# On topological obstructions to the existence of non-periodic Wannier bases

Yu. Kordyukov, V. Manuilov  Institute of Mathematics, Ufa Federal Research
Centre, Russian Academy of Sciences, 112 Chernyshevsky str., 450008 Ufa,
Russia  Moscow Center for Fundamental and Applied Mathematics, Moscow State
University, Leninskie Gory 1, Moscow, 119991, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Recently, M. Ludewig and G. C. Thiang introduced a notion of a uniformly
localized Wannier basis with localization centers in an arbitrary uniformly
discrete subset $D$ in a complete Riemannian manifold $X$. They show that,
under certain geometric conditions on $X$, the class of the orthogonal
projection onto the span of such a Wannier basis in the $K$-theory of the Roe
algebra $C^{*}(X)$ is trivial. In this short note, we clarify the geometric
conditions on $X$, which guarantee triviality of the $K$-theory class of any
Wannier projection. We show that this property is equivalent to triviality of
the unit of the uniform Roe algebra of $D$ in the $K$-theory of its Roe
algebra, and provide a geometric criterion for that. As a consequence, we
prove triviality of the $K$-theory class of any Wannier projection on a
connected proper measure space $X$ of bounded geometry with a uniformly
discrete set of localization centers, coarsely equivalent to $X$.

The results of Section 2 were obtained jointly by both authors, the results of
Section 3 were obtained by V. Manuilov and supported by the RSF grant
23-21-00068.

##  1\. Introduction

Wannier functions, introduced by G. Wannier in 1937, have become a fundamental
tool in theoretical and computational solid-state physics. Given a Schrödinger
operator $\mathcal{H}$ with periodic potential acting on
$L^{2}(\mathbb{R}^{d})$, the Wannier functions form an orthonormal basis of a
spectral subspace of $\mathcal{H}$ constructed from a finite set of functions
along with their translations in $\mathbb{Z}^{d}$. In [7], M. Ludewig and G.
C. Thiang initiated the study of Wannier bases for (magnetic) Schrödinger
operators on general Riemannian manifolds, invariant under non-abelian
discrete groups of isometries. Furthermore, in [8], they introduced the
general notion of a uniformly localized Wannier basis with localization
centers to come from an arbitrary uniformly discrete subset $D$ in a complete
Riemannian manifold $X$. They asked the following question:

######  Question 1.

Does a given subspace $H\subset L^{2}(X)$ admit a uniformly localized Wannier
basis, whose localization centers come from some uniformly discrete set
$D\subset X$? If not, what is the obstruction?

In [8], it is proved that if $H\subset L^{2}(X)$ admits a uniformly localized
Wannier basis with localization centers in some uniformly discrete set
$D\subset X$, then the orthogonal projection $p_{H}$ onto $H$ (which will be
called the Wannier projection in this case) lies in the Roe $C^{*}$-algebra
$C^{*}(X)$. The main result of [8], Theorem 3.6, states that, under certain
geometric conditions on $X$, the class $[p_{H}]\in K_{0}(C^{*}(X))$ of any
Wannier projection $p_{H}$ in the $K$-theory of the Roe algebra $C^{*}(X)$ is
trivial. In other words, if $[p_{H}]$ is non-trivial, then $H$ does not admit
a uniformly localized Wannier basis with localization centers $D\subset X$,
for any choice of uniformly discrete set $D$. We refer to the papers [3, 9]
and references therein for various approaches to the analysis of Wannier bases
for non-periodic systems.

In this short note we clarify the geometric conditions on $X$, which guarantee
triviality of the $K$-theory class of any Wannier projection. We consider a
more general setting of metric measure spaces. We show that triviality of the
$K$-theory class of a Wannier projection with localization centers in $D$ is
equivalent to triviality of the unit of the uniform Roe algebra in the $K_{0}$
group of the Roe algebra of $D$ (Theorem 3). We also provide a geometric
criterion for the latter property (Theorems 5 and 6). As a consequence, we
prove triviality of the $K$-theory class of any Wannier projection on a
connected proper measure space $X$ of bounded geometry with a uniformly
discrete set of localization centers, coarsely equivalent to $X$ (Corollary
11).

