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text:
\section{Introduction}

Orbifolds have generated a lot of interest in the recent
mathematical and physical literature. As first defined in the
paper of Satake \cite{Sat}, under the name of $V$-manifolds, they
generalise the notion of smooth manifolds, by being slightly
singular. More precisely, they are locally homeomorphic to the
space of orbits of a finite group action on some Euclidian space.
The original definition of Satake is equivalent to the modern
definition of an effective orbifold.

The problem of generalising the definition of an orbifold to
incorporate ineffective group actions in local charts is
cumbersome. More convenient way is to use the language of Lie
groupoids, as shown in the work of Moerdijk and Pronk
\cite{Mo,MoPr}. Although the theory of groupoids might seem
abstract and lacking of geometric intuition at first, it provides
a powerfull tool to extend the differential geometric ideas to the
(singular) spaces such as the spaces of leaves of a foliation
\cite{CdSW,Con94}, spaces of orbits of Lie group actions and, in
our case, orbifolds. Orbifold groupoids \cite{Adem,Mo,MoMr,MoPr}
have been effectively used to represent orbifolds in the language
of Lie groupoids. The space of orbits of such a groupoid carries a
natural structure of an orbifold. Moreover, it is easy to describe
effective orbifold groupoids as those groupoids that correspond to
the effective orbifolds. In this way the definition of an
ineffective orbifold comes for free in the framework of orbifold
groupoids.

It is a well known result (see \cite{Adem} or \cite{MoMr} for
details) that the space of orbits of a smooth almost free action
of a compact Lie group on a smooth manifold, such that the slice
representations are effective, carries a natural structure of an
effective orbifold. Conversely, each effective orbifold is
isomorphic to the space of orbits of an almost free action of a
unitary group on the bundle of frames of the orbifold. In the
language of Lie groupoids this statement can be reformulated to
saying that each effective orbifold is Morita equivalent to the
translation groupoid of an almost free action of a compact group
on a smooth manifold. It is conjectured (global quotient
conjecture, see \cite{Adem} for the formulation of the
conjecture), but unknown at present, that similar statement holds
for ineffective orbifolds as well. A partial result was obtained
by Henriques and Metzler in \cite{Hen}, where they proved the
statement for the class of orbifolds, whose ineffective groups
have trivial centre.

The problem of presenting an orbifold groupoid as a translation
groupoid of an almost free action of a compact Lie group is
equivalent to finding a faithfull unitary representation of the
groupoid on some hermitian vector bundle over the space of objects
of the groupoid. We show in this paper (Theorem
\ref{Representation of orbifold groupoids}) that such a
representation (faithfull and unitary) exists on a continuous
family of finite dimensional Hilbert spaces over the space of
objects of the groupoid. As a result (see Theorem \ref{Orbifold
groupoids Morita equivalence}) we can show that each orbifold
groupoid is Morita equivalent to a translation groupoid of an
almost free action of a proper bundle of topological groups, with
each group being a finite product of unitary groups.

\section{Preliminaries}

\subsection{The Morita category of Lie groupoids}

In this section we review the basic definitions and facts that
will be used throughout the paper.

The notion of a topological groupoid is a combination and a
generalization of both topological spaces and topological groups.
The topological part is reflected in the space of orbits of the
groupoid, which carries information of its transversal structure.
On the other hand, the isotropy groups of the groupoid represent
the algebraic part of the groupoid and make it a topological space
with extra algebraic structure. Roughly, two groupoids represent
the same geometric space if they have isomorphic transversal and
algebraic structures. From the differential geometric viewpoint
Lie groupoids form the most interesting class of topological
groupoids and allow a natural extension of many of the operations
on smooth manifolds.

For the convenience of the reader we first recall the notion of a
topological groupoid (see \cite{Mrc96} for more details) and
proceed to the definition of Lie groupoids and generalised maps
between them. Detailed exposition with many examples of Lie
groupoids can be found in one of the books \cite{Mac,MoMr,MoMr2}
and references cited there.

A topological groupoid $G$ over the Hausdorff topological space
$G_{0}$ is given by a structure of a category on the topological
space $G$ with objects $G_{0}$, in which all arrows are invertible
and all the structure maps
\[
\xymatrix{G\times^{s,t}_{G_{0}}G \ar[r]^-{\mathord{\mathrm{mlt}}} &
G \ar[r]^-{\mathord{\mathrm{inv}}} & G \ar@<2pt>[r]^{s} \ar@<-2pt>[r]_{t} &
G_{0} \ar[r]^-{\mathord{\mathrm{uni}}} & G
}
\]
are continuous. The maps $s,t$ and $\mathord{\mathrm{mlt}}$ are required to be open,
while the map $\mathord{\mathrm{uni}}$ is an embedding. If $g\in G$ is any arrow
with source $s(g)=x$ and target $t(g)=y$, and $g'\in G$ is another
arrow with $s(g')=y$ and $t(g')=y'$, then the product
$g'g=\mathord{\mathrm{mlt}}(g',g)$ is an arrow from $x$ to $y'$. The map $\mathord{\mathrm{uni}}$
assigns to each $x\in G_{0}$ the identity arrow $1_{x}=\mathord{\mathrm{uni}}(x)$ in
$G$, and we often identify $G_{0}$ with $\mathord{\mathrm{uni}}(G_{0})$. The map
$\mathord{\mathrm{inv}}$ maps each $g\in G$ to its inverse $g^{-1}$. We use the
notation $G(x,y)=s^{-1}(x)\cap t^{-1}(y)$ for the set of arrows
from $x$ to $y$ and we denote by $G_{x}=G(x,x)$ the isotropy group
of the element $x$.

Each groupoid $G$ induces an equivalence relation on its space of
objects $G_{0}$ by identifying two points if and only if there is
an arrow between them. The resulting quotient map $q:G_{0}\rightarrow
G_{0}/G$ onto the space of orbits is an open surjection. The
subset $O$ of $G_{0}$ is $G$-invariant if it is saturated with
respect to this natural equivalence relation. If $O\subset G_{0}$
is an open subset, the space $G|_{O}=t^{-1}(O)\cap s^{-1}(O)$ has
a natural structure of a topological groupoid over $O$.

We say that the groupoid $G$ is Hausdorff if the space of arrows
$G$ is a Hausdorff topological space. In this paper we will be
mostly interested in proper topological groupoids. A topological
groupoid $G$ is proper if it is Hausdorff and if the map
$(s,t):G\rightarrow G_{0}\times G_{0}$ is a proper continuous map.

A Lie groupoid is a topological groupoid $G$ over $G_{0}$, such
that both $G$ and $G_{0}$ are smooth manifolds and where all the
structure maps are smooth. The maps $s$ and $t$ are required to be
submersions with Hausdorff fibers, to insure the existence of a
smooth manifold structure on the space $G\times^{s,t}_{G_{0}}G$,
while the manifold of objects $G_{0}$ is usually taken to be
Hausdorff and second countable. Here are some basic examples of
topological groupoids.

\begin{ex}\rm \label{Ex_Groupoids}
(i) Each smooth (Hausdorff, second countable) manifold $M$ can be
seen as a Lie groupoid with no nontrivial arrows, where
$G=G_{0}=M$ and where all the structure maps equal the identity
map on the manifold $M$. On the other hand, each Lie group is a
Lie groupoid with only one object and the structure maps induced
from the Lie group structure.

(ii) Let a Lie group $K$ act smoothly from the left on a smooth
(Hausdorff, second countable) manifold $M$. The translation
groupoid $K\ltimes M$ of this action has the manifold $M$ as the
space of objects and the space of arrows equal to $K\times M$. The
source and target maps of the translation groupoid are given by
the formulas $s(k,x)=x$ respectively $t(k,x)=k\cdot x$, while the
multiplication is given by $(k',x')(k,x)=(k'k,x)$ for $x'=k\cdot
x$. The identity and inverse maps are then induced from the group
structure of the Lie group $K$. Translation groupoids associated
to right actions of Lie groups on smooth manifolds can be defined
analogously.

(iii) Let $Q$ be a Hausdorff topological space. A bundle of
topological groups over $Q$ is given by a topological space $U$,
together with an open surjection $r:U\rightarrow Q$, such that each fiber
of the map $r$ has a structure of a topological group and these
structures vary continuously across $Q$. Each such bundle can be
naturally seen as a topological groupoid $G=U$ over the space
$G_{0}=Q$ with the structure maps $s=t=r$ and the maps $\mathord{\mathrm{uni}}$,
$\mathord{\mathrm{mlt}}$ and $\mathord{\mathrm{inv}}$ induced by the group structures on the fibers of
the map $r$. The bundle of topological groups $U$ is locally
trivial if the map $r$ is locally trivial.

(iv) A bundle of topological groups $U$ over $Q$ is proper if it
is proper as a topological groupoid. In this case each fiber is
automatically a compact topological group. The converse is not
always true. Let $U$ be a trivial bundle of nontrivial finite
groups over $\mathbb{R}-\{0\}$ together with the trivial group at
$0\in\mathbb{R}$. This bundle of groups over $\mathbb{R}$ has
compact fibers but it is not a proper bundle of topological
groups.

(v) Let $P$ be a Hausdorff topological space and let a bundle of
topological groups $r:U\rightarrow Q$ act on $P$ from the right along the
map $\phi:P\rightarrow Q$ (see below for the definition of the groupoid
action). The translation groupoid $P\rtimes U$ is a topological
groupoid with the space of arrows $P\times_{Q}U$ over the space of
objects $P$. The structure maps are given by: $t(p,u)=p$,
$s(p,u)=p\cdot u$, $(p,u)(p',u')=(p,uu')$,
$\mathord{\mathrm{uni}}(p)=(p,1_{\phi(p)})$ and $(p,u)^{-1}=(p\cdot u,u^{-1})$ for
$\phi(p)=\phi(p')=r(u)=r(u')$ and $p\cdot u=p'$. If $U$ is a
proper bundle of topological groups it follows that $P\rtimes U$
is a proper topological groupoid.
\end{ex}

Morphisms between Lie groupoids are smooth functors. Two Lie
groupoids $G$ and $H$ are isomorphic if there exist morphisms
$i:G\rightarrow H$ and $j:H\rightarrow G$ of Lie groupoids such that $j\circ
i=id_{G}$ and $i\circ j=id_{H}$. However, in the context of the
representation theory of groupoids the notion of a generalised
morphism or a Hilsum-Skandalis map \cite{MoMr2,Mrc96}, which we
review in the sequel, is more suitable.

A smooth left action of a Lie groupoid $G$ on a smooth manifold
$P$ along a smooth map $\pi\!:P\rightarrow G_{0}$ is a smooth map
$\mu\!:G\times^{s,\pi}_{G_{0}}P\rightarrow P$, $(g,p)\mapsto g\cdot p$,
which satisfies $\pi(g\cdot p)=t(g)$, $1_{\pi(p)}\cdot p=p$ and
$g'\cdot(g\cdot p)=(g'g)\cdot p$, for all $g',g\in G$ and $p\in P$
with $s(g')=t(g)$ and $s(g)=\pi(p)$. We define right actions of
Lie groupoids on smooth manifolds in a similar way.

Let $G$ and $H$ be Lie groupoids. A principal $H$-bundle over $G$
is a smooth manifold $P$, equipped with a left action $\mu$ of $G$
along a smooth submersion $\pi\!:P\rightarrow G_{0}$ and a right action
$\eta$ of $H$ along a smooth map $\phi\!:P\rightarrow H_{0}$, such that
(i) $\phi$ is $G$-invariant, $\pi$ is $H$-invariant and both
actions commute: $\phi(g\cdot p)=\phi(p)$, $\pi(p\cdot h)=\pi(p)$
and $g\cdot(p\cdot h)=(g\cdot p)\cdot h$ for every $g\in G$, $p\in
P$ and $h\in H$ with $s(g)=\pi(p)$ and $\phi(p)=t(h)$, (ii)
$\pi:P\rightarrow G_{0}$ is a principal right $H$-bundle:
$(\mathord{\mathrm{pr}}_{1},\eta)\!:P\times^{\phi,t}_{H_{0}}H\rightarrow
P\times^{\pi,\pi}_{G_{0}}P$ is a diffeomorphism.

A map $f\!:P\rightarrow P'$ between principal $H$-bundles $P$ and $P'$
over $G$ is equivariant if it satisfies $\pi'(f(p))=\pi(p)$,
$\phi'(f(p))=\phi(p)$ and $f(g\cdot p\cdot h)=g\cdot f(p)\cdot h$,
for every $g\in G$, $p\in P$ and $h\in H$ with $s(g)=\pi(p)$ and
$\phi(p)=t(h)$. Any such map is automatically a diffeomorphism.
Principal $H$-bundles $P$ and $P'$ over $G$ are isomorphic if
there exists an equivariant diffeomorphism between them. A
generalised map (sometimes called Hilsum-Skandalis map) from $G$
to $H$ is an isomorphism class of principal $H$-bundles over $G$.

