# Breaking symmetries for equivariant coarse homology theories

Ulrich Bunke  Fakultät für Mathematik, Universität Regensburg, 93040
Regensburg, Germany<EMAIL_ADDRESS>and  Matthias
Ludewig  Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg,
Germany<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We describe a symmetry breaking construction in coarse geometry which allows
to obtain information about equivariant coarse homology classes by restriction
to smaller groups and spaces. In the case of equivariant coarse $K$-homology
theory we give an analytic interpretation of this construction. As a
consequence we obtain applications to the spectral theory of invariant
differential operators.

###### Contents

  1. 1 Introduction
  2. 2 A homotopy theoretic construction
  3. 3 Topological coarse $K$-homology
  4. 4 Operators in Roe algebras
  5. 5 Differential operators
  6. 6 Incoorporating local positivity
  7. 7 Spectral projections of magnetic laplacians

##  1\. Introduction

We consider a differential operator $D$ of Dirac or Laplace type acting on
sections of a Hermitean vector bundle on a complete Riemannian manifold $M$.
We assume the operator is invariant under the action of a discrete group of
symmetries $G$. According to Definition 4.1 a spectral interval for $D$ is an
interval in $\mathbb{R}$ whose endpoints are located in spectral gaps of $D$.
The spectral projection $E_{D}(I)$ associated to a bounded spectral interval
$I$ is an element of the $G$-invariant Roe algebra $C^{G}(M)$ associated to
the underlying coarse space of $M$, see Proposition 5.2.

We now consider a subset $Z$ with smooth boundary $\partial^{\infty}Z$ which
are both preserved by the action of a subgroup $K$ of $G$. If we define a
selfadjoint extension of the restriction $D_{Z}$ to $Z$ using elliptic local
boundary conditions at $\partial^{\infty}Z$, then for a bounded spectral
interval $I$, $E_{D_{Z}}(I)$ is an element of the $K$-invariant Roe algebra
$C^{K}(Z)$ associated to the underlying coarse space of $Z$, see Proposition
5.2.

The operator $D_{Z}$ is “locally $G$-variant” as a differential operator, but
as a functional analytic object its invariance is distorted by the boundary
condition near $\partial^{\infty}Z$. However, it turns out that the influence
of this distortion on the spectral projections decays with the distance from
$\partial^{\infty}Z$. In particular, they become “more and more $G$-invariant”
if we move far away from the boundary. In order to capture this behavior in a
precise manner, under a wideness assumption (see Definition 3.2) on $Z$ in
Theorem 3.3 we introduce an extension of Roe algebras

(1.1) $0\to C^{K}(\partial Z)\to
C^{G,K}(X,Z)\stackrel{{\scriptstyle\sigma}}{{\to}}C^{G}(M)\to 0\ ,$

where $C^{K}(\partial Z)$ is the subalgebra of $C^{K}(Z)$ generated by
$K$-invariant operators supported near $\partial^{\infty}Z$, and
$C^{G,K}(X,Z)$ is a certain subalgebra of $C^{K}(Z)$ consisting of
asymptotically $G$-invariant operators. In the special case that $K$ is
trivial, this extension has previously been considered in [LT20]. The precise
statement is now that $E_{D_{Z}}(I)$ (apriori belonging to $C^{K}(Z)$) belongs
to the subalgebra $C^{G,K}(X,Z)$, and that

(1.2) $\sigma(E_{D_{Z}}(I))=E_{D}(I)\ ,$

see Proposition 5.3.

Applying $K$-theory to the exact sequence (1.1) of $C^{*}$-algebras we get a
long exact sequence

(1.3) $\cdots\to K_{*}(C^{G,K}(X,Z))\stackrel{{\scriptstyle
K(\sigma)}}{{\to}}K_{*}(C^{G}(M))\stackrel{{\scriptstyle\delta}}{{\to}}K_{*-1}(C^{K}(\partial
Z))\to\cdots$

of $K$-theory groups. A projection $P$ in a $C^{*}$ algebra represents a
$K$-theory class $[P]$ in the degree zero $K$-theory group. As first observed
in [LT20, Thm. 3.4] (see also [Thi20, §2.2.3]), the exactness of (1.3) has the
following simple consequence (see Proposition 6.5).

######  Corollary 1.1.

If $I$ is a spectral interval of $D$ and $\delta[E_{D}(I)]\not=0$, then $I$ is
not a spectral interval of $D_{Z}$.

Note that the conclusion of the corollary means that at least one of the
endpoints of $I$ belongs to the spectrum of $D_{Z}$. The corollary can
therefore be employed to show that certain numbers must belong to the spectrum
of $D_{Z}$. Such statements are important in the theory of topological
insulators, where one is interested in the existence of certain “boundary-
localised” states that arise precisely from a spectral gap in the bulk
Hamiltonian that is filled when introducing a boundary. For a coarse geometric
discussion of these phenomena, see [EM19], [Thi20], [LT20] and [LT21].

We will demonstrate this sort of application in the case of magnetic
Hamiltonians in Corollary 7.4.

In order to apply Corollary 1.1 we must be able to calculate the boundary
operator $\delta$ of the long exact sequence in (1.3). The main idea of the
present paper is to consider the construction of the sequence (1.3) as a
special case of a general construction in equivariant coarse homotopy theory.

The main objects of coarse homotopy theory as developed in [BEKW20b] are
equivariant coarse homology theories

$E^{G}:G\mathbf{BornCoarse}\to\mathbf{C}\ ,$

where $G\mathbf{BornCoarse}$ is the category of $G$-bornological coarse spaces
and the target $\mathbf{C}$ is some stable $\infty$-category such as the
category of spectra. We consider a $G$-bornological coarse space $X$ with a
$K$-invariant subset $Z$. In Definition 2.4 we define the obstruction morphism

$r:E^{G}(X)\stackrel{{\scriptstyle
c^{*}}}{{\to}}E^{K}(\operatorname{Res}^{G}_{K}(X))\stackrel{{\scriptstyle\delta^{MV}}}{{\to}}\Sigma
E^{K}(\partial Z)\ .$

Here $\operatorname{Res}^{G}_{K}(X)$ is the space $X$ considered as a
$K$-bornological coarse space via the restriction along $K\to G$. The symbol
$E^{K}$ denotes the $K$-equivariant coarse homology theory which is naturally
derived from $E^{G}$ using a transfer structure, see (2.1). The map
$\delta^{MV}$ is the Mayer-Vietoris boundary map associated to the
decomposition of $X$ into $Z$ and its complement $\\{X\setminus Z\\}$, the big
family consisting of the coarse thickenings of $X\setminus Z$. Finally, the
big family $\partial Z:=Z\cap\\{X\setminus Z\\}$ is the coarse version of the
boundary of $Z$ (we refer [BEKW20b] for the basic definitions from equivariant
coarse homotopy theory). Since $\mathbf{C}$ is a stable $\infty$-category we
can extend the obstruction morphism $r$ to a fibre sequence

(1.4) $E^{G,K}(X,Z)\to E^{G}(X)\stackrel{{\scriptstyle
r}}{{\to}}E^{K}(\partial Z)$

defining the relative object $E^{G,K}(X,Z)$, see Definition 2.2.

We can apply these constructions to the equivariant coarse $K$-homology
$K\mathcal{X}^{G}$ introduced in [BEc] in place of $E^{G}$ and the manifold
$M$ from above in place of $X$. Except for degenerate cases we have natural
isomorphisms of $K$-theory groups

$K\mathcal{X}^{G}_{*}(M)\cong K_{*}(C^{G}(M))\ ,\quad
K\mathcal{X}^{K}_{*}(\partial Z)\cong K_{*}(C^{K}(\partial Z))\ ,$

see [BEb, Thm. 6.1]. The following theorem is a reformulation of Theorem 3.4
in terms of homotopy groups.

######  Theorem 1.2.

There exists a canonical isomorphism of long exact sequences

$\textstyle{K_{*}(C^{G,K}(M,Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{*}(C^{G}(M))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\delta}$$\textstyle{K_{*-1}(C^{K}(\partial
Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K\mathcal{X}^{G,K}_{*}(M,Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\mathcal{X}^{G}_{*}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{*}}$$\textstyle{K\mathcal{X}^{K}_{*-1}(\partial
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We interpret this theorem and in particular the commutativity of the right
square as a kind of index theorem relating the purely analytic construction of
$\delta$ in terms of Roe algebras with the purely homotopy theoretic
construction of $r_{*}$. The theorem reduces the calculation of $\delta$ to
the calculation of the obstruction morphism $r$.

Acknowledgements: U. Bunke and M. Ludewig were supported by the SFB 1085
“Higher Invariants” funded by the Deutsche Forschungsgemeinschaft DFG.

##  2\. A homotopy theoretic construction

We consider a group $G$. The category of $G$-bornological coarse spaces and
the notion of a $G$-equivariant coarse homology theory was introduced in
[BEKW20b]. Given $G$-equivariant coarse homology theory

$E^{G}:G\mathbf{BornCoarse}\to\mathbf{C}$

and a $G$-bornological coarse space $X$, in the present section we describe a
construction which may produce information about $E^{G}(X)$ by breaking the
symmetry. Our main result is the construction of the relative object
$E^{G,K}(X,Z)$ in Definition 2.2 and the fibre sequence (2.7).

Let $K$ be a subgroup of $G$ and consider the induction functor

$\operatorname{Ind}_{K}^{G}:K\mathbf{BornCoarse}\to G\mathbf{BornCoarse}$

[BEKW20b, Sec. 6.5]. We define an associated $K$-equivariant coarse homology
$E^{K}$ by

(2.1)
$E^{K}:=E^{G}\circ\operatorname{Ind}_{K}^{G}:K\mathbf{BornCoarse}\to\mathbf{C}$

[BEKW20c, Lemma 4.19]. In the case of the trivial group $K=\\{1\\}$ we will
omit this symbol from the notation. In this case the induction functor is
given in terms of the symmetric monoidal structure of $G\mathbf{BornCoarse}$
by $\operatorname{Ind}^{G}(X)\cong G_{min,min}\otimes X$, and we therefore
have the equivalence

(2.2) $E(X)\simeq E^{G}(G_{min,min}\otimes X)\ .$

######  Remark 2.1.

Often we are in the situation that we already have a family of equivariant
coarse homology theories $(E^{H})_{H}$ (with a fixed target) for all groups
$H$, which at first sight clashes with the notation (2.1). However, in all
relevant examples the homology theory $E^{G}\circ\operatorname{Ind}_{K}^{G}$
derived from the member $E^{G}$ of the family using (2.1) is equivalent to the
member of the family denoted by the same symbol $E^{K}$. In this situation we
will say that the family has induction equivalences.

Examples of families with induction equivalences are the family of equivariant
coarse ordinary homology theories $(H\mathcal{X}^{{H}})_{{H}}$ [BEKW20b, Prop.
7.12] and the family of coarse algebraic $K$-theories
$(K\mathbf{A}\mathcal{X}^{{H}})_{{H}}$ associated to an additive category
$\mathbf{A}$ (with trivial action) [BEKW20b, Prop. 8.27]. For the present
paper, the relevant example is the coarse topological $K$-homology associated
to a $C^{*}$-category with a $G$-action obtained in [BEc, Thm. 6.3] (for the
$K$-theory of $C^{*}$-categories in place of homological functor
$\operatorname{Hg}$), see Section 3. That this example has induction
equivalences has been shown in [BEc, Cor. 10.5.1]. ∎

We also have a restriction functor

$\operatorname{Res}^{G}_{K}:G\mathbf{BornCoarse}\to K\mathbf{BornCoarse}$

which sends a $G$-bornological space $X$ to the $K$-bornological space
obtained by restricting the action from $G$ to $K$. If $X$ is a
$G$-bornological coarse space, then we have a natural map of coarse spaces

(2.3) $c_{X}:G\times_{K}X\to X\ ,\quad[g,x]\mapsto gx\ .$

In general this morphism in $G\mathbf{Coarse}$ is not proper (and hence not a
morphism in $G\mathbf{BornCoarse}$), but it is a bounded covering [BEKW20a,
Def. 2.14].

