}(X)))$ is an injection.

Passing to inductive limit as $d\to\infty$, the top horizontal arrow is an
isomorphism for the following reason. An element in the sum, as a finite
sequence, is supported on summands below some fixed $m$ and, as $d\to\infty$,
will eventually be absorbed into the first term on a single simplex. Thus, to
prove $\mu$ is injective, it suffices to prove $\mu_{\infty}$ is injective by
diagram chasing.

###  5.2 The case of spaces which admit an infinite coarse component

The next notion is introduced by M. Finn-Sell in [8].

######  Definition 5.4 ([8]).

A bounded geometry proper metric space $X$ is said to have an infinite coarse
component if there exists $R>0$ such that $P_{R}(X)$ has an unbounded
connected component. Otherwise, we say $X$ only has finite coarse components.

######  Lemma 5.5.

A bounded geometry proper metric space $X$ has only finite coarse components
if and only if it is a coarse disjoint union of a sequence of finite metric
spaces.

###### Proof.

Let $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$ be a coarse disjoint union of the
sequence of finite metric spaces $\\{X_{n}\\}$. For each $R>0$, there exists
$N>0$ such that $d(X_{i},X_{j})>R$ for all $i,j>N$. Let
$X_{N_{R}}=\bigsqcup_{n=1}^{N}X_{n}$ and we have that
$P_{R}(X)=P_{R}(X_{N_{R}})\sqcup\bigsqcup_{n>N}P_{R}(X_{n})$. Thus $X$ has
only finite coarse components.

On the other hand, let $X$ be a metric space which has only finite coarse
components. Fix $x_{0}\in X$, denote $P_{d}(X)_{x_{0}}$ the connected
component of $P_{d}(X)$ which contains $x_{0}$. Then let

$\begin{split}X_{1}&=P_{2^{1}}(X)_{x_{0}}\cap X;\\\
X_{2}&=(P_{2^{2}}(X)_{x_{0}}\cap X)\backslash X_{1};\\\ \cdots&\\\
X_{n}&=(P_{2^{n}}(X)_{x_{0}}\cap X)\backslash X_{n-1};\\\ \cdots&.\end{split}$

Then we have that $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$. As $X$ has bounded
geometry and finite coarse components, $X_{n}$ is a finite metric space for
each $n\in\mathbb{N}$. Assume that $i,j\in\mathbb{N}$ ($i>j$) and any two
points $x_{i}\in X_{i}$, $x_{j}\in X_{j}$. As $x_{i}\notin
P_{2^{j}}(X)_{x_{0}}$, we have that

$d(x_{i},x_{j})>2^{j},$

which means that $d(X_{i},X_{j})\to\infty$ as $i,j\to\infty$. Thus $X$ is the
coarse disjoint union of the sequence $\\{X_{n}\\}_{n\in\mathbb{N}}$. ∎

######  Lemma 5.6.

Let $X$ be a bounded geometry metirc space which admits an infinite coarse
component. Then

  * (1)

the $K$-theory map $i_{*}:K_{*}(\mathcal{K})\to K_{*}(C^{*}(X))$ induced by
the canonical inclusion $i:\mathcal{K}\to C^{*}(X)$ is the zero map, where
$\mathcal{K}=\mathcal{K}(H_{X})$ is the algebra of all compact operators on
the geometric module $H_{X}$;

  * (2)

the map $\lim_{d\to\infty}RK_{*}(P_{d}(X))\to\lim_{d\to\infty}K_{*}(P_{d}(X))$
induced by the canonical inclusion $RC_{L}^{*}(P_{d}(X))\to
C^{*}_{L}(P_{d}(X))$ is also the zero map.

###### Proof.

As $X$ has an infinite coarse component and bounded geometry property, then
there exists $R>0$ and $x_{0}\in X$ for which there is a unbounded component
contianing $x_{0}\in P_{R}(X)$. Thus there exists a sequence
$\\{x_{i}\\}_{i=1}^{\infty}\subset X$ such that
$\lim_{i\to\infty}d(x_{i},x_{0})=\infty$ and $d(x_{i},x_{i+1})\leq R$ for all
$i\in\mathbb{N}$. Let $K_{n}=\\{x\in X\mid d(x_{0},x)\leq n\\}$ for all
$n\in\mathbb{N}$. As $X$ is proper metric space, for any compact subset
$K\subset X$, there exists $n\in\mathbb{N}$ such that $K\subset K_{n}$. We can
identify $\mathcal{K}(H_{X})$ by $\lim_{n\to\infty}C^{*}(K_{n})$, where
$C^{*}(K_{n})$ is the Roe algebra of $K_{n}$.

