# The Primitive Ideal Space of Groupoid $C^{*}$-Algebras for Groupoids with
Abelian Isotropy

Daniel W. van Wyk  Department of Mathematics
Dartmouth College
Hanover, NH 03755-3551 USA<EMAIL_ADDRESS>and  Dana P. Williams
Department of Mathematics
Dartmouth College
Hanover, NH 03755-3551 USA<EMAIL_ADDRESS>
(Date: 24 August 2021)

###### Abstract.

We study the topology of the primitive ideal space of groupoid
$C^{*}$-algebras for groupoids with abelian isotropy. Our results include the
known results for action groupoids with abelian stabilizers. Furthermore, we
obtain complete results when the isotropy map is continuous except for jump
discontinuities, and also when $G$ is a unit space fixing extension of a
proper groupoid by an abelian group bundle. We hope that our methods will be a
springboard to further results of this type.

## Introduction

If $(\mathcal{G},X)$ is a second countable locally compact transformation
group, then we can form the action groupoid $G=G(\mathcal{G},X)$. The term
transformation groupoid is also used as in this case the groupoid
$C^{*}$-algebra, $C^{*}(G)$, is naturally isomorphic to the transformation
group $C^{*}$-algebra $C_{0}(X)\rtimes_{\operatorname{lt}}\mathcal{G}$. If all
the stability groups $S_{x}=\\{\,g\in\mathcal{G}:g\cdot x=x\,\\}$ are
contained in a fixed abelian subgroup of $\mathcal{G}$, and if
$(\mathcal{G},X)$ is EH-regular, then it has been known for a long time how
describe both the primitive ideal space, $\operatorname{Prim}C^{*}(G)$, of
$C^{*}(G)$ as well as its Jacobson topology [wil:tams81]*Theorem 5.8. This
immediately suggests the question as to what one can say about
$\operatorname{Prim}C^{*}(G)$ if we merely assume that each stability group is
abelian. Since transformation groups provide fundamental examples of
groupoids, this naturally leads to considering locally compact groupoids with
abelian isotropy groups. At present, a general result of this type is well out
of reach even for action groupoids. This is amply illustrated by
[simwil:art15]*Theorem 3.2 where considerable technology is required just to
describe $\operatorname{Prim}C^{*}(G)$ as a set when $G$ is a second countable
locally compact Deaconu–Renault groupoid.

The purpose of this paper is to suggest an attack on the problem of describing
$\operatorname{Prim}C^{*}(G)$ as a topological space when $G$ is a second
countable locally compact Hausdorff groupoid and all the isotropy groups
$G(u)=\\{\,\gamma\in G:s(\gamma)=u=r(\gamma)\,\\}$ are abelian. We are able to
obtain results that subsume the known results for action groupoids with
abelian stability groups, and to extend these results to the groupoid setting.
We hope that this work will motivate even more successful attacks on the
problem in the future. In an expository vein, we also give an alternate
description of $\operatorname{Prim}C^{*}(G)$ in cases where the answer is
already known. For our approach to work, we need to know that our groupoid $G$
is EH-regular in the sense that every primitive ideal in
$\operatorname{Prim}C^{*}(G)$ is induced from a character $\pi$ of a isotropy
group $G(u)$ with $u\in G^{(0)}$. Fortunately, this is known to be the case if
if every $G$-orbit $[u]:=G\cdot u$ is locally closed in $G^{(0)}$
[wil:toolkit]*Theorem 5.35, or more generally if $G$ is amenable
[ionwil:iumj09]. Then we can parameterize $\operatorname{Prim}C^{*}(G)$ as a
quotient of $\operatorname{Stab}(G)=\\{\,(u,\pi):\text{$u\in G^{(0)}$ and
$\pi\in G(u)^{\wedge}$}\,\\}$ where $G(u)^{\wedge}$ is the Pontryagin dual of
the abelian group $G(u)$. Having described this quotient, the crucial step is
to equip it with a topology such that it is homeomorphic to
$\operatorname{Prim}C^{*}(G)$. We also want the description of this topology
to be sufficiently transparent so that it can be used in applications. It
should be said that our methods are unlikely to uncover explicit information
unless one can describe the isotropy groups in a coherent way.

In the case of an action groupoid $G=G(\mathcal{G},X)$, with $\mathcal{G}$
acting properly on $X$, Echterhoff and Emerson solve this problem elegantly in
[echeme:em11]*Theorem 4.6 without assuming that the isotropy is abelian
provided $(\mathcal{G},X)$ satisfies what they call the Palais slice property.
(This additional assumption is automatically satisfied when $\mathcal{G}$ is a
Lie group or a discrete group.) A related, but slightly different, solution
for all proper group actions is presented in Katharina Neumann’s Ph.D. thesis
[neu:phd11]. Neumann replaces the Palais slice property with a theorem of
Abels [abe:mz78]*Theorem 3.3 for proper actions. Her description of the
topology requires knowledge of the Jacobson topology on the spectrum of the
Fell subgroup algebra $\mathcal{S}(\mathcal{G})$. In the extreme case that the
isotropy map $u\mapsto G(u)$ is continuous, Geoff Goehle gives a description
of $\operatorname{Prim}C^{*}(G)$ as a topological space for any groupoid with
abelian isotropy in [goe:rmjm12]*Theorem 3.5—in fact, Goehle’s results apply
to the much more general setting of groupoid crossed products
[goe:rmjm12]*Theorem 2.23.

In this paper, we equip $\operatorname{Stab}(G)$ with a second countable
topology so that the natural $G$-action is continuous. As does Neumann, this
description is in terms of an auxillary space, but our space is the Gelfand
dual of an abelian $C^{*}$-algebra, and hence is much more easily and
concretely described. In the case that $G\backslash G^{(0)}$ is $T_{0}$—that
is, each orbit $[u]$ is locally closed in $G^{(0)}$—induction induces a
continuous injection $\underline{\operatorname{Ind}}$ of
$G\backslash\operatorname{Stab}(G)$ onto $C^{*}(G)^{\wedge}$ (Corollary 4.2).
In the general case, we need to consider the quotient
$(G\backslash\operatorname{Stab}(G))^{\sim}$ where we identify orbits with the
same closure. Then we obtain a continuous map $\tilde{\kappa}$ of
$(G\backslash\operatorname{Stab}(G))^{\sim}$ into
$\operatorname{Prim}C^{*}(G)$ which is surjective if $G$ is EH-regular
(Proposition 4.1). Naturally, we expect that both
$\underline{\operatorname{Ind}}$ and $\tilde{\kappa}$ are homeomorphisms in
most circumstances. Unfortunately, at present, we can only prove this under
somewhat restrictive hypotheses. In fact, we aren’t even sure whether
$\tilde{\kappa}$ is always injective. Our hope is that we, or others, will be
able to push the envelope in future work.

Nevertheless, we are able to make some interesting new contributions. In
particular, we consider groupoids $G$ where the isotropy map is continuous
except for jump discontinuities at discrete points. Our first main result,
Theorem 7.3, is that $\tilde{\kappa}$ is a homeomorphism in this case. Thus we
obtain Goehle’s continuously varying isotropy result as a special case, and
also give an elegant description of Goehle’s topology on
$(G\backslash\operatorname{Stab}(G))^{\sim}$. We show by example that our
result applies to the groupoids associated to many directed graphs.

Our second main result, Theorem 9.8, shows that
$\underline{\operatorname{Ind}}$ is a homeomorphism whenever $G$ is what we
call “proper modulo its isotropy”. This class of groupoids not only includes
proper groupoids but also extensions of proper groupoids by abelian group
bundles as studied in, for example, [iksw:jot19, ikrsw:jfa20, ikrsw:xx21]. In
particular, the isotropy groups need no longer be compact in this case. This
result recovers the Echterhoff–Emerson and Neumann results in the case
$G=G(\mathcal{G},X)$ with $\mathcal{G}$ acting properly with abelian
stabilizers.

The paper is organized as follows. In Sections 1 and 2, we set out our
assumptions and review some standard constructions including induced
representations of groupoids. In Section 3, we introduce our topology on
$\operatorname{Stab}(G)$. In addition to giving a criterion for convergence in
this topology, we provide a basis for our topology and investigate some of
$\operatorname{Stab}(G)$’s elementary properties.

In Section 4 we introduce the maps $\underline{\operatorname{Ind}}$ and
$\tilde{\kappa}$ described above, and establish continuity. Even though these
are not (yet) homeomorphism results, we allow ourselves an aside in Section 5
to see that our methods already recover a well known simplicity result of
Renault’s for the special case of amenable groupoids with abelian isotropy.

In Section 6, we introduce our basic tool, what we call a subgroup section,
which allows us to deal with certain discontinuities of the isotropy map. We
use this in Section 7 to prove Theorem 7.3 and thereby recover Goehle’s result
as a special case.

In Section 8, we show that our description includes the known results for
action groupoids. At the same time, we obtain a separate description of the
Jacobson topology in these cases which, depending on the application at hand,
might be a more useful description.

In Section 9 we prove our Theorem 9.8 which generalizes the
Echterhoff–Emerson/Neumann result to proper groupoids in the case of abelian
isotropy and also applies to groupoids that are proper modulo their isotropy.
We provide some interesting examples in Section 10 of groupoids (other than
action groupoids) with abelian isotropy that is continuous except for jump
discontinuities.

### Assumptions

Throughout, $G$ will be a second countable locally compact Hausdorff groupoid
with abelian isotropy and a Haar system $\lambda=\\{\,\lambda^{u}\,\\}_{u\in
G^{(0)}}$. Similarly, locally compact spaces are assumed to be second
countable and Hausdorff. We assume that our $C^{*}$-algebras are separable
with the exception of multiplier algebras which obviously aren’t.
Homomorphisms between $C^{*}$-algebras are assumed to be $*$-preserving and
representations are assumed to be nondegenerate. We rely on [wil:toolkit] as a
convenient references for basic groupoid technology and notation. We also rely
heavily on the Rieffel machine for inducing representations as laid out in
[rw:morita]*§2.4.

##  1\. Preliminiaries

As in [wil:toolkit]*§3.4, we let $\Sigma_{0}$ be the locally compact Hausdorff
space of closed subgroups of $G$ with the Fell topology. If $H\in\Sigma_{0}$,
then there is a $u=p_{0}(H)\in G^{(0)}$ such that $H$ is a subgroup of the
isotropy group $G(u):=\\{\,\gamma\in G:s(\gamma)=u=r(\gamma)\,\\}$. Then
$p_{0}:\Sigma_{0}\to G^{(0)}$ is continuous and proper in the sense that
$p_{0}^{-1}(K)$ is compact for all compact $K\subset G^{(0)}$.

Let $\pi$ be a representation of $H\in\Sigma_{0}$. As is usual for groups, we
do not distinguish between the unitary representation $\pi:G\to
U(\mathcal{H}_{\pi})$ and its integrated form $\pi:C_{c}(H)\to
B(\mathcal{H}_{\pi})$ given by

(1.1) $\pi(b)h=\int_{H}b(t)\pi(t)h\,d\beta^{H}(t)\quad\text{$b\in C_{c}(H)$,
and $h\in\mathcal{H}_{\pi}$.}$

There is a well known method for inducing a representation $\pi$ of
$H\in\Sigma_{0}$ to a representation $\operatorname{Ind}_{H}^{G}\pi$ of
$C^{*}(G)$ based on the Rieffel machine. The details are spelled out in
[ionwil:pams08]*§2, and we recall some of these to motivate the constructions
in Section 2. For convenience, let $u=p_{0}(H)$ and let $G_{u}:=s^{-1}(u)$. We
equip $C_{c}(G_{u})$ with a $C_{c}(H)$-valued pre-inner product given by

(1.2)
$\langle\psi\nobreak\mathrel{,}\nobreak\phi\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}(t)=\int_{G}\overline{\psi(\gamma)}\phi(\gamma
t)\,d\lambda_{u}(\gamma).$

Then we can complete $C_{c}(G_{u})$ to a right Hilbert $C_{c}(H)$-module
$\mathsf{X}$. We get an action of $C^{*}(G)$ on $\mathsf{X}$ by adjointable
operators where $f\in C_{c}(G)$ acts on $\phi\in C_{c}(G_{u})$ by

(1.3)
$f\cdot\phi(\gamma)=\int_{G}f(\eta)\phi(\eta^{-1}\gamma)\,d\lambda^{r(\gamma)}(\eta).$

Then the induced representation $\operatorname{Ind}_{H}^{G}\pi$ of $C^{*}(G)$
acts on $\mathcal{H}_{\operatorname{Ind}\pi}$ which is the completion of
$C_{c}(G_{u})\odot\mathcal{H}_{\pi}$ with respect to the pre-inner product

(1.4) $\displaystyle(\phi\otimes h\mid\psi\otimes k)$
$\displaystyle=\bigl{(}\pi\bigl{(}\langle\psi\nobreak\mathrel{,}\nobreak\phi\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}\bigr{)}h\mid
k\bigr{)}\,d\beta^{H}(t)$ (1.5)
$\displaystyle=\int_{H}\int_{G}\overline{\psi(\gamma)}\phi(\gamma
t)\bigl{(}\pi(t)h\mid k\bigr{)}\,d\lambda_{u}(\gamma)\,d\beta^{H}(t).$

If $f\in C_{c}(G)$, then $\operatorname{Ind}_{H}^{G}\pi(f)$ acts on
$C_{c}(G_{u})\odot\mathcal{H}_{\pi}$ by
$\operatorname{Ind}_{H}^{G}\pi(f)(\phi\otimes h)=f\cdot\phi\otimes h$.