We now assume that $E^{G}$ admits transfers [BEKW20a, Def. 1.2]. This means
that $E^{G}$ has a contravariant functoriality with respect to bounded
coverings in addition to the covariant functoriality for morphisms in
$G\mathbf{BornCoarse}$. The compatibility of these operations is encoded in
the category $G\mathbf{BornCoarse}_{\operatorname{tr}}$ of $G$-bornological
coarse spaces with transfers introduced in [BEKW20a, Sec.2.2]. The family of
bounded coverings $c=(c_{X})$ can be considered as natural transformation of
functors

(2.4)
$c:\operatorname{Ind}_{K}^{G}\circ\operatorname{Res}^{G}_{K}\to\iota:G\mathbf{BornCoarse}\to
G\mathbf{BornCoarse}_{\operatorname{tr}}\ ,$

where $\iota:G\mathbf{BornCoarse}\to G\mathbf{BornCoarse}_{\operatorname{tr}}$
is the canonical embedding [BEKW20a, Sec.2.33]. The transfers along the values
of $c$ in (2.4) provide a natural transformation

(2.5) ${c^{G}_{K}}:E^{G}\to
E^{G}\circ\operatorname{Ind}_{K}^{G}\circ\operatorname{Res}^{G}_{K}\stackrel{{\scriptstyle\eqref{NotationEK}}}{{\simeq}}E^{K}\circ\operatorname{Res}^{G}_{K}\
.$

For a $K$-invariant subset $Y$ of $X$ we let
$\\{Y\\}:=(U[Y])_{U\in\mathcal{C}^{K}_{X}}$ denote the $K$-invariant big
family [BEKW20b, Def. 3.5] consisting of all of $U$-thickenings of $Y$ for
$K$-invariant coarse entourages $U$ in $\mathcal{C}^{K}_{X}$. Let now $Z$ be a
$K$-invariant subset of $X$ with the induced $K$-bornological coarse
structure. Observe that $X\setminus Z$ is also $K$-invariant. We get a
$K$-invariant complementary pair $(Z,\\{X\setminus Z\\})$ on
$\operatorname{Res}^{G}_{K}(X)$ [BEKW20b, Def. 3.7]. We let $i:E^{K}(Z)\to
E^{K}(\operatorname{Res}^{G}_{K}(X))$ and $j:E^{K}(\\{X\setminus Z\\})\to
E^{K}(\operatorname{Res}^{G}_{K}(X))$ denote the maps induced by the canonical
inclusions.

######  Definition 2.2.

We define the object $E^{G,K}(X,Z)$ in $\mathbf{C}$ by the pull-back

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}\hbox{$\scriptstyle{c^{G}_{K}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qed$

We consider the big family

(2.6) $\partial Z:=Z\cap\\{X\setminus Z\\}$

defined by intersecting the members of the $K$-invariant big family
$\\{X\setminus Z\\}$ with $Z$. It plays the coarse geometric role of the
boundary of $Z$, hence the notation.

######  Lemma 2.3.

We have a canonical fibre sequence

(2.7) $\dots\to E^{K}(\partial Z)\to E^{G,K}(X,Z)\stackrel{{\scriptstyle
s}}{{\to}}E^{G}(X)\stackrel{{\scriptstyle r}}{{\to}}\Sigma E^{K}(\partial
Z)\to\dots$

###### Proof.

Since $(Z,\\{X\setminus Z\\})$ is an $K$-invariant complementary pair on
$\operatorname{Res}^{G}_{K}(X)$, by the excision axiom for $E^{K}$ [BEKW20b,
Def. 3.10 (2)], we have a cartesian square

${E^{K}(\partial Z)}$${E^{K}(Z)}$${E^{K}(\\{X\setminus
Z\\})}$${E^{K}(\operatorname{Res}^{G}_{K}(X)),}$$\scriptstyle{i}$$\scriptstyle{j}$

leading to a fibre sequence

(2.8) $E^{K}(\partial Z)\to E^{K}(Z)\oplus E^{K}(\\{X\setminus Z\\})\to
E^{K}(\operatorname{Res}^{G}_{K}(X))\stackrel{{\scriptstyle\delta}}{{\to}}\Sigma
E^{K}(\partial Z)$

in $\mathbf{C}$. Equivalently, the right square in the following diagram is
cartesian:

(2.9) ${E^{G,K}(X,Z)}$${E^{K}(Z)\oplus E^{K}(\\{X\setminus
Z\\})}$${0}$${E^{G}(X)}$${E^{K}(\operatorname{Res}^{G}_{K}(X))}$${\Sigma
E^{K}(\partial Z).}$$\scriptstyle{e_{Z}\oplus e_{X\setminus
Z}}$$\scriptstyle{s}$$\scriptstyle{i+j}$$\scriptstyle{c^{G}_{K}}$$\scriptstyle{\delta}$

######  Definition 2.4.

We define the obstruction morphism as the composition

(2.10) $r\simeq\delta\circ{c^{G}_{K}}:E^{G}(X)\to\Sigma E^{K}(\partial Z)\
.\qed$

Since the left square of (2.9) is cartesian by Definition 2.2 we conclude that
the outer square is cartesian. This is the desired fibre sequence (2.7). ∎

Let $f:X^{\prime}\to X$ be a morphism of $G$-bornological coarse spaces and
define the $K$-invariant subset $Z^{\prime}:=f^{-1}(Z)$ of $X^{\prime}$.

######  Lemma 2.5.

There exists a morphism

$\widetilde{f}_{*}:E^{G,K}(X^{\prime},Z^{\prime})\longrightarrow E^{G,K}(X,Z)$

fitting into a morphism of fibre sequences

(2.11) ${\dots}$${E^{K}(\partial
Z^{\prime})}$${E^{G,K}(X^{\prime},Z^{\prime})}$${E^{G}(X^{\prime})}$${\Sigma
E^{K}(\partial Z^{\prime})}$${\dots}$${\dots}$${E^{K}(\partial
Z)}$${E^{G,K}(X,Z)}$${E^{G}(X)}$${\Sigma E^{K}(\partial
Z)}$${\dots}$$\scriptstyle{\partial
f_{*}}$$\scriptstyle{s^{\prime}}$$\scriptstyle{\widetilde{f}_{*}}$$\scriptstyle{f_{*}}$$\scriptstyle{r^{\prime}}$$\scriptstyle{\partial
f_{*}}$$\scriptstyle{s}$$\scriptstyle{r}$

###### Proof.

First of all, the map $f$ induces morphisms $f_{*}:E^{G}(X^{\prime})\to
E^{G}(X)$ and
$\operatorname{Res}^{G}_{K}(f)_{*}:E^{K}(\operatorname{Res}^{G}_{K}(X^{\prime}))\to
E^{K}(\operatorname{Res}^{G}_{K}(X))$, and the square

(2.12)
${E^{G}(X^{\prime})}$${E^{K}(\operatorname{Res}^{G}_{K}(X^{\prime}))}$${E^{G}(X)}$${E^{K}(\operatorname{Res}^{G}_{K}(X))}$$\scriptstyle{f_{*}}$$\scriptstyle{c^{G}_{K}}$$\scriptstyle{\operatorname{Res}^{G}_{K}(f)_{*}}$$\scriptstyle{c^{G}_{K}}$

commutes by naturality of the transfer.

Moreover, since $f(X^{\prime}\setminus Z^{\prime})\subseteq X\setminus Z$, for
every $U^{\prime}$ in $\mathcal{C}_{X^{\prime}}^{G}$ we have
$f(U^{\prime}[X^{\prime}\setminus Z^{\prime}])\subseteq
f(U^{\prime})[X\setminus Z]$ and $f(Z^{\prime}\cap
U^{\prime}[X^{\prime}\setminus Z^{\prime}])\subseteq Z\cap
f(U^{\prime})[X\setminus Z]$. Therefore $f$ induces morphisms of big families
$f_{X^{\prime}\setminus Z^{\prime}}:\\{X^{\prime}\setminus
Z^{\prime}\\}\to\\{X\setminus Z\\}$ and $\partial f:\partial
Z^{\prime}\to\partial Z$, and the diagram

(2.13) $\textstyle{E^{K}(Z^{\prime})\oplus E^{K}(\\{X^{\prime}\setminus
Z^{\prime}\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}+j^{\prime}}$$\scriptstyle{f_{Z^{\prime},*}\oplus
f_{X^{\prime}\setminus
Z^{\prime},*}}$$\textstyle{E^{K}(\operatorname{Res}^{G}_{K}(X^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Res}^{G}_{K}(f)_{*}}$$\scriptstyle{\delta}$$\textstyle{\Sigma
E^{K}(\partial
Z^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial
f_{*}}$$\textstyle{E^{K}(Z)\oplus E^{K}(\\{X\setminus
Z\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i+j}$$\textstyle{E^{K}(\operatorname{Res}^{G}_{K}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\Sigma
E^{K}(\partial Z)}$

commutes by the naturality of the Mayer-Vietoris fibre sequence. The
horizontal concatenation of the square in (2.12) and the right square in
(2.13) yields the commutative square

${E^{G}(X^{\prime})}$${\Sigma E^{K}(\partial Z^{\prime})}$${E^{G}(X)}$${\Sigma
E^{K}(\partial
Z)}$$\scriptstyle{f_{*}}$$\scriptstyle{r}$$\scriptstyle{\partial
f_{*}}$$\scriptstyle{r}$

which extends to the desired morphism (2.11) of fibre sequences. ∎

######  Lemma 2.6.

If $\partial Z^{\prime}$ consists of empty sets, then the composition $r\circ
f_{*}:E^{G}(X^{\prime})\to E^{G}(X)\to E^{K}(\partial Z)$ vanishes.

###### Proof.

Since $\partial Z^{\prime}$ consists of empty sets, we have $E^{K}(\partial
Z^{\prime})\simeq 0$, hence $r^{\prime}\simeq 0$. The commutativity of the
right square in (2.11) implies the assertion. ∎

######  Example 2.7.

If $X^{\prime}$ is obtained from $X$ by replacing the coarse structure by a
smaller one, then the identity map is a morphism $X^{\prime}\to X$. In
particular, we can take $X^{\prime}=X_{\mathrm{disc}}$, the bornological
coarse space with the same underlying space and bornology as $X$ but the
minimal coarse structure, generated by the empty set. This means that all
entourages $U$ of $X^{\prime}$ are subsets of the diagonal, so that
$U[X\setminus Z]\subseteq X\setminus Z$. This implies $Z\cap U[X\setminus
Z]=\varnothing$ for each entourage $U$ of $X_{\mathrm{disc}}$, hence $\partial
Z=\\{\varnothing\\}$, so the condition of Lemma 2.6 is satisfied in this case.
∎

######  Example 2.8.

The following idea is inspired by [HPS15, Sec. 3.2]. Let $a:Y\to Z$ be the
embedding of a $K$-invariant subset of $Z$. Following [HPS15, Def. 3.9] we
could call $Y$ coarsely $E^{K}$-negligible if the map $E^{K}(a):E^{K}(Y)\to
E^{K}(Z)$ is zero. The following more restrictive notion captures a reason for
this triviality.

######  Definition 2.9.

$Y$ admits a flasque exit in $Z$ if the embedding $a:Y\to Z$ extends to a
morphism $h:[0,\infty)\otimes Y\to Z$ in $K\mathbf{BornCoarse}$ such that
$h_{|\\{0\\}\times Y}=a$. ∎

For example, if $K=\\{1\\}$, $Y$ is $U$-bounded for some entourage $U$ of $X$
and there exists a coarse ray in $Z$ (a map from $[0,\infty)$ to $Z$) starting
in a point of $Y$, then $Y$ admits a flasque exit in $Z$ [HPS15, Prop. 3.10].