Let $X$ be a proper metric measure space, that is, $X$ is a set, which is
equipped with a metric $d$ and a measure $m$ defined on the Borel
$\sigma$-algebra defined by the topology on $X$ induced by the metric, and all
balls are compact. Let $D\subset X$ be a discrete subspace such that the
inclusion $D\subset X$ is a coarse equivalence. The latter means that there
exists $C>0$ such that for any $x\in X$ there exists $y\in D$ with $d(x,y)<C$.
We assume that $D$ is uniformly discrete (i.e. $\inf_{g,h\in D,g\neq
h}d(g,h)>0$) and has bounded geometry (i.e. for any $R>0$ the number of points
in each ball of radius $R$ is uniformly bounded). We may consider $D$ as a
measure space with the measure of any point equal to one. It is natural to
think of $D$ as a discretization of $X$.

For a Hilbert space $H$ we write $\mathbb{B}(H)$ (resp., $\mathbb{K}(H)$) for
the algebra of all bounded (resp., all compact) operators on $H$. Recall the
definition of the Roe algebra of $X$ [11]. Let $H_{X}$ be a Hilbert space with
an action of the algebra $C_{0}(X)$ of continuous functions on $X$ vanishing
at infinity (i.e. a $*$-homomorphism $\psi:C_{0}(X)\to\mathbb{B}(H_{X})$). We
will assume that $\\{\psi(f)\xi:f\in C_{0}(X),\xi\in H_{X}\\}$ is dense in
$H_{X}$ and $\psi(f)\in\mathbb{K}(H_{X})$ implies that $f=0$. An operator
$T\in\mathbb{B}(H_{X})$ is locally compact if the operators $T\psi(f)$ and
$\psi(f)T$ are compact for any $f\in C_{0}(X)$. It has finite propagation if
there exists some $R>0$ such that $\psi(f)T\psi(g)=0$ whenever the distance
between the supports of $f,g\in C_{0}(X)$ is greater than $R$. The Roe algebra
$C^{*}(X,H_{X})$ is the norm completion of the $*$-algebra of locally compact,
finite propagation operators on $H_{X}$.

Let $H_{X}=L^{2}(X)\otimes H$ for some Hilbert space $H$ (possibly finite-
dimensional). In this case there is a standard action of $C_{0}(X)$ on $H_{X}$
by multiplication, and we shall skip $\psi$ from the notation. The Roe algebra
$C^{*}(X,H_{X})$ in this case will be denoted by $C^{*}_{H}(X)$.

Often one may forget about $H$, namely one may take $H$ one-dimensional. This
happens when the operator of multiplication by any non-zero $f\in C_{0}(X)$ in
$L^{2}(X)$ is not compact, i.e. when the measure on $X$ has no atoms. In this
case the algebras $C^{*}_{H}(X)$ and $C^{*}_{\mathbb{C}}(X)$ are isomorphic.

But for discrete space $D$ this is not true: $C^{*}_{H}(D)$ is not isomorphic
to $C^{*}_{\mathbb{C}}(D)$, so for discrete spaces we have two algebras: the
Roe algebra $C^{*}_{H}(D)$ with an infinite-dimensional Hilbert space $H$,
usually denoted by $C^{*}(D)$, and the uniform Roe algebra
$C^{*}_{\mathbb{C}}(D)$, usually denoted by $C^{*}_{u}(D)$.

##  2\. Wannier projections as the image of the unit of
$C^{*}_{\mathbb{C}}(D)$

Let $\\{\phi_{x}:x\in D\\}$ be a set of functions in $L^{2}(X)$ such that
$\operatorname{supp}\phi_{x}\cap\operatorname{supp}\phi_{y}=\emptyset$ when
$x\neq y$, $x\in\operatorname{supp}\phi_{x}$, the diameters of
$\operatorname{supp}\phi_{x}$, $x\in D$, are uniformly bounded, and
$\|\phi_{x}\|_{2}=1$, $x\in D$. Let $H_{\phi}\subset L^{2}(X)$ be the closure
of the linear span of the set $\\{\phi_{x}:x\in D\\}$, and let $p_{\phi}$
denote the orthogonal projection onto $H_{\phi}$. The set of functions
$\\{\phi_{x}:x\in D\\}$ is called a $D$-compactly supported Wannier basis for
$H_{\phi}$, and the projection $p_{\phi}$ a Wannier projection.

Further on we shall need a construction from [6, Section 4], which allows us
to induce maps between Roe algebras from maps between spaces. Given two metric
measure spaces, $X$ and $Y$, and two Hilbert spaces, $H_{X}$ and $H_{Y}$, with
respective actions $\psi_{X}$ and $\psi_{Y}$ of $C_{0}(X)$ and $C_{0}(Y)$,
respectively, a coarse map $F:X\to Y$ induces a $*$-homomorphism
$C^{*}(X,H_{X})\to C^{*}(Y,H_{Y})$, $T\mapsto VTV^{*}$, where $V:H_{X}\to
H_{Y}$ is an isometry that covers $F$, which means that there exists $C>0$
such that $\psi_{X}(f)V\psi_{Y}(g)=0$ when
$d(\operatorname{supp}f,\operatorname{supp}(g\circ F))>C$, $f\in C_{0}(X)$,
$g\in C_{0}(Y)$.