In the sequel, we will be interested primarily in the case that $\pi$ is a
character in the dual $\hat{H}$. Then $\mathcal{H}_{\pi}=\mathbf{C}$ and we
can identify $C_{c}(G_{u})\odot\mathcal{H}_{\pi}$ with $C_{c}(G_{u})$ equipped
with the inner product

(1.6) $(\phi\mid\psi)=\int_{H}\int_{G}\overline{\psi(\gamma)}\phi(\gamma
t)\pi(t)\,d\lambda_{u}(\gamma)\,d\beta^{H}(t).$

We can form a locally compact abelian group bundle $p:\Sigma\to\Sigma_{0}$
where

(1.7) $\Sigma=\\{\,(H,t)\in\Sigma_{0}\times G:t\in H\,\\}.$

Then $\Sigma$ always has a Haar system
$\beta=\\{\,\beta^{H}\,\\}_{H\in\Sigma_{0}}$ by [wil:toolkit]*Proposition
3.23.111In [wil:toolkit]*§3.5, the elements of $\Sigma$ were written as
triples for pedagogical reasons. Here we have opted for a more compact
notation. It will be useful to recall the following.

######  Lemma 1.1 ([wil:toolkit]*Lemma 3.25).

Let $G*\Sigma_{0}=\\{\,(\gamma,H):s(\gamma)=p_{0}(H)\,\\}$. There is a
continuous function $\omega:G*\Sigma_{0}\to(0,\infty)$ such that

(1.8)
$\int_{H}f(\sigma\gamma\sigma^{-1})\,d\beta^{H}(\gamma)=\omega(\sigma,H)\int_{\sigma\cdot
H}f(\gamma)\,d\beta^{\sigma\cdot H}(\gamma)$

for all $f\in C_{c}(G)$. Moreover, for all $\sigma,\tau\in G$ and
$H\in\Sigma_{0}$, we have

(1.9) $\omega(\sigma\tau,H)=\omega(\tau,H)\omega(\sigma,\tau\cdot
H)\quad\text{and}\quad\omega(\sigma,H)^{-1}=\omega(\sigma^{-1},\sigma\cdot
H).$

Since $\Sigma$ is an abelian group bundle, $C^{*}(\Sigma)$ is a commutative
$C^{*}$-algebra. As shown in [mrw:tams96]*§3, its Gelfand spectrum,
$\widehat{\Sigma}$, is an abelian group bundle over $\Sigma_{0}$ such that the
fibre over $H$ can be identified with $\hat{H}$. Thus we can assume that
$\widehat{\Sigma}=\\{\,(H,\pi):\text{$H\in\Sigma_{0}$ and
$\pi\in\hat{H}$}\,\\}$. Then $(H,\pi)$ represents the complex homomorphism
given on $b\in C_{c}(\Sigma)$ by

(1.10) $(H,\pi)(b)=\hat{b}(H,\pi)=\int_{H}b(H,t)\pi(t)\,d\beta^{H}(t).$

Whenever convenient, we identify $C^{*}(\Sigma)$ with
$C_{0}(\widehat{\Sigma})$ via the Gelfand transform $b\mapsto\hat{b}$, where
$\hat{b}$ is defined as in (1.10).

One of the benefits of restricting to abelian isotropy is that the Gelfand
topology on $\widehat{\Sigma}$ is well understood with a convenient and
elegant description in terms of sequences.

######  Proposition 1.2 ([mrw:tams96]*Proposition 3.3).

A sequence $\bigl{(}(H_{n},\pi_{n})\bigr{)}$ in $\widehat{\Sigma}$ converges
to $(H_{0},\pi_{0})$ in the Gelfand topology on $\widehat{\Sigma}$ if and only
if

  1. (a)

$H_{n}\to H_{0}$ in $\Sigma_{0}$, and

  2. (b)

whenever $a_{n}\in H_{n}$ is such that $a_{n}\to a_{0}$, then we have
$\pi_{n}(a_{n})\to\pi_{0}(a_{0})$.

##  2\. The Subgroup Module

Let $\mathsf{X}_{0}=C_{c}(G*\Sigma_{0})$. It is straightforward to check that

(2.1) $\langle F\nobreak\mathrel{,}\nobreak
G\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}(H,t)=\int_{G}\overline{F(\gamma,H)}G(\gamma
t,H)\,d\lambda_{p_{0}(H)}(\gamma)$

is a $C_{c}(\Sigma)$-valued pre-inner product on $\mathsf{X}_{0}$ so that the
completion is a right Hilbert $C^{*}(\Sigma)$-module $\mathsf{X}$ (see
[rw:morita]*Lemma 2.16). Furthermore, we get a left action of $C_{c}(G)$ on
$\mathsf{X}_{0}$ by

(2.2) $f\cdot
F(\gamma,H)=\int_{G}f(\eta)F(\eta^{-1}\gamma,H)\,d\lambda^{r(\gamma)}(\eta).$

Just as in [ionwil:pams08]*§2, this action extends to a homomorphism of
$C^{*}(G)$ into the algebra of adjointable operators
$\mathcal{L}(\mathsf{X})$. Therefore we can use the Rieffel machine to induce
a representation $L$ of $C^{*}(\Sigma)$ to a representation
$\operatorname{Ind}_{\Sigma}^{G}(L)$ of $C^{*}(G)$. In particular, if
$H\in\Sigma_{0}$ and $\pi\in\hat{H}$, then we can let $L=(H,\pi)$ and form the
representation $\operatorname{Ind}_{\Sigma}^{G}(H,\pi)$. Then we have the
following equivalence result. The proof is routine, so we omit it.

######  Lemma 2.1.

If $H\in\Sigma_{0}$ and $\pi\in\hat{H}$, then $\operatorname{Ind}_{H}^{G}\pi$
and $\operatorname{Ind}_{\Sigma}^{G}(H,\pi)$ are equivalent representations of
$C^{*}(G)$.

Recall that if $A$ is a $C^{*}$-algebra, then we can equip the collection
$\mathcal{I}(A)$ of closed two-sided ideals of $A$ with the topology generated
by the sets

(2.3) $\mathcal{O}_{J}=\\{\,I\in\mathcal{I}(A):J\not\subset I\,\\}$

where $J\in\mathcal{I}(A)$. Therefore a basis for this topology consists of
finite intersections of $\\{\,\mathcal{O}_{J}:J\in\mathcal{I}(A)\,\\}$ with
the whole space arising as in the intersection over the empty set. One of the
few redeeming properties of this topology is that it restricts to the usual
Jacobson topology on $\operatorname{Prim}A\subset\mathcal{I}(A)$, and then the
collection $\\{\,\mathcal{O}_{J}:J\in\mathcal{I}(A)\,\\}$ is actually a basis
for the Jacobson topology on $\operatorname{Prim}A$.

If $\mathsf{X}$ is a right Hilbert $B$-module and
$\phi:A\to\mathcal{L}(\mathsf{X})$ is a homormorphism, then the Rieffel
machine gives us a means of inducting a representation $\pi$ of $B$ to a
representation
$\mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\\!\operatorname{Ind}\nolimits_{B}^{A}\pi$
of $A$. Furthermore, we get a well-defined continuous map from
$\mathcal{I}(B)$ to $\mathcal{I}(A)$ given by
$\ker\pi\mapsto\ker\bigl{(}\mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\\!\operatorname{Ind}\nolimits_{B}^{A}\pi\bigr{)}$
[rw:morita]*Corollary 3.35. Since $\operatorname{Ind}_{\Sigma}^{G}$ arises
from such a Rieffel machine, and since $\widehat{\Sigma}$ and
$\operatorname{Prim}\bigl{(}C^{*}(\Sigma)\bigr{)}$ are homeomorphic, we have
the following.

######  Lemma 2.2.

The map
$(H,\sigma)\mapsto\ker\bigl{(}\operatorname{Ind}_{\Sigma}^{G}(H,\sigma)\bigr{)}$
is a continuous map from $\widehat{\Sigma}$ to $\mathcal{I}(C^{*}(G))$.

The following observation will be of use in the sequel.

######  Lemma 2.3 ([wil:crossed]*Lemma 8.38).

Suppose that $X$ is a second countable locally compact Hausdorff space. If $F$
is closed in $X$, let $I(F)$ be the ideal in $C_{0}(X)$ of functions vanishing
on $F$. If a sequence $\bigl{(}I(F_{n})\bigr{)}$ converges to $I(F)$ in
$\mathcal{I}\bigl{(}C_{0}(X)\bigr{)}$ and if $x\in F$, then there is a
subsequence $(I(F_{n_{k}}))$ and $x_{k}\in F_{n_{k}}$ such that $x_{k}\to x$
in $X$.222This is special case of [wil:crossed]*Lemma 8.38. Hence the result
remains true without a separability assumption if nets are used in place of
sequences. It should be noted that the converse statement in
[wil:crossed]*Lemma 8.38 is false as stated. It should be amended to say that
$I(F_{i})\to I(F)$ if every subnet has the given property.

###### Proof.

Suppose that $I(F_{n})\to I(F)$ and $x\in F$. Let $\\{U_{k}\\}_{k=1}^{\infty}$
be a countable neighborhood basis at $x$ consisting of open sets such that
$U_{k+1}\subset U_{k}$. Let $J_{k}=I(X\setminus U_{k})$. Then
$I(F)\in\mathcal{O}_{J_{k}}$ for all $k$. Thus there is a $n_{1}$ such that
$n\geq n_{1}$ implies $I(F_{n})\in\mathcal{O}_{J_{1}}$. In particular, there
is a $x_{1}\in F_{n_{1}}\cap U_{1}$. Then if $k\geq 2$ and if we have choosen
$x_{j}\in F_{n_{j}}$ such that $x_{j}\in F_{n_{j}}\cap U_{j}$ for all $j<k$,
then we can find $n_{k}>n_{k-1}$ such that
$I(F_{n_{k}})\in\mathcal{O}_{J_{k}}$. Hence we can find $x_{k}\in
F_{n_{k}}\cap U_{k}$. Then $x_{k}\to x$ as required. ∎

##  3\. $\operatorname{Stab}(G)$

######  Definition 3.1.

We let

(3.1) $\operatorname{Stab}(G)=\\{\,(u,\pi):\text{$u\in G^{(0)}$ and $\pi\in
G(u)^{\wedge}$}\,\\}.$

######  Remark 3.2.

The map $(u,\pi)\mapsto\bigl{(}G(u),\pi\bigr{)}$ identifies
$\operatorname{Stab}(G)$ with a subset of $\widehat{\Sigma}$, but we need a
more subtle topology on $\operatorname{Stab}(G)$ than the relative topology.

We let $G^{\prime}=\\{\,\gamma\in G:r(\gamma)=s(\gamma)\,\\}$ be the isotropy
subgroupoid of $G$. Then $G^{\prime}$ is closed in $G$. Let $U$ be a pre-
compact open subset of $G^{(0)}$ and $K$ a compact subset of $G^{\prime}$. If
$V$ is open in the unit circle $\mathbf{T}$, then we let

(3.2) $\mathcal{O}(U,K,V)=\\{\,(u,\pi)\in\operatorname{Stab}(G):\text{$u\in U$
and $\pi(G(u)\cap K)\subset V$}\,\\}$

with the understanding that if $G(u)\cap K=\emptyset$, then $\pi(G(u)\cap
K)\subset V$ holds vacuously. Then we let $\rho_{S}$ be the collection of all
$\mathcal{(}U,K,V)$ for $U$, $K$, and $V$ as above. Since $\rho_{S}$ covers
$\operatorname{Stab}(G)$, $\rho_{S}$ generates a topology $\tau_{S}$ for
$\operatorname{Stab}(G)$ with a basis consisting of finite intersections of
elements of $\rho_{S}$.

######  Lemma 3.3.

The topology $\tau_{S}$ is second countable.

###### Proof.

Let $\\{\,U_{n}\,\\}$ and $\\{\,V_{m}\,\\}$ be countable bases for $G^{(0)}$
and $\mathbf{T}$, respectively. Let $\\{\,W_{j}\,\\}$ be a countable basis of
pre-compact open subsets of $G^{\prime}$ and let $K_{j}=\overline{W_{j}}$.
Then the collection $\mathcal{C}(G^{\prime})_{f}$ of finite unions of the
$K_{j}$ is also countable.

Given $(u,\pi)\in\mathcal{O}(U_{n},K,V_{m})$, it suffices to find
$K^{\prime}\in\mathcal{C}(G^{\prime})_{f}$ such that $K\subset K^{\prime}$ and
$\pi\bigl{(}G(u)\cap K^{\prime}\bigr{)}\subset V_{m}$ as this implies

(3.3)
$(u,\pi)\in\mathcal{O}(U_{n},K^{\prime},V_{m})\subset\mathcal{O}(U_{n},K,V_{m}).$

For each $a\in K$, we claim there is a $j$ such that $a\in W_{j}$ and
$\pi\bigl{(}G(u)\cap K_{j}\bigr{)}\subset V_{m}$. If $a\notin G(u)$, then by
regularity there is a $U\in G^{\prime}$ such that $a\in
U\subset\overline{U}\subset G^{\prime}\setminus G(u)$. Then we can find
$W_{j}$ such that $a\in W_{j}\subset U$ and $K_{j}$ will do. If $a\in G(u)$,
then the continuity of $\pi$ implies there is a neighborhood $V$ of $a$ in
$G^{\prime}$ such that $\pi\bigl{(}G(u)\cap V\bigr{)}\subset V_{m}$. Again
regularity implies there is a $W_{j}$ such that $a\in W_{j}\subset
K_{j}\subset V$. This proves the claim.

Since $K$ is compact, there are $j_{1},\dots,j_{l}$ such that $K\subset\bigcup
K_{j_{s}}$ and $\pi\bigl{(}G(u)\cap K_{j_{s}}\bigr{)}\subset V_{m}$. Thus we
can let $K^{\prime}=\bigcup K_{j_{s}}$. ∎

We can now give an elegant sequential characterization of $\tau_{S}$ in terms
of our sequential characterization of the Gelfand topology on
$\widehat{\Sigma}$ in Proposition 1.2.