If $Y$ admits a flasque exit in $Z$, then the inclusion induces a zero map in
coarse homology:

$0\simeq E^{K}(a):E^{K}(Y)\to E^{K}(Z)\ .$

Indeed, ${E^{K}(a)}$ factorizes as $E^{K}(Y)\to E^{K}([0,\infty)\otimes
Y)\stackrel{{\scriptstyle h_{*}}}{{\to}}E^{K}(Z)$, and
$E^{K}([0,\infty)\otimes Y)\simeq 0$ by flasqueness of $[0,\infty)\otimes Y$.
So the choice of $h$ provides a preferred choice of an equivalence ${0\simeq
E^{K}(a)}$.

######  Definition 2.10.

We say that the big family $\partial Z$ admits a two-sided flasque exit, if
for every $U$ in $\mathcal{C}^{K}_{X}$ the member $Z\cap U[X\setminus Z]$ of
$\partial Z$ admits a flasque exit in both $Z$ and $U[X\setminus Z]$. ∎

If $\partial Z$ admits a two-sided flasque exit, then the fibre sequence (2.8)
splits. We therefore get a decomposition

$E^{K}(\operatorname{Res}^{G}_{K}(X))\simeq E^{K}(Z)\oplus E^{K}(\\{X\setminus
Z\\})\oplus\Sigma E^{K}(\partial Z),$

where $\delta$ corresponds to the projection onto the last summand. ∎

######  Example 2.11.

In this example we describe a situation where $\partial Z$ has a two-sided
flasque exit. Let $m,n$ be in $\mathbb{N}$ and consider the group
$G:=\mathbb{Z}^{m+n}$. Then $X:=\mathbb{R}^{m+n}$ is a
$\mathbb{Z}^{m+n}$-bornological coarse space with the structures induced by
the standard metric with the usual action by translations. We let
$K:=\mathbb{Z}^{m}$ be the first summand of
$\mathbb{Z}^{m+n}\cong\mathbb{Z}^{m}\oplus\mathbb{Z}^{n}$. We furthermore take
the $\mathbb{Z}^{m}$-invariant subset

$Z:=\mathbb{R}^{m+n}_{+}:=\\{(x,y)\in\mathbb{R}^{m+n}\>|\>y\in[0,\infty)^{n}\\}\
,$

where identify $\mathbb{R}^{m+n}\cong\mathbb{R}^{m}\times\mathbb{R}^{n}$ in
order to write its elements as pairs. Note that $\partial Z$ admits a two-
sided flasque exit (see Definition 2.10).

But actually in this case the situation is even simpler. Namely, $Z$ and all
members of $\\{X\setminus Z\\}$ are flasque. This implies that the boundary
map $\delta$ in the cofibre sequence (2.8) is an equivalence. The obstruction
morphism

$r:E^{\mathbb{Z}^{m+n}}(\mathbb{R}^{m+n})\longrightarrow{\Sigma}E^{\mathbb{Z}^{m}}(\partial
Z)$

in the fiber sequence (2.7) is therefore equivalent to the transfer morphism

${c^{\mathbb{Z}^{m+n}}_{\mathbb{Z}^{m}}}:E^{\mathbb{Z}^{m+n}}(\mathbb{R}^{m+n})\to
E^{\mathbb{Z}^{m}}(\operatorname{Res}^{\mathbb{Z}^{m+n}}_{\mathbb{Z}^{m}}(\mathbb{R}^{m+n})).\qed$

##  3\. Topological coarse $K$-homology

If $\mathbf{C}$ is a $C^{*}$-category with strict $G$-action admitting all
orthogonal AV-sums [BEa, Def. 7.1], then we have the equivariant coarse
$K$-homology functor

$K\mathcal{X}_{\mathbf{C}}^{G}:G\mathbf{BornCoarse}\to\mathbf{Sp}$

with transfers [BEc, Def. 6.2.1 & Thm. 6.3 & 9.7] (with $K$-theory functor for
$C^{*}$-categories [BEa, Def. 14.3] in place of $\operatorname{Hg}$). If we
specialize this construction to the category
$\mathbf{C}=\mathbf{Hilb}_{c}(\mathbb{C})$ of Hilbert spaces and compact
operators with the trivial $G$-action, then the resulting functor will be
denoted by $K\mathcal{X}^{G}$. This is the usual equivariant topological
coarse $K$-homology. For simplicity, in the present paper we will stick to
this example.

We can apply the constructions described in Section 2 to $K\mathcal{X}^{G}$.
In particular, for a subgroup $K$ of $G$ and a $K$-invariant subset $Z$ of $X$
we can construct the spectrum $K\mathcal{X}^{G,K}(X,Z)$ as in Definition 2.2
and obtain the fibre sequence (2.7):

(3.1) $\dots\to K\mathcal{X}^{K}(\partial Z)\to K\mathcal{X}^{G,K}(X,Z)\to
K\mathcal{X}^{G}(X)\stackrel{{\scriptstyle r}}{{\to}}\Sigma
K\mathcal{X}^{K}(\partial Z)\to\dots\ .$

Under certain restrictions on $G$ and $X$ the values of $K\mathcal{X}^{G}(X)$
can be described in terms of the topological $K$-theory of Roe algebras. The
main result of the present subsection is a presentation of
$K\mathcal{X}^{G,K}(X,Z)$ and the sequence (3.1) in terms of $C^{*}$-algebras.
This interpretation will be used for application to the spectral theory of
differential operators in Section 5.

We assume that $G$ is countable, and that $X$ is very proper [BEb, Def. 3.7].
Then by [BEb, Prop. 4.2] we can choose an ample equivariant $X$-controlled
Hilbert space $(H,\phi)$ [BEb, Def. 4.1]. Here $H$ is a complex Hilbert space
with $G$-action and $\phi$ is an invariant, finitely additive projection-
valued measure on $X$ defined on all subsets. For any such equivariant
$X$-controlled Hilbert space one can define the Roe algebra $C(X,H,\phi)$ (see
Definitions 2.3, 3.6 & 3.8 of [BEb]).

By [BEb, Thm. 6.1] we have a canonical (up to equivalence) morphism of spectra

(3.2)
$K(C(X,H,\phi))\stackrel{{\scriptstyle\kappa_{(X,H,\phi)}}}{{\simeq}}K\mathcal{X}^{G}(X)\
.$

Note that the assumption that $X$ is very proper as a $G$-bornological coarse
space implies that $\operatorname{Res}^{G}_{K}(X)$ is a very proper
$K$-bornological coarse space. If we assume that $Z$ is a nice subset of
$\operatorname{Res}^{G}_{K}(X)$ for some $K$-invariant entourage of $X$ [BEb,
Def. 8.2], then $Z$ is a very proper $K$-bornological coarse space [BEb, Prop.
8.3].

We consider the subspace $H_{Z}:=\phi(Z)(H)$ of $H$, and we let $\phi_{Z}$
denote the restriction of the projection-valued measure $\phi$ to subsets of
$Z$ and $H_{Z}$. Then $(H_{Z},\phi_{Z})$ is a $K$-equivariant $Z$-controlled
Hilbert space. It is easy to check that it is ample.

The localized Roe algebra is the subalgebra $C(\partial Z,H_{Z},\phi_{Z})$ of
$C(Z,H_{Z},\phi_{Z})$ generated by operators supported on the members of the
big family $\partial Z$ (2.6). It is a closed ideal [BE20b, Lem. 7.57]. We
define a linear map

$q:C(X,H,\phi)\to C(Z,H_{Z},\phi_{Z})\ ,\quad q(A):=\phi(Z)A\phi(Z)\ .$

This map is $*$-preserving, but in general $q$ is not a homomorphism of
$C^{*}$-algebras.

######  Definition 3.1.

We define the $C^{*}$-algebra $C^{G,K}(X,Z)$ as the sub-$C^{*}$-algebra of
$C(Z,H_{Z},\phi_{Z})$ generated by $C(\partial Z,H_{Z},\phi_{Z})$ and
$\operatorname{im}(q)$. ∎

Recall that for an entourage $U$ of $X$, a subset $B$ of $X$ is called
$U$-bounded, if $B\times B\subseteq U$ [BE20b, Def. 2.15].

######  Definition 3.2.

We call $Z$ wide if for every entourage $U$ of $X$ and each $U$-bounded subset
$Y$ of $X$, there exists $g$ in $G$ such that $gY\subseteq Z\setminus
U[X\setminus Z]$. ∎

######  Theorem 3.3.

If $Z$ is wide, then we have a short exact sequence of $C^{*}$-algebras

(3.3) $0\to C(\partial
Z,H_{Z},\phi_{Z})\stackrel{{\scriptstyle\iota}}{{\to}}C^{G,K}(X,Z)\stackrel{{\scriptstyle\sigma}}{{\to}}C(X,H,\phi)\to
0\ ,$

where the map $\iota$ is the natural inclusion.

###### Proof.

Since $C(\partial Z,H_{Z},\phi_{Z})$ is a closed ideal in
$C(Z,H_{Z},\phi_{Z})$ it is also a closed ideal in $C^{G,K}(X,Z)$.

We need to construct the map $\sigma$. As in the proof of [BE20b, Lemma 7.58],
one shows that for $A,A^{\prime}$ in $C(X,H,\phi)$ we have

(3.4) $q(A)q(A^{\prime})-q(AA^{\prime})\in C(\partial Z,H_{Z},\phi_{Z})\ .$

Since $q$ is $*$-preserving, it then follows that $C(\partial
Z,H_{Z},\phi_{Z})+\operatorname{im}(q)$ is a $*$-subalgebra of
$C(Z,H_{Z},\phi_{Z})$, and that the composition

$\bar{q}:C(X,H,\phi)\stackrel{{\scriptstyle q}}{{\to}}C^{G,K}(X,Z)\to
C^{G,K}(X,Z)/C(\partial Z,H_{Z},\phi_{Z})$

is a homomorphism of $C^{*}$-algebras. We claim that $\bar{q}$ is injective.
Assuming the claim, since $\bar{q}$ has dense range by construction, it is an
isomorphism. Hence we can define $\sigma$ as the composition

$\sigma:C^{G,K}(X,Z)\to C^{G,K}(X,Z)/C(\partial
Z,H_{Z},\phi_{Z})\stackrel{{\scriptstyle\bar{q}^{-1}}}{{\to}}C(X,H,\phi)$

and obtain the desired exact sequence (3.3).

We now show the claim. Let $A$ be in $C(X,H,\phi)$ and assume that
$\bar{q}(A)=0$, i.e., that

$q(A)=\phi(Z)A\phi(Z)\in C(\partial Z,H_{Z},\phi_{Z})\ .$

Then for every $\epsilon$ in $(0,\infty)$ we can then choose an entourage $U$
and a $U$-bounded subset $Y$ of $X$ such that the following two inequalities
are satisfied:

$\|\phi(Y)A\phi(Y)\|\geq\frac{1}{2}\|A\|\ ,\quad\|\phi(Z\setminus U[X\setminus
Z])A\phi(Z\setminus U[X\setminus Z])\|<\epsilon\ .$

Since $Z$ is wide we can choose an element $g$ in $G$ such that $gY\subseteq
Z\setminus U[X\setminus Z]$. Then using the equality

$\phi(Z\setminus U[X\setminus Z])\phi(gY)=\phi((Z\setminus U[X\setminus
Z])\cap gY)=\phi(gY)\ ,$

the equivariance $\phi(Y)=g\phi(gY)g^{-1}$ of $\phi$ and the equivariance
$g^{-1}Ag=A$ of $A$ we get

(3.5)
$\phi(Y)A\phi(Y)=g\phi(gY)g^{-1}Ag\phi(gY)g^{-1}=g\phi(gY)A\phi(gY)g^{-1}\\\
=g\phi(gY)\phi(Z\setminus U[X\setminus Z])A\phi(Z\setminus U[X\setminus
Z])\phi(gY)g^{-1}$

Hence

$\frac{1}{2}\|A\|\leq\|\phi(Y)A\phi(Y)\|\leq\epsilon\ .$

Since $\epsilon$ is arbitrary this implies that $A=0$. ∎

The sequence of $C^{*}$-algebras (3.3) induces a fibre sequence of $K$-theory
spectra

(3.6) ${\cdots}$${K(C(\partial
Z,H_{Z},\phi_{Z}))}$${K(C^{G,K}(X,Z))}$${K(C(X,H,\phi))}$${\Sigma K(C(\partial
Z,H_{Z},\phi_{Z}))}$${\cdots\ .}$${\scriptstyle{\delta^{C^{*}}}}$

The following is the main theorem of this section.