If $P$ is a principal $H$-bundle over $G$ and $P'$ is a principal
$K$-bundle over $H$, for another Lie groupoid $K$, one can define
the composition $P\mathbin{\otimes}_{H}P'$ \cite{MoMr2,Mrc96,Mrc99}, which is a
principal $K$-bundle over $G$. It is the quotient of
$P\times^{\phi,\pi'}_{H_{0}}P'$ with respect to the diagonal
action of the groupoid $H$. Lie groupoids form a category $\mathsf{GPD}$
\cite{MoMr2,Mrc96} with generalised maps from $G$ to $H$ as
morphisms between groupoids $G$ and $H$. A principal $H$-bundle
$P$ over $G$ is called a Morita equivalence if it is also left
$G$-principal. The isomorphisms in the category $\mathsf{GPD}$ correspond
precisely to equivalence classes of Morita equivalences.

Actions of topological groupoids on topological spaces and the
generalised maps between topological groupoids can be defined in a
similar way. In the topological category all the maps are required
to be continuous, while the condition that the projection map
$\pi:P\rightarrow G_{0}$ of the principal $H$-bundle $P$ is a surjective
submersion is replaced by the condition that $\pi$ is an open
surjective map.

Orbifold groupoids, which are defined in the next subsection, are
examples of Lie groupoids. In Section
\ref{Section_Representation}, where the representation theorem for
orbifold groupoids (Theorem \ref{Representation of orbifold
groupoids}) is proven, we do not need the notion of a more general
topological groupoid. However, presentation of an orbifold
groupoid by a Morita equivalent translation groupoid (Theorem
\ref{Orbifold groupoids Morita equivalence}), associated to an
almost free action of a proper bundle of topological groups, needs
to be done in the topological category.

\subsection{Orbifolds and Lie groupoids}

Orbifolds are topological spaces which generalise the notion of
smooth manifolds in a way that they locally look like quotients of
smooth manifolds by a finite group action. They were first
introduced by Satake in \cite{Sat} under the name of
$V$-manifolds. That original definition is equivalent to the
definition of effective (also called reduced) orbifolds, found in
the modern literature. A certain class of Lie groupoids, called
orbifold groupoids \cite{Mo,MoMr,MoPr}, can be used to represent
effective orbifolds and at the same time provide a way to define
ineffective orbifolds.

Let $Q$ be a topological space. An orbifold chart of dimension $n$
on the space $Q$ is given by a triple $(\tilde{U},G,\phi)$, where
$\tilde{U}$ is a connected open subset of $\mathbb{R}^{n}$, $G$ is a
finite subgroup of the group $\text{Diff}(\tilde{U})$ of smooth
diffeomorphisms of $\tilde{U}$ and $\phi:\tilde{U}\rightarrow Q$ is an
open map that induces a homeomorphism between $\tilde{U}/G$ and
$U=\phi(\tilde{U})$. An embedding of an orbifold chart
$(\tilde{U},G,\phi)$ into an orbifold chart $(\tilde{V},H,\psi)$
is a smooth embedding $\lambda:\tilde{U}\rightarrow\tilde{V}$ that
satisfies $\psi\circ\lambda=\phi$. The charts $(\tilde{U},G,\phi)$
and $(\tilde{V},H,\psi)$ are compatible, if for any $z\in U\cap V$
there exists an orbifold chart $(\tilde{W},K,\nu)$ with $z\in W$
and embeddings of the chart $(\tilde{W},K,\nu)$ into the charts
$(\tilde{U},G,\phi)$ respectively $(\tilde{V},H,\psi)$. An
orbifold atlas (of dimension $n$) on $Q$ is given by a family
$\mathcal{U}=\{(U_{i},G_{i},\phi_{i})\}_{i\in I}$ of pairwise
compatible orbifold charts (of dimension $n$) that cover $Q$. An
atlas $\mathcal{U}$ refines the atlas $\mathcal{V}$ if every chart
of $\mathcal{U}$ can be embedded into some chart of $\mathcal{V}$.
Two orbifold atlases are equivalent if there exists an atlas that
refines both of them. Effective orbifold of dimension $n$ is a
paracompact Hausdorff topological space $Q$ together with an
equivalence class of $n$-dimensional orbifold atlases on $Q$.

Primary examples of effective orbifolds are the orbit spaces of
effective actions of finite groups on smooth manifolds, where the
charts are given by connected components of (small enough)
invariant open subsets. More generally (see \cite{Adem,MoMr}), let
a compact Lie group $K$ act smoothly and almost freely (with
finite isotropy groups) on a smooth manifold $M$. Since the
actions of compact Lie groups are proper, there exist local
slices, equipped with the actions of the isotropy groups. If these
actions are assumed to be effective, the slices can be used as the
local orbifold charts on the space of orbits $M/K$.

The group actions, defined by the charts, and especially the
isotropy groups form an important part of the orbifold structure.
Namely, two orbifolds can be non-isomorphic, despite being
homeomorphic, when seen as topological spaces. Since Lie groupoids
have a natural built-in algebraic structure, they provide a
suitable framework for the study of orbifolds \cite{MoMr,MoPr}.
Furthermore, the notion of a generalised morphism between Lie
groupoids representing orbifolds turns out to be the proper notion
of a map between the corresponding orbifolds.

Let $G$ be a Lie groupoid. If the maps $s$ and $t$ (and therefore
all structure maps) are local diffeomorphisms we call $G$ an
\'{e}tale Lie groupoid. A bisection of an \'{e}tale Lie groupoid
$G$ is an open subset $U$ of $G$ such that both $s|_{U}$ and
$t|_{U}$ are injective. Any such bisection $U$ gives a local
diffeomorphism $\tau_{U}\!:s(U)\rightarrow t(U)$, by $\tau_{U}=t|_{U}\mathbin{{\scriptstyle \circ }}
(s|_{U})^{-1}$. For any arrow $g\in G(x,y)$ there exists a
bisection $U_{g}$ containing $g$; the germ at $x$ of the induced
local diffeomorphism is independent of the choice of the
bisection.

Orbifold groupoid is a proper \'{e}tale Lie groupoid. An orbifold
groupoid $G$ is effective if for each $x\in G_{0}$ and each
nontrivial $g\in G_{x}$ the germ at $x$ of some (and therefore
every) local diffeomorphism $\tau_{U_{g}}$, defined by a bisection
through $g$, is nontrivial.

Crucial theorem in the connection between effective orbifolds and
effective orbifold groupoids states that there is a natural
structure of an effective orbifold \cite{MoMr,MoPr} on the space
of orbits of an effective orbifold groupoid. In this way an
ineffective orbifold groupoid can be seen as one possible way to
define an ineffective orbifold.

\subsection{Continuous families of Hilbert spaces}

Representation theory of topological groupoids extends the
classical representation theory of groups on vector spaces, where
the latter are replaced by families of vector spaces, indexed by
the space of objects of the groupoid. We first recall the
definition and basic properties of a continuous family of Hilbert
spaces over a topological space, as given in \cite{Dix} (see also
\cite{Bos} for further examples).

\begin{dfn}\label{Family of Hilbert spaces}
Let $B$ be a locally compact Hausdorff topological space. A
continuous family of Hilbert spaces over $B$ is given by a pair
$(\{E_{x}\}_{x\in B},\Gamma)$, where $E_{x}$ is a Hilbert space
for each $x\in B$ and $\Gamma\subset\prod_{x\in B}E_{x}$ is a
vector subspace that satisfies:
\begin{enumerate}
\item For each $x\in B$ and each $v\in E_{x}$ there exists
$s\in\Gamma$ such that $s(x)=v$;

\item For every $s_{1},s_{2}\in\Gamma$ the function $x\mapsto
\scp{s_{1}(x),s_{2}(x)}_{x}$ is a continuous function on $B$;

\item If $w\in\prod_{x\in B}E_{x}$ satisfies: for each $x\in B$
and each $\epsilon>0$ there exists a neighbourhood $U$ of $x$ and
$s\in\Gamma$ such that $\norm{s(x')-w(x')}_{x'}<\epsilon$ for all
$x'\in U$, then $w\in\Gamma$.
\end{enumerate}
Family $(\{E_{x}\}_{x\in B},\Gamma)$ is a continuous family of
finite dimensional Hilbert spaces if all the Hilbert spaces
$E_{x}$ are finite dimensional.
\end{dfn}

From the topological viewpoint the following consequence of
Definition \ref{Family of Hilbert spaces} is useful and allows us
to think of continuous families of Hilbert spaces as
generalizations of hermitian vector bundles.

\begin{prop}
Let $(\{E_{x}\}_{x\in B},\Gamma)$ be a continuous family of
Hilbert spaces over a locally compact Hausdorff space $B$. Denote
by $E=\coprod_{x\in B} E_{x}$ the disjoint union of the spaces
$\{E_{x}\}_{x\in B}$. There exists a topology on the space $E$
that makes the projection map $p:E\rightarrow B$ (which maps each Hilbert
space $E_{x}$ to the point $x$) a continuous open surjection and
such that the space $\Gamma$ equals the space of continuous
sections of the map $p$.
\end{prop}

\begin{proof}
We first define a basis for the topology on the total space $E$.
For each open subset $V\subset B$, each $s\in\Gamma$ and each
$\epsilon>0$ define the tubular set $B(V,s,\epsilon)=\{v\in
E|p(v)\in V,\norm{s(p(v))-v}_{p(v)}<\epsilon\}$. Condition $(1)$
in Definition \ref{Family of Hilbert spaces} insures that the
family of all such tubular sets covers the space $E$. Now let
$W_{1}=B(V_{1},s_{1},\epsilon_{1})$ and
$W_{2}=B(V_{2},s_{2},\epsilon_{2})$ be two such tubular sets and
choose arbitrary element $v\in W_{1}\cap W_{2}$. For any such $v$
the inequalities $\norm{s_{i}(p(v))-v}_{p(v)}<\epsilon_{i}$ hold
for $i=1,2$ and there exists a section $s\in\Gamma$ such that
$s(p(v))=v$. Denote
$\delta=\min\{\epsilon_{1}-\norm{s_{1}(p(v))-v}_{p(v)},
\epsilon_{2}-\norm{s_{2}(p(v))-v}_{p(v)}\}$. Since $\Gamma$ is a
vector subspace of $\prod_{x\in B}E_{x}$, $s_{1}-s$ and $s_{2}-s$
are elements of $\Gamma$ as well. Using condition $(2)$ in
Definition \ref{Family of Hilbert spaces} we can find open
neighbourhoods $U_{1}$ and $U_{2}$ of the point $p(v)$ such that
$\norm{s_{i}(x)-s(x)}_{x}<\epsilon_{i}-\frac{\delta}{2}$ for
$i=1,2$ and all $x\in U_{1}$ respectively $x\in U_{2}$. The
tubular set $B(U_{1}\cap U_{2},s,\frac{\delta}{2})$ then satisfies
$B(U_{1}\cap U_{2},s,\frac{\delta}{2})\subset W_{1}\cap W_{2}$ and
contains the point $v$.

With the above topology the map $p$ becomes a continuous open
surjection. It remains to be proven that the space $\Gamma$ equals
the space of the continuous sections of the map $p$. Choose any
section $s\in\Gamma$, sending $x\in B$ to $v\in E$. We want to
show that $s$ is a continuous section of the map $p$. For any
basic open neighbourhood $B(V,s',\epsilon)$ of the element $v$ we
have $s-s'\in\Gamma$ and $\norm{s(x)-s'(x)}_{x}<\epsilon$.
Continuity of the map $y\mapsto\norm{s(y)-s'(y)}_{y}$ gives us a
neighbourhood $U$ of the point $x$ such that
$\norm{s(y)-s'(y)}_{y}<\epsilon$ on $U$. The neighbourhood $U\cap
V$ then satisfies $s(U\cap V)\subset B(V,s',\epsilon)$, which
proves that $s$ is a continuous section of the map $p$.
Conversely, let $s:B\rightarrow E$ be any continuous section of the map
$p$. We will show that $s$ satisfies condition $(3)$ in Definition
\ref{Family of Hilbert spaces}. Choose an element $x\in B$ and
$\epsilon>0$. By condition $(1)$ in Definition \ref{Family of
Hilbert spaces} we can find $s'\in\Gamma$ such that $s(x)=s'(x)$.
Since $s:B\rightarrow E$ is a continuous map, the set
$V=s^{-1}(B(B,s',\epsilon))$ is an open neighbourhood of the point
$x$ such that $\norm{s(y)-s'(y)}_{y}<\epsilon$ for every $y\in V$.
The neighbourhood with this property exists for every $x\in B$ and
every $\epsilon>0$, therefore $s\in\Gamma$.
\end{proof}

From now on we will denote the continuous family of Hilbert spaces
$(\{E_{x}\}_{x\in B},\Gamma)$ over $B$ simply by $E$, according to
the notations from the preceding proposition, and refer to
$\Gamma$ as the space of the continuous sections of the map $p$.
It is not hard to check that the space $\Gamma$ is in fact a
module over the algebra of the continuous functions on the space
$B$. The dimension $d(x)$ of the fiber $E_{x}$ of a continuous
family of Hilbert spaces is not necesarilly constant along $B$,
but it is a lower semi-continuous function on $B$, as can be seen
by using properties $(1)$ and $(2)$ of Definition \ref{Family of
Hilbert spaces}. Denote by $\mathord{\mathrm{supp_{o}}}(E)=\{x\in B|d(x)>0\}$ the
support of the family of Hilbert spaces $E$. Notice that
$\mathord{\mathrm{supp_{o}}}(E)$ is an open subset of $B$ since the dimension
function $d$ is lower semi-continuous.