######  Proposition 3.4.

A sequence $\bigl{(}(u_{n},\pi_{n})\bigr{)}$ converges to $(u_{0},\pi_{0})$ in
$\bigl{(}\operatorname{Stab}(G),\tau_{S}\bigr{)}$ if and only if every
subsequence $\bigl{(}(u_{n_{k}},\pi_{n_{k}})\bigr{)}$ has a subsequence
$\bigl{(}(u_{n_{k_{j}}},\pi_{n_{k_{j}}})\bigr{)}$ such that

  1. (a)

$u_{n_{k_{j}}}\to u_{0}$ in $G^{(0)}$,

  2. (b)

there is a $H\in\Sigma_{0}$ such that $G(u_{n_{k_{j}}})\to H$ in $\Sigma_{0}$,
and

  3. (c)

$\bigl{(}G(u_{n_{k_{j}}}),\pi_{n_{k_{j}}}\bigr{)}\to(H,\pi_{0}|_{{H}})$ in
$\widehat{\Sigma}$.

######  Remark 3.5.

Note that the $H$ in part (b) must satisfy $H\subset G(u_{0})$ and can depend
on the subsequence.

###### Proof.

Suppose that $(u_{n},\pi_{n})\to(u_{0},\pi_{0})$ in $\operatorname{Stab}(G)$.
If $U$ is any pre-compact neighborhood of $u_{0}$, then
$\mathcal{O}(U,\\{u_{0}\\},\mathbf{T})$ is a neighborhood of
$(u_{0},\pi_{0})$. Hence $u_{n}\to u_{0}$ in $G^{(0)}$. Hence (a) holds for
any subsequence. We can assume that we have already replaced
$\bigl{(}(u_{n},\pi_{n})\bigr{)}$ with a subsequence. Since
$p_{0}:\Sigma_{0}\to G^{(0)}$ is a proper map, we can pass to another
subsequence, relabel, and assume that $G(u_{n})\to H\subset G(u_{0})$. Thus
part (b) will hold for any further subsequence.

Suppose now that $a_{n}\in G(u_{n})$ is such that $a_{n}\to a_{0}$ in $G$.
Since $G(u_{n})\to H$, we have $a_{0}\in H$. If it is not the case that
$\pi_{n}(a_{n})\to\pi_{0}(a_{0})$, then we can pass to a subsequence, relabel,
and assume that there is a neighborhood $V$ of $\pi_{0}(a_{0})$ such that
$\pi_{n}(a_{n})\notin V$ for all $n$. Let $W$ be a pre-compact neighborhood of
$a_{0}$ in $G$ such that
$\pi_{0}\bigl{(}G(u_{0})\cap\overline{W}\bigr{)}\subset V$. Then $r(W)$ is a
neighborhood of $u_{0}$ in $G^{(0)}$, and there is a pre-compact open set $U$
in $G^{(0)}$ such that $u_{0}\in U\subset\overline{U}\subset r(W)$. Then
$r^{-1}(\overline{U})\cap\overline{W}$ is compact as is
$K=r^{-1}(\overline{U})\cap\overline{W}\cap G^{\prime}$. Since $G(u_{0})\cap
K\subset G(u_{0})\cap\overline{W}$, we have
$(u_{0},\pi_{0})\in\mathcal{O}(U,K,V)$. Since
$(u_{n},\pi_{n})\to(u_{0},\pi_{0})$, we eventually have $a_{n}\in G(u_{n})\cap
K$ and $(u_{n},\pi_{n})\in\mathcal{O}(U,K,V)$. But then
$\pi_{n}\bigl{(}G(u_{n})\cap K\bigr{)}\subset V$ contradicts
$\pi_{n}(a_{n})\notin V$. Therefore $\pi_{n}(a_{n})\to\pi_{0}(a_{0})$ and
$\bigl{(}G(u_{n}),\pi_{n}\bigr{)}\to(H,\pi_{0}|_{{H}})$ by Proposition 1.2.

Conversely, suppose that $\bigl{(}(u_{n},\pi_{n})\bigr{)}$ is a sequence such
that every subsequence has a subsequence satisfying (a), (b), and (c) with
respect to $(u_{0},\pi_{0})$. Suppose that
$(u_{0},\pi_{0})\in\mathcal{O}(U,K,V)$. It will suffice to see that
$\bigl{(}(u_{n},\pi_{n})\bigr{)}$ is eventually in $\mathcal{O}(U,K,V)$.
Suppose not. Then we can pass to a subsequence that is never in $O(U,K,V)$.
Moreover, we can pass to additional subsequences if necessary, relabel, and
assume that $u_{n}\to u_{0}$, $G(u_{n})\to H\in\Sigma_{0}$, and that property
(c) holds. We can also assume $u_{n}\in U$ for all $n$.

Since $(u_{n},\pi_{n})\notin\mathcal{O}(U,K,V)$ and $u_{n}\in U$, we must have
$\pi_{n}\bigl{(}G(u_{n})\cap K\bigr{)}\not\subset V$. Thus there is a
$a_{n}\in G(u_{n})\cap K$ such that $\pi(a_{n})\notin V$. Since $K$ is
compact, we can pass to another subsequence, relabel, and assume that
$a_{n}\to a_{0}$. Then by assumption $\pi_{n}(a_{n})\to\pi_{0}(a_{0})\in V$.
This leads to a contradiction and completes the proof. ∎

As a corollary of the proof, we have the following observation.

######  Corollary 3.6.

If $(u_{n},\pi_{n})\to(u_{0},\pi_{0})$ in
$\bigl{(}\operatorname{Stab}(G),\tau_{S}\bigr{)}$, then $u_{n}\to u_{0}$.

We have a natural action of $G$ on $\widehat{\Sigma}$ given by

(3.4) $\gamma\cdot(H,\pi)=(\gamma\cdot H,\gamma\cdot\pi)$

where $p_{0}(H)=s(\gamma)$ and $\gamma\cdot\pi(a)=\pi(\gamma^{-1}a\gamma)$.

######  Lemma 3.7.

The $G$-action on $\widehat{\Sigma}$ defined by (3.4) is continuous.

###### Proof.

Suppose $\gamma_{n}\to\gamma$ in $G$ and $(H_{n},\pi_{n})\to(H,\pi)$ in
$\widehat{\Sigma}$ with $p_{0}(H_{n})=s(\gamma_{n})$. Using
[wil:toolkit]*Lemma 3.22 for example, it is clear that $\gamma_{n}\cdot
H_{n}\to\gamma\cdot H$. Suppose that $a_{n}\in\gamma\cdot H_{n}$ and that
$a_{n}\to a_{0}$. Then $\gamma_{n}^{-1}a_{n}\gamma_{n}\in H_{n}$ and converges
to $\gamma^{-1}a_{0}\gamma$. Then Proposition 1.2 implies that
$\gamma_{n}\cdot\pi_{n}(a_{n})=\pi_{n}(\gamma_{n}^{-1}a_{n}\gamma_{n})\to\pi(\gamma^{-1}a_{0}\gamma)=\gamma\cdot\pi(a_{0})$.
Another application of Proposition 1.2 implies that
$\gamma_{n}\cdot(H_{n},\pi_{n})\to\gamma\cdot(H,\pi)$ as required. ∎

Since $\gamma\cdot G(s(\gamma))=G(r(\gamma))$, we get a similar action of $G$
on $\operatorname{Stab}(G)$:

$\gamma\cdot(s(\gamma),\pi)=(r(\gamma),\gamma\cdot\pi).$

######  Corollary 3.8.

The $G$-action on $\operatorname{Stab}(G)$ is continuous.

###### Proof.

Suppose that $\gamma_{n}\to\gamma$ in $G$ while
$(s(\gamma_{n}),\pi_{n})\to(s(\gamma),\pi)$ in $\operatorname{Stab}(G)$. We
need to show that
$(r(\gamma_{n}),\gamma_{n}\cdot\pi_{n})\to(r(\gamma),\gamma\cdot\pi)$ in
$\operatorname{Stab}(G)$. We can assume that we have already passed to a
subsequence and relabeled. Then we can pass to a subsequence, relabel, and
assume that $G(s(\gamma_{n}))\to H$ and that
$\bigl{(}G(s(\gamma_{n})),\pi_{n}\bigr{)}\to(H,\pi|_{{H}})$ in
$\widehat{\Sigma}$. But then Lemma 3.7 implies
$\gamma_{n}\cdot\bigl{(}G(s(\gamma_{n})),\pi_{n}\bigr{)}\to\bigl{(}\gamma\cdot
H,\gamma\cdot\pi|_{{H}}\bigr{)}=(\gamma\cdot H,(\gamma\cdot\pi)|_{{\gamma\cdot
H}})$, and the result follows. ∎

Recall that if $X$ is a topological space, then we write $(X)^{\sim}$ for the
“$T_{0}$-ization” of $X$ as defined in [wil:crossed]*Definition 6.9. Thus
$(X)^{\sim}=X/\\!\\!\sim$ where $x\sim y$ if and only if
$\overline{\\{x\\}}=\overline{\\{y\\}}$. We give $(X)^{\sim}$ the quotient
topology with respect to the natural map $q:X\to(X)^{\sim}$. This is the
largest topology making $q$ continuous. Since

(3.5) $\displaystyle\tau:=\\{\,V\subset(X)^{\sim}:\text{$q^{-1}(V)$ is open in
$X$}\,\\}$

is a topology on $(X)^{\sim}$ making $q$ continuous, it follows that
$V\subset(X)^{\sim}$ is open if and only if $q^{-1}(V)$ is open in $X$. The
space $(X)^{\sim}$ is always $T_{0}$, and if $f:X\to Y$ is a continuous map
into a $T_{0}$-space $Y$, then $f$ factors through a continuous map
$f^{\prime}:(X)^{\sim}\to Y$ given by $f^{\prime}(q(x))=f(x)$
[wil:crossed]*Lemma 6.10.

In the sequel, we will focus on the $T_{0}$-izaion,
$(G\backslash\operatorname{Stab}(G))^{\sim}$, of the orbit space
$G\backslash\operatorname{Stab}(G)$. Therefore the next result will be useful.

######  Lemma 3.9.

Let $X$ by a $G$-space and let $k:X\to(G\backslash X)^{\sim}$ be the natural
map. Then $k$ is continuous and open (cf., [wil:crossed]*Lemma 6.12).

###### Proof.

Since $k$ is the composition of continuous maps, it is clearly continuous. To
see that $k$ is open, it suffices to see that

(3.6) $\displaystyle k^{-1}\bigl{(}k(U)\bigr{)}=G\cdot U$

when $U$ is open in $X$. Since $k(x)=k(\gamma\cdot x)$, we clearly have

(3.7) $\displaystyle G\cdot U\subset k^{-1}\bigl{(}k(U)\bigr{)}.$

Suppose $k(x)\in k(U)$. Then $k(x)=k(y)$ with $y\in U$. Then $\overline{G\cdot
x}=\overline{G\cdot y}$. Therefore there are $(\gamma_{i})\subset G$ such that
$\gamma_{i}\cdot x\to y$. Since $U$ is open, we eventually have
$\gamma_{i}\cdot x$ in $U$. But then $x=\gamma_{i}^{-1}\cdot(\gamma_{i}\cdot
x)$ is eventually in $G\cdot U$ and $x\in G\cdot U$. ∎

##  4\. $\operatorname{Prim}C^{*}(G)$

In this section we start our examination of $\operatorname{Prim}C^{*}(G)$ and
the spectrum $C^{*}(G)^{\wedge}$ by defining a continuous map of
$\operatorname{Stab}(G)$ into $C^{*}(G)^{\wedge}$ and examining its
properties.

######  Proposition 4.1.

Suppose that $G$ is a second countable locally compact Hausdorff groupoid with
abelian isotropy and with a Haar system. If
$(u,\pi)\in\operatorname{Stab}(G)$, then
$\operatorname{Ind}(u,\pi)=\operatorname{Ind}_{\Sigma}^{G}\bigl{(}G(u),\pi\bigr{)}$
is an irreducible represenaton of $C^{*}(G)$. Then $\operatorname{Ind}$
induces a continuous map of $\operatorname{Stab}(G)$ into $C^{*}(G)^{\wedge}$
that is constant on $G$-orbits and factors through a continuous map
$\underline{\operatorname{Ind}}$ of $G\backslash\operatorname{Stab}(G)$ into
$C^{*}(G)^{\wedge}$. If
$\kappa(u,\pi)=\ker\bigl{(}\operatorname{Ind}(u,\pi)\bigr{)}\bigr{)}$, then
$\kappa$ is continuous as a map of $\operatorname{Stab}(G)$ into
$\operatorname{Prim}C^{*}(G)$ and factors through a continuous map
$\tilde{\kappa}$ of $(G\backslash\operatorname{Stab}(G))^{\sim}$ into
$\operatorname{Prim}C^{*}(G)$. If $G$ is amenable, then $\kappa$, and hence
$\tilde{\kappa}$, is surjective.

We believe that $\tilde{\kappa}$ is often a homeomorphism and one of our
immediate goals is to establish general conditions for which it is. But in
general, we do not even know if $\tilde{\kappa}$ is always injective. However,
in the GCR case, we can sharpen this result considerably.

######  Corollary 4.2.

Suppose that $G$ is as in the proposition, and that $G\backslash G^{(0)}$ is a
$T_{0}$ topological space. Then $G\backslash\operatorname{Stab}(G)$ is $T_{0}$
and $\operatorname{Ind}$ factors through a continuous bijection
$\underline{\operatorname{Ind}}$ of $G\backslash\operatorname{Stab}(G)$ onto
$C^{*}(G)^{\wedge}$.