######  Theorem 3.4.

The fibre sequences (3.6) and (3.1) are canonically equivalent.

###### Proof.

We use the comparison maps $\kappa_{\bullet}$ as in (3.2) and consider the
following diagram:

(3.7) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{r}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
176.81636pt\raise-41.24664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
176.81636pt\raise-41.24664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Sigma
K\mathcal{X}^{K}(\partial Z)}$}}}}}}}\ignorespaces}}}}\ignorespaces\ .$

It suffices to show that the right square commutes. Then we use the universal
properties of the fibre sequences in order to obtain the dotted arrow and the
fact that it is an equivalence.

We expand the right square as follows:

(3.8)
$\textstyle{K(C(X,H,\phi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K(i)}$$\scriptstyle{\delta^{C^{*}}}$$\scriptstyle{\simeq}$$\scriptstyle{\kappa_{(X,H,\phi)}}$$\textstyle{K(C(\operatorname{Res}^{G}_{K}(X,H,\phi)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\kappa_{\operatorname{Res}^{G}_{K}(X,H,\phi)}}$$\scriptstyle{\delta^{MV}}$$\textstyle{\Sigma
K(C(\partial
Z,H_{Z},\phi_{Z}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\kappa_{(\partial
Z,H_{Z},\phi_{Z})}}$$\textstyle{K\mathcal{X}^{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{c^{G}_{K}}}$$\textstyle{K\mathcal{X}^{K}(\operatorname{Res}^{G}_{K}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{\Sigma}K\mathcal{X}^{K}(\partial
Z)}$

In order to see that the right square and the upper triangle commute, we
consider the following morphisms of short exact sequences of $C^{*}$-algebras

(3.9) ${0}$${C(\partial
Z,H_{Z},\phi_{Z})}$${C^{G,K}(X,Z)}$${C(X,H,\phi)}$${0}$${0}$${C(\partial
Z,H_{Z},\phi_{Z})}$${C(Z,H_{Z},\phi_{Z})}$${\frac{C(Z,H_{Z},\phi_{Z})}{C(\partial
Z,H_{Z},\phi_{Z})}}$${0}$${0}$${C(\\{X\setminus Z\\},H_{X\setminus
Z},\phi_{X\setminus
Z})}$${C(\operatorname{Res}^{G}_{K}(X,H,\phi))}$${\frac{C(\operatorname{Res}^{G}_{K}(X,H,\phi))}{C(\\{X\setminus
Z\\},H_{X\setminus Z},\phi_{X\setminus
Z})}}$${0}$$\scriptstyle{\iota}$$\scriptstyle{\sigma}$$\scriptstyle{i}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{b}$$\scriptstyle{\cong}$$\scriptstyle{a}$

The vertical maps are induced by the natural inclusions.

Applying the $K$-theory functor for $C^{*}$-algebras to this diagram and
extending to the right, we get a morphism of fibre sequences

(3.10) ${K(C^{G,K}(X,Z)}$${K(C(X,H,\phi))}$${\Sigma K(C(\partial
Z,H_{Z},\phi_{Z}))}$${K(C(Z,H_{Z},\phi_{Z})}$${K(\frac{C(Z,H_{Z},\phi_{Z})}{C(\partial
Z,H_{Z},\phi_{Z})})}$${\Sigma K(C(\partial
Z,H_{Z},\phi_{Z}))}$${K(C(\operatorname{Res}^{G}_{K}(X,H,\phi)))}$${K\bigl{(}\frac{C(\operatorname{Res}^{G}_{K}(X,H,\phi))}{C(\\{X\setminus
Z\\},H_{X\setminus Z},\phi_{X\setminus Z})}\bigr{)}}$${\Sigma
K(C(\\{X\setminus Z\\},H_{X\setminus Z},\phi_{X\setminus Z}))\
.}$$\scriptstyle{K(\sigma)}$$\scriptstyle{\delta^{C^{*}}}$$\scriptstyle{K(i)}$$\scriptstyle{K(q)}$$\scriptstyle{K(p)}$$\scriptstyle{\delta^{C^{*}\prime}}$$\scriptstyle{K(b)}$$\scriptstyle{\simeq}$$\scriptstyle{K(a)}$$\scriptstyle{\delta^{MV}}$$\scriptstyle{\delta^{C^{*}\prime\prime}}$

By definition, the Mayer-Vietoris boundary map for the complementary pair
$(Z,\\{X\setminus Z\\})$ on $X$ is given by

(3.11) $\delta^{MV}:=\delta^{C^{*}\prime}\circ{K(b)^{-1}}\circ
K(a):K(C(\operatorname{Res}^{G}_{K}(X,H,\phi)))\to\Sigma K(C(\partial
Z,H_{Z},\phi_{Z}))\ .$

Similarly, the boundary map $\delta$ is obtained from the diagram

(3.12)
${K\mathcal{X}^{K}(Z)}$${\frac{K\mathcal{X}^{K}(Z)}{K\mathcal{X}^{H}(\partial
Z)}}$${\Sigma K\mathcal{X}^{K}(\partial
Z)}$${K\mathcal{X}^{K}(\operatorname{Res}^{G}_{K}(X))}$${\frac{K\mathcal{X}^{K}(\operatorname{Res}^{G}_{K}(X))}{K\mathcal{X}^{K}(\\{X\setminus
Z\\})}}$${\Sigma K\mathcal{X}^{K}(\\{X\setminus
Z\\})}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{\simeq}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\delta}$$\scriptstyle{\delta^{\prime\prime}}$

as

$\delta:=\delta^{\prime}\circ(b^{\prime})^{-1}\circ
a^{\prime}:K\mathcal{X}^{K}(\operatorname{Res}^{G}_{K}(X))\to{\Sigma}K\mathcal{X}^{H}(\partial
Z)\ .$

Here we write cofibres as quotients. In view of [BEb, Thm. 6.1] the
transformation $\kappa_{\bullet}$ induces a morphism from the lower two rows
of the diagram (3.10) to the diagram (3.12). Now the right square of (3.8) is
contained in the commutative cube obtained this way and hence commutative.

The boundary map $\delta^{C^{*}}$ is induced by the first sequence in (3.9).
The naturality of the boundary maps for the map from the first to the second
sequence in (3.9) and the existence of the dotted lift therefore imply that
the upper triangle in (3.8) commutes.

Finally, the left square in (3.8) commutes in view of [BEc, Rem. 10.6]. ∎

##  4\. Operators in Roe algebras

Assume that $A$ is a selfadjoint operator in some $C^{*}$-algebra
$\mathcal{A}$ with spectrum $\sigma(A)$, and let $I$ be an interval in
$\mathbb{R}$.

######  Definition 4.1.

We say that $I$ is a spectral interval of $A$ if $\partial
I\cap\sigma(A)=\varnothing$. ∎

If $I$ is a spectral interval for $A$, then the spectral projection $E_{A}(I)$
also belongs to $\mathcal{A}$, where $E_{A}$ is the spectral measure of $A$
and hence defines a class $[E_{A}(I)]$ in $K_{0}(\mathcal{A})$. The point here
is that under the assumption of Definition 4.1, we can write $E_{A}(I)$ as a
continuous function of $A$ instead of using the characteristic function.

As in Section 3 we consider a countable group $G$ and let $X$ be a very proper
$G$-bornological coarse space. We choose an $X$-controlled ample Hilbert space
$(H,\phi)$. We furthermore consider a subgroup $K$ and a wide $K$-invariant
subset $Z$ of $X$ (see Definition 3.2).

We consider elements $A$ in $C(X,H,\phi)$ and $B$ in $C(Z,H_{Z},\phi_{Z})$. In
the following we use the maps

$p:C(Z,H_{Z},\phi_{Z})\to\frac{C(Z,H_{U},\phi_{Z})}{C(\partial
Z,H_{Z},\phi_{Z})}\ ,\quad
q:C(X,H,\phi)\to\frac{C(Z,H_{U},\phi_{Z})}{C(\partial Z,H_{Z},\phi_{Z})}$

from the diagram (3.9).

######  Definition 4.2.

We say that $A$ and $B$ are affiliated if $q(A)=p(B)$. ∎

Using $\kappa_{(X,H,\phi)}$ in (3.2) we consider $[E_{A}(I)]$ as a class in
$K\mathcal{X}^{G}(X)$. In order to shorten the notation we will drop the
identification (3.2) from the notation. Recall the obstruction morphism
$r:K\mathcal{X}^{G}(X)\to K\mathcal{X}^{K}(\partial Z)$ from (3.1).

######  Proposition 4.3.

Assume:

  1. (1)

$A$ and $B$ are affiliated.

  2. (2)

$I$ is a spectral inverval for both $A$ and $B$.

Then $r([E_{A}(I)])=0$.

###### Proof.

We have the equality

(4.1) $p(E_{B}(I))=E_{p(B)}(I)=E_{q(A)}(I)=q(E_{A}(I))\ .$

Recall the boundary map $\delta^{C^{*}\prime}$ in (3.10). We calculate

$\displaystyle r([E_{A}(I)])$
$\displaystyle\stackrel{{\scriptstyle\eqref{eoirgjowergregwergerg}}}{{=}}$
$\displaystyle\delta^{C^{*}}([E_{A}(I)])\stackrel{{\scriptstyle\eqref{vewvewrvervevervwevervrev}}}{{=}}\delta^{MV}(K(i)([E_{A}(I)]))$
$\displaystyle\stackrel{{\scriptstyle\eqref{ervwrekpwervewveverv}}}{{=}}$
$\displaystyle\delta^{C^{*}\prime}(K(b^{-1})(K(a)(K(i)([E_{A}(I)]))))\stackrel{{\scriptstyle\eqref{asdvkjnkqfv}}}{{=}}\delta^{C^{*}\prime}(K(q)([E_{A}(I)]))$
$\displaystyle\stackrel{{\scriptstyle\eqref{f32f23f3f23f3}}}{{=}}$
$\displaystyle\delta^{C^{*}\prime}(K(p)([E_{B}(I)]))=0\ .$

The last equivalence follows from $\delta^{C^{*}\prime}\circ K(p)\simeq 0$
since the upper horizontal part in (3.10) is a segment of a fibre sequence. ∎

Proposition 4.3 gives a $K$-theoretic condition for checking that certain
elements in the resolvent set of $A$ belong to the spectrum of $B$. More
precisely we have the following corollary.

######  Corollary 4.4.

Assume:

  1. (1)

$A$ and $B$ are affiliated.

  2. (2)

$I$ is a spectral inverval for $A$.

  3. (3)

$r([E_{A}(I)])\not=0$.

Then $\partial I\cap\sigma(B)\not=\varnothing$.

##  5\. Differential operators

Let $M$ be a complete Riemannian manifold with an isometric action of $G$. Let
furthermore $D$ be an elliptic selfadjoint differential operator on $M$ acting
on sections of some equivariant Hermitean vector bundle $E\to M$. We will
consider the following two cases simultaneously.

######  Assumption 5.1.

  1. (1)

(Dirac case:) $D$ is a first order Dirac type operator of degree $n$. The
degree is implemented by a grading and the action an appropriate Clifford
algebra. We will hide both structures from the notation, see [BEb, Sec. 7] for
details.

  2. (2)

(Laplace case): $D$ is a second order Laplace type operator with non-negative
spectrum.

∎

We consider $M$ as a bornological coarse space with the metric coarse
structure and the minimal compatible bornology. We consider the Hilbert space
$H_{0}:=L^{2}(M,E)$ with the control $\phi_{0}$ as described in [BEb, Sec. 9].
We can assume that $(H_{0},\phi_{0})$ embeds into an ample $M$-controlled
Hilbert space $(H,\phi)$. Using this embedding we can and will view the Roe
algebras associated to $(H_{0},\phi_{0})$ as subalgebras of the corresponding
Roe algebras associated to $(H,\phi)$.