Let $V:\mathbb{C}\to H$ be an isometry, i.e. an inclusion of $\mathbb{C}$ onto
a one-dimensional subspace of $H$, and let $V_{X}=\operatorname{id}\otimes
V:L^{2}(X)=L^{2}(X)\otimes\mathbb{C}\to L^{2}(X)\otimes H$. Clearly, $V_{X}$
covers the identity map of $X$, and we get the maps

$i_{X}:C^{*}_{\mathbb{C}}(X)\to C^{*}_{H}(X)\quad\mbox{and}\quad
i_{D}:C^{*}_{\mathbb{C}}(D)\to C^{*}_{H}(D)$

given by $i_{X}(T)=V_{X}TV^{*}_{X}$ and $i_{D}(T)=V_{D}TV_{D}^{*}$
respectively. These two maps induce maps in $K$-theory

$(i_{X})_{*}:K_{0}(C^{*}_{\mathbb{C}}(X))\to
K_{0}(C^{*}_{H}(X))\quad\mbox{and}\quad(i_{D})_{*}:K_{0}(C^{*}_{\mathbb{C}}(D))\to
K_{0}(C^{*}_{H}(D)),$

where the first map is an isomorphism when the measure on $X$ has no atoms.
These maps are independent of the choice of $V$.

Given a Wannier basis $\\{\phi_{x}\\}_{x\in D}$, let $U:l^{2}(D)\to L^{2}(X)$
be the isometry defined by $U(\delta_{x})=\phi_{x}$, so that the range of $U$
is the subspace $H_{\phi}\subset L^{2}(X)$. We shall use also the isometry
$U_{H}=U\otimes 1_{H}:l^{2}(D)\otimes H\to L^{2}(X)\otimes H$. Clearly, $U$
covers the inclusion map $D\subset X$, so it gives the map

$j_{\mathbb{C}}:C^{*}_{\mathbb{C}}(D)\to C^{*}_{\mathbb{C}}(X),\quad
j_{\mathbb{C}}(T)=UTU^{*},\ T\in C^{*}_{\mathbb{C}}(D).$

Similarly, using $U_{H}$ instead of $U$, we get the map

$j_{H}:C^{*}_{H}(D)\to C^{*}_{H}(X),\quad j_{H}(T)=U_{H}TU_{H}^{*},\ T\in
C^{*}_{H}(D).$

Note that $C^{*}_{\mathbb{C}}(D)$ is unital, and $j_{\mathbb{C}}(1)=p_{\phi}$
is the Wannier projection. Set $q=i_{D}(1)\in C^{*}_{H}(D)$.

The above maps can be organized into the commutative diagram of
$C^{*}$-algebras

$\textstyle{C^{*}_{\mathbb{C}}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\mathbb{C}}}$$\scriptstyle{i_{D}}$$\textstyle{C^{*}_{\mathbb{C}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{X}}$$\textstyle{C^{*}_{H}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{H}}$$\textstyle{C^{*}_{H}(X),}$

and of their $K_{0}$ groups

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(1)

######  Lemma 2.

One has $j_{\mathbb{C}}(1)\otimes 1_{H}=p_{\phi}\otimes 1_{H}=j_{H}(q)$.

###### Proof.

As $j_{H}(q)=p_{\phi}\otimes 1_{H}$, the conclusion follows from commutativity
of the diagram (1).

∎

######  Theorem 3.

Let $p_{\phi}$ be a Wannier projection with a uniformly discrete set $D$ of
localization centers, coarse equivalent to $X$. The following are equivalent:

  * •

$[p_{\phi}]=0$ in $K_{0}(C^{*}_{\mathbb{C}}(X))$;

  * •

$[i_{D}(1)]=0$ in $K_{0}(C^{*}_{H}(D))$.

###### Proof.

The map $(i_{X})_{*}$ is an isomorphism, and $i_{X}(p_{\phi})=p_{\phi}\otimes
1_{H}$. Coarse equivalence of $D$ and $X$ implies that $(j_{H})_{*}$ is an
isomorphism as well. Thus, all entries in (1), except the upper left one, are
isomorphic.