We need a number of preliminary results before proving the proposition and its
corollary. Recall that our standing assumptions are that $G$ is second
countable, has abelian isotropy, and has a Haar system.

######  Lemma 4.3 ([ionwil:pams08]).

If $(u,\pi)\in\operatorname{Stab}(G)$, then
$\operatorname{Ind}_{\Sigma}^{G}\bigl{(}G(u),\pi\bigr{)}$ is an irreducible
representation of $C^{*}(G)$.

###### Proof.

If $(u,\pi)\in\operatorname{Stab}(G)$, then $\operatorname{Ind}_{G(u)}^{G}\pi$
is irreducible by [ionwil:pams08]*Theorem 5. Hence
$\operatorname{Ind}_{\Sigma}^{G}\bigl{(}G(u),\pi\bigr{)}$ is irreducible by
Lemma 2.1. ∎

######  Lemma 4.4.

If $G\backslash G^{(0)}$ is $T_{0}$, then $G$ is amenable and $C^{*}(G)$ is
GCR.

###### Proof.

If $G\backslash G^{(0)}$ is $T_{0}$, then since the isotropy groups are
abelian, and hence GCR, $C^{*}(G)$ is GCR by [cla:iumj07]*Theorem 1.4.

On the other hand, if $G\backslash G^{(0)}$ is $T_{0}$, then since all the
isotropy groups are amenable, $G$ is amenable by [wil:toolkit]*Theorem 9.86. ∎

######  Lemma 4.5.

Let $\bigl{(}(u_{n},\pi_{n})\bigr{)}$ be a sequence in
$\operatorname{Stab}(G)$ such that
$\bigl{(}G(u_{n}),\pi_{n}\bigr{)}\to(H,\sigma)$ in $\widehat{\Sigma}$. If
$\gamma\in G(u)^{\wedge}$ is such that $\gamma|_{{H}}=\sigma$, then
$\operatorname{Ind}(u_{n},\pi_{n})\to\operatorname{Ind}(u,\gamma)$ in
$C^{*}(G)^{\wedge}$.

###### Proof.

Since induction is continuous by Lemma 2.2,

(4.1)
$\ker\bigl{(}\operatorname{Ind}_{\Sigma}^{G}(G(u_{n}),\pi_{n})\bigr{)}\to\ker\bigl{(}\operatorname{Ind}_{\Sigma}^{G}(H,\sigma)\bigr{)}$

in $\mathcal{I}(C^{*}(G))$.

Since $G(u)$ is abelian, and hence amenable, we can apply [gre:jfa69]*Theorem
5.1 to conclude that as representations of the group $G(u)$,

(4.2)
$\ker\bigl{(}\operatorname{Ind}_{H}^{G(u)}\sigma\bigr{)}=\ker\bigl{(}\operatorname{Ind}_{H}^{G(u)}\gamma|_{{H}}\bigr{)}\subset\ker\gamma.$

Therefore we can use induction in stages (cf., [ionwil:pams08]*Theorem 4) as
well as Lemma 2.1 to conclude that

(4.3)
$\displaystyle\ker\bigl{(}\operatorname{Ind}_{\Sigma}^{G}(H,\sigma)\bigr{)}$
$\displaystyle=\ker\bigl{(}\operatorname{Ind}_{H}^{G}\sigma\bigr{)}=\ker\bigl{(}\operatorname{Ind}_{G(u)}^{G}\bigl{(}\operatorname{Ind}_{H}^{G(u)}\sigma\bigr{)}\bigr{)}$
(4.4)
$\displaystyle\subset\ker\bigl{(}\operatorname{Ind}_{G(u)}^{G}(\gamma)\bigr{)}=\ker\bigl{(}\operatorname{Ind}_{\Sigma}^{G}(G(u),\gamma)\bigr{)}=\ker\bigl{(}\operatorname{Ind}(u,\gamma)\bigr{)}.$

Thus, in classical terms, $\operatorname{Ind}(u,\gamma)$ is weakly contained
in $\operatorname{Ind}_{\Sigma}^{G}(H,\sigma)$. Now the result follows by
untangling definitions as in [echeme:em11]*Lemma 4.7(ii). ∎

######  Corollary 4.6.

The map $\kappa:\operatorname{Stab}(G)\to\operatorname{Prim}C^{*}(G)$ is
continuous.

###### Proof.

Suppose that $(u_{n},\pi_{n})\to(u_{0},\pi_{0})$ in $\operatorname{Stab}(G)$.
We need to verify that $\kappa(u_{n},\pi_{n})\to\kappa(u_{0},\pi_{0})$ in
$\operatorname{Prim}C^{*}(G)$. If this fails, then after passing to a
subsequence and relabeling, there is a $J\in\mathcal{I}(C^{*}(G))$ such that
$\kappa(u_{0},\pi_{0})\in\mathcal{O}_{J}$, but
$\kappa(u_{n},\pi_{n})\notin\mathcal{O}_{J}$ for all $n$.

But then we can pass to another subsequence, relabel, and assume that
$u_{n}\to u_{0}$, $G(u_{n})\to H\subset G(u_{0})$, and that
$(G(u_{n}),\pi_{n})\to(H,\pi_{0}|_{{H}})$. By Lemma 4.5, this implies
$\kappa(u_{n},\pi_{n})=\ker\bigl{(}\operatorname{Ind}(u_{n},\pi_{n})\bigr{)}\to\ker\bigl{(}\operatorname{Ind}(u_{0},\pi_{0})\bigr{)}=\kappa(u_{0},\pi_{0})\in\mathcal{O}_{J}$.
This leads to a contradiction and completes the proof. ∎

If $u\in G^{(0)}$, we let $[u]$ be the orbit $G\cdot u$ in $G^{(0)}$.

######  Lemma 4.7.

Let $G$ and $\Sigma$ be as in Proposition 4.1.

  1. (a)

If $\gamma\in G$ and $(H,\sigma)\in\widehat{\Sigma}$, then
$\operatorname{Ind}_{\Sigma}^{G}(\gamma\cdot(H,\sigma))$ is equivalent to
$\operatorname{Ind}_{\Sigma}^{G}(H,\sigma)$.

  2. (b)

If $\operatorname{Ind}(u,\pi)$ is equivalent to
$\operatorname{Ind}(v,\sigma)$, then $[u]=[v]$.

  3. (c)

If $G\backslash G^{(0)}$ is $T_{0}$ and if $\operatorname{Ind}(u,\pi)$ is
equivalent to $\operatorname{Ind}(u,\sigma)$, then $\pi=\sigma$.

###### Proof.

(a) If $H=G(u)$, then this is exactly [wil:toolkit]*Lemma 5.48. The general
case is proved similarly.

(b) This is [wil:toolkit]*Proposition 5.50.

(c) Let $j:C^{*}(G)\to C^{*}(G|_{{\overline{[u]}}})$ be the usual map
[wil:toolkit]*Theorem 5.1. Then $\operatorname{Ind}(u,\pi)$ is equivalent to
$\operatorname{Ind}_{G(u)}^{G|_{{\overline{[u]}}}}\pi\circ j$ by
[wil:toolkit]*Corollary 5.29. Hence
$\operatorname{Ind}_{G(u)}^{G|_{{\overline{[u]}}}}\pi$ and
$\operatorname{Ind}_{G(u)}^{G|_{{\overline{[u]}}}}\sigma$ are equivalent.
Since $G\backslash G^{(0)}$ is $T_{0}$, $[u]$ is open in $\overline{[u]}$ by
the Mackey–Glimm–Ramsay Dichcotomy [wil:toolkit]*Theorem 2.27. Therefore
$\operatorname{Ind}_{G(u)}^{G|_{{\overline{[u]}}}}\pi$ is the canonical
extension of $\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\pi$ to
$C^{*}(G|_{{\overline{[u]}}})$ by [wil:toolkit]*Lemma 5.31. Therefore
$\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\pi$ and
$\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\sigma$ are equivalent. Since
$G|_{{[u]}}$ and $G(u)$ are equivalent groupoids,
$\pi\mapsto\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\pi$ induces a homeomorphism
of $C^{*}(G(u))^{\wedge}$ onto $C^{*}(G|_{{[u]}})^{\wedge}$ by
[rw:morita]*Corollary 3.33. The result follows. ∎

###### Proof of Proposition 4.1.

We saw that $\operatorname{Ind}_{\Sigma}^{G}\bigl{(}G(u),\pi\bigr{)}$ is
irreducible in Lemma 4.3. Since the topology on $C^{*}(G)^{\wedge}$ is pulled
back from $\operatorname{Prim}C^{*}(G)$, the continuity of
$\operatorname{Ind}$ follows from Corollary 4.6. Then $\operatorname{Ind}$ is
constant on $G$-orbits by Lemma 4.7(a) and $\underline{\operatorname{Ind}}$ is
well-defined and necessarily continuous (since
$G\backslash\operatorname{Stab}(G)$ has the quotient topology).

The continuity of $\kappa$ is immediate as is the observation that it factors
through $G\backslash\operatorname{Stab}(G)$. Since
$\operatorname{Prim}C^{*}(G)$ is $T_{0}$, then $\kappa$ factors through a
continuous map $\tilde{\kappa}$ by the univeral property of the
$T_{0}$-ization [wil:crossed]*Lemma 6.10.

If $G$ is amenable, then the surjectivity of $\kappa$ follows from the Effros-
Hahn Conjecture for groupoid $C^{*}$-algebras [ionwil:iumj09]*Theorem 2.1. ∎

###### Proof of Corollary 4.2.

Since $G\backslash G^{(0)}$ is $T_{0}$, we have $G$ amenable and $C^{*}(G)$
GCR by Lemma 4.4. Since $C^{*}(G)$ is GCR, we can identify $C^{*}(G)^{\wedge}$
and $\operatorname{Prim}C^{*}(G)$ by [ped:cs-algebras]*Theorem 6.1.5. Since
$G$ is also amenable, $\underline{\operatorname{Ind}}$ is continuous and
surjective by the last part of Proposition 4.1.

It will suffice to see that $\underline{\operatorname{Ind}}$ is injective
since $C^{*}(G)^{\wedge}$ is $T_{0}$ if $C^{*}(G)$ is GCR. But if
$\operatorname{Ind}(u,\pi)$ is equivalent to $\operatorname{Ind}(v,\sigma)$,
we must have $[u]=[v]$ by Lemma 4.7(b). Then we can assume $v=\gamma\cdot u$.
Then $\operatorname{Ind}(u,\pi)$ and $\operatorname{Ind}(u,\gamma^{-1}\sigma)$
are equivalent by Lemma 4.7(a). Since $G\backslash G^{(0)}$ is $T_{0}$,
$\gamma^{-1}\cdot\sigma=\pi$ by Lemma 4.7(c). But then
$G\cdot(u,\pi)=G\cdot(v,\sigma)$ and $\underline{\operatorname{Ind}}$ is
injective as required. ∎

##  5\. Renault’s Simplicity Result

As something of an aside, we want to see that Proposition 4.1 already implies
a nice simplicity result in our setting for amenable $G$. It is a special case
of Renault’s fundamental [ren:jot91]*Corollary 4.9.

First we need to recall some terminology. In [ren:jot91]*Definition 4.1,
Renault says that the isotropy of $G$ is _discretely trivial_ at $v\in
G^{(0)}$ if for each compact set $K$ in $G$ there is a neighborhood $V$ of $v$
in $G^{(0)}$ such that $u\in V$ implies $G(u)\cap K\subset\\{u\\}$. Of course
this also implies $G(v)=\\{v\\}$. This notion is of most import when $G$ is
étale in which case this isotropy is discretely trivial at $v$ if and only if
$G(v)=\\{v\\}$.

######  Proposition 5.1.

Suppose that $G$ is an _amenable_ second countable locally compact Hausdorff
groupoid with abelian isotropy and a Haar system. Suppose that the action of
$G$ on $G^{(0)}$ is minimal; that is, we assume $[u]$ is dense in $G^{(0)}$
for all $u\in G^{(0)}$. If in addition there is a $u_{0}\in G^{(0)}$ such that
the isotropy is discretely trivial at $u_{0}$, then $C^{*}(G)$ is simple.

###### Proof.

In view Proposition 4.1, it will suffice to see that
$(G\backslash\operatorname{Stab}(G))^{\sim}$ is reduced to a single point.
Suppose $(u,\pi)\in\operatorname{Stab}(G)$. Since the orbit $[u_{0}]$ is
dense, there are $\gamma_{n}\in G$ such that $\gamma_{n}\cdot u_{0}\to u$. But
$G(\gamma_{n}\cdot u_{0})=\\{\gamma_{n}\cdot u_{0}\\}$ converges to
$H=\\{u\\}$ in $\Sigma_{0}$. Then it is not hard to see that
$\bigl{(}G(\gamma_{n}\cdot
u_{0}),1\bigr{)}=\gamma_{n}\cdot\bigl{(}G(u_{0}),1\bigr{)}$ converges to
$(H,1)$ in $\widehat{\Sigma}$. It follows from Proposition 3.4 that
$\gamma_{n}\cdot(u_{0},1)\to(u,\pi)$ in $\operatorname{Stab}(G)$. Therefore

(5.1) $(u,\pi)\in\overline{G\cdot(u_{0},1)}.$

On the other hand, $[u]$ is dense and there are $\eta_{n}\in G$ such that
$\eta_{n}\cdot u\to u_{0}$ in $G^{(0)}$. Since $G(u_{0})=\\{u_{0}\\}$ it
follows that $G(\eta_{n}\cdot u)\to G(u_{0})$ in $\Sigma_{0}$
[wil:toolkit]*Lemma 6.10. We claim that $\bigl{(}G(\eta_{n}\cdot
u),\eta_{n}\cdot\pi\bigr{)}$ converges to $\bigl{(}G(u_{0}),1\bigr{)}$ in
$\widehat{\Sigma}$. We will apply Proposition 1.2. Suppose that $a_{n}\in
G(\eta_{n}\cdot u)\to u_{0}$. Then $K=\\{\,\eta_{n}\cdot
u\,\\}\cup\\{u_{0}\\}$ is compact. Since the isotropy is discretely trivial at
$u_{0}$, we eventually have $a_{n}=\eta_{n}\cdot u$. Then
$\eta_{n}\cdot\pi(a_{n})=1$ and the claim follows. Therefore
$\eta_{n}\cdot(u,\pi)$ converges to $(u_{0},1)$ in $\operatorname{Stab}(G)$.
Then

(5.2) $(u_{0},1)\in\overline{G\cdot(u,\pi)}.$

Since $(u,\pi)\in\operatorname{Stab}(G)$ is arbitrary,
$(G\backslash\operatorname{Stab}(G))^{\sim}$ is a single point and the result
is proved. ∎

##  6\. Subgroup Sections

######  Definition 6.1.

A continuous map $\mathbb{H}:G^{(0)}\to\Sigma_{0}$ is called a _subgroup
section_ if $p_{0}(\mathbb{H}(u))=u$ for all $u\in G^{(0)}$. We call
$\mathbb{H}$ _equivariant_ if $\mathbb{H}(\gamma\cdot
u)=\gamma\cdot\mathbb{H}(u)$.