Here are some examples of continuous families of Hilbert spaces
that will be used later on in the paper.

\begin{ex}\rm \label{Ex_Families of Hilbert spaces}
(i) Every $n$-dimensional hermitian vector bundle $E$ over a
locally compact Hausdorff space $B$ is an example of a family of
finite dimensional Hilbert spaces with fibers of constant
dimension. Conversely, if the dimension of the fibers of the
continuous family of Hilbert spaces $E$ over $B$ is a constant
function on $B$, then $E$ is actually a hermitian vector bundle
over $B$.

(ii) Let $B$ be a locally compact Hausdorff topological space,
$O\subset B$ an open subset and $E_{O}$ a hermitian vector bundle
over the space $O$. The trivial extension of the bundle $E_{O}$ is
the continuous family of Hilbert spaces $E_{O}^{B}$ over $B$,
defined as follows. The fiber of $E_{O}^{B}$ over the point $x\in
B$ is by definition
\begin{displaymath}
(E^{B}_{O})_{x}=\left\{ \begin{array} {ll}
(E_{O})_{x}, & \textrm{$x\in O$,}\\
\{0\}, & \textrm{otherwise.}\\
\end{array}\right.
\end{displaymath}
We define the vector space $\Gamma(E^{B}_{O})$ of sections of
$E^{B}_{O}$ to be the trivial extensions of those sections of
$E_{O}$ that tend to zero at the boundary of the space $O$ in $B$.
By definition, the section $s\in\Gamma(E_{O})$ tends to zero at
the boundary of the space $O$ in $B$ if for every $x\in\partial O$
and every $\epsilon>0$ there exists a neighbourhood $U$ of $x$ in
$B$ such that $\norm{s(y)}_{y}<\epsilon$ for all $y$ in $U\cap O$.
It is straightforward to check that the space $\Gamma(E^{B}_{O})$
satisfies the conditions in Definition \ref{Family of Hilbert
spaces}.

(iii) Let $\{E^{i}\}_{i\in I}$ be a collection of families of
finite dimensional Hilbert spaces over the locally compact
Hausdorff space $B$ and assume that the family of open sets
$\{\mathord{\mathrm{supp_{o}}}(E^{i})\}_{i\in I}$ is locally finite over $B$. The sum
$E=\bigoplus_{i\in I}E^{i}$ of families $\{E^{i}\}_{i\in I}$ is
defined as follows. First define $E_{x}=\bigoplus_{i\in
I}E^{i}_{x}$ for each $x\in B$. Since the family
$\{\mathord{\mathrm{supp_{o}}}(E^{i})\}_{i\in I}$ is locally finite, this sum is
actually a finite sum, so $E_{x}$ is a finite dimensional Hilbert
space for every $x\in B$. The space of sections $\Gamma(E)$ is
defined to be the product of the spaces $\Gamma(E^{i})$. The
induced topology on the space $E$ coincides with the topology of
the fibrewise product of the spaces $E^{i}$ along the space $B$.
\end{ex}

\section{Representations of topological groupoids}

Let $G$ be a topological groupoid with locally compact space of
objects $G_{0}$ and let $p:E\rightarrow G_{0}$ be a continuous family of
Hilbert spaces over $G_{0}$. A continuous representation of the
groupoid $G$ on the family $E$ is given by a continuous left
action of $G$ on the space $E$, along the map $p$, such that each
$g\in G(x,y)$ acts as a linear isomorphism $g:E_{x}\rightarrow E_{y}$.
Representation of the groupoid $G$ on the family of Hilbert spaces
$E$ is unitary if each $g\in G$ acts as a unitary map between the
corresponding Hilbert spaces.

\begin{ex}\rm
(i) Let $K$ be a topological group. Then the continuous
representations of $K$, viewed as a topological groupoid, coincide
with the continuous representations of $K$ on Hilbert spaces. Each
continuous family of Hilbert spaces over the locally compact space
$X$ is naturally a representation of the space $X$, seen as a
topological groupoid.

(ii) Combining previous two examples we get the representations of
the translation groupoid $K\ltimes X$ of a continuous action of
the topological group $K$ on the locally compact Hausdorff space
$X$. These are precisely $K$-equivariant continuous families of
Hilbert spaces over $X$, i.e. there is a fibrewise linear action
of the group $K$ on the total space $E$ such that the projection
map $p$ is $K$-equivariant.
\end{ex}

Generalised maps between groupoids can be used to pull back
representations in the same sense as vector bundles can be pulled
back by continuous maps. Let $G$ and $H$ be Lie groupoids and let
$P$ be a principal $H$-bundle over $G$. Assume that $E$ is a
hermitian vector bundle over $H_{0}$, equipped with a unitary
representation of the groupoid $H$, and denote by $\pi:P\rightarrow G_{0}$
respectively $\phi:P\rightarrow H_{0}$ the moment maps of the principal
bundle $P$. The pull back bundle $\phi^{\ast}E=P\times_{H_{0}}E$
has a natural structure of a vector bundle over $P$ with
projection onto the first factor as the projection map. The
groupoid $H$ acts from the right on the space $\phi^{\ast}E$ by
the formula: $(p,v)\cdot h=(p\cdot h,h^{-1}\cdot v)$. Since the
action of $H$ on $P$ is along the fibers of the map $\pi$, it is
easy to see that the map $\pi_{G}:\phi^{\ast}E/H\rightarrow G_{0}$,
$\pi_{G}([p,v])=\pi(p)$ is well defined and continuous. We will
show that the space $P^{\ast}E=\phi^{\ast}E/H$ has a natural
structure of a hermitian vector bundle over $G_{0}$ with
projection map $\pi_{G}$. Furthermore, the action of the groupoid
$G$ on the space $P$ induces a unitary representation of the
groupoid $G$ on the bundle $P^{\ast}E$.

\begin{prop}\label{Pull back bundles}
The space $P^{\ast}E$ is a hermitian vector bundle over $G_{0}$
with a natural unitary representation of the Lie groupoid $G$.
\end{prop}

\begin{proof}
First consider the induced vector bundle $\phi^{\ast}E$ over $P$.
Its fiber over the point $p\in P$ can be canonically identified to
the fiber of the vector bundle $E$ over the point $\phi(p)\in
H_{0}$, while the formula
$\scp{(p,v_{1}),(p,v_{2})}_{\phi^{\ast}E}=\scp{v_{1},v_{2}}_{E}$
induces a scalar product on the fiber $(\phi^{\ast}E)_{p}$. So
defined structures of Hilbert spaces on the fibers of the bundle
$\phi^{\ast}E$ over $P$ turn it into a hermitian vector bundle.

The groupoid $H$ acts from the right on the space $\phi^{\ast}E$
by $(p,v)\cdot h=(p\cdot h,h^{-1}\cdot v)$ for $\phi(p)=t(h)$ and
$v\in E_{t(h)}$. Let $r:\phi^{\ast}E\rightarrow G_{0}$ be the
$H$-invariant projection defined with the formula
$r((p,v))=\pi(p)$ and denote by $\pi_{G}:\phi^{\ast}E/H\rightarrow G_{0}$
the induced map from the quotient space. First observe that the
fibers of the map $r:\phi^{\ast}E\rightarrow G_{0}$ equal the restrictions
$P^{\ast}E|_{\pi^{-1}(x)}$ of the bundle $P^{\ast}E$ to the fibers
of the map $\pi$ over the points $x\in G_{0}$. Furthermore, since
$P$ is a principal $H$-bundle over $G_{0}$, $H$ acts freely and
transitively along the fibers of the map $\pi$. Combining these
two observations with the fact that the action of $H$ on $E$ is
linear we get natural structures of Hilbert spaces on the fibers
of the map $\pi_{G}$, which we now describe. Let
$\delta=pr_{2}\circ(\mathord{\mathrm{pr}}_{1},\eta)^{-1}\!: P\times_{G_{0}}P\rightarrow H$
be the continuous (actually smooth in our case) map, uniquely
defined by the condition $p\cdot\delta(p,p')=p'$, for $p$ and $p'$
that satisfy $\pi(p)=\pi(p')$. Using the map $\delta$ we can
define the maps $+:P^{\ast}E\times_{G_{0}}P^{\ast}E\rightarrow P^{\ast}E$,
$\cdot:\mathbb{C}\times P^{\ast}E\rightarrow P^{\ast}E$ and
$\scp{-,-}_{P^{\ast}E}:P^{\ast}E\times_{G_{0}}P^{\ast}E\rightarrow\mathbb{C}$ by
the formulas
\[
[p,v]+[p',v']=[p,v+\delta(p,p')v'],
\]
\[
\lambda[p,v]=[p,\lambda v],
\]
\[
\scp{[p,v],[p',v']}_{P^{\ast}E}=\scp{v,\delta(p,p')v'}_{E}.
\]
It is straightforward to verify that these maps are well defined,
continuous and that they induce structures of Hilbert spaces on
the fibers of the map $\pi_{G}$. To show that the map
$\pi_{G}:P^{\ast}E\rightarrow G_{0}$ carries a structure of a vector
bundle we have to show that it is locally trivial. Choose a point
$x\in G_{0}$ and an element $p\in P$ with $\pi(p)=x$. Let
$\psi:\phi^{\ast}E|_{V}\rightarrow V\times\mathbb{C}^{n}$ be a trivialization of
the bundle $\phi^{\ast}E$ on the neighbourhood $V$ of the point
$p$. Since the map $\pi$ is a submersion, there exists a local
section $s:U\rightarrow V$ of the map $\pi|_{V}$, defined on some
neighbourhood $U$ of the point $x$, such that $s(x)=p$. Using the
maps $\psi$ and $s$ we can define the trivialization
$\psi':P^{\ast}E|_{U}\rightarrow U\times\mathbb{C}^{n}$ by the formula
$\psi'([p',v'])=(\pi(p'),pr_{2}(\psi(s(\pi(p')),\delta(s(\pi(p')),p')v'))$.
Finally, by defining $g\cdot[p,v]=[g\cdot p,v]$, we get a unitary
representation of the Lie groupoid $G$ on the hermitian vector
bundle $P^{\ast}E$.
\end{proof}

A representation of the groupoid $G$ on the family of Hilbert
spaces $E$ is faithfull if for each $x\in G_{0}$ the isotropy
group $G_{x}$ acts faithfully on the Hilbert space $E_{x}$. This
condition is equivalent to the requirement that for each $x,y\in
G_{0}$ and $g_{1},g_{2}\in G(x,y)$, with $g_{1}\neq g_{2}$, the
elements $g_{1}$ and $g_{2}$ induce different isomorphisms from
$E_{x}$ to $E_{y}$.

\begin{prop}\label{Faithfullness}
Let $G$ and $H$ be Lie groupoids and let $P$ be a Morita
equivalence between $G$ and $H$. If the representation of the
groupoid $H$ on $E$ is faithfull the representation of the
groupoid $G$ on $P^{\ast}E$ is faithfull as well.
\end{prop}

\begin{proof}
We have to prove that for each $x\in G_{0}$ the isotropy group
$G_{x}$ acts faithfully on the vector space $(P^{\ast}E)_{x}$.
Choose $x\in G_{0}$ and any $p\in\pi^{-1}(x)\subset P$. Since $P$
is a Morita equivalence, the Lie groups $G_{x}$ and $H_{\phi(p)}$
act freely and transitively on the space
$P(p)=\pi^{-1}(x)\cap\phi^{-1}(\phi(p))$ from the left
respectively from the right. Denote by $i_{p}:G_{x}\rightarrow
H_{\phi(p)}$ the induced bijection (which is in fact a group
isomorphism), implicitely defined by the equation $g\cdot p=p\cdot
i_{p}(g)$. Suppose that an arrow $g\in G_{x}$ acts as the identity
transformation on the space $(P^{\ast}E)_{x}$, i.e.
$g\cdot[p,v]=[p,v]$ for all $v\in E_{\phi(p)}$, where we have
identified $[p,v]\in (P^{\ast}E)_{x}$ with $v\in E_{\phi(p)}$.
Then the equality
\[
[p,v]=g\cdot[p,v]=[g\cdot p,v]=[p\cdot i_{p}(g),v]=[p,i_{p}(g)v]
\]
holds for all $v\in E_{\phi(p)}$. This shows that $i_{p}(g)$ acts
as the identity on the space $E_{\phi(p)}$ and is therefore by the
assumption of faithfullness of the representation of $H$ on $E$
equal to $1_{\phi(p)}$. Since $i_{p}$ is a group isomorphism $g$
must be equal to $1_{x}$, which shows that the representation of
$G_{x}$ on $(P^{\ast}E)_{x}$ is faithfull as well.
\end{proof}

\section{Representations of orbifold groupoids}\label{Section_Representation}

The problem of representing an orbifold groupoid $G$ faithfully
and unitarily on a hermitian vector bundle is equivalent to
finding a smooth almost free action of a compact Lie group $K$ on
a smooth manifold $M$ such that the translation groupoid $M\rtimes
K$ is Morita equivalent to the groupoid $G$.