We now assume that $Z$ is a $K$-invariant codimension-zero submanifold of $M$
with smooth boundary111We add a superscript $\infty$ in order to distinguish
this boundary from the coarse boundary of $Z$ introduced above.
$\partial^{\infty}Z$. We further assume that $D^{\prime}$ is a selfadjoint
extension of $D$ determined by imposing a local $K$-invariant elliptic
boundary condition at $\partial^{\infty}Z$. In the Dirac case we assume that
this condition is in addition compatible with the grading and the Clifford
action, while in the Laplace case we assume that $D^{\prime}$ is still non-
negative. While local elliptic boundary conditions always exist in the Laplace
case (e.g., Dirichlet boundary conditions), the existence of boundary
conditions compatible with grading and Clifford action may be obstructed in
the Dirac case. In this case, an operator $D^{\prime}$ as above does not
exist.

Let $\varphi\in C_{0}(\mathbb{R})$.

######  Proposition 5.2.

We have $\varphi(D)\in C(M,H,\phi)$ and $\varphi(D^{\prime})\in
C(Z,H_{Z},\phi_{Z})$.

###### Proof.

For the Dirac case the assertion $\varphi(D)\in C(M,H_{0},\phi_{0})$ is [HR00,
Prop. 10.5.6]. The case of $D^{\prime}$ has the same proof taking into account
that the wave equation still has finite propagation speed in view of the
locality of the boundary condition.

For the Laplace case, by positivity of $D$ and $D^{\prime}$ we can assume that
$\varphi$ is even. We then use the finite propagation speed of the wave
operator family $\cos(t\sqrt{D})$ and argue similarly as in the Dirac case
(see e.g., Lemma 1.7 in [LT21]). ∎

######  Proposition 5.3.

The operators $\varphi(D)$ and $\varphi(D^{\prime})$ are affiliated.

###### Proof.

In the Laplace case we set $\varphi_{1}:=\varphi$, and in the Dirac case use
$\varphi_{1}(t):=\varphi(t^{2})$. It suffices to show the assertion for
$\varphi_{1}$ in $C_{0}(\mathbb{R})$ with $\widehat{\varphi}_{1}\in
C_{c}(\mathbb{R})$. This implies the general case since these functions are
dense in $C_{0}(\mathbb{R})$.

Let $U_{R}:=\\{d\leq R\\}$ be the metric entourage of size $R$ on $M$. Then by
finite propagation speed

$\phi(Z)(\varphi(D)-\varphi(D^{\prime}))\phi(Z\setminus U_{R}[X\setminus
Z])=0\ .$

It follows that

$\phi(Z)\varphi(D)\phi(Z)-\varphi(D^{\prime})=\phi(Z)\varphi(D)\phi(Z\cap
U_{R}[X\setminus Z])+\varphi(D^{\prime})U_{R}(Z\cap U_{R}[X\setminus Z])\ .$

Both summands on the right-hand side belong to $C(\partial Z,H_{Z},\phi_{Z})$.
Consequently $q(\varphi(D))=p(\varphi(D^{\prime}))$. ∎

Let $\sigma(D)$ denote the spectrum of $D$. Let $\lambda$ be in
$\mathbb{R}\setminus\sigma(D)$.

######  Proposition 5.4.

Assume that $D$ is of Laplace type.

  1. (1)

We have $E_{D}(-1,\lambda)\in C(M,H,\phi)$.

  2. (2)

If $r([E_{D}(-1,\lambda)])\not=0$, then $\lambda\in\sigma(D^{\prime})$.

###### Proof.

The assertions are obvious if $\lambda$ is negative. In the following we
assume that $\lambda$ is non-negative. We can choose $\varphi$ in
$C_{0}(\mathbb{R})$ such that it is strictly monotoneously decreasing on
$[-1,\infty)$. Since $D$ is positive, by the spectral mapping theorem we have
the equality

$E_{D}(-1,\lambda)=E_{\varphi(D)}((\phi(\lambda),\varphi(-1))\ .$

Since $\varphi(-1)$ and $\phi(\lambda)$ are not in the spectrum of
$\varphi(D)$, the interval $(\phi(\lambda),\varphi(-1))$ is a spectral
interval for $\phi(D)$. Hence
$E_{\varphi(D)}((\varphi(\lambda),\varphi(-1))\in C(M,H,\phi)$. This implies
(1).

In order to show (2) we assume by contradiction that
$\lambda\not\in\sigma(D^{\prime})$. Then similarly as above
$(\varphi(\lambda),\varphi(-1))$ is a spectral interval for
$\varphi(D^{\prime})$, too. We conclude by Proposition 4.3 that
$r_{X,Z}([E_{D}(-1,\lambda)])=0$ which contradicts our Assumption (2). ∎

If $D$ is of Dirac type (of degree $n$), then we have an index class
${\mathtt{index}}^{G}(D)$ in $K\mathcal{X}_{n}^{G}(M)$.

######  Proposition 5.5.

We assume that $D$ is of Dirac type. If $D^{\prime}$ exists, then we have
$r({\mathtt{index}}(D))=0$.

###### Proof.

The index class ${\mathtt{index}}^{G}(D)$ is represented by the homomorphism

$C_{0}(\mathbb{R})\ni\varphi\mapsto\varphi(D)\in C(M,H,\phi)\ ,$

where we use the homomorphism picture of $K$-theory, see e.g. [Zei16, Sec.
4.1].

Similarly, if $D^{\prime}$ exists, then ${\mathtt{index}}^{K}(D^{\prime})$ in
$K\mathcal{X}^{K}_{n}(Z)$ is represented by the homomorphism

$C_{0}(\mathbb{R})\ni\varphi\mapsto\varphi(D^{\prime})\in C(Z,H_{Z},\phi_{Z})\
.$

Since $q(\varphi(D))=p(\varphi(D^{\prime}))$ by Proposition 5.3 we conclude
that $q({\mathtt{index}}^{G}(D))=p({\mathtt{index}}^{K}(D^{\prime}))$. The
assertion now follows from the long exact sequence associated to the pair
$(Z,\partial Z)$, see also the proof of Proposition 4.3. ∎

######  Corollary 5.6.

If $r({\mathtt{index}}^{G}(D))\not=0$, then $D^{\prime}$ does not admit a
$K$-invariant, local, selfadjoint, elliptic boundary condition which is
compatible with the Clifford action.

For $G$-equivariant Dirac operators $\not{D}$ we can calculate
$r({\mathtt{index}}^{G}(\not{D}))$ as follows. The choice of $Z$ induces an
orientation and hence trivialization of the normal bundle of
$\partial^{\infty}Z$ by the out-going normal. The Dirac operator $\not{D}$
then naturally induces a $K$-invariant Dirac operator $\partial\not{D}$ on
$\partial^{\infty}Z$ which is well-defined up to zero order perturbations (if
we would require a geometric product structure, then the operator would be
canonical). This restriction procedure works such that if $\not{D}$ is the
Dirac operator on $M$ associated to a $Spin^{c}$-structure or
$Spin$-structure, then $\partial\not{D}$ is the Dirac operator associated to
the induced $Spin^{c}$\- or $Spin$-structure induced on $\partial^{\infty}Z$.

The following result is well-known and an equivariant version of Roe’s
partitioned manifold index theorem, see e.g. [Zei15], [NSZ21].

######  Lemma 5.7.

We have
$r({\mathtt{index}}^{G}(\not{D}))={\mathtt{index}}^{K}(\partial\not{D})$.

###### Proof.

We will sketch a version of Zeidler’s proof [Zei15] using the formalism
developed in [Bun18]. Using the coarse cone construction
$\mathcal{O}^{\infty}$ [BEKW20b, Sec. 9.1] and the (equivariant version of)
[BE20a, Lemma 9.6] we define the closed local homology functor
$K\mathcal{X}^{G}\mathcal{O}^{\infty}:G\mathbf{UBC}\to\mathbf{Sp}$ satisfying
(equivariant generalizations of) the axioms in [BE20a, 3.12]. By [Bun18, Def.
4.14] the Dirac operator $\not{D}$ has a symbol class $\sigma(\not{D})$ in
$\Sigma^{-1}K\mathcal{X}^{G}\mathcal{O}^{\infty}(M)$. The symbol class
determines the index via

(5.1) $\partial\sigma^{G}(\not{D})={\mathtt{index}}^{G}(\not{D})\ ,$

where $\partial$ is the cone boundary, see [BE20a, (9.1]. One can compare the
Mayer-Vietoris boundary $\delta$ for $K\mathcal{X}^{G}\mathcal{O}^{\infty}$
applied to $M$ and the decomposition $(Z,M\setminus\mathrm{int}(Z))$ with that
for $\mathbb{R}\otimes Z$ and $((-\infty,0]\otimes Z,[0,\infty)\otimes Z)$.
Using the relative index theorem and the suspension formula [BEb, Th, 10.4 &
Thm. 11.1] we get222This is the sketchy step of the argument. the equality

$\sigma^{G}(\partial\not{D})=\delta\operatorname{Res}^{G}_{K}\sigma^{G}(\not{D})\
.$

Since the cone boundary is a morphism of local homology theories it is
compatible with the Mayer-Vietoris boundaries. We conclude that

$\displaystyle r({\mathtt{index}}^{G}(\not{D}))$ $\displaystyle=$
$\displaystyle\delta(\operatorname{Res}^{G}_{K}({\mathtt{index}}^{G}(\not{D})))=\delta\operatorname{Res}^{G}_{K}\partial\sigma^{G}(\not{D})=\delta\partial\operatorname{Res}^{G}_{K}\sigma^{G}(\not{D})$
$\displaystyle=$
$\displaystyle\partial\delta\operatorname{Res}^{G}_{K}\sigma^{G}(\not{D})=\partial\sigma^{K}(\partial\not{D})={\mathtt{index}}^{K}(\partial\not{D})\
,$

as claimed. ∎

######  Example 5.8.

Assume that $D$ is of Laplace type and given by $D=\not{D}^{2}$ for some
graded Dirac operator. Then we can decompose $D=D^{+}+D^{-}$, and we have the
equality

(5.2)
$[E_{D^{+}}(-1,\lambda)]-[E_{D^{-}}(-1,\lambda)]={\mathtt{index}}^{G}(\not{D})\
.$

In this case we can use Lemma 5.7 in order to calculate at least the image of
the difference $[E_{D^{+}}(-1,\lambda)]-[E_{D^{-}}(-1,\lambda)]$ under $r$. ∎

######  Example 5.9.

We assume that $\not{D}$ is the Dirac operator associated to a spin structure
on $M$. By the Schrödinger-Lichenerowicz formula we have

$\not{D}^{2}=\nabla^{*}\nabla+\frac{1}{4}s\ ,$

where $s$ in $C^{\infty}(M)$ is the scalar curvature function.

######  Corollary 5.10.

If ${\mathtt{index}}^{K}(\partial\not{D})\not=0$, then $\inf_{Z}s\leq 0$.

###### Proof.

Assume that $c:=\inf_{Z}s>0$. Then the spectra of the Dirichlet extensions of
$D^{\prime,\pm}$ are bounded below by $c$. This implies
$r([E_{D^{\pm}}(-1,c/2)])=0$. From (5.2) we conclude that
$r({\mathtt{index}}^{G}(\not{D}))=0$. By Lemma 5.7 we get
${\mathtt{index}}^{K}(\partial\not{D})=0$. ∎

In view of ${\mathtt{index}}^{K}(\partial\not{D})=\partial\sigma^{K}(\not{D})$
we see that ${\mathtt{index}}^{K}(\partial\not{D})$ only depends on the
$K$-uniform bornological coarse structure associated to the $K$-invariant
metric on $\partial Z$. If ${\mathtt{index}}^{K}(\partial\not{D})\not=0$, then
by Corollary 5.10 there does not exist a complete $G$-invariant metric on $M$
whose restriction to $Z$ has uniformly positive scalar curvature and induces
the given $K$-uniform bornological coarse structure on $Z$. ∎

######  Remark 5.11.

In this remark we explain the relation of Example 5.9 with the classical
codimension-one obstruction against positive scalar curvature in the version
of [Zei15, Thm. 1.7].