######  Remark 6.2.

In the extreme case that $u\mapsto G(u)$ is continuous, then
$\mathbb{H}(u)=G(u)$ is an equivariant subgroup section. At the other extreme,
$\mathbb{H}(u)=\\{u\\}$ for all $u\in G^{(0)}$ is always an equivariant
subgroup section.

Given a subgroup section $\mathbb{H}$, we define a left action of
$C_{c}(\Sigma)$ on $C_{c}(G)$ by

(6.1) $b\cdot
f(\gamma)=\int_{\mathbb{H}(r(\gamma))}b\bigl{(}\mathbb{H}(r(\gamma)),t\bigr{)}f(t^{-1}\gamma)\,d\beta^{\mathbb{H}(r(\gamma))}(t).$

It is not hard to check that if $b\in C_{c}(\Sigma)$ and $f\in C_{c}(G)$, then
$b\cdot f\in C_{c}(G)$ (for example, see [wil:toolkit]*Lemma 3.29).

In the sequel, we will realize the multiplier algebra $M(C^{*}(G))$ as the
algebra of adjointable operators $\mathcal{L}(C^{*}(G))$ where $C^{*}(G)$ is
viewed a right Hilbert module over itself (see [rw:morita]*§2.3).

######  Proposition 6.3.

Suppose that $\mathbb{H}$ is a subgroup section. Then there is a homomorphism
$P_{\mathbb{H}}:C^{*}(\Sigma)\to M(C^{*}(G))$ such that for $b\in
C_{c}(\Sigma)$ and $f\in C_{c}(G)$ we have $P_{\mathbb{H}}(b)f=b\cdot f$ where
the latter is given by (6.1).

######  Remark 6.4.

When the choice of $\mathbb{H}$ is clear, we simply write $P$ in place of
$P_{\mathbb{H}}$. In the case $u\mapsto G(u)$ is continuous and
$\mathbb{H}(u)=G(u)$, this result is a special case of
[goe:rmjm12]*Proposition 2.19.

###### Sketch of the Proof.

If $f,g\in C_{c}(G)$, then we let $\langle f\nobreak\mathrel{,}\nobreak
g\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}=f^{*}*g$. Then we can use
Lemma 1.1 and the observation that $\omega(\eta t,H)=\omega(\eta,H)$ if $t\in
H$ to conclude that for all $b\in C_{c}(\Sigma)$ we have

(6.2) $\langle P(b)f\nobreak\mathrel{,}\nobreak
g\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}=\langle
f\nobreak\mathrel{,}\nobreak P(b^{*})g\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}.$

Similar calculations show that
$P(b*b^{\prime})f=P(b)\bigl{(}P(b^{\prime})f\bigr{)}$ and that
$b\mapsto\langle P(b)f\nobreak\mathrel{,}\nobreak
g\rangle_{{\mskip-3.0mu}\hbox{\vrule
height=8.5pt,width=0.0pt}\scriptscriptstyle{\star}}$ is continuous when
$C_{c}(G)$ is given the inductive limit topology. Then we can use
[kmqw:nyjm10]*Proposition 1.7 to see that $P$ extends to a homomorphism of
$C^{*}(\Sigma)$ into $\mathcal{L}(C^{*}(G))=M(C^{*}(G))$. (We can dispense
with condition (iii) of [kmqw:nyjm10]*Proposition 1.7 using
[wil:toolkit]*Corollary 8.4.) ∎

######  Corollary 6.5.

Let $\mathbb{H}$ be a subgroup section and let $P=P_{\mathbb{H}}$ be the
corresponding map. Then there is a continuous map
$P^{*}:\mathcal{I}(C^{*}(G))\to\mathcal{I}(C_{0}(\widehat{\Sigma}))$ given by

(6.3) $P^{*}(J)=\\{\,\hat{b}\in C_{0}(\widehat{\Sigma}):P(b)\cdot
C^{*}(G)\subset J\,\\}.$

###### Proof.

This is [wil:crossed]*Lemma 8.35 combined with the Gelfand isomorphism of
$C^{*}(\Sigma)$ with $C_{0}(\widehat{\Sigma})$. ∎

######  Proposition 6.6.

Suppose that $\mathbb{H}$ is a subgroup section and that
$P^{*}:\mathcal{I}(C^{*}(G))\to\mathcal{I}(C_{0}(\widehat{\Sigma}))$ is the
associated map. If $J(u,\pi)=\ker(\operatorname{Ind}(u,\pi))$, then
$P^{*}(J(u,\pi))$ is the ideal of functions in $C_{0}(\widehat{\Sigma})$ that
vanish on the closure of

(6.4) $\\{\,\bigl{(}\mathbb{H}(\gamma\cdot
u),(\gamma\cdot\pi)|_{{\mathbb{H}(\gamma\cdot
u)}}\bigr{)}\in\widehat{\Sigma}:\gamma\in G_{u}\,\\}$

in $\widehat{\Sigma}$.

######  Remark 6.7.

If $\mathbb{H}$ is equivariant then
$(\gamma\cdot\pi)|_{{\mathbb{H}(\gamma\cdot
u)}}=\gamma\cdot(\pi|_{{\mathbb{H}(u)}})$ and $P^{*}(J(u,\pi))$ is the ideal
of functions vanishing on the orbit closure
$\overline{G\cdot(\mathbb{H}(u),\pi|_{{\mathbb{H}(u)}})}$.

We will need some preliminaries before embarking on the proof of the
proposition. Let $L=\operatorname{Ind}_{G(u)}^{G}\pi$. Since $L$ is equivalent
to $\operatorname{Ind}(u,\pi)$, $\ker L=J(u,\pi)$. Recall that $L$ acts on
$\mathcal{H}$ which is the completion of $C_{c}(G)$ with respect to the inner
product $\bigl{(}\cdot\mid\cdot\bigr{)}_{u}$ given in (1.6) by taking
$H=G(u)$. (There is no harm in substituting $C_{c}(G)$ for $C_{c}(G_{u})$
here.) We let $\underline{L}$ be the representation of $C^{*}(\Sigma)$ on
$\mathcal{H}$ given by $\bar{L}\circ P$ where $\bar{L}$ is the canonical
extension of $L$ to $M(C^{*}(G))$.

######  Lemma 6.8.

Let $b\in C^{*}(\Sigma)$. Then $\hat{b}\in P^{*}(J(u,\pi))$ if and only if

(6.5) $\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}=0\quad\text{for all $f,g\in
C_{c}(G)$.}$

###### Proof.

We have $\hat{b}\in P^{*}(J(u,\pi))$ if and only if

(6.6) $\bigl{(}\underline{L}(b)L(e)f\mid g\bigr{)}_{u}=\bigl{(}L(P(b)e)f\mid
g\bigr{)}_{u}=0\quad\text{for all $e,f,g\in C_{c}(G)$.}$

Since $\bigl{(}\underline{L}(b)L(e)f\mid
g\bigr{)}_{u}=\bigl{(}\underline{L}(b)(e*f)\mid g\bigr{)}_{u}$, it follows
that (6.5) implies $\hat{b}\in P^{*}(J(u,\pi))$.

On the other hand, if $(e_{i})$ is an approximate identity for $C_{c}(G)$ in
the inductive limit topology ([wil:toolkit]*Proposition 1.49), then

(6.7) $\displaystyle\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}$
$\displaystyle=\bigl{(}f\mid\underline{L}(b\bigr{)}^{*}g)_{u}=\lim_{i}\bigl{(}L(e_{i})f\mid\underline{L}(b\bigr{)}^{*}g)_{u}=\lim_{i}\bigl{(}e_{i}*f\mid\underline{L}(b\bigr{)}^{*}g)_{u}$
(6.8) $\displaystyle=\bigl{(}\underline{L}(b)(e_{i}*f)\mid g\bigr{)}_{u}.$

Therefore if $\hat{b}\in P^{*}(J(u,\pi))$, we have
$\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}=0$ for all $f,g\in C_{c}(G)$. ∎

We need a technical observation. As in the previous proof, we employ an
approximate identity $(e_{i})$ for $C_{c}(G)$ in the inductive limit topology.

######  Lemma 6.9.

If $b\in C_{c}(\Sigma)$, then

(6.9) $\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}=\bigl{(}b\cdot f\mid
g\bigr{)}_{u}.$

(This is equivalent to saying that $\underline{L}(b)f=b\cdot f$ as elements of
the Hilbert space $\mathcal{H}$.)

###### Proof.

Since $P(b)\in\mathcal{L}(C^{*}(G)_{C^{*}(G)})$, it respects the right action
of $C^{*}(G)$. Thus if $e,f\in C_{c}(G)$, $P(b)(e*f)=(P(b)e)*f$. Thus in
$C^{*}(G)$, $b\cdot(e*f)=(b\cdot e)*f$. Since the universal norm is a norm on
$C_{c}(G)$, this means $b\cdot(e*f)=(b\cdot e)*f$ in $C_{c}(G)$. (This may
also be verified directly.) Then

(6.10) $\displaystyle\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}$
$\displaystyle=\lim_{i}\bigl{(}\underline{L}(b)L(e_{i})f\mid
g\bigr{)}_{u}=\lim_{i}\bigl{(}L(P(b)e_{i})f\mid
g\bigr{)}_{u}=\lim_{i}\bigl{(}(b\cdot e_{i})*f\mid g\bigr{)}_{u}$ (6.11)
$\displaystyle=\lim_{i}\bigl{(}b\cdot(e_{i}*f)\mid
g\bigr{)}_{u}=\lim_{i}\bigl{(}b\cdot f\mid g\bigr{)}_{u}$

since $b\cdot(e_{i}*f)\to b\cdot f$ in the inductive limit topology. ∎

###### Proof of Proposition 6.6.

Using Lemma 6.9, observe that if $b\in C_{c}(\Sigma)$, then

(6.12) $\displaystyle\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}=(b\cdot f\mid
g)_{u}\hskip-113.81102pt$ (6.13)
$\displaystyle=\int_{G(u)}\int_{G}\overline{g(\gamma)}b\cdot f(\gamma
t)\pi(t)\,d\lambda_{u}(\gamma)\,d\beta^{G(u)}(t)$ (6.14)
$\displaystyle=\int_{G}\int_{G(u)}\int_{\mathbb{H}(r(\gamma))}\overline{g(\gamma)}b\bigl{(}\mathbb{H}(r(\gamma)),s\bigr{)}f(s^{-1}\gamma
t)\pi(t)$ (6.15) $\displaystyle\hskip
216.81ptd\beta^{\mathbb{H}(r(\gamma))}(s)\,d\beta^{G(u)}(t)\,d\lambda_{u}(\gamma)$
which, after $t\mapsto\gamma^{-1}t\gamma$, is (6.16)
$\displaystyle=\int_{G}\omega(\gamma,G(u))\int_{G(r(\gamma))}\int_{\mathbb{H}(r(\gamma))}\overline{g(\gamma)}b\bigl{(}\mathbb{H}(r(\gamma)),s\bigr{)}f(s^{-1}t\gamma)\pi(\gamma^{-1}t\gamma)$
(6.17) $\displaystyle\hskip
216.81ptd\beta^{\mathbb{H}(r(\gamma))}(s)\,d\beta^{G(r(\gamma))}(t)\,d\lambda_{u}(\gamma)$
which, after Fubini and $t\mapsto st$, is (6.18)
$\displaystyle=\int_{G}\omega(\gamma,G(u))\int_{\mathbb{H}(r(\gamma))}\int_{G(r(\gamma))}\overline{g(\gamma)}b\bigl{(}\mathbb{H}(r(\gamma)),s\bigr{)}f(t\gamma)\pi(\gamma^{-1}st\gamma)$
(6.19) $\displaystyle\hskip
216.81ptd\beta^{G(r(\gamma))}(t)\,d\beta^{\mathbb{H}(r(\gamma))}(s)\,d\lambda_{u}(\gamma)$
(6.20)
$\displaystyle=\int_{G}\omega(\gamma,G(u))\overline{g(\gamma)}\Bigl{(}\int_{\mathbb{H}(r(\gamma))}b\bigl{(}\mathbb{H}(r(\gamma)),s\bigr{)}\pi(\gamma^{-1}s\gamma)\,d\beta^{\mathbb{H}(r(\gamma))}(s)\Bigr{)}$
(6.21) $\displaystyle\hskip
72.26999pt\int_{G(r(\gamma))}f(t\gamma)\pi(\gamma^{-1}t\gamma)\,d\beta^{G(r(\gamma))}(t)\,d\lambda_{u}(\gamma)$
(6.22)
$\displaystyle=\int_{G}\omega(\gamma,G(u))\overline{g(\gamma)}\hat{b}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}}\bigr{)}\int_{G(r(\gamma))}f(t\gamma)\pi(\gamma^{-1}t\gamma)$
(6.23) $\displaystyle\hskip
216.81ptd\beta^{G(r(\gamma)}(t)\,d\lambda_{u}(\gamma)$ (6.24)
$\displaystyle=\int_{G}\int_{G(u)}\hat{b}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}}))\overline{g(\gamma)}f(\gamma
t)\pi(t)\,d\beta^{G(u)}(t)\,d\lambda_{u}(\gamma)$ (6.25)
$\displaystyle=\int_{G}\hat{b}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}})\bigr{)}\overline{g(\gamma)}\hat{f}(\gamma,\pi)\,d\lambda_{u}(\gamma)$

where

(6.26) $\displaystyle\hat{f}(\gamma,\pi):=\int_{G(u)}f(\gamma
t)\pi(t)\,d\beta^{G(u)}(t).$

