A Twisted Version of Controlled K-theory

Abstract

There are a number of (co-)homology theories on coarse spaces. Controlled operator K-theory is by far the most popular one of them. Our approach is geometric. We study when does the Roe-algebra of a space restrict to a subspace. Then we show the Roe-algebra is a cosheaf on the coarse topology. A result is a Mayer–Vietoris exact sequence in the presence of a coarse cover. We compute examples as an application.

1 Introduction

The K-theory of the Roe-algebra is one of the most popular homological invariants on coarse metric spaces. Meanwhile a new cohomological invariant on coarse spaces recently appeared in [3] which studies sheaf cohomology on coarse spaces.

In this paper,we study the K-theory of the Roe-algebra of a proper metric space X which is introduced in [4, Chapter 6.3]. Note that this theory does not appear as a derived functor as far as we know.

In [5] is studied a coarse excisive property on coarse spaces which we recall now. If $Y\subseteq X$ is a closed subspace then $C^*(Y,X)$ denotes the ideal in $C^*(X)$ which is the norm closure of operators with support near Y. Let $A,B\subseteq X$ be two closed subsets of a proper metric space which are $\omega$-excisive. Then

is a six-term Mayer–Vietoris exact sequence by [5, Sect. 5].

Note our approach can be compared with [8] where it was shown that a quotient $D(X)/C^*(X)$ is a sheaf on the underlying topological space of X.

The paper [3] introduced a Grothendieck topology $X_{\text {ct}}$ associated to a coarse space X. The underlying category of $X_{\text {ct}}$ is the poset of subsets of X and the coverings are finite collections of subsets called coarse covers.

Theorem 13 shows if X is a proper metric space then the association

$$\begin{aligned} U\mapsto C^*({\bar{U}})/\mathbb {K}(\mathcal {H}_U), \end{aligned}$$

for every subset $U\subseteq X$ with restriction maps is a cosheaf on $X_{\text {ct}}$.

Note that in a general setting cosheaves with values in the category of abelian groups $\texttt {Ab}$ do not give rise to a derived functor. In [1] is explained that the dual version of sheafification, cosheafification, does not work in general. Moreover, the category of $C^*$-algebras $\texttt {CStar}$ is not abelian.

Our result gives rise to new computational tools one of which is a new Mayer–Vietoris six-term exact sequence which is Corollary 14: If $U_1,U_2\subseteq X$ are subsets of a proper metric space that coarsely cover a subspace $U\subseteq X$ then

is exact. Here ${\hat{C}}^*(Y)=C^*({\bar{Y}})/\mathbb {K}(\mathcal {H}_Y)$ for $Y\subseteq X$ a subset.

The outline of this paper is as follows: The Chapter 2 discusses cosheaves on coarse spaces. The main part of the study is in Chapters 3 and  4 computes examples.

2 Cosheaves

If X is a metric space, a subset $E\subseteq X\times X$ is called an entourage if

$$\begin{aligned} \sup _{(x,y)\in E}d(x,y)<\infty . \end{aligned}$$

A subset B is bounded if $\sup _{x,y\in B}$d$(x,y)<\infty$. A map $\varphi : X\rightarrow Y$ between metric spaces is called coarse if $\varphi \times \varphi$ maps entourages to entourages and $\varphi ^{-1}$ maps bounded sets to bounded sets. Two maps $\phi ,\psi :X\rightarrow Y$ between metric spaces are close if $\phi \times \psi$ maps the diagonal to an entourage. The coarse category consists of metric spaces as objects and coarse maps modulo close as morphisms.

We recall [3, Definition 45]:

Definition 1

(coarse cover) If X is a metric space and $U\subseteq X$ a subset then a finite family of subsets $U_1,\ldots ,U_n\subseteq U$ is said to coarsely cover U if for every entourage $E\subseteq X^2$ there is a bounded set $B\subseteq X$ such that

$$\begin{aligned} U^2\cap \left( \bigcup _i U_i^2\right) ^c\cap E\subseteq B^2. \end{aligned}$$

Coarse covers determine a Grothendieck topology $X_{ct}$ associated to a metric space X. If $f:X\rightarrow Y$ is a coarse map between metric spaces then there is a morphism of Grothendieck topologies $f^{-1}:Y_{ct}\rightarrow X_{ct}$.