We let $\bar{M}$ be a closed $n$-dimensional spin manifold with a codimension-
one submanifold $\bar{N}$ with trivialized normal bundle such that
$\pi_{1}(\bar{N})\to\pi_{1}(\bar{M})$ is injective. We then set
$G:=\pi_{1}(\bar{M})$ and $K:=\pi_{1}(\bar{N})$. We let $M$ be the universal
covering of $\bar{M}$. For $\partial^{\infty}Z$ we choose a connected
component of the preimage of $\bar{N}$ under the projection $M\to\bar{M}$.
Since the normal bundle of $\partial^{\infty}Z$ is trivial, the submanifold
$\partial^{\infty}Z$ separates $M$ into two components, and we let $Z$ be the
component such that the normal vectors are outward pointing. The subgroup $K$
preserves $Z$.

Since $G$ acts freely and cocompactly on $M$, and similarly, $K$ acts freely
and cocompactly on $N$, we have equivalences

(5.3) $\displaystyle K\mathcal{X}^{G}(M)$ $\displaystyle\simeq
K\mathcal{X}^{G}(G_{can,min})\simeq K(C_{r}(G))\ ,$ $\displaystyle
K\mathcal{X}^{K}(\partial^{\infty}Z)$ $\displaystyle\simeq
K\mathcal{X}^{K}(K_{can,min})\simeq K(C_{r}(K))\ .$

The $\alpha^{G}$-invariant $\alpha^{G}(\bar{M})$ of $\bar{M}$ in
$K_{n}(C_{r}(G))$ is defined as the image of
${\mathtt{index}}^{G}(\not{D}_{M})$ under the identification (5.3), where
$\not{D}_{M}$ is the spin Dirac operator. Similarly, $\alpha^{G}(\bar{N})$ in
$K_{n-1}(C_{r}(G))$ is the image of ${\mathtt{index}}^{K}(\not{D}_{\partial
Z})$ under (5.3). Under the identifications (5.3) the map

$r_{*}:K\mathcal{X}^{G}_{*}(M)\to K\mathcal{X}_{*-1}^{K}(\partial Z)$

corresponds to the map

$\Psi:K_{*}(C_{r}(G))\to K_{*-1}(C_{r}(K))$

from the proof of [Zei15, Thm. 1.7], and the equality

$\Psi(\alpha^{G}(\bar{M}))=\alpha^{K}(\bar{N})$

shown in this reference is a special case of Lemma 5.7.

Note that [Zei15] has also a version for the maximal group $C^{*}$-algebras
which at the moment is not accessible by our general coarse methods. ∎

##  6\. Incoorporating local positivity

Let $M$ and $D$ be as in Section 5. Let $P$ be a $G$-invariant subset of $M$.

######  Definition 6.1.

We say that $D$ is uniformly locally positive on $P$ if there exists a $c$ in
$(0,\infty)$ such that the following holds.

  1. (1)

(Dirac case) We have $D^{2}=\nabla^{*}\nabla+R$ for some selfadjoint bundle
endomorphism $R$ such that $R(m)\geq c^{2}$ for all $m$ in $P$.

  2. (2)

(Laplace case) $D=\nabla^{*}\nabla+R$ for some selfadjoint bundle endomorphism
$R$ such that $R(m)\geq c$ for all $m$ in $P$.

The constant $c$ is called the positivity bound.∎

We let $M_{P}$ denote the $G$-bornological coarse space $M$ with the metric
coarse structure and the bornology

$\mathcal{B}_{P}:=\\{B\subseteq M\>|\>\mbox{$B\cap(M\setminus P)$ is
bounded}\\}\ .$

In this subsection we let $Z_{P}$ be the $K$-invariant subset $Z$ of $M$ with
the induced $K$-bornological coarse structure from $M_{P}$.

######  Proposition 6.2.

For every $\varphi$ in $C_{0}((-c,c))$ we have $\varphi(D)\in C(M_{P},H,\phi)$
and $\varphi(D^{\prime})\in C(Z_{P},H_{Z},\phi_{Z})$.

###### Proof.

For the Dirac case the assertion $\varphi(D)\in C(\\{M\setminus P\\},H,\phi)$
is [Roe16, Lemma 2.3]. We now use the obvious inclusion

$C(\\{M\setminus P\\},H,\phi)\subseteq C(M_{P},H,\phi)\ .$

Again the case of $D^{\prime}$ has the same proof.

For the Laplace case, by positivity of $D$ and $D^{\prime}$ we can assume that
$\varphi$ is even. We then use the finite propagation speed of the wave
operator family $\cos(t\sqrt{D})$ and argue similarly as in the Dirac case. ∎

######  Proposition 6.3.

For every $\varphi$ in $C_{0}((-c,c))$ the operators $\varphi(D)$ and
$\varphi(D^{\prime})$ are affiliated.

###### Proof.

The proof is the same as for Proposition 5.3. ∎

In the following we use the notation $r_{P}:K\mathcal{X}^{G}(M_{P})\to\Sigma
K\mathcal{X}^{K}(\partial Z_{P})$ for the obstruction morphism (2.10) with the
new bornology incorporated, where $\partial Z_{P}$ is the big family $\partial
Z$ whose members have the bornological coarse structure induced from $M_{P}$.
The identity maps of the underlying sets of $M$ and the members of the family
induce the vertical maps in the following commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
25.02692pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K\mathcal{X}^{G}(M_{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 49.02692pt\raise 0.0pt\hbox{\hbox{\kern
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49.02692pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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Z_{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
79.90463pt\raise-29.5pt\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{$\textstyle{K\mathcal{X}^{G}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.27605pt\raise-36.7397pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{r}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
51.21346pt\raise-41.24664pt\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
51.21346pt\raise-41.24664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Sigma
K\mathcal{X}^{K}(\partial Z)}$}}}}}}}\ignorespaces}}}}\ignorespaces\ .$

Let $\sigma(D)$ denote the spectrum of $D$. Let $\lambda$ be in
$(-\infty,c)\setminus\sigma(D)$.

######  Proposition 6.4.

Assume that $D$ is of Laplace type.

  1. (1)

We have $E_{D}(-1,\lambda)\in C(M_{P},H,\phi)$.

  2. (2)

If $r_{{P}}([E_{D}(-1,\lambda)])\not=0$, then $\lambda\in\sigma(D^{\prime})$.

###### Proof.

Again this has the same proof as Proposition 5.4. ∎

If $D$ is of Dirac type, then we have an index class ${\mathtt{index}}^{G}(D)$
in $K\mathcal{X}_{n}^{G}(M_{P})$.

######  Proposition 6.5.

We assume that $D$ is of Dirac type. If $D^{\prime}$ exists, then we have
$r_{{P}}({\mathtt{index}}^{G}(D))=0$.

###### Proof.

The index class ${\mathtt{index}}^{G}(D)$ is represented by the homomorphism

$C_{0}((-c,c))\ni\varphi\mapsto\varphi(D)\in C(M_{P},H,\phi)\ .$

Similarly, if $D^{\prime}$ exists, then ${\mathtt{index}}^{K}(D^{\prime})$ in
$K\mathcal{X}^{K}_{n}(Z_{P})$ is represented by the homomorphism

$C_{0}((-c,c))\ni\varphi\mapsto\varphi(D^{\prime})\in C(Z_{P},H_{Z},\phi_{Z})\
.$

Since $q(\varphi(D))=p(\varphi(D^{\prime}))$ by Proposition 6.3 we conclude
that $q({\mathtt{index}}^{G}(D))=p({\mathtt{index}}^{K}(D^{\prime}))$. We now
argue as in the proof of Proposition 5.5. ∎

######  Corollary 6.6.

If $r_{{P}}({\mathtt{index}}^{G}(D))\not=0$, then $D^{\prime}$ does not admit
a local selfadjoint elliptic boundary condition which is compatible with the
Clifford action.

Since in general the map $K\mathcal{X}_{*}^{K}(\partial Z_{P})\to
K\mathcal{X}_{*}^{K}(\partial Z)$ may have a kernel, the condition
$r_{P}(...)\not=0$ in Proposition 6.4, Proposition 6.5, and Corollary 6.6 is
expected to be satisfied in more cases than the condition $r(...)\not=0$.

##  7\. Spectral projections of magnetic laplacians

In this section we discuss the example of the two-dimensional magnetic
Laplacian.

We consider the smooth manifold $\mathbb{R}^{2}$ with coordinates $(x,y)$ and
the standard Riemannian metric $dx^{2}+dy^{2}$. The group $\mathbb{Z}^{2}$
acts on $\mathbb{R}^{2}$ by translations

$((m,n),(x,y))\mapsto(x+m,y+n)\ .$

In this way $\mathbb{R}^{2}$ becomes a $\mathbb{Z}^{2}$-bornological coarse
space. The embedding $\mathbb{Z}^{2}\to\mathbb{R}^{2}$ induces an equivalence
of $\mathbb{Z}^{2}$-bornological coarse spaces
$\mathbb{Z}^{2}_{can,min}\to\mathbb{R}^{2}$. By [BEc, Prop.8.2] for any group
$G$ we have an equivalence $K\mathcal{X}^{G}(G_{can,min})\simeq
K(C^{*}_{r}(G))$, where $C^{*}_{r}(G)$ denotes the reduced group
$C^{*}$-algebra of $G$. The case of $G=\mathbb{Z}$ gives the second
equivalence in the following chain

(7.1)
$K\mathcal{X}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})\stackrel{{\scriptstyle\simeq}}{{\leftarrow}}K\mathcal{X}^{\mathbb{Z}^{2}}(\mathbb{Z}^{2}_{can,min})\simeq
K(C_{r}^{*}(\mathbb{Z}^{2}))\stackrel{{\scriptstyle
Fourier}}{{\simeq}}K(C(\widehat{T}^{2}))\stackrel{{\scriptstyle\begin{subarray}{c}Serre\\\
Swan\end{subarray}}}{{\simeq}}KU^{\widehat{T}^{2}}\ ,$

where $\widehat{T}^{2}$ denotes the dual group of $\mathbb{Z}^{2}$. The
decomposition $\Sigma^{\infty}_{+}\widehat{T}^{2}\simeq S\oplus\Sigma
S\oplus\Sigma S\oplus\Sigma^{2}S$ (where $S$ is the sphere spectrum) provides
the decomposition

(7.2) $KU^{{\widehat{T}^{2}}}\simeq
KU\oplus\Sigma^{-1}KU\oplus\Sigma^{-1}KU\oplus\Sigma^{-2}KU\ .$

If we combine (7.1) and (7.2) and apply $\pi_{0}$, then we get an isomorphism

(7.3)
$(\dim,c_{1}):K\mathcal{X}_{0}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})\cong\mathbb{Z}\oplus\mathbb{Z}\
,$

where $\dim$ corresponds to the summand $\mathbb{Z}\cong\pi_{0}({KU})$
contributed by the first summand in (7.2), and $c_{1}$ corresponds to the
summand $\mathbb{Z}\cong\pi_{0}(\Sigma^{-2}KU)$ contributed by the fourth
summand in (7.2).

For a natural number $k\geq 1$ we let $\mathbb{Z}^{2}$ act on the total space
of the trivial line bundle $\mathbb{R}^{2}\times\mathbb{C}\to\mathbb{R}^{2}$
by

(7.4) $((m,n),(x,y,z))\mapsto(x+m,y+n,e^{2\pi ik(my-nx)}z)\ .$

Let $W$ in $C^{\infty}(\mathbb{R}^{2})^{\mathbb{Z}^{2}}$ be a real periodic
function and set $w:=\|W\|_{\infty}$. Then the partial differential operator

(7.5) $D_{k}:=\Delta-4\pi
ik(y\partial_{x}-x\partial_{y})+4\pi^{2}k^{2}(x^{2}+y^{2})-4\pi k+W$

is invariant under this action, where
$\Delta:=-(\partial_{x}^{2}+\partial_{y}^{2})$.