To prove Property (1), we will only need to show that
$(i_{n})_{*}:K_{*}(C^{*}(K_{n}))\to C*(X)$ is the zero map for all
$n\in\mathbb{N}$, where $(i_{n})_{*}$ is induced by the canonical inclusion
$i_{n}:K_{n}\to X$. By the definition of $\\{x_{i}\\}$, we have that
$K_{n}\cup\\{x_{i}\\}$ is coarse equivalent to $\mathbb{R}_{+}$. Thus, $i_{n}$
can be seen the composition $K_{n}\to K_{n}\cup\\{x_{n}\\}\to X$. Notice that
$K_{*}(C^{*}(\mathbb{R}_{+}))=0$, then the following diagram commutes

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116.18979pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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52.39803pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(K_{n}\cup\\{x_{i}\\}))\cong
K_{*}(C^{*}(\mathbb{R}_{+}))=0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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Then $(i_{n})_{*}$ is the zero map. The canonical inclusion $K_{n}\to K_{n+1}$
induces an isomorphism $K_{*}(\mathcal{K})\to K_{*}(\mathcal{K})$. Thus we
have that

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.97581pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\raise-1.75pt\hbox{$\scriptstyle{(i_{n})_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 55.97581pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(K_{n+1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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55.97581pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
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55.97581pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(X))}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Therefore, $i_{*}:K_{*}(\mathcal{K})=\lim_{n\to\infty}K_{*}(K_{n})\to
K_{*}(C^{*}(X))$ is the zero map.

We will next prove Property (2). Notice that
$RK_{*}(P_{d}(X))=\lim_{n\to\infty}K_{*}(P_{d}(K_{n}))$. Then it suffices to
show

$\lim_{d\to\infty}\lim_{n\to\infty}K_{*}(P_{d}(K_{n}))\to\lim_{d\to\infty}K_{*}(P_{d}(X))$

is the zero map for all $n\in\mathbb{N}$, where the map is induced by the
canonical inclusion $P_{d}(K_{n})\to P_{d}(X)$ for each $n\in\mathbb{N}$ and
$d>0$. Notice that we have the following commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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52.86865pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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for all $d^{\prime}>d$, $n^{\prime}>n$, where the arrows are all given by the
inclusions. Thus we have that

$\lim_{d\to\infty}RK_{*}(P_{d}(X))\cong\lim_{d\to\infty}\lim_{n\to\infty}K_{*}(P_{d}(K_{n}))\cong\lim_{n\to\infty}\lim_{d\to\infty}K_{*}(P_{d}(K_{n})).$

Now it suffices to show that, for each $n\in\mathbb{N}$,
$K_{*}(P_{d}(K_{n}))\to K_{*}(P_{d}(X))$ is the zero map for sufficiently
large $d$.

For given $n\in\mathbb{N}$, let $R_{n}$ be a real number such that
$\textup{diam}(K_{n})<R_{n}$ and $d(x_{i},x_{i+1})<R_{n}$ for all $i\geq 0$.
For each $d>R_{n}$, the inclusion $P_{d}(K_{n})\to P_{d}(X)$ can be seen as
the composition

$P_{d}(K_{n})\to P_{d}(K_{n}\cup\\{x_{i}\\})\to P_{d}(X).$

Notice that $P_{d}(K_{n}\cup\\{x_{i}\\})$ is homotopy equivalent to
$\mathbb{R}_{+}$ and $K_{*}(\mathbb{R}_{+})=0$. Thus we have the following
commuting diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
52.42867pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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K_{*}(\mathbb{R}_{+})=0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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This shows that
$\lim_{d\to\infty}K_{*}(P_{d}(K_{n}))\to\lim_{d\to\infty}K_{*}(X)$ is the zero
map for each $n\in\mathbb{N}$. Taking the direct limit as $n$ tends to
infinity, this completes the proof. ∎

Let $g(t)\in RC^{*}_{L}(X)$ be a function such that there exists a compact
subset $K\subset X$ such that $g(t)=\chi_{K}g(t)\chi_{K}\in\mathcal{K}$ for
all $t\in[0,\infty)$. Consider the evaluation map $ev:C^{*}_{L}(X)\to
C^{*}(X)$ defined by $ev(g)=g(0)$. Notice that the restriction of $ev$ to
$RC^{*}_{L}(X)$ is a $*$-homomorphism $RC^{*}_{L}(X)\to\mathcal{K}$.

Define $\Phi:C^{*}(X)\to C^{*}_{\infty}(X)$ by $T\mapsto[T]$ and the
evaluation map at infinity

$ev_{\infty}:C^{*}_{L,\infty}(P_{d}(X))\to C^{*}_{\infty}(P_{d}(X))\quad\text{
by }\quad[g]\mapsto[g(0)].$

Then for each $d\geq 0$, we have the following commuting diagram:

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RC^{*}_{L}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\textstyle{C^{*}_{L}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev}$$\textstyle{C^{*}_{L,\infty}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ev_{\infty}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(P_{d}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{C^{*}_{\infty}(P_{d}(X))}$

and the commuting diagram of $K$-theory:

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