Definition 2

(precosheaf) A precosheaf on ${X}_{ct}$ with values in a category C is a covariant functor $Cat({X}_{ct})\rightarrow C$.

Definition 3

(cosheaf) Let C be a category with finite limits and colimits. A precosheaf $\mathcal {F}$ on ${X}_{ct}$ with values in C is a cosheaf on ${X}_{ct}$ with values in C if for every coarse cover $\{U_i\rightarrow U\}_i$ there is a coequalizer diagram:

$$\begin{aligned} \bigoplus _{ij} \mathcal {F}(U_i\cap U_j)\rightrightarrows \bigoplus _i \mathcal {F}(U_i) \rightarrow \mathcal {F}(U). \end{aligned}$$

(1)

Here the two arrows on the left side relate to the following 2 diagrams:

and

where $\bigoplus$ denotes the coproduct over the index set.

Notation 4

If we write

• $\sum _i a_i \in \bigoplus _i \mathcal {F}(U_i)$ then $a_i$ is supposed to be in $\mathcal {F}(U_i)$

• $\sum _{ij} b_{ij}\in \bigoplus _{ij} \mathcal {F}(U_i\cap U_j)$ then $b_{ij}$ is supposed to be in $\mathcal {F}(U_i\cup U_j)$

Proposition 5

If $\mathcal {F}$ is a precosheaf on ${X}_{ct}$ with values in a category C with finite limits and colimits and for every coarse cover $\{U_i\rightarrow U\}_i$

• (1) and every $a\in \mathcal {F}(U)$ there is some $\sum _i a_i\in \bigoplus _i \mathcal {F}(U_i)$ such that $\sum _i a_i|_U=a$

• (2) and for every $\sum _i a_i\in \bigoplus _i \mathcal {F}(U_i)$ such that $\sum _i a_i|_U=0$ there is some $\sum _{ij} b_{ij}\in \bigoplus _{ij} \mathcal {F}(U_i\cap U_j)$ such that $(\sum _j b_{ij}-b_{ji})|_{U_i}=a_i$ for every i.

then $\mathcal {F}$ is a cosheaf.

Proof

We have to prove that conditions (1) and (2) are equivalent to exactness of the diagram 1. Call the right map $\beta$ and the left map $\alpha$. Then exactness at $\mathcal {F}(U)$ means the map $\alpha$ is surjective. That is condition (1).

Now ${{\,\textrm{im}\,}}(\beta )\subseteq \ker (\alpha )$ always holds. If $\sum _i a_i\in \bigoplus _i \mathcal {F}(U_i)$ then $\sum _ia_i|_U=0$ is equivalent to $\sum a_i\in \ker \alpha$. If $\ker (\alpha )\subseteq {{\,\textrm{im}\,}}(\beta )$ then there exists some $\sum _{ij}b_{ij}\in \bigoplus _{ij}\mathcal {F}(U_i\cap U_j)$ with $\sum _i(\sum _j b_{ij}-b_{ji})|_{U_i}=\beta (\sum _{ij}b_{ij})=\sum _ia_i$. This is condition 2). $\square$

Remark 6

Denote by $\texttt {CStar}$ the category of $C^*$-algebras. According to [6], all finite limits and finite colimits exist in $\texttt {CStar}$.

3 Roe–Calkin Algebra

This exposition uses notation from [4, Chapter 6] which is a standard reference for K-theory of the Roe-algebra. We recall a few of the definitions:

Let X be a proper metric space. A presentation $\rho :C_0(X)\rightarrow \mathbb {B}(\mathcal {H}_X)$ of $C_0(X)$ on a separable Hilbert space $\mathcal {H}_X$ is called an X-module if it is non-degenerate and ample. The support of a vector $v\in \mathcal {H}_X$ is the complement in X of the union of all open subsets $U \subseteq X$ such that $\rho (f)v=0$ for all $f\in C_0(X)$. An operator $T\in \mathbb {B}(\mathcal {H}_X)$ is called locally compact on X if $\rho (f)T$ and $T\rho (f)$ are compact operators for all $f\in C_0(X)$. The support of an operator $T\in \mathbb {B}(\mathcal {H}_X,\mathcal {H}_Y)$ is the complement in $Y\times X$ of the union of all open subsets $U\times V\subseteq Y\times X$ such that $\rho (f)T\rho (g)=0$ for every $f\in C_0(U)$ and $g\in C_0(V)$. An operator $T\in \mathbb {B}(\mathcal {H}_X)$ is said to be controlled if ${{\,\textrm{supp}\,}}(T)$ is an entourage. The Roe algebra $C^*(X)$ is the norm closure of the algebra of locally compact, controlled operators on $\mathcal {H}_X$. If $C_0(X)$ is represented by an X-module then the K-theory of the Roe-algebra is a functor on coarse proper metric spaces. If $\varphi :X\rightarrow Y$ is a coarse map between metric spaces then a bounded operator $V:\mathcal {H}_X\rightarrow \mathcal {H}_{Y}$ covers $\varphi$ if the two maps $\pi _1$ and $\varphi \circ \pi _2$ from ${{\,\textrm{supp}\,}}(V)\subseteq Y\times X$ are close.

Lemma 7

If X is a proper metric space and $Y\subseteq X$ is a closed subspace then

• The subset $I(Y)=\{f\in C_0(X):f|_Y=0\}$ is an ideal of $C_0(X)$ and we have

  $$\begin{aligned} C_0(Y)=C_0(X)/I(Y) \end{aligned}$$

• There exists a sub-Hilbert space $\mathcal {H}_Y\subseteq \mathcal {H}_X$ and a non-degenerate representation $\rho _Y:C_0(Y)\rightarrow \mathbb {B}(\mathcal {H}_Y)$ that is a natural restriction of a non-degenerate representation $\rho _X:C_0(X)\rightarrow \mathbb {B}(\mathcal {H}_X)$.

• The inclusion $i_Y:\mathcal {H}_Y\rightarrow \mathcal {H}_X$ covers the inclusion $i:Y\rightarrow X$.

Proof

• This one follows by Gelfand duality.

• We define $\mathcal {H}_{I(Y)}=\overline{\rho _X(I(Y))\mathcal {H}_X}$. Then

  $$\begin{aligned} \mathcal {H}_X=\mathcal {H}_{I(Y)}\oplus \mathcal {H}_{I(Y)}^\perp \end{aligned}$$

  is the direct sum of reducing subspaces for $\rho _X(C_0(X))$. We define

  $$\begin{aligned} \mathcal {H}_Y=\mathcal {H}_{I(Y)}^\perp \end{aligned}$$

  and a representation of $C_0(Y)$ on $\mathcal {H}_Y$ by

  $$\begin{aligned} \rho _Y([a])=\rho _X(a)|_{\mathcal {H}_Y} \end{aligned}$$

  for every $[a]\in C_0(Y)$. Note that $\rho _X(\cdot )|_{\mathcal {H}_Y}$ annihilates I(Y) so this is well defined.

• Note that the support of $i_Y$ is

  $$\begin{aligned} {{\,\textrm{supp}\,}}(i_Y)&=\Delta _Y\\&\subseteq X\times Y \end{aligned}$$

$\square$

Remark 8

Note that we can not conclude the following: If the representation $\rho _X:C_0(X)\rightarrow \mathbb {B}(\mathcal {H}_X)$ is ample and $Y\subseteq X$ is a closed subspace then the induced representation $\rho _Y:C_0(Y)\rightarrow \mathbb {B}(\mathcal {H}_Y)$ is ample. Thus if we want to restrict representations to subspaces, we have to check the ample property each time.

Lemma 9

If X is a proper metric space,

• $B\subseteq X$ is a compact subset and $T\in C^*(X)$ is an operator with

  $$\begin{aligned} {{\,\textrm{supp}\,}}T\subseteq B^2 \end{aligned}$$

  then T is a compact operator.

• The converse does not hold. If $T\in C^*(X)$ is a compact operator then there does not necessarily exist a bounded set $B\subseteq X^2$ such that ${{\,\textrm{supp}\,}}T\subseteq B^2$

• The $C^*$-algebra of compact operators $\mathbb {K}(\mathcal {H}_X)$ is an ideal in $C^*(X)$.