Note that $\hat{f}$ is continuous on $G_{u}$. Hence $\gamma\mapsto
g(\gamma)\hat{f}(\gamma,\pi)$ is in $C_{c}(G_{u})$. If $b\in C^{*}(\Sigma)$,
then there is a sequence $(b_{i})$ in $C_{c}(\Sigma)$ converging to $b\in
C^{*}(\Sigma)$ and bounded in norm in $C^{*}(\Sigma)$. Then $(\hat{b}_{i})$ is
bounded in $C_{0}(\widehat{\Sigma})$ and $\hat{b}_{i}\to\hat{b}$ uniformly.
Thus

(6.27)
$\lim_{i}\int_{G}\hat{b}_{i}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}})\bigr{)}\overline{g(\gamma)}\hat{f}(\gamma,\pi)\,d\lambda_{u}(\gamma)\\\
=\int_{G}\hat{b}(\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}}))\overline{g(\gamma)}\hat{f}(\gamma,\pi)\,d\lambda_{u}(\gamma)$

by the dominated converge theorem. On the other hand

(6.28) $\displaystyle\lim_{i}\bigl{(}\underline{L}(b_{i})f\mid
g\bigr{)}_{u}=\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}.$

It follows that $b\in C^{*}(\Sigma)$, then

(6.29) $\bigl{(}\underline{L}(b)f\mid
g\bigr{)}_{u}=\int_{G}\hat{b}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}})\bigr{)}\overline{g(\gamma)}\hat{f}(\gamma,\pi)\,d\lambda_{u}(\gamma)$

for all $f,g\in C_{c}(G)$. Thus if (6.4) holds, then $\hat{b}\in P^{*}(J)$.

Suppose that $b\in C^{*}(\Sigma)$ is such that
$\phi(\gamma)=\hat{b}\bigl{(}\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}})\bigr{)}$
is non-zero at $\gamma_{0}\in G_{u}$. Then there is a $h\in C_{c}(G)$ such
that

(6.30) $\displaystyle\phi(\gamma)h(\gamma)\geq 0\quad\text{if $\gamma\in
G_{u}$}$

and such that $\phi(\gamma_{0})h(\gamma_{0})=1$. There is a $\zeta\in
C_{c}(G(u))$ such that

(6.31) $\displaystyle\int_{G(u)}\zeta(t)\pi(t)\,d\beta^{G(u)}(t)=1.$

Then there is a $f\in C_{c}(G)$ such that $f(\gamma_{0}t)=\zeta(t)$ for all
$t\in G(u)$. Thus $\hat{f}(\gamma_{0},\pi)=1$. Since $\hat{f}(\gamma,\pi)$ is
non-zero near $\gamma_{0}$, we can find $k\in C_{c}(G)$ such that

(6.32) $\displaystyle k(\gamma)\hat{f}(\gamma,\pi)\geq 0\quad\text{if
$\gamma\in G_{u}$, and}$

such that $k(\gamma_{0})\hat{f}(\gamma_{0},\pi)=1$. Let
$g(\gamma)=\overline{h(\gamma)k(\gamma)}$ and let $f$ be as above. Since
$\beta^{G(u)}$ has full support, it follows that

(6.33) $\displaystyle\bigl{(}\underline{L}(b)f\mid g\bigr{)}_{u}>0.$

Consequently, $\hat{b}\notin P^{*}(J)$. Since
$(\mathbb{H}(r(\gamma)),(\gamma\cdot\pi)|_{{\mathbb{H}(r(\gamma))}})=(\mathbb{H}(\gamma\cdot
u),(\gamma\cdot\pi)|_{{\mathbb{H}(\gamma\cdot u)}})$, this completes the
proof. ∎

The following corollary will be useful in the sequel.

######  Corollary 6.10.

As above, let $J(u,\pi)=\ker\bigl{(}\operatorname{Ind}(u,\pi)\bigr{)}$. If
$J(u_{n},\pi_{n})\to J(u,\pi)$ in $\operatorname{Prim}C^{*}(G)$, then after
passing to a subsequence and relabeling, there are $\gamma_{n}\in G$ such that
$\gamma_{n}\cdot u_{n}\to u$.

###### Proof.

There is always a trivial (equivariant) subsgroup section
$\mathbb{H}:G^{(0)}\to\Sigma_{0}$ given by $\mathbb{H}(u)=\\{u\\}$. Then
$P^{*}(J(u,\pi))$ is the ideal of functions in $C_{0}(\widehat{\Sigma})$
vanishing on $\overline{G\cdot(\\{u\\},1)}$ in $\widehat{\Sigma}$. Since
$P^{*}(J(u_{n},\pi_{n}))\to P^{*}(J(u,\pi))$ in
$\mathcal{I}(C_{0}(\widehat{\Sigma}))$, we can use Lemma 2.3 to pass to a
subsequence, relabel, and find $a_{n}\in\overline{G\cdot(\\{u_{n}\\},1)}$
converging to $(\\{u\\},1)$ in $\widehat{\Sigma}$. Since $\widehat{\Sigma}$ is
a metric space, we can take $a_{n}\in G\cdot(\\{u_{n}\\},1)$. The result now
follows easily from Proposition 1.2. ∎

######  Corollary 6.11.

Suppose that
$\ker\bigl{(}\operatorname{Ind}(u,\pi)\bigr{)}=\ker\bigl{(}\operatorname{Ind}(v,\sigma)\bigr{)}$
Then $\overline{[u]}=\overline{[v]}$.

###### Proof.

By Corollary 6.10, we may as well assume that there are $\gamma_{n}\in G$ such
that $\gamma_{n}\cdot u\to v$. Thus $v\in\overline{[u]}$. The result follows
by symmetry. ∎

##  7\. Continuous Isotropy

In this section, we want to consider examples were the isotropy map $u\mapsto
G(u)$ continuous except at isolated points. Specifically, we make the
following definition.

######  Definition 7.1.

We say that $G$ has continuous isotropy except for jump discontinuities if $G$
has an equivariant subgroup section $\mathbb{H}:G^{(0)}\to\Sigma_{0}$ such
that the set $D=\\{\,u\in G^{(0)}:\mathbb{H}(u)\not=G(u)\,\\}$ is discrete.

######  Remark 7.2.

Let $G$ be such that $G(u)=\\{u\\}$ for all $u$ off a discrete set $D\subset
G^{(0)}$. Then $G$ has continuous isotropy except for jump discontinuities.
For a specific example of this, see Example 10.2. Other more subtle examples
of jump discontinuities are given in Examples 10.1 and 10.3.

######  Theorem 7.3.

Suppose that $G$ is a second countable locally compact Hausdorff groupoid with
a Haar system and abelian isotropy. Suppose that the isotropy is continuous
except for jump discontinuities. If $G$ is amenable, then
$(u,\pi)\mapsto\kappa(u,\pi):=\ker\bigl{(}\operatorname{Ind}(u,\pi)\bigr{)}$
is an open surjection inducing a homeomorphism $\tilde{\kappa}$ of
$(G\backslash\operatorname{Stab}(G))^{\sim}$ onto
$\operatorname{Prim}C^{*}(G)$. If $G\backslash G^{(0)}$ is $T_{0}$, then $G$
is amenable and $C^{*}(G)$ is GCR. Furthermore, in that case,
$(u,\pi)\mapsto\operatorname{Ind}(u,\pi)$ induces a homeomorphism
$\underline{\operatorname{Ind}}$ of $G\backslash\operatorname{Stab}(G)$ onto
$C^{*}(G)^{\wedge}$.

######  Remark 7.4 (Goehle).

If the isotropy map $u\mapsto G(u)$ is continuous, then Theorem 7.3 applies.
This allows us to recover Goehle’s [goe:rmjm12]*Theorem 3.5 describing
$\operatorname{Prim}C^{*}(G)$ as a special case. At the same time we can give
a cleaner description of the topology of
$(G\backslash\operatorname{Stab}(G))^{\sim}$ using results from Section 3.
Specifically, consider Lemma 7.6 and Remark 7.7.

For the proof, it will be useful to note that the equivariance of the subgroup
section easily implies the following and we omit the proof.

######  Lemma 7.5.

If $G$ has continuous isotropy except for jump discontinuities and if
$\mathbb{H}:G^{(0)}\to\Sigma_{0}$ is an equivariant subgroup section such that
$D=\\{\,u\in G^{(0)}:\mathbb{H}(u)\not=G(u)\,\\}$ is discrete, then $D$ is
$G$-invariant and $[u]$ is closed for all $u\in D$.

###### Proof of Theorem 7.3.

Let $\mathbb{H}:G^{(0)}\to\Sigma_{0}$ be a subgroup section such that
$D=\\{\,u\in G^{(0)}:\mathbb{H}(u)\not=G(u)\,\\}$ is discrete. Let
$P^{*}:\mathcal{I}(C^{*}(G))\to\mathcal{I}(C_{0}(\widehat{\Sigma}))$ be the
corresponding continuous map from Corollary 6.5.

By Proposition 4.1, we know that $\kappa$ and $\tilde{\kappa}$ are continuous
surjections.

Suppose that $\kappa(u,\pi)=\kappa(v,\sigma)$. Then $[v]$ and $[u]$ have the
same closure by Corollary 6.11. Since $u\in D$ implies $[u]$ is closed, $u\in
D$ implies that $v\in[u]$. Thus we may as well assume $v=u$. Since
$G|_{{[u]}}$ and $G(u)$ are equivalent groupoids and since $G(u)$ is abelian,
$C^{*}(G|_{{[u]}})$ is GCR. Since $\operatorname{Ind}(u,\pi)$ and
$\operatorname{Ind}(u,\sigma)$ factor through $C^{*}(G|_{{[u]}})$ and have the
same kernel, it follows that $\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\pi$ and
$\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\sigma$ are equivalent [ped:cs-
algebras]*Theorem 6.1.5. Since $G|_{{[u]}}$ is equivalent to $G(u)$,
$\pi=\sigma$. Thus $G\cdot(u,\pi)$ and $G\cdot(v,\sigma)$ are equal. In
particular, $(u,\pi)$ and $(v,\sigma)$ define the same class in
$(G\backslash\operatorname{Stab}(G))^{\sim}$.

On the other hand, if $u\notin D$, then we can use Proposition 6.6 and Remark
6.7 to conclude that $P^{*}(\kappa(u,\pi))$ is the ideal of functions
vanishing on $\overline{G\cdot(G(u),\pi)}\subset\widehat{\Sigma}$. Thus if
$\kappa(u,\pi)=\kappa(v,\sigma)$ we also have $v\notin D$ and
$\overline{G\cdot(u,\pi)}=\overline{G\cdot(v,\sigma)}$. Thus $(u,\pi)$ and
$(v,\sigma)$ define the same class in
$(G\backslash\operatorname{Stab}(G))^{\sim}$ in this case as well. This shows
that $\tilde{\kappa}$ is injective.

In view of Lemma 3.9, to see that $\tilde{\kappa}$ is a homeomorphism, it will
suffice to see that $\kappa$ is open. Let $V$ be a neighborhood of $(u,\pi)$
in $\operatorname{Stab}(G)$. Suppose that
$\kappa(u_{n},\pi_{n})\to\kappa(u,\pi)$. Since this sequence is arbitrary, it
will suffice to see that $\kappa(u_{n},\pi_{n})$ has a subsequence which is
eventually in $\kappa(V)$.

Suppose that $u\notin D$. Then we may as well assume that $u_{n}\notin D$ for
all $n$. However, Corollary 6.5 implies that $P^{*}(\kappa(u_{n},\pi_{n}))\to
P^{*}(\kappa(u,\pi))$ in $\mathcal{I}(C_{0}(\widehat{\Sigma}))$. By Lemma 2.3,
we can pass to a subsequence, relabel, and assume that there there are
$a_{n}\in\overline{G\cdot(G(u_{n}),\pi_{n})}$ converging to $(G(u),\pi)$ in
$\widehat{\Sigma}$. Since $\widehat{\Sigma}$ is a metric space, there are
$\gamma_{n}\in G$ such that $\gamma_{n}\cdot(G(u_{n}),\pi_{n})\to(G(u),\pi)$
in $\widehat{\Sigma}$. Of course, we also have
$\gamma_{n}\cdot(u_{n},\pi_{n})\to(u,\pi)$ in $\operatorname{Stab}(G)$ and
$\gamma_{n}\cdot(u_{n},\pi_{n})$ is eventually in $V$. Since
$\kappa\bigl{(}\gamma_{n}\cdot(u_{n},\pi_{n})\bigr{)}=\kappa(u_{n},\pi_{n})$
by Lemma 4.7(b), we have shown that $\kappa(u_{n},\pi_{n})$ is eventually in
$\kappa(V)$ as required.