A faithfull unitary representation of the groupoid $G$ on an
$n$-dimensional hermitian vector bundle $E$ over $G_{0}$ induces a
free left action of the groupoid $G$ on the principal
$U(n)$-bundle $\mathrm{UFr}(E)$ of unitary frames of the bundle $E$, which
commutes with the natural right action of the Lie group $U(n)$ on
the bundle $\mathrm{UFr}(E)$. Since the action of $G$ on $\mathrm{UFr}(E)$ is proper
and free, the orbit space $G\backslash \mathrm{UFr}(E)$ inherits a natural
smooth structure. Moreover, since the actions of $G$ and $U(n)$ on
$\mathrm{UFr}(E)$ commute, there exists an induced action of the group
$U(n)$ on the manifold $G\backslash \mathrm{UFr}(E)$, which is almost free
as a consequence of the fact that $G$ is an orbifold groupoid. It
is then straightforward to check that $G$ is Morita equivalent to
the translation groupoid $(G\backslash \mathrm{UFr}(E))\rtimes U(n)$.

On the other hand, let the compact Lie group $K$ act smoothly and
almost freely from the right on the smooth manifold $M$ and let
$P$ denote the Morita equivalence between the groupoids $G$ and
$M\rtimes K$. By the Peter-Weyl theorem for compact Lie groups
there exists a finite dimensional Hilbert space $V$ and a
faithfull unitary representation of the group $K$ on $V$. This
representation induces a faithfull unitary representation of the
groupoid $M\rtimes K$ on the trivial vector bundle $M\times V$,
where the action is given by $(x,g)\cdot (x',v)=(x,g\cdot v)$ for
$x'=x\cdot g$. Combining Propositions \ref{Pull back bundles} and
\ref{Faithfullness} we get a faithfull unitary representation of
the groupoid $G$ on the hermitian vector bundle $P^{\ast}(M\times
V)$ over $G_{0}$.

The question whether every orbifold groupoid admits a faithfull
unitary representation on a hermitian vector bundle is believed to
have a positive answer, but it is unproven at the moment. It has
been long known to be true for effective orbifold groupoids, as
sketched below. Each \'{e}tale Lie groupoid $G$ has a natural
representation on the tangent bundle $TG_{0}$, where an arrow
$g\in G(x,y)$ acts via the differential of the local
diffeomorphism $\tau_{U_{g}}$, induced by some bisection $U_{g}$
containing the arrow $g$. Straight from the definition it follows
that this representation of the groupoid $G$ on $TG_{0}$ is
faithfull if and only if $G$ is an effective orbifold groupoid.
This canonical representation can be extended to the
representation of $G$ on the complexified tangent bundle
$T^{\mathbb{C}}G_{0}$ and made unitary by averaging an arbitrary
hermitian metric on $T^{\mathbb{C}}G_{0}$. More recently, in the paper
\cite{Hen} by Henriques and Metzler, the authors proved the
statement for the class of ineffective orbifold groupoids, whose
ineffective isotropy groups have trivial centre.

However, in the broader framework of unitary representations on
continuous families of finite dimensional Hilbert spaces we are
able to prove the following result.

\begin{theo}\label{Representation of orbifold groupoids}
Let $G$ be an orbifold groupoid over $G_{0}$. Then there exists a
faithfull unitary representation of the groupoid $G$ on a
continuous family of finite dimensional Hilbert spaces over
$G_{0}$.
\end{theo}

We start by proving some propositions that will be needed in the
proof of Theorem \ref{Representation of orbifold groupoids}.

\begin{prop}\label{Invariant}
Let $G$ be an orbifold groupoid over $G_{0}$. For each $x\in
G_{0}$ there exist a $G$-invariant open neighbourhood $O_{x}$ of
$x$ and a faithfull unitary representation of the groupoid
$G|_{O_{x}}$ on a hermitian vector bundle $E_{O_{x}}$ over
$O_{x}$.
\end{prop}

\begin{proof}
In the proof of the proposition we use the following
characterization of the local structure of orbifold groupoids
\cite{MoMr,MoPr}. For each $x\in G_{0}$ there exist a
neighbourhood $U_{x}$ of $x$ and a natural isomorphism of Lie
groupoids $G|_{U_{x}}\cong G_{x}\ltimes U_{x}$, where each $g\in
G_{x}$ acts on $U_{x}$ by the diffeomorphism corresponding to the
suitable bisection through $g$. Let $\mathbb{C}[G_{x}]$ denote the
Hilbert space of complex functions on the finite group $G_{x}$
with the orthonormal basis $\{\delta_{g}\}_{g\in G_{x}}$. The left
regular representation of the group $G_{x}$ on the space
$\mathbb{C}[G_{x}]$ induces a faithfull unitary representation of the
groupoid $G|_{U_{x}}\cong G_{x}\ltimes U_{x}$ on the trivial
vector bundle $U_{x}\times\mathbb{C}[G_{x}]$ by the formula
$(g,x)\cdot(x,f)=(g\cdot x,g\cdot f)$.

The saturation $O_{x}=s(t^{-1}(U_{x}))$ of the open set $U_{x}$ is
again an open set since $s$ is a submersion and hence an open map.
It is straightforward to check that the manifold
$P=t^{-1}(U_{x})$, together with the left action of the groupoid
$G|_{U_{x}}$ and the right action of the groupoid $G|_{O_{x}}$,
defines a Morita equivalence between the groupoids $G|_{U_{x}}$
and $G|_{O_{x}}$. Denote by
$E_{O_{x}}=(P^{-1})^{*}(U_{x}\times\mathbb{C}[G_{x}])$ the pullback
bundle over $O_{x}$, together with the induced unitary
representation of the groupoid $G|_{O_{x}}$. Since the
representation of the groupoid $G|_{U_{x}}$ on
$U_{x}\times\mathbb{C}[G_{x}]$ was faithfull and since $P^{-1}$ is a
Morita equivalence the representation of $G|_{O_{x}}$ on
$E_{O_{x}}$ is faithfull by Proposition \ref{Faithfullness}.
\end{proof}

\begin{prop}\label{Extension}
Let $G$ be an orbifold groupoid over $G_{0}$ and let $O\subset
G_{0}$ be a $G$-invariant open subset of $G_{0}$. Every unitary
representation of the groupoid $G|_{O}$ on a hermitian vector
bundle $E_{O}$ over $O$ can be extended to a unitary
representation of the groupoid $G$ on the continuous family of
finite dimensional Hilbert spaces $E^{G_{0}}_{O}$ over $G_{0}$.
\end{prop}

\begin{proof}
Let $E_{O}$ be a Hermitian vector bundle over the space $O$,
equipped with a unitary representation of the groupoid $G|_{O}$.
Denote by $p:E^{G_{0}}_{O}\rightarrow G_{0}$ the trivial extension of the
hermitian vector bundle $E_{O}$ over $O$ to a family of finite
dimensional Hilbert spaces over $G_{0}$ as in Example
\ref{Ex_Families of Hilbert spaces}. Recall that
\begin{displaymath}
(E^{G_{0}}_{O})_{x}=\left\{ \begin{array} {ll}
(E_{O})_{x}, & \textrm{if $x\in O$,}\\
\{0\}, & \textrm{otherwise.}\\
\end{array}\right.
\end{displaymath}
For any arrow $g\in G$ we define the action as follows:

(i) If $g\in G|_{O}$ let $g$ act on $E^{G_{0}}_{O}$ as it acts on
$E_{O}$;

(ii) If $g\notin G|_{O}$ and $g\in G(x,y)$ then $g$ acts in the
only possible way, sending the vector $0_{x}$ to the vector
$0_{y}$.

This defines a unitary representation of the groupoid $G$ on the
family of finite dimensional Hilbert spaces $E^{G_{0}}_{O}$, which
extends the representation of the groupoid $G|_{O}$ on the vector
bundle $E_{O}$. To prove the claim of the proposition we have to
check that this defines a continuous representation, i.e. the map
$\mu:G\times_{G_{0}}E^{G_{0}}_{O}\rightarrow E^{G_{0}}_{O}$ is continuous.

First decompose the space $G_{0}$ as a disjoint union of
$G$-invariant subspaces $O$, $V=\overline{O}^{c}$ and $\partial
O$. Since $O$ and $V$ are open subsets of the space $G_{0}$, the
spaces $W_{1}=G\times_{G_{0}}p^{-1}(V)$ respectively
$W_{2}=G\times_{G_{0}}p^{-1}(O)$ are open subspaces of the space
$G\times_{G_{0}}E^{G_{0}}_{O}$. Observing that $\mu|_{W_{1}}$ is
basically the left action of the groupoid $G|_{V}$ on $V$, while
$\mu|_{W_{2}}$ equals the action map of the representation of the
groupoid $G|_{O}$ on the bundle $E_{O}$, we see that
$\mu|_{W_{1}}$ respectively $\mu|_{W_{2}}$ are continuous maps.
Now let $g\in G(x,y)$ be an arrow such that $x\in\partial O$ and
therefore $y\in\partial O$. For such $g$ there exists only one
element in $G\times_{G_{0}}E^{G_{0}}_{O}$ with first coordinate
$g$, namely $(g,0_{x})$ and we have $\mu(g,0_{x})=g\cdot
0_{x}=0_{y}$. We need to show that the map $\mu$ is continuous at
the point $(g,0_{x})$. To this extent choose arbitrary
neighbourhood $W$ of the point $0_{y}$ in $E^{G_{0}}_{O}$. By the
definition of the topology on the space $E^{G_{0}}_{O}$ we can
find a smaller tubular open neighbourhood $B(U_{y},0,\epsilon)$ of
the point $0_{y}$, where $U_{y}$ is a neighbourhood of the point
$y$ in $G_{0}$ and $0$ is the zero section of $E^{G_{0}}_{O}$.
Shrinking the set $U_{y}$ if necessary we can assume that there
exists a bisection $U$ of the groupoid $G$ through the arrow $g$
such that $t(U)=U_{y}$. The unitarity of the representation of $G$
on $E^{G_{0}}_{O}$ now implies that
$\mu(U\times_{G_{0}}B(s(U),0,\epsilon))\subset
B(U_{y},0,\epsilon)$, which proves that $\mu$ is continuous at the
point $(g,0_{x})$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Representation of orbifold groupoids}]
Let $G$ be an orbifold groupoid over $G_{0}$. The quotient
projection $q:G_{0}\rightarrow G_{0}/G$ is an open surjective map, which
insures that the space $Q=G_{0}/G$ is second countable and locally
compact. Since $G$ is a proper groupoid, the map $(s,t):G\rightarrow
G_{0}\times G_{0}$ is a proper map between Hausdorff topological
spaces and hence a closed map. This shows that $(s,t)(G)\subset
G_{0}\times G_{0}$ is a closed equivalence relation, so $Q$ is a
Hausdorff space. It follows that $Q$ is paracompact.