Proof

• Suppose there is a non-degenerate representation $\rho :C_0(X)\rightarrow \mathbb {B}(\mathcal {H}_X)$. For every $f\in C_0(B^c),g\in C_0(X)$ the equations $\rho (f)T\rho (g)=0$ and $\rho (g)T\rho (f)=0$ hold. This implies $T(I(B))=0$ and ${{\,\textrm{im}\,}}T\cap I(B)=0$. Thus $T:\mathcal {H}_B\rightarrow \mathcal {H}_B$ is the same map. Thus $T\in C^*(B)$ already. Now T is locally compact, B is compact thus T is a compact operator.

• Note the set of ghost operator as defined in [9, Definition 1.2] contains the compact operators. The space X has property A if and only if every ghost operators is compact by [9, Theorem 1.3]. Not all of them have bounded support. The paper [2] shows that there exists a metric space ${\mathcal {O}}$ with non-compact ghost projections in $\mathbb {B}(L_2({\mathcal {O}}))$. This implies that ${\mathcal {O}}$ does not coarsely embed into Hilbert space. More precisely if X is the space $\mathbb {Z}$, the Hilbert space $\mathcal {H}_X:=\ell ^2(X)\otimes \ell ^2(\mathbb {N})$ the standard ample X-module, D is a diagonal operator with coefficients in $C_0(X)$ and e is a finite rank projection then $T:=D\otimes e\in C^*(X,\mathcal {H}_X)$ is compact and has support the full diagonal which cannot be contained in $B\times B$ with B bounded.

• This is already [7, Lemma 4.12]. For the convenience of the reader, we recall the proof: Let $K\subseteq X$ be a set and $v\in \mathcal {H}_X$ be a vector with ${{\,\textrm{supp}\,}}v\subseteq K$. Then for every $f\in I(K)$, we obtain $\rho (f)v=0$. Now a vector in $\mathcal {H}_{I(K)}$ can be written as $\rho (f)w$ where $f\in I(K),w\in \mathcal {H}_X$. Then

  $$\begin{aligned} \langle v,\rho (f)w\rangle&=\langle \rho (f^*)v,w\rangle \\&=0 \end{aligned}$$

  Thus $v\in \mathcal {H}_K$. If on the other hand $v'\in \mathcal {H}_K$ is any vector then

  $$\begin{aligned} 0&=\langle \rho (f)w,v'\rangle \\&=\langle w,\rho (f^*)v'\rangle \end{aligned}$$

  for every $f\in I(K),w\in \mathcal {H}_X$. This implies $\rho (f^*)v'=0$ for every $f\in I(K)$. Thus ${{\,\textrm{supp}\,}}v'\subseteq K$. Since X can be written as a union of bounded sets, $X=\bigcup B_i$ with $B_i$ bounded for every i, the vectors with compact support form an orthonormal basis of $\mathcal {H}_X$. A finite rank operator T with respect to this basis belongs to $C^*(X)$: First of all, T is locally compact since it is compact. We can write

  $$\begin{aligned} T:h\mapsto \sum _{i=1}^n\alpha _i\langle h,v_i\rangle u_i \end{aligned}$$

  here $\alpha _i\ge 0$ and $v_i,u_i$ are vectors with compact support ${{\,\textrm{supp}\,}}v_i\subseteq B_i,{{\,\textrm{supp}\,}}u_i\subseteq A_i$ for $1\le i\le n$. Let $v\in \mathcal {H}_X$ be a vector. If $g\in I(B_i)$ and $f\in C_0(X)$ are two functions then

  $$\begin{aligned} \alpha _i\langle \rho (g)v,v_i\rangle \rho (f) u_i&=\alpha _i\langle v,\rho (g^*)v_i\rangle \rho (f)u_i\\&=0 \end{aligned}$$

  Now let $g\in C_0(X),f\in I(A_i)$ be functions. Then $\alpha _i\langle \rho (g)v,v_i\rangle \rho (f) u_i=0$ since ${{\,\textrm{supp}\,}}u_i\subseteq A_i$. Thus ${{\,\textrm{supp}\,}}T\subseteq \bigcup _{i=1}^n A_i\times B_i$ is controlled. Since the finite rank operators are dense in the compact operators $\mathbb {K}(\mathcal {H}_X)$, we obtain the inclusion $\mathbb {K}(\mathcal {H}_X)\subseteq C^*(X)$. Since the composition with a compact operator yields a compact operator,the subset $\mathbb {K}(\mathcal {H}_X)$ is an ideal in $C^*(X)$.