∎

Thus triviality of the $K$-theory class $[p_{\phi}]$ of $p_{\phi}$ is not
related to $X$, but only to $D$.

######  Remark 1.

Recall that a partial translation is a bijection $f:A\to B$ between subsets
$A,B\subset D$ such that $\sup_{x\in A}d(x,f(x))<\infty$. The space $D$ is
paradoxical if there exist a decomposition $D=D_{+}\sqcup D_{-}$ and partial
translations $f_{\pm}:D\to D_{\pm}$. Theorem 4.9 in [1] shows that if $D$ is
paradoxical then $[1]$ is zero already in $K_{0}(C^{*}_{\mathbb{C}}(D))$.

######  Remark 2.

Theorem 3 is still true if we weaken the condition $\phi_{x}\phi_{y}=0$ for
$x,y\in D$, $x\neq y$. Let $V:l^{2}(D)\to L^{2}(X)$ be defined by
$V(\delta_{x})=\phi_{x}$, $x\in D$, as before. Instead of requiring it to be
an isometry we may require only that $V^{*}V$ is invertible. In this case
$\\{\phi_{x}\\}_{x\in D}$ is not a basis for $H_{\phi}$, but only a frame. Let
$V=WH$ be the polar decomposition. Then the range of $W$ is still $H_{\phi}$.
The isometry $W$ does not cover the inclusion $D\subset X$, nevertheless the
map $T\mapsto WTW^{*}$ still maps $C^{*}_{\mathbb{C}}(D)$ to
$C^{*}_{\mathbb{C}}(X)$, and Lemma 2 and Theorem 3 hold.

##  3\. Geometric criterion for triviality of $[p_{\phi}]$

######  Definition 4.

We say that a discrete metric space $D$ has a ray structure if there exists a
family $\\{D_{i}\\}_{i\in I}$, of subsets of $D$ such that $D=\sqcup_{i\in
I}D_{i}$, and a family of bijective uniformly Lipschitz maps
$\beta_{i}:\mathbb{N}\to D_{i}$, $i\in I$.

In this case we call the subsets $D_{i}$, $i\in I$, rays.

Let $C_{b}(D)$ denote the commutative $C^{*}$-algebra of bounded functions on
$D$. The inclusion $C_{b}(D)\subset C^{*}_{\mathbb{C}}(D)$ induces a map
$\gamma:K_{0}(C_{b}(D))\to K_{0}(C^{*}_{\mathbb{C}}(D))$.

######  Theorem 5.

Let $D$ be a uniformly discrete space of bounded geometry. If $D$ has a ray
structure then $(i_{D})_{*}(p)=0$ for any $p\in\gamma(K_{0}(C_{b}(D)))$. In
particular, $(i_{D})_{*}([1])=0$, hence $[p_{\phi}]=0$ for any Wannier basis
on $X$ with localization centers in $D$.

###### Proof.

Fix a basis in $H$, and let $p_{k}\in\mathbb{K}(H)$ be the projection onto the
first $k$ vectors of the fixed basis of $H$. Note that $K_{0}(C_{b}(D))$ is
the group of bounded $\mathbb{Z}$-valued functions on $D$, and it suffices to
prove the claim for the case when $p$ is a bounded $\mathbb{N}$-valued
function $x\mapsto k_{x}$, $x\in D$, i.e. $p=[f]$, where $f(x)=p_{k_{x}}$.

Endow $Y=\mathbb{N}\times I$ with the wedge metric, i.e.
$d_{Y}((k,i),(l,i))=|k-l|$ and $d_{Y}((k,i),(l,j))=k+l+1$ for $i\neq j$, where
$k,l\in\mathbb{N}$, $i,j\in I$. Then the family $(\beta_{i})_{i}\in I$
determines a bijective map $\beta:Y\to D$, $\beta((k,i))=\beta_{i}(k)$. As

$d(\beta(k,i),\beta(l,i))<|k-l|C=Cd_{Y}((k,i),(l,i)),$

we have $d_{Y}(y_{1},y_{2})\geq\frac{1}{C}d(\beta(y_{1}),\beta(y_{2}))$ for
any $y_{1},y_{2}\in Y$. Therefore, the bijection $\beta$ defines inclusions
$C^{*}_{\mathbb{C}}(Y)\subset C^{*}_{\mathbb{C}}(D)$ and $C^{*}_{H}(Y)\subset
C^{*}_{H}(D)$, the first of which is unital. If we show that the class of the
projection $(i_{D})_{*}(p)$ is zero in $K_{0}(C^{*}_{H}(Y))$ then it should be
zero in $K_{0}(C^{*}_{H}(D))$ too. Making the $C^{*}$-algebra smaller once
again, we can pass from $C^{*}_{H}(Y)$ to $\prod_{i\in
I}C^{*}_{H}(\mathbb{N})\subset C^{*}_{H}(Y)$, and then we have to show that
the class of the projection $(i_{D})_{*}(p)$ is zero in $K_{0}(\prod_{i\in
I}C^{*}_{H}(\mathbb{N}))$.