Now assume $u\in D$. If there are infinitely many $u_{n}$ such that
$u_{n}\notin D$, then we can pass to a subsequence, relabel, and assume that
$u_{n}\notin D$ for all $n$. Just as above, $P^{*}(\kappa(u_{n},\pi_{n}))\to
P^{*}(\kappa(u,\pi))$ in $\mathcal{I}(C_{0}(\widehat{\Sigma}))$ except now
$P^{*}(\kappa(u,\pi))$ is the ideal of functions vanishing on
$\overline{G\cdot(\mathbb{H}(u),\pi|_{{\mathbb{H}(u)}})}$. Then we can pass to
a subsequence, relabel, and assume that there are $\gamma_{n}\in G$ such that
$\gamma_{n}\cdot(G(u_{n}),\pi_{n})\to(\mathbb{H}(u),\pi|_{{\mathbb{H}(u)}})$
in $\widehat{\Sigma}$. But then $\gamma_{n}\cdot(u_{n},\pi_{n})\to(u,\pi)$ in
$\operatorname{Stab}(G)$ by Lemma 4.5. Then we have
$\kappa(u_{n},\pi_{n})=\kappa(\gamma_{n}\cdot(u_{n},\pi_{n}))$ eventually in
$\kappa(V)$.

This leaves the case where there are infinitely many $u_{n}\in D$. Then we can
reduce to the case where $u_{n}\in D$ for all $n$. By Corollary 6.10, we can
assume that there are $\gamma_{n}$ such that $\gamma_{n}\cdot u_{n}\to u$.
After replacing $(u_{n},\pi_{n})$ with $\gamma_{n}\cdot(u_{n},\pi_{n})$, we
can assume $u_{n}\ \to u$. Since $D$ is discrete, we can pass to a
subsequence, relabel, and assume $u_{n}=u$ for all $n$. But then we can assume
$\operatorname{Ind}_{G(u)}^{G|_{{[u]}}}\pi_{n}\to\operatorname{Ind}_{G(u)}^{G|_{{{[u]}}}}\pi$
in $C^{*}(G|_{{[u]}})^{\wedge}$. Since $G|_{{[u]}}$ is equivalent to $G(u)$
this implies $\pi_{n}\to\pi$ in $G(u)^{\wedge}$. Then
$(u_{n},\pi_{n})=(u,\pi_{n})\to(u,\pi)$ in $\operatorname{Stab}(G)$.

This completes the proof that $\kappa$ is open.

If $G\backslash G^{(0)}$ is $T_{0}$, the rest follows from Corollary 4.6. ∎

If the isotropy varies continuously, then we can sharpen our description of
the topology on $\operatorname{Stab}(G)$.

######  Lemma 7.6.

Let $\widehat{\Sigma}_{S}=\\{\,(G(u),\pi)\in\widehat{\Sigma}:u\in
G^{(0)}\,\\}$. Then $\widehat{\Sigma}_{S}$ is a closed subset of
$\widehat{\Sigma}$ and $(G(u),\pi)\mapsto(u,\pi)$ is a homeomorphism of
$\widehat{\Sigma}_{S}$ onto $\operatorname{Stab}(G)$. In particular, a
sequence $\bigl{(}(u_{n},\pi_{n})\bigr{)}$ converges to $(u,\pi)$ in
$\operatorname{Stab}(G)$ if and only if $u_{n}\to u$ and given $a_{n}\in
G(u_{n})$ converging to $a\in G(u)$ we have $\pi_{n}(a_{n})\to\pi(a)$.

######  Remark 7.7.

In the continuously varying isotropy case in particular, the openness of the
map of $\widehat{\Sigma}_{S}$ onto
$(G\backslash\operatorname{Stab}(G))^{\sim}$, and the proof of Theorem 7.3,
give us a rather robust description of the topology on
$\operatorname{Prim}C^{*}(G)$. If $[u,\pi]$ is the class of $(u,\pi)$ in
$(G\backslash\operatorname{Stab}(G))^{\sim}$, then we note that a sequence
$\bigl{(}[u_{n},\pi_{n}]\bigr{)}$ converges to $[u,\pi]$ if and only if every
subsequence has a subsequence $\bigl{(}[u_{n_{k}},\pi_{n_{k}}]\bigr{)}$ such
there are $\gamma_{k}\in G$ such that
$\gamma_{k}\cdot\bigl{(}G(u_{n_{k}}),\pi_{n_{k}}\bigr{)}$ converges to
$(G(u),\pi)$ in $\widehat{\Sigma}_{S}$.

##  8\. Action Groupoids for Abelian Groups

We pause to see what the known results for action groupoids imply about our
more general constructions. Recall that if $(\mathcal{G},X)$ is a second
countable locally compact transformation group, then the corresponding action
groupoid, or transformation groupoid, is $G=G(\mathcal{G},X)=\\{\,(x,g,y)\in
X\times\mathcal{G}\times X:x=g\cdot y\,\\}$ [wil:toolkit]*Example 1.12. Then
$C^{*}(G)$ can be identified with the transformation group $C^{*}$-algebra
$C_{0}(X)\rtimes_{\operatorname{lt}}\mathcal{G}$ [wil:toolkit]*Example 1.55.
If $\mathcal{G}$ is abelian, then we can use [wil:tams81]*Theorem 5.3 to
describe $\operatorname{Prim}C^{*}(G)$ and its topology. However this
description is very different than what we propose here. Nevertheless, we can
use this earlier work to verify our map $\tilde{\kappa}$ from Section 4 is a
homeomorphism in this case.

More generally, provided that $\mathcal{G}$ is amenable or that
$\mathcal{G}\backslash X$ is $T_{0}$, we can also deal with the case where
there is an abelian subgroup $\mathcal{H}$ of $\mathcal{G}$ such that each
stability subgroup, $S_{x}=\\{\,g\in\mathcal{G}:g\cdot x=x\,\\}$, is contained
in $\mathcal{H}$. Note that $G=G(\mathcal{G},X)$ is amenable if $\mathcal{G}$
is, or if $G\backslash X=\mathcal{G}\backslash X$ is $T_{0}$. Furthermore,
replacing $\mathcal{H}$ with $\bigcap_{g\in\mathcal{G}}g\mathcal{H}g^{-1}$
allows us to assume that $\mathcal{H}$ is normal in $\mathcal{G}$. Hence we
can apply [wil:tams81]*Theorem 5.8 and its corollaries.

It will be helpful to translate the results in [wil:tams81] to our setting. As
above, we write $S_{x}$ for the stability subgroup
$\\{\,h\in\mathcal{G}:h\cdot x=x\,\\}$ in $\mathcal{G}$ to distinguish it from
the isotropy group $G(x)=\\{\,(x,h,x)\in G:h\in S_{x}\,\\}$ in $G$. However,
if $\pi\in\hat{S}_{x}$, then we will use the same symbol for the corresponding
character on $G(x)$. It is not hard to see that the induced representation
$\operatorname{Ind}_{(x,S_{x})}^{\mathcal{G}}(\pi)$ of the transformation
group $C^{*}$-algebra, defined in [wil:tams81]*Definition 3.4, when viewed as
a representation of $C^{*}(G)$, is equivalent to our
$\operatorname{Ind}_{G(x)}^{G}(\pi)$. Thus if $(x,\rho)\in
X\times\hat{\mathcal{H}}$ and if we define
$\psi(x,\rho)=\ker\bigl{(}\operatorname{Ind}_{(x,S_{x})}^{\mathcal{G}}(\rho|_{{S_{x}}})\bigr{)}$
as in [wil:tams81], then $\psi(x,\rho)=\kappa(x,\rho|_{{G(x)}})$.

We let $\Lambda_{\mathcal{H}}$ be the quotient of $X\times\hat{\mathcal{H}}$
were we identify $(x,\rho)$ and $(y,\sigma)$ if $\overline{\mathcal{H}\cdot
x}=\overline{\mathcal{H}\cdot y}$ and $\sigma$ and $\rho$ agree on $S_{x}$
(which is necessarily the same as $S_{y}$ since $\overline{\mathcal{H}\cdot
x}=\overline{\mathcal{H}\cdot y}$ and $\mathcal{H}$ is abelian). We let
$[x,\rho]$ be the class of $(x,\rho)$ in $\Lambda_{\mathcal{H}}$. As discussed
following [wil:tams81]*Corollary 5.9, there is a continuous
$\mathcal{G}$-action on $\Lambda_{\mathcal{H}}$ given by
$g\cdot[x,\rho]=[g\cdot x,g\cdot\rho]$.

######  Proposition 8.1 ([wil:tams81]*§5).

Let $\mathcal{G}$, $\mathcal{H}$, and $\Lambda_{\mathcal{H}}$ be as above.

  1. (a)

The natural map of $X\times\hat{\mathcal{H}}$ onto $\Lambda_{\mathcal{H}}$ is
open.

  2. (b)

The natural map
$\alpha:X\times\hat{\mathcal{H}}\to(\mathcal{G}\backslash\Lambda_{\mathcal{H}})^{\sim}$
is an open map.

  3. (c)

$\alpha(x,\rho)\mapsto\kappa(x,\rho|_{{G(x)}})$ is a homemorphism of
$(\mathcal{G}\backslash\Lambda_{\mathcal{H}})^{\sim}$ onto
$\operatorname{Prim}C^{*}(G)$.

  4. (d)

If $\alpha(x,\rho)=\alpha(y,\sigma)$, then there are $g_{n}\in\mathcal{G}$ and
$\rho_{n}\in\hat{\mathcal{H}}$ such that $(g_{n}\cdot
x,\rho_{n})\to(y,\sigma)$ in $X\times\hat{\mathcal{H}}$ and such that
$\rho_{n}$ and $g_{n}\cdot\rho$ agree on $S_{g_{n}\cdot x}$.

###### Proof.

(a) This follows from [wil:tams81]*Theorem 5.3 and the discussion following
its proof.

(b) As in the first paragraph of the proof of [wil:tams81]*Corllary 5.11, this
follows from [wil:tams81]*Corollary 5.9 and part (a).

(c) This is [wil:tams81]*Corollary 5.10.

(d) If $\alpha(x,\rho)=\alpha(y,\sigma)$, then there are $g_{n}\in\mathcal{G}$
such that $[g_{n}\cdot x,g_{n}\cdot\rho]$ converges to $[y,\sigma]$ in
$\Lambda_{H}$. Since the natural map of $X\times\hat{\mathcal{H}}$ onto
$\Lambda_{H}$ is open by part (a), we can pass to a subseqence, relabel, and
assume that there are $(z_{n},\rho_{n})\in X\times\hat{\mathcal{H}}$ such that
$(z_{n},\rho_{n})\to(y,\sigma)$ in $X\times\hat{\mathcal{H}}$,
$\overline{H\cdot z_{n}}=\overline{H\cdot(g_{n}\cdot x)}$, and
$g_{n}\cdot\rho$ and $\rho_{n}$ agree on $S_{z_{n}}=S_{g_{n}\cdot x}$. Since
$X$ is a metric space, there are $h_{n}\in\mathcal{H}$ such that
$h_{n}g_{n}\cdot x\to y$. Since $\mathcal{H}$ is abelian and all the stability
groups are contained in $\mathcal{H}$, $h_{n}g_{n}\cdot\rho=g_{n}\cdot\rho$
and $S_{h_{n}g_{n}\cdot x}=S_{g_{n}\cdot x}$. Thus $(h_{n}g_{n}\cdot
x,\rho_{n})\to(y,\sigma)$ in $X\times\hat{\mathcal{H}}$ and
$h_{n}g_{n}\cdot\rho$ and $\rho_{n}$ agree on $S_{h_{n}g_{n}\cdot x}$. ∎

######  Lemma 8.2.

Suppose that $(x_{n},\rho_{n})\to(x_{0},\rho_{0})$ in
$X\times\hat{\mathcal{H}}$. Then
$(x_{n},\rho_{n}|_{{S_{x_{n}}}})\to(x,\rho_{0}|_{{S_{n}}})$ in
$\operatorname{Stab}(G)$.

###### Proof.

For convenience, let $\pi_{n}=\rho_{n}|_{{S_{x_{n}}}}$. We clearly have
$x_{n}\to x$. If we have passed to a subsequence and relabeled, we can pass to
another so that $S_{n_{k}}\to S\subset S_{x_{0}}$. But then
$(H(x_{n_{k}}),\pi_{n_{k}})\to(S,\pi_{0}|_{{S}})$ in $\widehat{\Sigma}$. So
the result follows from Proposition 3.4. ∎

######  Proposition 8.3.

Suppose that $G=G(\mathcal{G},X)$ is an amenable second countable action
groupoid such that the stability groups are all contained in a fixed abelian
subgroup of $\mathcal{G}$. If $\tilde{\kappa}$ is as in Proposition 4.1, then
$\tilde{\kappa}:(G\backslash\operatorname{Stab}(G))^{\sim}\to\operatorname{Prim}C^{*}(G)$
is a homeomorphism.

###### Proof.

Since $G$ is second countable and amenable, it is not hard to see that
$C_{0}(X)\rtimes_{\operatorname{lt}}\mathcal{G}$ is EH-regular (see
[ionwil:iumj09]). Of course, since $\mathcal{H}$ is abelian,
$C_{0}(X)\rtimes_{\operatorname{lt}}\mathcal{H}$ is necessarily EH-regular.
Furthermore, we may assume that $\mathcal{H}$ is normal and apply the results
from [wil:tams81] via Proposition 8.1.