$\square$

Definition 10

Let X be a proper metric space then

$$\begin{aligned} {\hat{C}}^*(X)=C^*(X)/\mathbb {K}(\mathcal {H}_X) \end{aligned}$$

where $\mathbb {K}(\mathcal {H}_X)$ denotes the compact operators of $\mathbb {B}(\mathcal {H}_X)$ is called the Roe–Calkin algebra of X.

We want to assign a $C^*$-algebra to every subset $U\subseteq X$.

Remark 11

If $U\subseteq X$ is a subset of a proper metric space then the inclusion $U\rightarrow {\bar{U}}$ is coarsely surjective which means that there is some $R\ge 0$ such that every point of ${\bar{U}}$ lies in an R-neighborhood of U. We define

$$\begin{aligned} {\hat{C}}^*(U):={\hat{C}}^*({\bar{U}}) \end{aligned}$$

This way we can use Lemma 7 to restrict representations and elements of the Roe–Calkin algebra to subspaces.

Lemma 12

If $Y\subseteq X$ is a closed subspace and $i_Y:\mathcal {H}_Y\rightarrow \mathcal {H}_X$ the inclusion operator of Lemma 7 then

• the operator

  $$\begin{aligned} Ad(i_Y):C^*(Y)&\rightarrow C^*(X)\\ T&\mapsto i_Y T i_Y^* \end{aligned}$$

  is well defined and maps compact operators to compact operators.

• Then the induced operator on quotients

  $$\begin{aligned} \hat{Ad}(i_Y):{\hat{C}}^*(Y)\rightarrow {\hat{C}}^*(X) \end{aligned}$$

  is the dual version of a restriction map, which means $U\mapsto {\hat{C}}^*(U)$ is a precosheaf on X.

Proof

• $i_Y$ covers the inclusion the other statement follows since composition with compact operators gives a compact operator.

• The assignment is a covariant functor.

$\square$

Theorem 13

If X is a proper metric space, then the assignment

$$\begin{aligned} U \mapsto {\hat{C}}^*(U) \end{aligned}$$

for every subspace $U\subseteq X$ is a cosheaf with values in $\texttt {CStar}$.

Proof

Let $U_1,\ldots ,U_n\subseteq U$ be subsets that coarsely cover $U\subseteq X$ and $V_i:\mathcal {H}_{U_i}\rightarrow \mathcal {H}_U$ and $V_{ij}:\mathcal {H}_{U_{ij}}\rightarrow \mathcal {H}_{U_i}$ the corresponding inclusion operators for $i,j=1,\ldots ,n$.

Let $T\in C^*(U)$ be a locally compact controlled operator. We need to construct $T_i\in C^*(U_i)$ such that

$$\begin{aligned} \sum _iV_iT_1V_i^*=T \end{aligned}$$

modulo compacts. Denote by $E={{\,\textrm{supp}\,}}(T)$ the support of T in U. Define

$$\begin{aligned} T_i:=V_i^*TV_i-\sum _{j=1}^{i-1}V_{ij}V_{ji}^*T_jV_{ji}V_{ij}^* \end{aligned}$$