We can use Proposition \ref{Invariant} to find for each $x\in
G_{0}$ a $G$-invariant open neighbourhood $O_{x}$ of the point $x$
and a faithfull unitary representation of the groupoid
$G|_{O_{x}}$ on a Hermitian vector bundle $E_{O_{x}}$ over
$O_{x}$. The family $\{q(O_{x})\}_{x\in G_{0}}$ is an open cover
of the second countable paracompact space $Q$, so we can choose a
countable, locally finite refinement $\{V'_{i}\}_{i\in\mathbb{N}}$ of the
cover $\{q(O_{x})\}_{x\in G_{0}}$. Pulling back the sets
$\{V'_{i}\}_{i\in\mathbb{N}}$ to $G_{0}$ we get a locally finite covering
$\{V_{i}\}_{i\in\mathbb{N}}$ of the space $G_{0}$ by $G$-invariant open
subsets, where we denoted $V_{i}=q^{-1}(V_{i}')$. For each
$i\in\mathbb{N}$ we can choose some $x_{i}$, such that $V_{i}\subset
O_{x_{i}}$, to get a faithfull unitary representation of the
groupoid $G|_{V_{i}}$ on the hermitian vector bundle
$E_{V_{i}}=E_{O_{x_{i}}}|_{V_{i}}$. By Proposition \ref{Extension}
we can extend the unitary representation of the groupoid
$G|_{V_{i}}$ on the bundle $E_{V_{i}}$ to the unitary
representation of the groupoid $G$ on the family of finite
dimensional Hilbert spaces $E^{i}=E_{V_{i}}^{G_{0}}$.

Let $E=\bigoplus_{i\in\mathbb{N}}E^{i}$ be the continuous family of
finite dimensional Hilbert spaces over $G_{0}$, defined as the sum
of the families $\{E^{i}\}_{i\in\mathbb{N}}$ as in Example
\ref{Ex_Families of Hilbert spaces}. The representations of the
groupoid $G$ on the families $\{E^{i}\}_{i\in\mathbb{N}}$ canonically
induce a continuous unitary representation of the groupoid $G$ on
$E$, defined by $g\cdot(v_{1},v_{2},\ldots)=(g\cdot v_{1},g\cdot
v_{2},\ldots)$. To see that the representation of $G$ on $E$ is
faithfull it is enough to show that for each $x\in G_{0}$ the
group $G_{x}$ acts faithfully on the Hilbert space $E_{x}$.
Straight from the definition of the representation of $G$ on $E$
it follows that the representation of the group $G_{x}$ on $E_{x}$
decomposes as the direct sum of the representations of the group
$G_{x}$ on the spaces $E^{i}_{x}$ for $i\in\mathbb{N}$. Since
$\{V_{i}\}_{i\in\mathbb{N}}$ is a cover of the space $G_{0}$, there
exists some $i\in\mathbb{N}$ such that $x\in V_{i}$. Faithfullness of the
representation of the groupoid $G|_{V_{i}}$ on the bundle
$E_{V_{i}}$ implies that the representation of the group $G_{x}$
on $E^{i}_{x}$ is faithfull and consequently the representation of
the group $G_{x}$ on $E_{x}$ is faithfull as well.
\end{proof}

\section{Orbifolds as global quotients}

\subsection{Families of unitary frames and proper bundles of topological groups}

Let $X$ be a locally compact Hausdorff space, $O\subset X$ an open
subset and $E_{O}$ an $n$-dimensional hermitian vector bundle over
$O$. Denote by $E_{O}^{X}$ the trivial extension of the hermitian
vector bundle $E_{O}$ to a continuous family of finite dimensional
Hilbert spaces over $X$ as in Example \ref{Ex_Families of Hilbert
spaces}. To the continuous family $E_{O}^{X}$ of Hilbert spaces
over $X$ one can assign a family $\mathrm{UFr}(E_{O}^{X})$ of unitary frames
over $X$ as follows.

We first recall the definition of the principal $U(n)$-bundle of
unitary frames $\mathrm{UFr}(E)$ of a hermitian vector bundle $E$ over $B$.
A unitary frame at a point $x\in B$ is an ordered orthonormal base
of the Hilbert space $E_{x}$. We can represent it as a unitary
isomorphism $e_{x}:\mathbb{C}^{n}\rightarrow E_{x}$, where $\mathbb{C}^{n}$ is equipped
with the standard scalar product. The set $\mathrm{UFr}(E)_{x}$ of all
frames of the bundle $E$ at $x$ is equipped with a natural right
action of the Lie group $U(n)$: a group element $A\in U(n)$ acts
on the frame $e_{x}\in\mathrm{UFr}(E)_{x}$ by $e_{x}\cdot A=e_{x}\circ A$ to
give a new frame at $x$. The bundle $\mathrm{UFr}(E)$ of unitary frames of
$E$ is the disjoint union of all the spaces $\mathrm{UFr}(E)_{x}$ with the
natural projection map $\pi:\mathrm{UFr}(E)\rightarrow B$, sending each of the sets
$\mathrm{UFr}(E)_{x}$ to their respective $x\in B$. A unitary local
trivialisation $\phi_{i}:E|_{U_{i}}\rightarrow U_{i}\times\mathbb{C}^{n}$ of the
hermitian vector bundle $E$ induces a local trivialisation
$\psi_{i}:\pi^{-1}(U_{i})\rightarrow U_{i}\times U(n)$ of the bundle
$\mathrm{UFr}(E)$, given by $\psi_{i}(e)=(\pi(e),\phi_{i,\pi(e)}\circ e)$,
where $\phi_{i,\pi(e)}$ is the unitary isomorphism from
$E_{\pi(e)}$ to $\mathbb{C}^{n}$. The topology on the space $\mathrm{UFr}(E)$ is
the finest topology which makes all of the maps $\psi_{i}^{-1}$
continuous.

The definition of the principal $U(n)$-bundle $\mathrm{UFr}(E_{O})$ of
unitary frames of the hermitian vector bundle $E_{O}$ can be
extended to define the family $\mathrm{UFr}(E_{O}^{X})$ of unitary frames of
the trivial extension $E_{O}^{X}$ of the bundle $E_{O}$. As a set
$\mathrm{UFr}(E_{O}^{X})$ is defined to be the disjoint union
\[
\mathrm{UFr}(E_{O}^{X})=\mathrm{UFr}(E_{O})\coprod (X\backslash O).
\]
Let $\pi=\pi_{E_{O}}\coprod id|_{X\backslash O}:\mathrm{UFr}(E_{O}^{X})\rightarrow
X$ denote the projection from the space $\mathrm{UFr}(E_{O}^{X})$ onto $X$,
where $\pi_{E_{O}}:\mathrm{UFr}(E_{O})\rightarrow O$ is the ordinary projection from
the bundle of the unitary frames of $E_{O}$ onto $O$. We will
define the topology on the space $\mathrm{UFr}(E_{O}^{X})$ by specifying its
basis $\mathcal{B}$. The basic open sets of the space $\mathrm{UFr}(E_{O}^{X})$ are
of two kinds:
\begin{enumerate}
\item For each open subset $O'$ of $X$ we have
$\pi^{-1}(O')\in\mathcal{B}$; \item If $O'\subset\mathrm{UFr}(E_{O})$ is an open
subset then $O'\in\mathcal{B}$.
\end{enumerate}
Equipped with the topology defined by the basis $\mathcal{B}$ the family
of frames $\mathrm{UFr}(E_{O}^{X})$ becomes a locally compact Hausdorff
space such that the map $\pi$ is a continuous open surjection.

Now let $G$ be an orbifold groupoid over $G_{0}$ and let $O$ be a
$G$-invariant open subset of $G_{0}$. To every $n$-dimensional
hermitian vector bundle $E_{O}$ over $O$ we associate the proper
bundle $U_{O}(n)$ of topological groups over the space $Q=G_{0}/G$
in the following way. The fiber of the bundle $U_{O}(n)$ at $x\in
Q$ is given by
\begin{displaymath}
(U_{O}(n))_{x}=\left\{ \begin{array} {ll}
U(n), & \textrm{$x\in q(O)$,}\\
\{0\}, & \textrm{otherwise,}\\
\end{array}\right.
\end{displaymath}
where $q:G_{0}\rightarrow Q$ is the quotient map. The topology on the
space $U_{O}(n)$ is the quotient topology from the space $Q\times
U(n)$, where the $U(n)$-fibers are fibrewise shrunk to a point for
the points outside of $q(O)$. The map $r_{O}:U_{O}(n)\rightarrow Q$ is a
proper continuous map from which it follows that $U_{O}(n)$ is a
proper bundle of topological groups.

We have a natural right action of the proper bundle of groups
$U_{O}(n)$ on the family of unitary frames
$\pi:\mathrm{UFr}(E_{O}^{G_{0}})\rightarrow G_{0}$ along the map
$q\circ\pi:\mathrm{UFr}(E_{O}^{G_{0}})\rightarrow Q$. It is explicitely given by the
formula $e_{x}\cdot A_{q(x)}=e_{x}\circ A_{q(x)}$, for $e_{x}\in
\pi^{-1}(O)$ and where $A_{q(x)}\in U(n)$ is seen as a unitary
isomorphism $A_{q(x)}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$. For $x$ outside of $O$
the action is defined in the only possible way. Note that the
proper bundle of topological groups $U_{O}(n)$ acts freely and
transitively along the fibers of the map $\pi$.

\subsection{Presenting orbifolds as translation groupoids}

Let $M$ be a smooth manifold and $K$ a compact Lie group acting
smoothly and almost freely on the manifold $M$ from the right. The
translation groupoid $M\rtimes K$ is then Morita equivalent to an
orbifold groupoid. The following proposition shows that the same
is true if we replace the compact group $K$ with some proper
bundle of Lie groups $U$.

\begin{prop}\label{Prop_Translation groupoids of proper bundles of groups}
Let $M$ be a smooth manifold and let $U$ be a proper bundle of Lie
groups over $N$, acting smoothly and almost freely from the right
on the space $M$ along the smooth map $\phi:M\rightarrow N$. Then the
translation groupoid $M\rtimes U$ is Morita equivalent to an
orbifold groupoid.
\end{prop}

\begin{proof}
The idea of the proof is similar to the case of an almost free
action of a compact Lie group. A proper bundle of Lie groups $U$
over $N$ is a bundle of topological groups over $N$ with a
structure of a Lie groupoid. By definition the action of $U$ on
$M$ is almost free if and only if the isotropy groups of the
groupoid $M\rtimes U$ are finite and thus discrete. The groupoid
$M\rtimes U$ is a proper Lie groupoid as a translation groupoid of
a proper Lie groupoid. By Proposition $5.20$ in \cite{MoMr} the
groupoid $M\rtimes U$ is Morita equivalent to an \'etale Lie
groupoid $G$. Since properness is invariant under Morita
equivalence the groupoid $G$ is proper and \'etale, thus an
orbifold groupoid.
\end{proof}

As proved in Theorem \ref{Representation of orbifold groupoids}
each orbifold groupoid $G$ admits a faithfull unitary
representation on a continuous family of finite dimensional
Hilbert spaces over $G_{0}$. We can use Theorem
\ref{Representation of orbifold groupoids} to prove the following
partial converse of Proposition \ref{Prop_Translation groupoids of
proper bundles of groups}.

\begin{theo} \label{Orbifold groupoids Morita equivalence}
Let $G$ be an orbifold groupoid. Then $G$ is Morita equivalent to
a translation groupoid associated to a continuous almost free
action of a proper bundle of topological groups on a topological
space. The bundle can be chosen such that the fibers are finite
products of unitary groups.
\end{theo}

We will prove Theorem \ref{Orbifold groupoids Morita equivalence}
by constructing a space $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$ over $G_{0}$,
equipped with a free left action of the orbifold groupoid $G$
along the map $\pi$ and with a right action of a proper bundle of
topological groups that acts freely and transitively along the
fibers of the map $\pi$.

\begin{prop}\label{Prop_Unitary rep to continuous action}
Let $G$ be an orbifold groupoid over $G_{0}$, $O$ a $G$-invariant
open subset of $G_{0}$ and let $E_{O}^{G_{0}}$ denote the trivial
extension of the hermitian vector bundle $E_{O}$ over $O$ to a
continuous family of finite dimensional Hilbert spaces over
$G_{0}$. Every continuous unitary representation of the groupoid
$G$ on the family $E_{O}^{G_{0}}$ induces a continuous action of
the groupoid $G$ on the family of frames $\mathrm{UFr}(E_{O}^{G_{0}})$.
\end{prop}

\begin{proof}
Define the action of the groupoid $G$ on the space
$\mathrm{UFr}(E_{O}^{G_{0}})$ as follows:
\begin{enumerate}
\item For $e_{x}\in\mathrm{UFr}(E_{O})_{x}$ and $g\in G(x,y)$ define $g\cdot
e_{x}=g\circ e_{x}$, where $g$ on the right is interpreted as a
unitary map from $E_{x}$ to $E_{y}$, coming from the
representation of the groupoid $G|_{O}$ on the bundle $E_{O}$.