Restricting $f$ to one copy of $D_{i}$, $i\in I$, and identifying $D_{i}$ with
$\mathbb{N}$, we get the function $f_{i}:D_{i}\to\mathbb{K}(H)$,
$f_{i}(n)=p_{k_{n}}$ for any $n\in\mathbb{N}\cong D_{i}$. Set
$l_{n}=\sum_{j=1}^{n}k_{n}$, $n\in\mathbb{N}$, and let
$h_{i}:D_{i}\to\mathbb{K}(H)$ be the function defined by $h_{i}(n)=p_{l_{n}}$.
Summing up operators of multipliction by $h_{i}$ over $I$, we get an operator
of multiplication by $h$ on $D$. Set $q=\gamma(h)$. We claim that $p+q=q$ in
$K_{0}(C^{*}_{H}(\mathbb{N}))$. Clearly, $p+q$ is the class of the operator of
multiplication by $h^{\prime}$ such that $h^{\prime}_{i}(n)=p_{l(n+1)}$,
$n\in\mathbb{N}$. Define $T^{(i)}i\in C^{*}(D_{i},H)$ by
$T^{(i)}_{nm}=\left\\{\begin{array}[]{cl}p_{l(n)}&\mbox{if\ }m=n+1;\\\
0&\mbox{otherwise.}\end{array}\right.$ Then $T^{(i)}(T^{(i)})^{*}=h_{i}$,
$(T^{(i)})^{*}T^{(i)}=h^{\prime}_{i}$. Setting $T=\oplus_{i\in I}T^{(i)}$, we
obtain Murray–von Neumann equivalence between the projections $p\oplus q$ and
$q$. Thus $(i_{D})_{*}(p)=0$.

Finally, note that $[1]\in\gamma(K_{0}(C_{b}(D)))$.

∎

For $\alpha>0$ let $D(\alpha)$ be the graph whose vertices are points of $D$,
and two vertices, $x,y\in D$ are connected by an edge whenever
$d(x,y)\leq\alpha$.

######  Theorem 6.

Let $D$ be a uniformly discrete metric space of bounded geometry. The
following are equivalent:

  1. (1)

there exists $\alpha>0$ such that the graph $D(\alpha)$ has no finite
connected components;

  2. (2)

there exists a discrete metric space $D^{\prime}$ with a ray structure and an
isometric inclusion $D\subset D^{\prime}$ which is a coarse equivalence.

###### Proof.

Bounded geometry of $D$ implies existence of some $N\in\mathbb{N}$ such that
each vertex in $D(\alpha)$ has no more than $N$ neighbors. By the Nash-
Williams Theorem [10], $D_{\alpha}$ is the union of not more than $N$
subforests. Moreover, in each connected component one can connect all trees of
a subforest into a single tree.

Consider first the case when $D(\alpha)=T$ is a tree. The simplest case is
when $T$ is an infinite tree without no dead ends, i.e. when each finite path
from the root vertex to any vertex can be extended infinitely, i.e. when each
vertex (with possible exception of the root) is the end-point of at least two
edges. Choose an infinite ray starting at the root. They give , and denote it
by $T_{1}$. Then the graph $T\setminus T_{1}$ is a forest. It contains
finitely many trees with roots at the minimal distance from the root of $T$.
Choose infinite rays starting at these new roots. They give rays
$T_{2},\ldots,T_{m_{1}}$. Going further from the root of $T$, we obtain, by
induction, a decomposition $T=\sqcup_{j\in J}T_{j}$, where each $T_{j}$ is
coarsely equivalent to $\mathbb{N}$.