then the $T_i$ are locally compact controlled operators, thus elements in $C^*(U_i)$. We now show that $T-\sum _i V_iT_iV_i^*=0$ on $U_1\times U_1\cup \cdots \cup U_n\times U_n$ which shows the result since $E\cap (U_1^2\times \cdots U_n^2)^c$ is bounded. Let $(x,y)\in (U_1^2\times U_n^2)\cap {{\,\textrm{supp}\,}}T$ be a point and choose k minimal with $(x,y)\in U_k\times U_k$. Then $T_k(x)=T(x)=y$. We now show $(x,y)\not \in {{\,\textrm{supp}\,}}T_i$ for $i\not =k$. For $i=1,\ldots ,k-1$, this is clear since k is minimal. We now use induction on $i=k+1,\ldots ,n$: If $i=k+1$ then $(x,y)\in {{\,\textrm{supp}\,}}(V_{k+1}^*TV_{k+1})$ exactly when $(x,y)\in {{\,\textrm{supp}\,}}(V_{k+1,k}V_{k,k+1}^*T_kV_{k,k+1}V_{k+1,k}^*)$. Thus $(x,y)\not \in {{\,\textrm{supp}\,}}T_{k+1}$. If the statement holds for $k+1,\ldots ,i$ then $(x,y)\in {{\,\textrm{supp}\,}}(V_{i+1}^*TV_{i+1})$ exactly when $(x,y)\in {{\,\textrm{supp}\,}}(V_{i+1,k}V_{k,i+1}^*T_kV_{k,i+1}V_{i+1,k}^*)$. Thus $(x,y)\not \in {{\,\textrm{supp}\,}}T_{i+1}$. This implies $\sum _i T_i|_U=T$, axiom 1).

Suppose $T_i\in C^*(U_i)$ are elements with

$$\begin{aligned} \sum _i V_i T_i V_i^*= 0 \end{aligned}$$

modulo compacts. Denote by $V_{ijk}:\mathcal {H}_{U_i\cap U_j \cap U_k}\rightarrow \mathcal {H}_{U_i\cap U_j}$ the covering isometry operator associated to the inclusion $U_i\cap U_j\cap U_k\rightarrow U_i\cap U_j$. Define for $1\le i<j\le n$:

$$\begin{aligned} T_{ij}:=V_{ij}^*T_iV_{ij}+\sum _{k<i}V_{ijk}V_{kij}^*T_{ki}V_{kij}V_{ijk}^*-\sum _{i<k<j}V_{ijk}V_{ikj}^*T_{ik}V_{ikj}V_{ijk}^*. \end{aligned}$$

Using $V_iV_{ij}=V_jV_{ji}$ and combinatorical information,we can show

$$\begin{aligned} \sum _j(T_{ij}-T_{ji})|_{U_i}=\sum _{i<j} T_{ij}|_{U_i}-\sum _{j<i}T_{ji}|_{U_i}=T_i \end{aligned}$$

modulo compacts. Thus axiom 2) of a cosheaf holds. $\square$

4 Computing Examples

Corollary 14

If $U_1,U_2$ coarsely cover a subset U of a proper metric space X then there is a six-term Mayer–Vietoris exact sequence

Proof

If $A\subseteq X$ is a subset define $C^*(A,X)$ to be the $C^*$-algebra generated by all locally compact operators with finite propagation on $\mathcal {H}_X$ whose support is contained in $E[A]\times E[A]$ for some entourage $E\subseteq X\times X$. It can be observed that $C^*(A,X)$ forms an ideal in $C^*(X)$. The inclusion $C^*(A)\rightarrow C^*(A,X)$ induces an isomorphism on the K-theory of the algebras obtained by modding out $\mathbb {K}( \mathcal {H}_X)$. That is because we have a commuting diagram with exact rows:

where the left vertical arrow is an isomorphism, the middle vertical arrow induces an isomorphism in K-theory by [4, Proposition 6.4.7]. By the five lemma, the right vertical arrow induces an isomorphism in K-theory.

We define $I_1:=C^*(U_1,X)/\mathbb {K}(\mathcal {H}_X),\,I_2:=C^*(U_2,X)/\mathbb {K}(\mathcal {H}_X)$ and $I_{12}=C^*(U_1\cap U_2,X)/\mathbb {K}(\mathcal {H}_X)$. They are the smallest ideals containing ${\hat{C}}^*(U_1),{\hat{C}}^*(U_2)$ and $C^*(U_1\cap U_2)$, respectively. Now Theorem 13 and the first cosheaf axiom imply ${\hat{C}}^*(U)={\hat{C}}^*(U_1)+{\hat{C}}^*(U_2)$ and the second cosheaf axiom implies ${\hat{C}}^*(U_1)\cap {\hat{C}}^*(U_2)={\hat{C}}^*(U_1\cap U_2)$. This implies $I_1+I_2=C^*(U,X)/\mathbb {K}(\mathcal {H}_X)$ and $I_1\cap I_2=I_{12}$. With those properties and [4, Exercise 4.10.21], the six-term exact sequence in K-theory is obtained. $\square$