\item For $x\in G_{0}\backslash O$ and $g\in G(x,y)$ define
$g\cdot x=y$.
\end{enumerate}
To show that this defines a continuous action
$\mu:G\times_{G_{0}}\mathrm{UFr}(E_{O}^{G_{0}})\rightarrow\mathrm{UFr}(E_{O}^{G_{0}})$ we use
similar techniques as in the proof of Proposition \ref{Extension}.
First decompose the space $\mathrm{UFr}(E_{O}^{G_{0}})$ as a disjoint union
of the subspaces $G_{0}\backslash\overline{O}$, $\partial O$ and
$\mathrm{UFr}(E_{O})$.

The sets $G_{0}\backslash\overline{O}$ and $\mathrm{UFr}(E_{O})$ are basic
open subsets of the space $\mathrm{UFr}(E_{O}^{G_{0}})$. First note that the
restriction of the map $\mu$ to the open set
$G\times_{G_{0}}(G_{0}\backslash\overline{O})$ is equal to the
natural left action of the groupoid $G$ on
$G_{0}\backslash\overline{O}$ and thus continuous. Choose now any
element $(g,e)\in G\times_{G_{0}}\mathrm{UFr}(E_{O})$, where $g\in G|_{O}$
is an arrow from $x$ to $y$, and unitary local trivializations of
the vector bundle $E_{O}$ around $x$ respectively $y$. The unitary
representation of the groupoid $G$ on $E_{O}$ induces a continuous
map $m_{g}$ from a small neighbourhood of the arrow $g$ into the
group $U(n)$, with respect to these two local trivializations. In
the associated principal bundle charts the action of $G$ on
$\mathrm{UFr}(E_{O})$ then looks like multiplication by the map $m_{g}$ and
is hence continuous.

It remains to be proven that $\mu$ is continuous at the points of
the form $(g,x)\in G\times_{G_{0}}\mathrm{UFr}(E_{O}^{G_{0}})$ where
$x\in\partial O\subset \mathrm{UFr}(E_{O}^{G_{0}})$ and $g\in G(x,y)$. We
then have $g\cdot x=y$ and $y\in\partial O$ as well. Let $W$ be
any neighbourhood of the point $y\in\partial O
\subset\mathrm{UFr}(E_{O}^{G_{0}})$. By the definition of the topology on
the space $\mathrm{UFr}(E_{O}^{G_{0}})$ there exists a neighbourhood $V$ of
the point $y\in G_{0}$ such that $\pi^{-1}(V)\subset W$, where
$\pi:\mathrm{UFr}(E_{O}^{G_{0}})\rightarrow G_{0}$ is the projection map onto
$G_{0}$. Choose a bisection $V_{g}$ of the arrow $g\in G$ such
that $t(V_{g})\subset V$. The set
$V_{g}\times_{G_{0}}\mathrm{UFr}(E_{O}^{G_{0}})$ is then an open
neighbourhood of the point $(g,x)$ such that
$\mu(V_{g}\times_{G_{0}}\mathrm{UFr}(E_{O}^{G_{0}}))\subset\pi^{-1}(V)\subset
W$.
\end{proof}

Now choose an orbifold groupoid $G$ over $G_{0}$ and let $E$ be a
continuous family of finite dimensional Hilbert spaces over
$G_{0}$ together with a faithfull unitary representation of the
groupoid $G$ as constructed in the proof of Theorem
\ref{Representation of orbifold groupoids}. Here we use the same
notations. The family $E$ can be decomposed as a direct sum
$E=\bigoplus_{i\in\mathbb{N}}E^{i}$, where each $E^{i}=E_{V_{i}}^{G_{0}}$
is a trivial extension of the hermitian vector bundle $E_{V_{i}}$
over $V_{i}$. We define the family of unitary frames $\mathrm{UFr}(E)$ of
the family $E$, with respect to the decomposition
$E=\bigoplus_{i\in\mathbb{N}}E^{i}$, to be the fibrewise product of the
families $\{\mathrm{UFr}(E^{i})\}_{i\in\mathbb{N}}$ along the projection maps
$\pi_{i}$,
\[
\mathrm{UFr}(E)=\{(e_{1},e_{2},\ldots)\in\prod_{i\in\mathbb{N}}\mathrm{UFr}(E^{i})|\pi_{1}(e_{1})=\pi_{2}(e_{2})
=\ldots\}.
\]
The space $\mathrm{UFr}(E)$ has a natural projection $\pi$ onto the space
$G_{0}$, induced from any of the projections $\pi_{i}$. The fiber
of the space $\mathrm{UFr}(E)$ over a point $x\in G_{0}$ can be canonically
identified with the product of the spaces of frames
$\prod_{i\in\mathbb{N}}\mathrm{UFr}(E^{i})_{x}$. Since the family
$\{V_{i}\}_{i\in\mathbb{N}}$ is a locally finite cover of the space
$G_{0}$ this product is in fact finite and therefore homeomorphic
to a finite product of unitary groups.

\begin{prop}
The faithfull unitary representation of the orbifold groupoid $G$
on the continuous family of Hilbert spaces $E$ over $G_{0}$
induces a continuous free action of the groupoid $G$ on the family
of unitary frames $\mathrm{UFr}(E)$.
\end{prop}

\begin{proof}
The action of the orbifold groupoid $G$ on the space $\mathrm{UFr}(E)$ along
the map $\pi$ can be defined coordinatewise by
$g\cdot(e_{1},e_{2},\ldots)=(g\cdot e_{1},g\cdot e_{2},\ldots)$.
The continuity of this action follows from the fact that all the
actions of the groupoid $G$ on the spaces $\mathrm{UFr}(E^{i})$ are
continuous by Proposition \ref{Prop_Unitary rep to continuous
action} and from the fact that the topology on the space $\mathrm{UFr}(E)$
is the one induced from the product topology on $\prod_{i\in\mathbb{N}}
\mathrm{UFr}(E^{i})$.

Choose an arrow $g\in G$ and an element
$e=(e_{1},e_{2},\ldots)\in\mathrm{UFr}(E)_{x}$ such that $g\cdot e=e$. Then
$g$ must be an isotropy element, $g\in G_{x}$, and there exists
some $i\in\mathbb{N}$ such that $x\in V_{i}$. From the definition of the
bundle $E^{i}$ it follows that $G_{x}$ acts faithfully on
$E^{i}_{x}$ and therefore $g\cdot e_{i}=e_{i}$ implies $g=1_{x}$.
This shows that the action of the groupoid $G$ on the space
$\mathrm{UFr}(E)$ is free.
\end{proof}

Denote by $U_{i}=U_{V_{i}}(n_{i})$ the proper bundle of
topological groups over the space $Q=G_{0}/G$, associated to the
hermitian vector bundle $E_{V_{i}}$ over the $G$-invariant open
subset $V_{i}$ of $G_{0}$. The fibrewise product $U$ of the
bundles $U_{i}$ has a natural structure of a proper bundle of
topological groups over $Q$, with each fiber being isomorphic to a
finite product of unitary groups. The right actions of the bundles
$U_{i}$ on the families of unitary frames $\mathrm{UFr}(E^{i})$ induce a
right action of the bundle $U$ on the space $\mathrm{UFr}(E)$, defined by
the formula
$(e_{1},e_{2},\ldots)\cdot(A_{1},A_{2},\ldots)=(e_{1}\cdot
A_{1},e_{2}\cdot A_{2},\ldots)$. It is not hard to see that the
proper bundle of groups $U$ acts freely and transitively along the
fibers of the map $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$.

\begin{proof}[Proof of Theorem \ref{Orbifold groupoids Morita equivalence}]
For the convenience of the reader we first recall the data we have
so far. Let $G$ be an orbifold groupoid over $G_{0}$ and let
$Q=G_{0}/G$ be the space of orbits of the groupoid $G$. We have
constructed the space of frames $\mathrm{UFr}(E)$, together with the moment
maps $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$ (the projection map) and the map
$u=q\circ\pi:\mathrm{UFr}(E)\rightarrow Q$, where $q:G_{0}\rightarrow Q$ is the quotient
projection. There are actions of the groupoid $G$ and of the
proper bundle of groups $r:U\rightarrow Q$ on the space $\mathrm{UFr}(E)$ from the
left along the map $\pi$ respectively from the right along the map
$u$. Both of these actions are free and moreover $U$ acts on the
space $\mathrm{UFr}(E)$ transitively along the fibers of the map $\pi$.

Now observe that both the actions are basically compositions of
linear maps from the left respectively from the right. The
associativity of the composition implies that the actions of $G$
and $U$ on the space $\mathrm{UFr}(E)$ commute. Combining this with the fact
that the map $u:\mathrm{UFr}(E)\rightarrow Q$ is $G$-invariant, as shown by the
equalities
\[
u(g\cdot e)=q(\pi(g\cdot e))=q(t(g))=q(s(g))=q(\pi(e))=u(e),
\]
we can define a right action of the proper bundle of groups $U$ on
the quotient space $G\backslash\mathrm{UFr}(E)$, along the induced map
$u':G\backslash\mathrm{UFr}(E)\rightarrow Q$, by the formula $[e]\cdot A=[e\cdot A]$
for $[e]\in G\backslash\mathrm{UFr}(E)$ and $u'([e])=r(A)$. This action is
almost free since the action of $U$ on $U(E)$ was free and since
$G$ has finite isotropy groups.

Let $H$ be the (proper) translation groupoid associated to this
action (see Example \ref{Ex_Groupoids}). It has the quotient
$H_{0}=G\backslash\mathrm{UFr}(E)$ as the space of objects and the space of
arrows equal to $(G\backslash\mathrm{UFr}(E))\times_{Q}U$. Note that $H_{0}$
is a Hausdorff space since $\mathrm{UFr}(E)$ is Hausdorff and $G$ is a
proper groupoid. The multiplication in the groupoid $H$ is defined
by the formula $([e],A)([e'],A')=([e],AA')$ for $[e\cdot A]=[e']$
and $u'([e])=u'([e'])=r(A)=r(A')$. The source and the target maps
of the groupoid $H$ are given by $s([e],A)=[e\cdot A]$
respectively $t([e],A)=[e]$. We have a natural action of the
translation groupoid $H$ on the space $\mathrm{UFr}(E)$, induced from the
action of the proper bundle of groups $U$ on $\mathrm{UFr}(E)$ and defined
by $e\cdot([e],A)=e\cdot A$.

We will show that the space of frames $\mathrm{UFr}(E)$, together with the
moment maps $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$ and $\phi:\mathrm{UFr}(E)\rightarrow
H_{0}=G\backslash\mathrm{UFr}(E)$ (the quotient projection), and the actions
of groupoids $G$ respectively $H$, represents a Morita equivalence
between the orbifold groupoid $G$ and the translation groupoid
$H$.