Now consider an infinite tree with dead ends. Suppose that the tree $T$ has
the form $T=R\cup T_{f}$, where $R$ is an infinite ray with vertices
$x_{0},x_{1},\ldots,x_{n},x_{n+1},\ldots$, $T_{f}$ is a finite tree with the
root $y_{0}=x_{n}=R\cap T_{f}$. As the number of neighbors of any vertex does
not exceed $N$, there is a path $\lambda$ in $T_{f}$ that starts and ends at
$y_{0}$ and passes through each vertex of $T_{f}$ not more than $2N$ times.
Let $n_{y}$ be the number of times that the path $\lambda$ passes through the
vertex $y$. Clone all vertices of $T_{f}$, i.e. replace each vertex $y\in
T_{f}$ by $n_{y}$ vertices. Similarly, clone the edges so that the path
$\lambda$ passes each edge only once. Denote the resulting finite graph by
$T^{\prime}_{f}$. Then the path $\lambda$ runs through all vertices of
$T^{\prime}_{f}$, $\lambda=(z_{0},z_{1},\ldots,z_{m})$. In particular, the
vertex $y_{0}$ becomes two vertices, $z_{0}$ and $z_{m}$. Set
$T^{\prime}=R\cup T^{\prime}_{f}$, and let the map $\beta:\mathbb{N}\to
T^{\prime}$ be given by the sequence
$(x_{0},x_{1},\ldots,x_{n-1},z_{0},z_{1},\ldots,z_{m},x_{n+1},x_{n+2},\ldots)$.
The map $\beta$ clearly satisfies Definition 5. It is also clear that
$T^{\prime}$ is coarsely equivalent to $T$.

If $T=R\cup T_{f}^{1}\cup T_{f}^{2}\cup\cdots$ with the roots of finite trees
$x_{n_{1}},x_{n_{2}},\ldots$ then we apply the above procedure to each finite
subtree one after one to obtain a ray
$x_{0},x_{1},\ldots,x_{n_{1}-1},z^{1}_{0},\ldots,z^{1}_{m_{1}},x_{n_{1}+1},\ldots,x_{n_{2}-2},z^{2}_{0},\ldots,z^{2}_{m_{2}},x_{n_{2}+1},\ldots$,
where $z^{i}_{j}$, $j=1,\ldots,m_{j}$, is the list of vertices of the modified
tree $(T^{i}_{f})^{\prime}$.

If a tree is finite then we can connect it to a ray from another (infinite)
tree and proceed as above. Thus we can present each of (not more than) $N$
trees in each connected component of $D(\alpha)$ as a disjoint union of rays,
and we can pass to the general case. There is still a problem that the trees
may share some common edges (Nash-Williams Theorem does not assert that the
forests are disjoint). So, consider the case when several rays share some
edges. Each such edge can belong to not more than $N$ rays. Once again, we can
clone these edges (and the corresponding vertices as well) so that each ray
would pass through its own copy of these edges, and the resulting graph gives
$D^{\prime}$ coarsely equivalent to $D$.

In the opposite direction, suppose that there exists $D^{\prime}$ as in (2),
and (1) does not hold, i.e. for any $\alpha>0$ there exists a finite component
of $D(\alpha)$. Then there exists $C>0$ such that (a) for any $x\in
D^{\prime}$ ther exists $y\in D$ with $d(x,y)<C$, and (b)
$D^{\prime}=\sqcup_{i\in I}D^{\prime}_{i}$ and for each $i\in I$ there exists
a bijective map $\beta:\mathbb{N}\to D^{\prime}_{i}$ with
$d(\beta(k+1),\beta(k))<C$ for any $k\in\mathbb{N}$. Take $\alpha>3C$, and let
$F\subset D$ be a finite subset such that $d(F,D\setminus F)>3C$. Then
$d(F,D^{\prime}\setminus N_{C}(F))>2C$, where $N_{C}(F)$ is the
$C$-neighborhood of $F$. Let $x\in F$, and let $x\in D_{i}$ for some $i\in I$.
The orbit $D^{\prime}_{i}$ of $x$ under the action $\beta$ is infinite, and
this, together with finiteness of $F$, contradicts $d(\beta(k+1),\beta(k))<C$,
$k\in\mathbb{N}$.

∎

######  Corollary 7.

Let $D$ be a uniformly discrete space of bounded geometry satisfying either of
the conditions of Theorem 6. Then $(i_{D})_{*}([1])=0$.