Remark 15

Now for every proper metric space there is a short exact sequence

$$\begin{aligned} 0\rightarrow \mathbb {K}(\mathcal {H}_X)\rightarrow C^*(X)\rightarrow {\hat{C}}^*(X)\rightarrow 0 \end{aligned}$$

which induces a 6-term sequence in K-theory:

If X is flasque [10] then

$$\begin{aligned} K_i({\hat{C}}^*(X))={\left\{ \begin{array}{ll} 0 &{} i=0\\ \mathbb {Z}&{} i=1 \end{array}\right. } \end{aligned}$$

Remark 16

Note that the result of Corollary 14 is applicable when computing controlled K-theory if the property ample is preserved by restricting the representation of U to the representations of $U_1,U_2$.

If X is a Riemannian manifold then fixing a volume form $\nu$ the Hilbert space $\mathcal {H}_X=L^2(X,\nu )\otimes \ell ^2$ is an ample X-module with $\ell ^2$ the standard separable Hilbert space and $\rho _X:C_0(X)\rightarrow \mathbb {B}(\mathcal {H}_X)$ trivial on the second factor. In Example 17, Example 18, we will use this canonical representation on $\mathbb {R},\mathbb {R}^2$ and certain subspaces of them without mentioning it. In those cases, the property ample is preserved by restricting $\mathbb {R}$ to $\mathbb {R}_{\ge 0}$, the space $R^2$ to $V_1,V_2$ and $V_1\cap V_2$ to $U_1,U_2$, respectively.

Example 17

($\mathbb {R}$) Now $\mathbb {R}$ is the coarse disjoint union of two copies of $\mathbb {R}_{\ge 0}$ which is a flasque space. By Corollary 14, there is an isomorphism

$$\begin{aligned} K_i({\hat{C}}^*(\mathbb {R}))={\left\{ \begin{array}{ll} 0 &{} i=0\\ \mathbb {Z}\oplus \mathbb {Z}&{} i=1 \end{array}\right. } \end{aligned}$$

Applying Remark 15,we can compare

$$\begin{aligned} K_i(C^*(\mathbb {R}))={\left\{ \begin{array}{ll} 0 &{} i=0\\ \mathbb {Z}&{} i=1 \end{array}\right. }, \end{aligned}$$

in [4, Theorem 6.4.10], if it matches our computation. And indeed it does.

Example 18

($\mathbb {R}^2$) We coarsely cover $\mathbb {R}^2$ with $V_1=\mathbb {R}_{\ge 0}\times \mathbb {R}\cup \mathbb {R}\times \mathbb {R}_{\ge 0}$ and $V_2=\mathbb {R}_{<0}\times \mathbb {R}\cup \mathbb {R}\times \mathbb {R}_{<0}$. The space $V_1\cap V_2$ is coarsely covered by $U_1=\mathbb {R}_{\ge 0}\times \mathbb {R}_{\ge 0}$ and $U_2=\mathbb {R}_{<0}\times \mathbb {R}_{<0}$. The second cover and Corollary 14 gives

$$\begin{aligned} K_i({\hat{C}}^*(V_1\cap V_2))={\left\{ \begin{array}{ll} 0 &{} i=0\\ \mathbb {Z}\oplus \mathbb {Z}&{} i=1 \end{array}\right. } \end{aligned}$$

Since $V_1,V_2$ are coarsely equivalent to flasque spaces, the first cover and Corollary 14 imply that $K_0({\hat{C}}^*(\mathbb {R}^2))$ and $K_1({\hat{C}}^*(\mathbb {R}^2))$ have the same (free abelian) rank. Translating back using Remark 15, the groups

$$\begin{aligned} K_i(C^*(\mathbb {R}^2))={\left\{ \begin{array}{ll} \mathbb {Z}&{}i=0\\ 0 &{}i=1 \end{array}\right. } \end{aligned}$$

of [4, Theorem 6.4.10] also fit in the exact sequence.