Assertion (1) in the following proposition states that ${D_{k}+w}$ is an
example of an operator satisfying 5.1.2. The following has also been shown in
[DN80].

######  Proposition 7.1.

  1. (1)

The operator $D_{k}$ is of Laplace type, formally selfadjoint, and lower
bounded by $-w$.

  2. (2)

For every $\lambda\in{(w,{8}\pi k-w)}$ the interval $(-w{-1},\lambda)$ is a
spectral interval for $D_{k}$.

  3. (3)

For every $\lambda\in{(w,{8}\pi k-w)}$ we have
$[E_{D_{k}}(-w{-1},\lambda)]=(2k,{-}1)$ under the identification (7.3).

######  Remark 7.2.

In particular, if $W=0$, then the spectrum of $D_{k}$ has $\\{0\\}$ as
isolated point (it is an eigenvalue of infinite multiplicity), while the open
interval $(0,8\pi k)$ is a spectral gap. With a bootstrap argument (see [LT21,
§2.3]) one can in fact show that the entire spectrum is the set $\\{8\pi
kn\mid n=0,1,\dots\\}$, with each spectral value being an eigenvalue of
infinite multiplicity. Of course, the introduction of $W$ will change this
spectrum. However, by bounded perturbation theory, the perturbed spectrum will
still be contained in the set of all points with distance at most $w$ from the
unperturbed spectrum. ∎

###### Proof.

The idea is to identify $D_{k}$ with $\not{D}_{{L}}^{2,+}+W$ for a
$\mathbb{Z}^{2}$-invariant Dirac-type operator $\not{D}_{{L}}$ on
$\mathbb{R}^{2}$. The assertions are then deduced from the Weizenboeck formula
for $\not{D}_{{L}}$ and the index theorem. Note that $D_{k}$ is designed such
that this interpretation exists.

We trivialize the spinor bundle of $\mathbb{R}^{2}$ so that

$S(\mathbb{R}^{2})\cong\mathbb{R}^{2}\times\mathbb{C}^{2}\ .$

We let $c(\partial_{x})$ and $c(\partial_{y})$ be the Clifford multiplications
on $S^{2}(\mathbb{R}^{2})$, and we let

(7.6) $z:=ic(\partial_{x})c(\partial_{y})$

be the $\mathbb{Z}/2\mathbb{Z}$-grading operator. Then
$S^{2}(\mathbb{R}^{2})^{\pm}$ denote the $\pm 1$-eigenbundles of $z$. They are
both one-dimensional.

We let $\mathbb{Z}^{2}$ act on $S(\mathbb{R}^{2})$ by
$(m,n)(x,y,v)\mapsto(x+m,y+n,v)$. By taking the quotient we obtain the spinor
bundle $S(T^{2})\to T^{2}$ associated to the Spin structure in which this
bundle is trivialized.

The Dirac operator on $S^{2}(\mathbb{R}^{2})$ is given by

(7.7) $\not{D}=c(\partial_{x})\partial_{x}+c(\partial_{y})\partial_{y}\ .$

We will obtain a generalized Dirac operator ${\not{D}_{L}}$ by twisting
$\not{D}$ with an equivariant Hermitean line bundle $L\to\mathbb{R}^{2}$ with
connection $\nabla^{L}$. The underlying equivariant Hermitean vector bundle is
the trivial bundle $\mathbb{R}^{2}\times\mathbb{C}\to\mathbb{R}^{2}$ with
action of $\mathbb{Z}^{2}$ given by (7.4). The invariant connection is given
by the formula

(7.8) $\nabla^{L}:=d-2\pi ik(xdy-ydx)\ .$

Its curvature form is

(7.9) $R^{\nabla^{L}}=-4\pi ikdx\wedge dy\ .$

One furthermore calculates that the connection Laplacian is given by

(7.10) $\nabla^{L,*}\nabla^{L}=\Delta-4\pi
ik(y\partial_{x}-x\partial_{y})+4\pi^{2}k^{2}(x^{2}+y^{2})\ .$

We now form the twisted Dirac operator

(7.11)
$\not{D}_{L}:=c(\partial_{x})\nabla^{L}_{\partial_{x}}+c(\partial_{y})\nabla^{L}_{\partial_{y}}\
.$

The Weizenboeck formula states

$\not{D}_{L}^{2}=\nabla^{L,*}\nabla^{L}+c(\partial_{x})c(\partial_{y})R^{\nabla^{L}}(\partial_{x},\partial_{y})\
.$

Using the explicit calculation (7.9) of the curvature $R^{\nabla}$ and the
definition of the grading (7.6) we can rewrite the Weizenboeck term as

$c(\partial_{x})c(\partial_{y})R^{\nabla^{L}}(\partial_{x},\partial_{y})={-}4\pi
kz$

and get

(7.12) $\not{D}_{L}^{2}=\nabla^{L,*}\nabla^{L}-4\pi kz\ .$

The operator $\not{D}_{{L}}^{2}$ commutes with $z$, and we let
$\not{D}_{L}^{2,\pm}$ be the restrictions to the $\pm 1$-eigenbundles of $z$.
By combining (7.10) and (7.12) we see that under the canonical identification
of $S^{2}(\mathbb{R}^{2})^{\pm}$ with trivial complex line bundles, we have

$\displaystyle\not{D}_{L}^{2,+}$ $\displaystyle=D_{k}-W,$
$\displaystyle\not{D}_{L}^{2,-}$ $\displaystyle=D_{k}-W+8k\pi.$

It is clear that $D_{k}\cong\not{D}_{L}^{2,+}+W$ is lower bounded by $-w$. In
view of (7.12) it is of Laplace type. This shows assertion (1).

Since $\not{D}_{L}$ anticommutes with $z$ we have a bijection

$\sigma(\not{D}_{L}^{2,+})\setminus\\{0\\}=\sigma(\not{D}_{L}^{2,-})\setminus\\{0\\}\
.$

Since $\not{D}_{L}^{2,+}\geq 0$ we have $\not{D}_{L}^{2,-}\geq 8\pi k$.
Consequently, the interval $(0,8\pi k)$ belongs to the resolvent set of
$\not{D}_{L}^{2,+}$ By bounded perturbation theory, this implies that $(w,8\pi
k-w)$ belongs to the resolvent set of $D_{k}=\not{D}_{L}^{2,+}+W$. Hence we
conclude assertion (2).

We further conclude that $0$ is an isolated eigenvalue of $\not{D}_{L}^{2,+}$.
Using pertubation theory we conclude that

$[E_{D_{k}}(-w{-1},\lambda)]=[E_{\not{D}_{L}^{2,+}}(\\{0\\})]$

in $K\mathcal{X}_{0}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})$ for all $\lambda$ in
$(w,8\pi k-w)$.

In order show assertion (3) we must calculate the $K$-theory class of the
representation of $C^{*}_{r}(\mathbb{Z}^{2})$ on $\ker(\not{D}^{2,+}_{L})$.
Under the Fourier transformation isomorphism $C^{*}_{r}(\mathbb{Z}^{2})\cong
C(T^{2})$ this representation corresponds to the representation of $C(T^{2})$
on the space of sections of the kernel bundle of a family of Dirac operators
parametrized by $T^{2}$ which we will describe the following.

The $\mathbb{Z}^{2}$-Hilbert space
$L^{2}(\mathbb{R}^{2},S(\mathbb{R}^{2})\otimes L)$ can be identified with a
direct integral

$\int_{\widehat{T}^{2}}V_{\chi}d\chi$

for the continuous field of Hilbert spaces
$((V_{\chi})_{\chi\in\widehat{T}},\mathcal{V})$. Here $V_{\chi}$ is the
subspace of $L^{2}_{{\mathrm{loc}}}(\mathbb{R}^{2},S(\mathbb{R}^{2})\otimes
L)$ of functions on which $\mathbb{Z}^{2}$ acts with character $\chi$, i.e.,
of sections $f$ of $S(\mathbb{R}^{2})$ such that

$e^{2\pi ik(my-nx)}f(x-m,y-n)=\chi(m,n)f(x,y)$

for all $(m,n)$ in $\mathbb{Z}^{2}$, with the scalar product given by the
$L^{2}$-scalar product of the the restriction to a fundamental domain. The
continuous structure is given by the subspace $\mathcal{V}$ of sections
$g:\widehat{T}^{2}\ni\chi\mapsto g(\chi)\in V_{\chi}$ such that the
section$\int_{\widehat{T}^{2}}g(\chi)\chi$ of $S(\mathbb{R}^{2}\otimes L)$ is
smooth with compact support.

The operator $\not{D}_{L}$ is the given by a family of operators
$(\not{D}_{L,\chi})_{\chi\in\widehat{T}^{2}}$, where $\not{D}_{L,\chi}$ acts
on $V_{\chi}$ by the formula (7.11). Since $\not{D}_{L}$ preserves smooth
compactly supported sections the family
$(\not{D}_{L,\chi})_{\chi\in\widehat{T}^{2}}$ preserves the subspace
$\mathcal{V}$. We must calculate the kernel bundle of this family.

We will now describe this family of operators as a family of twisted Dirac
operators. To this end we consider the Poincaré bundle $\bar{P}\to
T^{2}\times\widehat{T}^{2}$. We let $(s,t)$ be coordinates on
$\widehat{T}^{2}$ such that

$\chi(s,t):\mathbb{Z}^{2}\to U(1)\ ,\quad\chi(s,t)(m,n)=e^{2\pi i(sm+tn)}\ .$

The bundle $\bar{P}\to T^{2}\times\widehat{T}^{2}$ is obtained as quotient of
the trivial bundle $P:=\mathbb{R}^{2}\times
T^{2}\times\mathbb{C}\to\mathbb{R}^{2}\times T^{2}$ by $\mathbb{Z}^{2}$, where
$\mathbb{Z}^{2}$ acts on the domain by

$\psi(m,n)(x,y,s,t,z):=(x+m,y+n,s,t,e^{2\pi i(sm+tn)}z)\ .$

On $P$ we consider the invariant connection

$\nabla^{P}:=d+2\pi i(xds+ydt)\ .$

It induces a connection $\nabla^{\bar{P}}$ on the quotient $\bar{P}$. Its
curvature is given by

(7.13) $R^{\nabla^{P}}=2\pi i(dx\wedge ds+dy\wedge dt)\ .$

We let $\bar{L}\to T^{2}$ be the quotient of the bundle $L\to\mathbb{R}^{2}$,
and $\not{D}_{\bar{L}}$ be the resulting twisted Dirac operator on $T^{2}$. We
can consider this operator as a constant family on the bundle
$T^{2}\times\widehat{T}^{2}\to\widehat{T}^{2}$. We now form the non-constant
family $\not{D}_{\bar{L}\otimes\bar{P}}$ by twisting $\not{D}_{\bar{L}}$
further with $\bar{P}$. On the fibre over $\chi$ in $\widehat{T}^{2}$ the
operator $\not{D}_{\bar{L}\otimes\bar{P}}$ identifies with $\not{D}_{L,\chi}$.
Hence the index bundle of $\not{D}_{\bar{L}\otimes\bar{P}}$ equals the kernel
bundle of the family $(\not{D}_{L,\chi})_{\chi\in\widehat{T}^{2}}$.