###### Proof.

Let $D^{\prime}$ be as in Theorem 6, and let $\iota:D\to D^{\prime}$ be the
corresponding inclusion. Then $\iota$ induces the maps
$\iota_{\mathbb{C}}:C^{*}_{\mathbb{C}}(D)\to C^{*}_{\mathbb{C}}(D^{\prime})$
and $\iota_{H}:C^{*}_{H}(D)\to C^{*}_{H}(D^{\prime})$ Set
$s=\iota_{\mathbb{C}}(1)\in C^{*}_{\mathbb{C}}(D^{\prime})$. As $\iota$ is a
coarse equivalence, the lower horizontal map in the commuting diagram

$\textstyle{K_{0}(C^{*}_{\mathbb{C}}(D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{\mathbb{C}})_{*}}$$\scriptstyle{(i_{D})_{*}}$$\textstyle{K_{0}(C^{*}_{\mathbb{C}}(D^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(i_{D^{\prime}})_{*}}$$\textstyle{K_{0}(C^{*}_{H}(D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{H})_{*}}$$\textstyle{K_{0}(C^{*}_{H}(D^{\prime}))}$

is an isomorphism. By Theorem 5, $(i_{D^{\prime}})_{*}(s)=0$, hence
$(i_{D})_{*}([1])=0$.

∎

######  Lemma 8.

Suppose that $D(\alpha)$ has finite components for any $\alpha>0$. Then
$(i_{D})_{*}([1])\neq 0$.

###### Proof.

By assumption, ther exists a sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ of finite
subsets of $D$ such that $d(F_{n},D\setminus F_{n})>n$. Then $D=E\sqcup F$,
where $F=\sqcup_{n\in\mathbb{N}}F_{n}$. There is a short exact sequence of
$C^{*}$-algebras

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{K}(l^{2}(D)\otimes
H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{H}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{C^{*}_{H}(E)}{\mathbb{K}(l^{2}(E)\otimes
H)}\oplus\frac{C^{*}_{H}(F)}{\mathbb{K}(l^{2}(F)\otimes
H)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

As $K_{1}$ for compact operators is zero, it suffices to show that the image
of $(i_{F})_{*}([1])$ in $K_{0}(\frac{C^{*}_{H}(F)}{\mathbb{K}(l^{2}(F)\otimes
H)})$ is non-zero. Note that $C^{*}_{H}(F)$ sits between
$\prod_{n\in\mathbb{N}}C_{b}(F_{n})\otimes\mathbb{K}(H)$ and
$\prod_{n\in\mathbb{N}}\mathbb{K}(l^{2}(F_{n})\otimes H)$. As
$(i_{F})_{*}([1])$ is non-zero both in
$K_{0}\left(\frac{\prod_{n\in\mathbb{N}}C_{b}(F_{n})\otimes\mathbb{K}(H)}{\oplus_{n\in\mathbb{N}}C_{b}(F_{n})\otimes\mathbb{K}(H)}\right)$
and in $K_{0}\left(\frac{\prod_{n\in\mathbb{N}}\mathbb{K}(l^{2}(F_{n})\otimes
H)}{\oplus_{n\in\mathbb{N}}\mathbb{K}(l^{2}(F_{n})\otimes H)}\right)$, it is
non-zero in $K_{0}\bigl{(}\frac{C^{*}_{H}(F)}{\mathbb{K}(l^{2}(F)\otimes
H)}\bigr{)}$ as well.

∎

Let $\Gamma$ be a graph with the set of vertices $\Gamma_{0}$ and the set of
edges $\Gamma_{1}$. The group $C_{k}^{BM}(\Gamma)$ of $k$-dimensional
Borel–Moore chains is the abelian group of all formal sums
$\sum_{x\in\Gamma_{k}}\lambda_{x}\cdot x$, $k=0,1$,
$\lambda_{x}\in\mathbb{Z}$, and the quotient
$H_{0}^{BM}(\Gamma)=C_{0}^{BM}(\Gamma)/\partial C_{1}^{BM}(\Gamma)$ with
respect to the standard boundary map is the 0-th Borel–Moore homology group of
$\Gamma$. Details on Borel–Moore homology can be found in [2]. Let
$c=\sum_{x\in\Gamma_{0}}x$ be the chain with all coefficients equal to 1. If
$\Gamma$ is infinite uniformly locally finite then $[c]=0$ in
$H_{0}^{BM}(\Gamma)$, while if $\Gamma$ is finite then $[c]\neq 0$. As
Borel–More homology is functorial, we have maps $H_{0}^{BM}(D(\alpha))\to
H_{0}^{BM}(D(\beta))$ when $\alpha<\beta$, and can pass to the direct limit.
Thus we obtain the following result.

######  Corollary 9.

The following are equivalent:

  1. (1)

$(i_{D})_{*}([1])=0$ in $K_{0}(C^{*}_{H}(D))$;

  2. (2)

$[c]=0$ in $\operatorname{dir}\lim_{\alpha}H_{0}^{BM}(D(\alpha))=0$.