The translation groupoid $H$ acts along the fibers of the map
$\pi$ because the bundle of groups $U$ does so, while the groupoid
$G$ acts along the fibers of the map $\phi$ by the definition of
$\phi$. Similarly, it is not hard to see that both actions
commute, so it remains to be proven that $\phi:\mathrm{UFr}(E)\rightarrow H_{0}$ is
a principal left $G$-bundle and that $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$ is a
principal right $H$-bundle. Both the maps $\phi$ and $\pi$ are
open, the first being the quotient map of a groupoid action and
the second one being open as a projection map of a fibrewise
product along a family of open maps. Since the action of the
groupoid $G$ on the space $\mathrm{UFr}(E)$ is free and transitive along the
fibers of the map $\phi$ the map
$i_{G}:G\times_{G_{0}}\mathrm{UFr}(E)\rightarrow\mathrm{UFr}(E)\times_{H_{0}}\mathrm{UFr}(E)$, given by
$i_{G}(g,e)=(g\cdot e,e)$, is a continuous bijection. Furthermore,
since the groupoid $G$ is proper, the action of $G$ on $\mathrm{UFr}(E)$ is
proper so $i_{G}$ is a closed map and hence a homeomorphism. This
proves that $\phi:\mathrm{UFr}(E)\rightarrow H_{0}$ is a principal left $G$-bundle.
Similarly, the map
$i_{H}:\mathrm{UFr}(E)\times_{H_{0}}H\rightarrow\mathrm{UFr}(E)\times_{G_{0}}\mathrm{UFr}(E)$, defined
by $i_{H}((e,([e],A))=(e,e\cdot A)$, defines a homeomorphism which
shows that $\pi:\mathrm{UFr}(E)\rightarrow G_{0}$ is a principal right $H$-bundle.
\end{proof}

\noindent {\bf Acknowledgements.} I would like to thank A.
Henriques, D. McDuff and J. Mr\v{c}un for discussions and advices
related to the paper.

redpajama_set_name: RedPajamaArXiv


text:
Home | Four Buffalo State Alumni Recognized at 149th Commencement
Four Buffalo State Alumni Recognized at 149th Commencement
Four of Buffalo State College's most accomplished alumni will be recognized during the college's virtual 149th Commencement ceremonies on Saturday, May 15. The Distinguished Alumnus Award will be given to Linda R. Appleby, '80, and Michael J. Garner, '83, and the Young Alumnus Achievement Award will go to Raven P. Baxter, '14, '17, and Nathan Mroz, '16. The ceremonies will be conducted virtually, starting at noon.
The Distinguished Alumnus Award recognizes graduates who have significant achievement and recognition in their field; dedication to community service, civic affairs, and volunteerism; and demonstrated ongoing support for and loyalty to Buffalo State College and its mission. The Young Alumnus Achievement Award recognizes graduates of the past 20 years who have achieved significant professional advancement and have demonstrated service to the community, as well as loyalty and commitment to the college.
YOUNG ALUMNUS ACHIEVEMENT AWARD
Raven P. Baxter, Class of 2014 and 2017
Raven "the Science Maven" Baxter is an award-winning, internationally acclaimed science educator and molecular biologist who works to advance science culture by creating spaces for learning that are inclusive, educational, and accessible. As an entertainer and content creator known for her unique style of combining science and music to teach and empower those in the STEM disciplines and beyond, she uses innovation in science education as a catalyst for social change.
Baxter is the founder of Science Haven, an organization that operates at the intersection of science, education, and the public sphere. Science Haven houses STEMbassy, a live web series that connects the public with science and technology professionals. She has reached more than 4 million viewers with over 200,000 followers, sharing best practices in science and education. Through her viral hit rap song "Wipe It Down," she has been instrumental in educating the community about myths and facts related to COVID-19.
As the founder and lead organizer of Black in Science Communication, a group that works to build relationships in the science community, she equips others with the knowledge and resources necessary to share science with the world in their own way. She also owns and operates Smarty Pants, a clothing company that sells whimsical and stylish STEM-themed apparel and accessories, while donating a portion of its sales to scholarships for STEM students.
Baxter has quickly developed a reputation as a strong voice in the science community. She has been recognized as a global influencer in several publications, including Fortune magazine's 40 Under 40 list for 2020. A recipient of the prestigious Arthur A. Schomburg Fellowship, she will complete her doctorate in science education at the University at Buffalo in May 2021, receiving the SUNY Chancellor's Award for Student Excellence in honor of her public service.
Nathan E. Mroz, Class of 2016
Nathan Mroz, owner and founder of the BFLO Store and BFLO Event Center, graduated from Buffalo State College with a bachelor of science in urban and regional planning. Already a successful entrepreneur in high school, he launched a photography business, Buffalo Scenic Prints, in 2007, selling his work online and to local gift shops. Upon graduation in 2012, he opened a stand-alone kiosk in the Eastern Hills Mall, where he sold his award-winning photos of the Buffalo Niagara region. By 2015, his business had expanded to a brick-and-mortar enterprise, the BFLO Store at Eastern Hills Mall.
Since then, the BFLO Store has grown to become the largest locally themed retailer in Western New York, specializing in designer apparel, home décor, gifts, and more. By 2020, Mroz had stores in the Eastern Hills Mall, the McKinley Mall, the Boulevard Mall, Walden Galleria, and Canalside Buffalo. In 2019, he fulfilled a lifelong dream when he signed with Kim and Terry Pegula, principal owners of the Buffalo Bills, to become the team's official retailer. Mroz is committed to Western New York and to promoting the beauty and uniqueness of Buffalo Niagara through his photography, artistry, and vision.
At Buffalo State, Mroz published his senior thesis on the redevelopment of the Eastern Hills Mall, and in 2019, took control of the site's former Sears department store. He then embarked on a multimillion-dollar redevelopment project of the 154,000-square-foot building, creating the BFLO District, a "lifestyle center" housing his flagship BFLO Store and BFLO Event Center, a high-end salon, décor retailer, bakery café, bar, restaurant, and soon-to-be state-of-the-art fitness center.
Mroz was awarded the Community Beautification Excellence Award from the Clarence Chamber of Commerce in 2019, and was named the chamber's Young Entrepreneur of the Year in 2020. He also received the Key Bank Sponsor's Award from the Amherst Chamber of Commerce in 2019 and was included on the Business First 30 Under 30 list in 2021. He is a member of the Clarence Chamber of Commerce board of directors.
Linda R. Appleby, Class of 1980
Linda Appleby has taught music and has served as the musical director for countless theatrical shows, both professionally and for school districts throughout the area, over her 40-plus-year career. She has taught at numerous schools, including Buffalo's Frederick Law Olmsted 56, City Honors, and South Park High School. Students under her direction have enjoyed a wide and varied list of accomplishments, including performances at Kleinhans Music Hall, Rockwell Hall, and Shea's Theater; in three professional Broadway shows; and for Buffalo Mayor Anthony Masiello's inauguration and heads of state in Albany.
Appleby has been an adjunct instructor of music at Villa Maria College since 2011 and was the director of the citywide Buffalo Select Chorus for six years, which brought together over 30 students representing various high schools throughout the city. The chorus was accompanied by some of Buffalo's finest musicians, including Joseph Wooten of the Steve Miller Band. She also curated the Artist in Residence Program that serviced over 40 public and charter schools in Buffalo from 2009 to 2014.
Before the pandemic, she worked as music director for the Buffalo Academy for Visual and Performing Arts production of Les Misérables, which won a Kenny Award for best musical in Western New York. She has served as a minister of music for area churches and is currently music minister for SS Columba-Brigid Church in Buffalo. She serves on the board of directors for Ujima Theater and as secretary for the historic Colored Musicians Club.
Appleby received a bachelor of arts in music education from Rosary Hill College in 1974, with a concentration in classical piano performance, and a master's degree in student personnel administration from Buffalo State College in 1980. She also accumulated over 30 hours beyond her master's in various courses offered throughout the city, with a concentration in special education, advanced teaching practices, and psychology.
In 2008, Buffalo Mayor Byron Brown presented Appleby with an Executive Chamber Proclamation for services rendered to the children of Buffalo. She is also the 2009 recipient of the National Federation of Just Communities Community Leader Award in the Arts. She was inducted into the Buffalo Music Hall of Fame in 2015.
Michael J. Garner, Class of 1983
Michael Garner, chief diversity and inclusion officer for the New York State Metropolitan Transportation Authority, is a champion of the diversity market, making it his personal and professional mission to create a more diverse business industry by empowering minority-owned and women-owned businesses throughout the state.
Since joining the MTA in 2009, Garner has designed the country's leading business development program for minority, women-owned, and disadvantaged business enterprises (MWDBE) based on dollars paid. Overall, through the MTA's programs, New York State certified MWDBEs have been awarded more than $6.5 billion.
As the architect of the MTA's Small Business Mentoring Program, Garner has transformed the largest transportation system in North America with a results-driven construction mentoring program. The program continues to break barriers for NYS certified MWDBEs that otherwise would not have contract opportunities in the transportation industry. To date, under Mr. Garner's direction, the MTA's Small Business Mentoring Program has awarded 451 contracts totaling $486 million to program participants.
Before joining the MTA, Garner held positions at the New York City Housing Authority and the New York City School Construction Authority. Over the course of his tenures there and at the MTA, Garner and his teams have been responsible for more than $13 billion in contracts and payments made to New York State and City certified MWDBEs and MTA certified DBEs.
He and his team have designed and implementing workforce strategies that have increased ethnic minority and women hires by 20 percent, which has further diversified the MTA's 70,000-plus employee workforce.
Garner is an active member of the Canaan Baptist Church in Harlem. He is the immediate past president and former chairman of the board of the founding chapter of One Hundred Black Men of New York City, an organization that provides scholarship, mentorship, and inspiration to hundreds of youths in the New York City area.
A frequent speaker on effective MWDBE programs across the nation, he has served on Governor Andrew Cuomo's MWBE Team, Mayor Bill DeBlasio's MWBE Advisory Committee, NYC Comptroller Scott Stringer's MWBE Advisory Council, and former Governor David Paterson's Small Business Advisory Council.
Garner often serves as an adviser to public agencies and the private sector. He is a former treasurer of the New York/New Jersey Minority Supplier Development Council and has served on the boards of Harlem Hospital and USA Rugby. In addition, Garner advises numerous U.S. cities and states on their minority business development and inclusion programs, including the City of Buffalo, the City of Chicago, the Chicago Transit Authority (CTA), and the State of New Jersey.

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text:
Bay Area Women Artists
Presented by O'Hanlon Center for the Arts and First Tuesday ArtWalk at O'Hanlon Center for the Arts, Mill Valley CA
Aug 01 - 22 2019
An annual exhibit at O'Hanlon since 2009, celebrating local area women artists!
First Tuesday ArtWalk Opening Reception:
* Aug 6, 5:30-7:30 pm
Artist Roundtable Dialogue:
* Aug 6, 4:00 pm
Exhibition dates: Aug 1 – 22
[Postcard Art: Karen Meadows| Design: Jeremy Thornton]
The emphasis for this exhibition is on exploration, process, and imaginative use of materials and content.
Open to artists who identify as female, residing in greater SF Bay Area, ages 16 & over.
Jurors: Donna Seager & Suzanne Gray, Seager Gray Gallery
A native of New Orleans, Donna Seager began in the art business in 1978. She was the director of Marlborough Gallery in Boston before moving to California in 1989. She continued to work in galleries in the bay area until 2005 when she opened her own gallery. Seager has distinguished herself for her discerning eye and commitment to the arts in her community. She has given lectures on the art of the book at international fairs in Miami and San Francisco. In six years, the gallery became a destination for artists and collectors all over the bay area.
Suzanne Gray, a resident of Mill Valley grew up in upstate New York and studied journalism and art history at Syracuse University and San Jose State. Her lifelong interest and background in art and art history landed her a position at Donna Seager Gallery where she became an integral part of the business and ultimately a valued partner. Gray's skills extend themselves to professional art consultant services and she has overseen both commercial and residential art programs. Her knowledge of the art market, sophisticated eye and understanding of art placement have gained her the trust and respect of her clients.
Entry Dates (online-only):
* June 18 – July 10 (online form closes 11:59 pm PT July 10)
Contemporary, expressionistic, abstract, exploratory, and/or experimental entries encouraged.
Email: info@ohanloncenter.org
First Tuesday ArtWalk Opening Reception: Aug 6, 5:30-7:30 pm
Artist Roundtable Dialogue: Aug 6, 4:00 pm
Tuesday to Friday: 11:00 am – 3:00 pm
O'Hanlon Center for the Arts
616 Throckmorton Ave., Mill Valley, CA 94941
Street and parking lot parking
Please phone in advance regarding accessibility for people with disabilities, and we will make every effort to accommodate you.
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TlsInteraction - Interaction with the user during TLS operations.
TlsInteraction provides a mechanism for the TLS connection and database code to interact with the user. It can be used to ask the user for passwords.
To use a TlsInteraction with a TLS connection use Gio::TlsConnection::set_interaction().
Callers should use the 'invoke' functions like invoke_ask_password() to run interaction methods. These functions make sure that the interaction is invoked in the main loop and not in the current thread, if the current thread is not running the main loop.
Derived classes can choose to implement whichever interactions methods they'd like to support by overriding those virtual methods. Any interactions not implemented will return Gio::TLS_INTERACTION_UNHANDLED. If a derived class implements an async method, it must also implement the corresponding finish method.
Run synchronous interaction to ask the user for a password.
In general, g_tls_interaction_invoke_ask_password() should be used instead of this function.
If the interaction is cancelled by the cancellation object, or by the user then Gio::TlsInteractionResult::FAILED will be returned with an error that contains a Gio::Error::CANCELLED error code. Certain implementations may not support immediate cancellation.
cancellable An optional Cancellable cancellation object.
Run asynchronous interaction to ask the user for a password.
Derived subclasses usually implement a password prompt, although they may also choose to provide a password from elsewhere. The password value will be filled in and then slot will be called. Alternatively the user may abort this password request, which will usually abort the TLS connection.
slot Will be called when the interaction completes.
Complete an ask password user interaction request.
This should be once the g_tls_interaction_ask_password_async() completion callback is called.
If Gio::TlsInteractionResult::HANDLED is returned, then the TlsPassword passed to g_tls_interaction_ask_password() will have its password filled in.
If the interaction is cancelled by the cancellation object, or by the user then Gio::TlsInteractionResult::FAILED will be returned with an error that contains a Gio::Error::CANCELLED error code.
result The result passed to the callback.
Invoke the interaction to ask the user for a password.
It invokes this interaction in the main loop, specifically the MainContext returned by Glib::main_context_get_thread_default() when the interaction is created. This is called by called by TlsConnection or TlsDatabase to ask the user for a password.
Invoke the interaction to ask the user to choose a certificate to use with the connection.
It invokes this interaction in the main loop, specifically the MainContext returned by Glib::main_context_get_thread_default() when the interaction is created. This is called by called by TlsConnection when the peer requests a certificate during the handshake.
flags Flags providing more information about the request.
Run synchronous interaction to ask the user to choose a certificate to use with the connection.
In general, g_tls_interaction_invoke_request_certificate() should be used instead of this function.
If Gio::TlsInteractionResult::HANDLED is returned, then the TlsConnection passed to g_tls_interaction_request_certificate() will have had its TlsConnection::property_certificate() filled in.
Run asynchronous interaction to ask the user for a certificate to use with the connection.
Derived subclasses usually implement a certificate selector, although they may also choose to provide a certificate from elsewhere. slot will be called when the operation completes. Alternatively the user may abort this certificate request, which will usually abort the TLS connection.
Complete an request certificate user interaction request.
This should be once the g_tls_interaction_request_certificate_async() completion callback is called.
If Gio::TlsInteractionResult::HANDLED is returned, then the TlsConnection passed to g_tls_interaction_request_certificate_async() will have had its TlsConnection::property_certificate() filled in.