We can now apply the Atiyah-Singer index theorem for families in order to
calculate the Chern character of the index bundle
${\mathbf{ch}}({\mathtt{index}}(\not{D}_{\bar{L}\otimes\bar{P}}))$ in
$H^{*}(\widehat{T}^{2})$ of this family. We get

${\mathbf{ch}}({\mathtt{index}}(\not{D}_{\bar{L}\otimes\bar{P}}))=\int_{T^{2}\times\widehat{T}^{2}/\widehat{T}^{2}}{\mathbf{ch}}(\bar{L}\otimes\bar{P})\
.$

We calculate in de Rham cohomology. Using the formulas for the curvatures
(7.9) and (7.13) and the formulas $c_{1}(\nabla)=\frac{i}{2\pi}R^{\nabla}$ and
${\mathbf{ch}}(\nabla)=e^{c_{1}(\nabla)}$ for a line bundle we have

$\displaystyle{\mathbf{ch}}(\nabla^{L}\otimes\nabla^{P})$ $\displaystyle=$
$\displaystyle 1+2k(dx\wedge dy)-(dx\wedge ds+dy\wedge dt)$
$\displaystyle-(dx\wedge dy\wedge ds\wedge dt)\ .$

We conclude that

$\int_{T^{2}\times\widehat{T}^{2}/\widehat{T}^{2}}{\mathbf{ch}}(\bar{L}\otimes\bar{P})=2k-[ds\wedge
dt]\ .$

This implies assertion (3). ∎

In the remainder of this subsection we show that the magnetic Laplacian
$D_{k}$ provides a non-trivial example for the theory developed in Section 5.
We consider a codimension-zero submanifold $Z$ of $\mathbb{R}^{2}$ with smooth
boundary $\partial Z$ such that $Z$ and $\mathbb{R}^{2}\setminus Z$ are
flasque, e.g., a half space. We then consider the morphism
$r:K\mathcal{X}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})\to K\mathcal{X}(\partial Z)$
from (2.4) for the trivial group $K$.

######  Lemma 7.3.

For $\lambda$ in $(w,8\pi k-w)$ we have
$r([E_{D_{k}}(-w{-1},\lambda)])\not=0$.

###### Proof.

By our flasqueness assumption on $Z$ and $\mathbb{R}^{2}\setminus Z$ the
Mayer-Vietoris boundary $\delta$ in (2.8) is an equivalence. It follows that
$r$ is equivalent to the transfer map

(7.14) $c^{\mathbb{Z}^{2}}:K\mathcal{X}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})\to
K\mathcal{X}(\operatorname{Res}^{\mathbb{Z}^{2}}(\mathbb{R}^{2}))\ .$

It thus suffices to show that the image of the class
$[E_{D_{k}}(-w{-1},\lambda)]$ in
$K\mathcal{X}_{0}^{\mathbb{Z}^{2}}(\mathbb{R}^{2})$ under (7.14) is non-
trivial.

In principle this immediately follows from Proposition 7.1.3 and the
isomorphism (7.3) if one could identify the transfer morphism with the
component $c_{1}$ up to sign. This fact is actually not completely obvious but
could be deduced a posteriori from our calculations. We prefer to give an
argument which is independent of the calculation in Proposition 7.1.3 and is
also applicable in other situations.

The first equality in the chain

$[E_{D_{k}}(-w{-1},\lambda)]=[E_{\not{D}_{L}^{2,+}}(\\{0\\})]\stackrel{{\scriptstyle!}}{{=}}{\mathtt{index}}^{\mathbb{Z}^{2}}(\not{D}_{L})$

has been shown in the proof of Proposition 7.1. The symbol
${\mathtt{index}}^{\mathbb{Z}^{2}}(\not{D}_{L})$ denotes the equivariant
coarse index in $K\mathcal{X}^{\mathbb{Z}^{2}}_{0}(\mathbb{R}^{2})$ of the
Dirac operator $\not{D}_{L}$ on $\mathbb{R}^{2}$ as in [BEb, Def. 9.5] (with
full support). For the equality marked by $!$ note that, as seen in the proof
of (7.1), the operator $\not{D}_{L}^{2,-}$ is invertible and $0$ is an
isolated point of the spectrum of $\not{D}_{L}$. We also use Bott periodicity
in order to identify the index of the Dirac operator which is a $K$-theory
class in degree $2$ with the degree-$0$ class represented by the spectral
projection.

The transfer map (7.14) sends ${\mathtt{index}}^{\mathbb{Z}^{2}}(\not{D}_{L})$
to the index ${\mathtt{index}}(\not{D}_{L})$ in
$K\mathcal{X}(\operatorname{Res}^{\mathbb{Z}^{2}}(\mathbb{R}^{2}))$ of the
same operator $\not{D}_{L}$ without equivariance, see [BEc, Rem. 10.6]. In
order to show that ${\mathtt{index}}(\not{D}_{L})\not=0$ we argue that

(7.15) ${\mathtt{index}}(\not{D}_{L})={\mathtt{index}}(\not{D})$

with $\not{D}$ as in (7.7) and then use the fact ${\mathtt{index}}(\not{D})$
is a generator of $K\mathcal{X}_{2}(\mathbb{R}^{2})\cong\mathbb{Z}$.

The idea for showing the equality (7.15) is that
${\mathtt{index}}(\not{D}_{L})$ is the image of the symbol of $\not{D}_{L}$
under the coarse assembly map, and that the symbol classes of $\not{D}_{L}$
and $\not{D}$ coincide as these operators are lower order perturbations of
each other. Since $\mathbb{R}^{2}$ is non-compact some care is needed in the
details. The problem is that $\not{D}_{L}-\not{D}$ is zero order, but
unbounded. Therefore it is not clear that the linear interpolation between
them preserves the index. We will give a completely formal argument using the
coarse interpretation of index theory developed in [Bun18, Def. 4.14].
Alternatively one could argue as in [LT21, Lem. 3.2].

In the following we omit the symbol $\operatorname{Res}^{\mathbb{Z}^{2}}$. We
will use the local homology theory
$K\mathcal{X}\mathcal{O}^{\infty}:\mathbf{UBC}\to\mathbf{Sp}$ from [BE20a,
Lem. 9.6], where $\mathbf{UBC}$ is the category of uniform bornological coarse
spaces. This theory is well-defined since $K\mathcal{X}$ is strong by [BEc,
Prop. 6.4]. The cone boundary

$\partial:\Sigma^{-1}K\mathcal{X}\mathcal{O}^{\infty}\to K\mathcal{X}$

(see [BE20a, (9.1)]) is a natural transformation of local homology theories,
where we consider $\Sigma^{-1}K\mathcal{X}$ as a local homology via the forget
functor $\mathbf{UBC}\to\mathbf{BornCoarse}$. According to [Bun18, Def. 4.14]
any generalized Dirac operator $\not{D}^{\prime}$ on $\mathbb{R}^{2}$ gives
rise to a symbol class $\sigma(\not{D}^{\prime})$ in
$\pi_{1}(K\mathcal{X}\mathcal{O}^{\infty}(\mathbb{R}^{2}))$ such that
${\mathtt{index}}(\not{D}^{\prime})=\partial(\sigma(\not{D}^{\prime}))$. By
[BE20a, Prop. 11.23] the functor $K\mathcal{X}\mathcal{O}^{\infty}$ behaves on
finite-dimensional manifolds like a locally finite homology theory. In
particular, if $B$ is the unit ball in $\mathbb{R}^{2}$, then the canonical
map

(7.16) $e:K\mathcal{X}\mathcal{O}^{\infty}(\mathbb{R}^{2})\to
K\mathcal{X}\mathcal{O}^{\infty}(\mathbb{R}^{2},B)\simeq\operatorname{Cofib}(K\mathcal{X}\mathcal{O}^{\infty}(\mathbb{R}^{2}\setminus
B)\to K\mathcal{X}\mathcal{O}^{\infty}(\mathbb{R}^{2}))$

is an equivalence.

Note that $\sigma(\not{D}_{L})$ is given by the index of a Dirac operator on a
geometric version of the cone which is locally derived from $\not{D}_{L}$. By
the coarse relative index theorem [BEb, Thm. 10.4] the class
$e(\sigma(\not{D}_{L}))$ only depends on the restriction of $\not{D}_{L}$ to
$B$. Let $\psi$ be in $C_{c}(\mathbb{R}^{2})$ such that $\psi_{|B}=1$ and
$\psi_{|\mathbb{R}^{2}\setminus 2B}=0$. Then we consider the family of
connections (compare with (7.8))

$\nabla^{L}_{t}:=d-(1-t\psi(x,y))2\pi ik(xdy-ydx)$

and the associated Dirac operator $\not{D}_{L_{t}}$. We have
$\not{D}_{L_{0}}=\not{D}_{L}$ and $(\not{D}_{L_{1}})_{|B}=\not{D}_{|B}$. Since
the family $D_{L_{t}}$ is constant outside of the compact subset $2B$ this
perturbation does not alter the coarse index, i.e. we have
${\mathtt{index}}(\not{D}_{L_{t}})={\mathtt{index}}(\not{D}_{L})$ for all $t$
in $[0,1]$. On the other hand, $e(\sigma(\not{D}_{L_{1}}))=e(\sigma(\not{D}))$
and hence $\sigma(\not{D}_{L_{1}})=\sigma(\not{D})$ by the injectivity of $e$
in (7.16). This implies the equality (7.15) in view of the chain

${\mathtt{index}}(\not{D}_{L})={\mathtt{index}}(\not{D}_{L_{1}})=\partial\sigma(\not{D}_{L_{1}})=\partial\sigma(\not{D})={\mathtt{index}}(\not{D})\
.\qed$

Combining Proposition 7.1, Lemma 7.3 and Proposition 5.4 we can now conclude:

######  Corollary 7.4.

If $D_{k}^{\prime}$ is the Dirichlet extension (or any other selfadjoint
extension determined by a local boundary condition) of the restriction of
$D_{k}$ to $Z$, then $[w,8\pi k-w]\subseteq\sigma(D_{k}^{\prime})$.

Examples for $Z$ are half spaces or spaces of the form $\\{(x,y)\>|\>y\leq
f(x)\\}$ for some smooth function $f:{\mathbb{R}}\to\mathbb{R}$. Furthermore,
we can allow any bounded perturbation of a space of this form, e.g. the half
space $\\{(x,y)\>|\>y\leq 0\\}$ together with the union of $1/3$-balls at all
points $(n,k)$ for all $n$ in $\mathbb{N}$ and fixed $k$ in $\mathbb{N}$.

######  Example 7.5.

We provide another example of a $K$-theory class which can be detected using
the restriction morphism. The unitary $u$ in $C^{*}_{r}(\mathbb{Z}^{2})$ given
by the element $(1,0)$ of $\mathbb{Z}^{2}$ represents a class $[u]\in
K_{1}(C_{r}^{*}(\mathbb{Z}^{2}))\simeq
K\mathcal{X}_{1}^{\mathbb{Z}^{2}}(\mathbb{Z}^{2}_{can,min})$. We consider the
group $\mathbb{Z}$ as a subgroup of $\mathbb{Z}^{2}$ embedded by
$n\mapsto(0,n)$ and the $\mathbb{Z}$-invariant subspace $Z=\\{(n,m)\mid n\geq
0\\}$ of $\mathbb{Z}^{2}_{can,min}$. The inclusion
$\mathbb{Z}_{can,min}\to\operatorname{Res}^{\mathbb{Z}^{2}}_{\mathbb{Z}}(\mathbb{Z}^{2}_{can,min})$
induces an equivalence $K\mathcal{X}^{\mathbb{Z}}(\mathbb{Z}_{can,min})\simeq
K\mathcal{X}^{\mathbb{Z}}(\partial Z)$. The morphism $r$ from Definition 2.4
can therefore be interpreted as a morphsim

$r:K(C_{r}^{*}(\mathbb{Z}^{2}))\simeq
K\mathcal{X}^{\mathbb{Z}^{2}}(\mathbb{Z}^{2}_{can,min})\to\Sigma
K\mathcal{X}^{\mathbb{Z}}(\partial Z)\simeq\Sigma
K\mathcal{X}^{\mathbb{Z}}(\mathbb{Z}_{can,min})\ .$

One can now check by an explicit calculation that $r([u])$ is a generator of
the cyclic group $K\mathcal{X}_{0}^{\mathbb{Z}}(\mathbb{Z}_{can,min})$.

This idea can be expanded in order to obtain a coarse-geometric proof of the
fact shown in [EN87] that for a general group $G$ the canonical map
$G_{\mathrm{ab}}\to K_{1}(C_{r}^{*}(G))$ is injective. 333This will be
discussed in detail in a forthcoming paper. ∎

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