It would be interesting to find a more direct proof of Corollary 9, avoiding
graph theory.

Note that the equivalent properties from Theorem 6 are coarsely invariant.

######  Lemma 10.

Let $D_{1},D_{2}\subset X$ be two uniformly discrete subsets of bounded
geometry coarsely equivalent to $X$. If $D_{1}$ satisfies the property (1) of
Theorem 6 then $D_{2}$ satisfies it too.

###### Proof.

Supoose that $D_{1}$ does not satisfy the property (1) of Theorem 6, and, for
any $\beta>0$, let $\Gamma_{1}(\beta)$ be a finite connected component of
$D_{1}(\beta)$. Take an arbitrary $\alpha>0$. Coarse equivalence between
$D_{1}$ and $D_{2}$ means that there exists $C>0$ such that for every $x\in
D_{1}$ there exists $y\in D_{2}$ with $d(x,y)<C$ and vice versa. Let
$Z=\\{z\in D_{2}:d(z,x)<C\mbox{\ for\ some\ }x\in\Gamma_{1}(\alpha+2C)\\}$. If
$z\in Z$, $y\in D_{2}\setminus Z$ then $d(z,y)\geq\alpha$. As
$\Gamma_{1}(\alpha)$ is finite for any $\alpha$, and as $D_{1}$ is of bounded
geometry, the set $Z$ is finite. Consider $Z$ as a subgraph of
$D_{2}(\alpha)$. It is not connected with any point from
$D_{2}(\alpha)\setminus Z$. Then any connected component of $Z$ is a finite
connected component of $D_{2}(\alpha)$. Thus, $D_{2}$ does not satisfy the
property (1) of Theorem 6.

∎

Recall that a metric space $X$ has bounded geometry if there exists $r>0$ such
that for any $R>0$ there exists $N\in\mathbb{N}$ such that any ball of radius
$R$ can be covered by not more than $N$ balls of radius $r$ (cf. [5], where it
is discussed that this definition for manifolds can be derived from the
traditional local definition via curvature).

It is shown in [8], Prop. 2.5, that if $X$ is a complete Riemannian manifold
admitting a decomposition $X=X_{1}\cup X_{2}$ with closed $X_{1}$ and $X_{2}$
such that $K_{0}(C^{*}(X_{1}))=K_{0}(C^{*}(X_{2}))=0$ then $[p_{\phi}]=0$ for
any Wannier projection $p_{\phi}$ with uniformly discrete set of localization
centers. The next corollary shows that, under the bounded geometry condition,
vanishing of $[p_{\phi}]$ is much more common. Importance of this condition is
explained by the Greene’s theorem: any smooth manifold admits a Riemannian
metric of bounded geometry [4].

######  Corollary 11.

Let $X$ be a connected proper measure space of bounded geometry. Then, for any
Wannier projection $p_{\phi}$ with a uniformly discrete set $D$ of
localization centers, coarsely equivalent to $X$, we have $[p_{\phi}]=0$.

###### Proof.

By Lemma 10, Corollary 7 and Theorem 3, it suffices to show that there exists
a uniformly discrete set $D$ of bounded geometry, coarsely equivalent to $X$,
which satisfies the property (1) of Theorem 6. Since $X$ has bounded geometry,
there exists $r>0$ such that any ball of radius $R$ is covered by at most $N$
balls of radius $r/2$. Given $c>0$, we say that a subset $A\subset X$ is
$c$-disjoint if $d(x,y)>c$ for any $x,y\in A$, $y\neq x$. By Zorn Lemma, there
exists a maximal discrete $r$-disjoint subset $D\subset X$. It is clear that
$D$ is uniformly discrete. Maximality of $D$ implies that for any $x\in X$
there exists $y\in D$ with $d(x,y)\leq r$, so $D$ is coarsely equivalent to
$X$. Since any ball of radius $r/2$ contains not more than one point of $D$,
$D$ has bounded geometry. We claim that the graph $D(3r)$ is connected and,
therefore, satisfies the property (1) of Theorem 6. Indeed, suppose the
contrary. If $D(3r)$ is not connected then we can write $D=A_{1}\sqcup A_{2}$,
where one has $d(x,y)\geq 3r$ if $x\in A_{1}$, $y\in A_{2}$. Set
$X_{i}=\\{x\in X:d(x,A_{i})\leq r\\}$, $i=1,2$. Then $X=X_{1}\cup X_{2}$ and
$X_{1}\cap X_{2}=\emptyset$, moreover, $d(X_{1},X_{2})\geq r$, which means
that $X$ is not connected.

∎

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