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Cold-chain technology company ZedBlox raises ₹2.25 Crores from angel investors
Cold chain tech start-up ZedBlox raises Rs. 2.25 Crores from angel investors. The fund will be used to scale up the company's operations to resolve the issues regarding the storage and transportation of sensitive healthcare products.
Cold-chain tech startup ZedBlox raises ₹2.25 Crores from angel investors
ZedBlox, a cutting-edge cold-chain technology company, has raised 2.25 Crores in the latest pre-Series A funding from FutureThoughts, Sanpra Investment, and other angel investors.
Earlier they had received seed investment from Atal Incubation Centre (AIC) - Centre for Cellular and Molecular Biology (CCMB), under the Nidhi Seed Support Scheme.
Most of pharmaceutical products are temperature sensitive. The vaccines, medicines, and samples become ineffective if the optimum temperature recommended by the manufacturer is NOT maintained. This can increase overheads, care delays, and operational inefficiencies. ActiPod, a patented product developed by ZedBlox is the perfect solution for these issues. Actipod is a smart portable battery-powered carrier that requires no cold packs or freezing infrastructure and can accurately maintain the desired temperature during transit. ActiPod is NABL-certified and has received ISO 13485 medical device certification.
ZedBlox will use this new funding to scale up the company's operations and expansion within India to address the issues regarding the storage and transportation of sensitive healthcare products. The Hyderabad-based company has been recognized by SiliconIndia magazine as among the " Top 10 Refrigeration Transport Service Companies".
ZedBlox is co-founded by Rao Korupolu, a Computer Science Master's Graduate from the University of Arizona, and has three patents to his name. Co-founder and CTO Viraaj Somayajula is an Electronics and Communication Engineer from IIT Roorkee. The innovative technology of ZedBlox has been incubated in CCMB-AIC, a premier research institute of Govt. of India, under the expert guidance of Dr. N Madhusudana Rao and Dr. Ramjee Pallela and received its patent in July 2022.
Tags : #fundingalert #coldchain #startup #zedblox #refrigerationtransportservice #healthcare #vaccine #medicine #actipod #medicircle #smitakumar
Rajeev Pillai
Rajeev Pillai has accumulated extensive experience in the Electronics and Telecom Service Industry with global companies in the Middle East. Being a continuous learner with a lifelong passion for writing, Rajeev writes on topics such as Healthcare, Finance, Travel and Technology.


These recipes will have your whole residence floor drooling outside your room. Photo: CC, Matt Mac Gillivray/Flickr.
At first, residence seems like a dream. You're living away from home for the first time and just a short walk away from your bed is the cafeteria—an endless supply of dozens of kinds of different foods for you to enjoy. But for some, the cafeteria can quickly lose its charm. Sooner or later, you may even start yearning for your kitchen again to prepare meals of your own.
But in a residence room, your culinary abilities are admittedly limited. There's no oven or stove for you to use, let alone decent counter space for food preparation. Often times, you'll be forced to turn to the basics for all of your stomach's needs. But cooking in a residence room can mean so much more than a soggy toaster strudel or a lukewarm bowl of ramen noodles.
As I browsed the festival, the Fadi's Fabulous Foods menu caught my eye, and when I noticed the General Tao Poutine I finally understood what Hannah Montana meant when she sang about the "best of both worlds". This dish had the perfect balance of two of my favourite foods—poutine, a Canadian staple, and General Tao chicken, my favourite kind of Chinese food.


Abraham, as a prophet and patriarch, is called "the friend of God" in Jewish, Christian, and Islamic scriptures, and has a place of eminence in the shared scriptures and histories of each of these three monotheistic, prophetic, and Middle Eastern religions.
In Abraham: The Friend of God, Dr. Dirks provides Jewish, Christian, and Muslim readers with a unique and original presentation of the life of Abraham.
In constructing a chronological biography of Abraham, the author integrated and synthesized information from a wide range of Judaeo-Christian and Islamic traditions, including the Bible, the Pseudeigrapha (I.e. Jubilees and the Genesis Apocryphon), the Qur'an, the authenticated sayings (Sahih Ahadith) of Prophet Muhammad, and the classic books of ancient history authored by Josephus and Al-Tabari.
This integrated biographical sketch is embedded in a framework drawn from the various historical and geographical contexts dealing with different aspects of Abraham's life.


Documents including US legislation relating to AGOA are housed within this section.
Presidential proclamation to take certain actions under AGOA and for other purposes.
Federal Register Announcement - Apparel Quotas 2015-2016, as published on 25 September 2015.
Summary and key points of the AGOA Extension Act 2015.
Limitations of duty- and quota-free imports of apparel articles assembled in beneficiary Sub-Saharan African countries from regional and third-country fabric. Publication of the new 12-month cap on duty-and quota-free benefits.


When it comes to new hope for the American cinema, filmcrit types are always in the market. New hope in 1980 took the form of a low-budget festival film with the misunderstandable title Return of the Secaucus 7. It wasn't a documentary, wasn't a tribute to sullen or snarling radicals, wasn't even a where-were-you-in-'72 American Graffiti–style slice of overpacked nostalgia. What it was was this genial, witty, low-key comedy, with just the right touch of rue, about a group of friends getting together for an informal reunion one summer weekend, and trying to get used to the idea of turning 30—and just a wee bit comfortably bourgeois. The screenplay was a beauty, ostensibly laidback and wide-open, yet carefully detailed without letting the pointedness show; the characters expertly drawn, no fuss, and so cleanly individualized (among other things, everyone's dialogue has a logic and texture all its own) that for the audience and for one another they step right out of any assigned boxes, free to explore a wide range of possibilities. The result was a droll ensemble portrait shot through with the cozy vitality the Sixties used to call natural, without any of the boring unintelligence that so often went along with it.
The film marked the directorial debut of John Sayles, himself age 30 and one of the most solidly talented writers of contemporary American fiction. About the time Secaucus 7 went into national release, Sayles accepted an invitation to meet with a scriptwriting class at the University of Washington and share some of his experiences. Virtually all the Hollywood personnel who graciously and generously gave of their time to support this course delivered themselves of frank and cogent remarks about the realities of the film biz at the dawn of the Eighties; but even in this company Sayles was conspicuous for the comprehensiveness and lucidity of his commentary. He talked for better than two hours, first supplying a general commentary on his background in film and the circumstances of Secaucus 7's making, then opening the floor for questions. Having never heard so much good sense about films and filmmaking collected in one place before, movietone news requested permission to share it with a larger public; the unassuming writer-director seemed surprised that anyone would think so highly of his off-the-cuff remarks, but he agreed. "We'll send you a transcript so you can check it out." He thought about that a moment, then said, "No. If I said it, I'll stand by it. Just go ahead." And that, with very little editing and rearranging, is what we did.
I'd always been interested in doing screenwriting, realized that there weren't too many ways into it. I didn't want to go out to Los Angeles and start knocking on people's doors trying to get an agent, so I went a route that isn't much help to most people, which is that I wrote two novels and got them published. I got a literary agent out of that, and his agency had a deal with a film agency on the West Coat, so they were automatically representing my novels as screen properties. I wrote a query letter to them saying, "I also write screenplays"—which I hadn't done at that time—"do you want to see one?" They said, "Sure, send one," so I wrote one and sent it off to them, and they said, "Sure, we'll represent you." So I moved out to the West Coast.


If you've been in an accident or opened up an insurance claim, you may be quickly contacted by an insurance investigator and asked to provide a statement. The insurance representative may initiate the conversation by asking how you are recovering from your accident. In reality, the question is motivated more by the company's bottom line than compassion. They will then ask you to provide a quick statement on what happened and should inform you the statement will be recorded.
Claims investigators are trained in making the statement seem low-stakes and just a matter of routine. But don't let their caring and casual demeanor fool you. In all actuality, they don't waste time when investigating claims for fear you will have spoken with an attorney, which usually makes their job much harder. If you've been asked to provide a statement about your accident, politely decline. You will most likely be required to provide one if you want to settle your claim out of court. But first, read through these tips and pointers and give yourself some time to speak with a personal injury claim attorney who can advise you of how to protect your best interests.
The statement you provide is not just for record-keeping purposes. Rather, the statement will remain a part of the record of the case and can be later relied on should your case go to court, which could be a year down the road, and most claimants don't think that far ahead.
The investigator will ask specific questions meant to get at the "truth" of what happened. Often, the truth they are looking for is that they are not on the hook to pay your claim. They may ask you trick questions or ask you the same question multiple times in different ways to test for any inconsistency. The investigator will also compare your statement to the accident or police report and the statement from the other party. Any discrepancies can lessen your credibility. You may also be asked about your injuries and how you are doing. The problem is many injuries from accidents have delayed onset. In fact, you might not feel the full weight of your injuries for a few days after the accident. If you report you are feeling OK before you know the extent of your injuries, that single admittance can be used as evidence you aren't severely hurt.
If not carefully crafted, a claimant can say goodbye to any hope of a claim payout because insurance companies use these statements as a way to mine for information they can leverage to deny your claim.
The insurance company could use your statement against you even if you weren't at fault. Don't assume because you aren't liable that the insurance company will come to the same conclusion. All too often claimants say the wrong thing and are shocked to get a letter stating the company is denying the claim. In part, it doesn't matter who was at fault. What matters more is what the evidence points to. And because the insurance investigator was not on the scene of the accident, they have to piece together a picture of what happened. Sometimes that picture is clearer and more accurate than other times.
Now that you know how an insurance investigator can use your statement against you, we hope you can more confidently handle your claim while protecting your rights. It's wise to have an attorney present with you when providing your statement. A California personal injury attorney can coach you on how to respond to curve-ball questions the investigator may ask to throw you off. Don't be fooled by their professional courtesy; it's their job to elicit information from you that will give them a legally valid reason to devalue or deny your claim. We advise you to never provide a statement to an insurance investigator before speaking with a personal injury attorney. But, if you do, immediately seek the advice of an attorney. An attorney can request a copy of the statement, advise you on how it might affect your claim, and help you navigate the claims process with that consideration in mind.
Are you involved in an insurance claim? Fresno area personal injury attorney Christopher Martens can help you protect your interests and rights while settling your claim. Attorney Christopher Martens has the skills and knowledge needed to help ensure you receive the compensation you deserve. Serving the Visalia and Fresno areas, The Law Offices of Christopher Martens can provide knowledgeable personal injury claim guidance. Call our office at 559-967-7386 or email us at MartensLaw@gmail.com for a free consultation.


UKRAINE coach Mykhaylo Fomenko has backed Andriy Yarmolenko to thrive in the Premier League.
Yarmolenko, who netted against Wales in an international friendly on Monday, is being tracked by Arsenal and Everton.
Dynamo Kiev value their attacking star at £25m but his asking-price could rocket if the 26-year-old enjoys a profitable Euro 2016.
Asked about Yarmolenko's potential switch to England, Fomenko said: "If he is wanted by the top English clubs then, yes. He can show his best in the Premier League and he must do this.
The news will come as a boost for Arsenal, who reportedly held discussions with Kiev about their coveted forward in January.
And they may show interest again after claims Arsene Wenger is looking to overhaul his squad ahead of next season.