To see that $\kappa$ is open, we proceed as in the proof of Theorem 7.3.
Suppose that $V$ is a neighborhood of $(x_{0},\pi_{0})$ in
$\operatorname{Stab}(G)$ and that
$\kappa(x_{n},\pi_{n})\to\kappa(x_{0},\pi_{0})$. Since we started with an
arbitrary sequence, it will suffice to show that $\kappa(x_{n},\pi_{n})$ has a
subsequence which is eventually in $\kappa(V)$.

Since every character on $S_{x_{n}}$ is the restriction of some
$\rho\in\hat{\mathcal{H}}$, we can replace $\pi_{n}$ with
$\rho_{n}|_{{G(x_{n})}}$ for some $\rho_{n}\in\hat{\mathcal{H}}$. Let
$\alpha:X\times\hat{\mathcal{H}}\to(\mathcal{G}\backslash\Delta)^{\sim}$ be
the natural map. Then in view of Proposition 8.1(c), we can assume that
$\alpha(x_{n},\rho_{n})\to\alpha(x_{0},\rho_{0})$. Since $\alpha$ is open, we
can pass to a subsequence, and relabel, so that there is a sequence
$(y_{n},\sigma_{n})\to(x_{0},\rho_{0})$ in $X\times\hat{\mathcal{H}}$ with
$\alpha(y_{n},\sigma_{n})=\alpha(x_{n},\rho_{n})$. Since $X\times\mathcal{H}$
is a metric space, we can use Proposition 8.1(d) to find $g_{n}\in\mathcal{G}$
such that $(g_{n}\cdot x_{n},\rho_{n}^{\prime})\to(x_{0},\rho_{0})$ and such
that for all $n$, $\rho_{n}^{\prime}$ and $g_{n}\cdot\rho_{n}$ agree on
$S_{g\cdot x_{n}}$. It follows from Lemma 8.2 that
$g_{n}\cdot(x_{n},\rho_{n}|_{{S_{x_{n}}}})\to(x,\rho_{0}|_{{S_{x}}})=(x,\pi_{0})$
in $\operatorname{Stab}(G)$. Then $g_{n}\cdot(x_{n},\rho_{n}|_{{S_{x_{n}}}})$
is eventually in $V$. Since $\kappa$ is $\mathcal{G}$-invariant,
$\kappa(x_{n},\rho_{n})$ is eventually in $\kappa(V)$. Thus, $\kappa$ is open.

To see that $\tilde{\kappa}$ is injective, suppose that
$\kappa(x,\rho|_{{S_{x}}})=\kappa(y,\sigma|_{{S_{y}}})$. Then
$\alpha(x,\rho)=\alpha(y,\sigma)$ and we can use Proposition 8.1(d) to find
$g_{n}$ such that $(g_{n}\cdot x,\rho_{n})\to(y,\sigma)$ in
$X\times\mathcal{H}$ such that $\rho_{n}$ and $g_{n}\cdot\rho$ agree on
$S_{g_{n}\cdot x}$. But then $g_{n}\cdot(x,\rho|_{{S_{n}}})$ converges to
$(y,\sigma|_{{S_{y}}})$ in $\operatorname{Stab}(G)$ by Lemma 8.2. Therefore
$(y,\sigma|_{{S_{y}}}\in\overline{G\cdot(x,\rho|_{{S_{x}}})}$. Since the
situation is symmetric, we are done. ∎

##  9\. Generalized Proper Groupoids

Let $\pi:G\to G^{(0)}\times G^{(0)}$ be given by
$\pi(\gamma)=(r(\gamma),s(\gamma))$. The image, $R_{G}=\pi(G)$, is an
equivalence relation on $G^{(0)}$ and a topological groupoid when $R_{G}$ is
given the relative product topology.333It is more common to equip $R_{G}$ with
the often finer quotient topology. However, it is the relative product
topology that is suitable for our needs here. Of course, in general, $R_{G}$
may not be locally compact. In fact, $R_{G}$ is closed in $G^{(0)}\times
G^{(0)}$ if and only if $G\backslash G^{(0)}$ is Hausdorff ([wil:toolkit]*Ex
2.1.10). Part of our interest in $\pi$ is due to the following straightforward
observation.

######  Lemma 9.1.

If $G$ has abelian isotropy and $H\in\Sigma_{0}$, then $\gamma\cdot H$ depends
only on $\pi(\gamma)$.

It follows we get a well-defined action of $R_{G}$ on $\Sigma_{0}$:
$\pi(\gamma)\cdot H=\gamma\cdot H$.

######  Lemma 9.2.

Suppose that the action of $R_{G}$ on $\Sigma_{0}$ is continuous and that
$H_{n}\to H_{0}$ in $\Sigma_{0}$. Let $u_{n}=p_{0}(H_{n})$. If $(\gamma_{n})$
is a sequence in $G$ such that $\gamma_{n}\cdot u_{n}\to\gamma_{0}\cdot
u_{0}$, then $\gamma_{n}\cdot H_{n}\to\gamma_{0}\cdot H_{0}$

###### Proof.

The convoluted hypotheses just imply that $\pi(\gamma_{n})\to\pi(\gamma_{0})$
in $R_{G}$. ∎

Recall that a locally compact groupoid is a proper if it acts properly on its
unit space. For example, the action groupoid $G=G(\mathcal{G},X)$ is proper if
and only if $\mathcal{G}$ acts properly on $X$. Note that if $G$ is a proper
groupoid, then all of its isotropy groups are compact.

######  Lemma 9.3.

Suppose that $G$ is a proper groupoid with abelian isotropy. Then $G\backslash
G^{(0)}$ is Hausdorff and $R_{G}$ acts continuously on $\Sigma_{0}$.

###### Proof.

If $G$ is proper, then $G\backslash G^{(0)}$ is Hausdorff
[wil:toolkit]*Proposition 2.18. To see that the $R_{G}$-action is continuous,
suppose that $H_{n}\to H_{0}$ in $\Sigma_{0}$ with $u_{n}=p_{0}(H_{n})$.
Suppose $\pi(\gamma_{n})=(v_{n},u_{n})\to\pi(\gamma_{0})=(v_{0},u_{0})$. We
need to verify that $\pi(\gamma_{n})\cdot H_{n}\to\pi(\gamma_{0})\cdot H_{0}$.
After passing to a random subsequence and relabeling, it will suffice to
produce a subsequence with this property. Since $u_{n}\to u_{0}$ and
$v_{n}=\gamma_{n}\cdot u_{n}\to v_{0}$, it follows from
[wil:toolkit]*Proposition 2.17 that we can pass to a subsequence and assume
$\gamma_{n}\to\eta_{0}$ with $\pi(\gamma_{0})=\pi(\eta_{0})$. Since the action
of $G$ on $\Sigma_{0}$ is continuous, $\gamma_{n}\cdot H_{n}\to\eta_{0}\cdot
H_{0}=\gamma_{0}\cdot H_{0}$. This completes the proof. ∎

Motivated in part by the previous lemma, we make the following definition.

######  Definition 9.4.

We say that $G$ is proper modulo its isotropy if $G\backslash G^{(0)}$ is
Hausdorff and $R_{G}$ acts continuously on $\Sigma_{0}$.

To both justify our terminology, and to see that our next theorem has wide
applicability, we want to exhibit the following class of examples based on the
unit space fixing extensions studied extensively in [iksw:jot19, ikrsw:jfa20,
ikrsw:xx21].

######  Example 9.5.

As usual, suppose that $G$ has abelian isotropy. Suppose that $\mathcal{G}$ is
a proper groupoid and that there is a continuous open groupoid epimorphism
$p:G\to\mathcal{G}$ that restricts to a homeomorphism of $G^{(0)}$ with
$\mathcal{G}^{(0)}$. Then $\mathcal{A}=\ker p=\\{\,\gamma\in
G:p(\gamma)\in\mathcal{G}^{(0)}\,\\}$ is contained in the isotropy subgroupoid
$G^{\prime}=\\{\,\gamma\in G:r(\gamma)=s(\gamma)\,\\}$. It follows that
$\mathcal{A}$ is an abelian group bundle and we have a diagram

(9.1)
${\mathcal{A}}$${G}$${\mathcal{G}}$${G^{(0)}}$$\scriptstyle{\iota}$$\scriptstyle{p}$

so that we can view $G$ as a unit space fixing extension of $\mathcal{G}$ by
$\mathcal{A}$. Our next lemma confirms that such groupoids are always examples
of groupoids that are proper modulo their isotropy.

######  Lemma 9.6.

Let $G$ be as in Example 9.5. Then $G$ is proper modulo its isotropy.

###### Proof.

We can use $p|_{{G^{(0)}}}$ to identify $G^{(0)}$ and $\mathcal{G}^{(0)}$.
Then $r(\gamma)=r(p(\gamma))$ and $s(\gamma)=s(p(\gamma))$. Then we can
identify $G\backslash G^{(0)}$ and $\mathcal{G}\backslash G^{(0)}$. Since
$\mathcal{G}$ is proper, it follows $G\backslash G^{(0)}$ is Hausdorff as in
Lemma 9.3.

Note that $p(\gamma)=p(\eta)$ if and only if there is a $a\in\mathcal{A}$ such
that $\eta=\gamma a$.

To see that $R_{G}$ acts continuously, we make the appropriate modifications
to the second part of the proof of Lemma 9.3. Suppose that $H_{n}\to H_{0}$ in
$\Sigma_{0}$ with $u_{n}=p_{0}(H_{n})$. Suppose
$\pi(\gamma_{n})=(v_{n},u_{n})\to\pi(\gamma_{0})=(v_{0},u_{0})$. We need to
see that after passing to a subsequence and relabeling, $\pi(\gamma_{n})\cdot
H_{n}\to\pi(\gamma_{0})\cdot H_{0}$.

Since $\mathcal{G}$ is proper, we can assume that $p(\gamma_{n})\to
p(\eta_{0})$ and
$\pi_{\mathcal{G}}(p(\gamma_{0}))=\pi_{\mathcal{G}}(p(\eta_{0}))$. Since $p$
is open, we can pass to subsequence, relabel, and assume that there are
$\eta_{n}\to\eta_{0}$ in $G$ such that $p(\eta_{n})=p(\gamma_{n})$. In
particular, there are $a_{n}\in\mathcal{A}$ such that
$\gamma_{n}a_{n}\to\eta_{0}$. Thus $\gamma_{n}\cdot
H_{n}=(\gamma_{n}a_{n})\cdot H_{n}\to\eta_{0}H_{0}=\gamma_{0}\cdot H_{0}$. ∎

######  Remark 9.7.

Notice that for all $u\in G^{(0)}$, $\mathcal{A}(u)$ is a subgroup of $G(u)$.
Since $\mathcal{G}(u)=p(G(u))$, we must have $G(u)/\mathcal{A}(u)$ compact.
While we can always form the groupoid $G^{\prime}\backslash G$, it may not
even be locally compact if the isotropy map is not continuous.

######  Theorem 9.8.

Suppose that $G$ is a second countable locally compact Hausdorff groupoid with
a Haar system and abelian isotropy. Suppose that $G$ is proper modulo its
isotropy. Then $\underline{\operatorname{Ind}}$ is a homeomorphism of
$G\backslash\operatorname{Stab}(G)$ onto $C^{*}(G)^{\wedge}$.

###### Proof of Theorem 9.8.

In view of Corollary 4.2, it suffices to see that $\operatorname{Ind}$ is an
open map. Since $C^{*}(G)$ is GCR, we can identify $C^{*}(G)^{\wedge}$ and
$\operatorname{Prim}C^{*}(G)$, so it will suffice to see that $\kappa$ (as
defined in Proposition 4.1) is open. As in the proof of Theorem 7.3, it will
suffice, given a sequence $\bigl{(}\kappa(u_{n},\pi_{n})\bigr{)}$ converging
to $\kappa(u_{0},\pi_{0})$, to find a subsequence
$\bigl{(}(u_{n_{k}},\pi_{n_{k}})\bigr{)}$ and a sequence
$\bigl{(}(v_{k},\sigma_{k})\bigr{)}$ in $\operatorname{Stab}(G)$ such that
$(v_{k},\sigma_{k})\to(u_{0},\pi_{0})$ and
$\kappa(v_{k},\sigma_{k})=\kappa(u_{n_{k}},\pi_{n_{k}})$.

Using Corollary 6.10, we can pass to a subsequence, relabel, and assume that
there are $\gamma_{n}\in G$ such that $\gamma_{n}\cdot u_{n}\to u_{0}$. Since
$\kappa(\gamma_{n}\cdot u_{n},\gamma_{n}\cdot\pi_{n})=\kappa(u_{n},\pi_{n})$,
we can replace $(u_{n},\pi_{n})$ with $(\gamma_{n}\cdot
u_{n},\gamma_{n}\cdot\pi_{n})$ and assume that $u_{n}\to u_{0}$.

Suppose that there are infinitely many $n$ such that $[u_{n}]=[v]$ for some
$v\in G^{(0)}$ (where $[u]$ denotes the orbit of $u$ in $G^{(0)}$). Then we
can pass to a subsequence, relabel, and assume that for all $n$ there are
$\gamma_{n}\in G$ such that $\gamma_{n}\cdot v=u_{n}$. Then $\gamma_{n}\cdot
v\to u_{0}$ and $u_{0}\in\overline{[v]}=[v]$. (Orbits are closed since
$G\backslash G^{(0)}$ is Hausdorff.) Hence for all $n$, $[u_{n}]=[u_{0}]=[v]$.
Replacing $(u_{n},\pi_{n})$ with $(\eta_{n}\cdot u_{n},\eta_{n}\cdot\pi_{n})$
for appropriate $\eta_{n}\in G$, we can assume $u_{n}=u_{0}$ for all $n$.

Now we can replace $G$ with $G|_{{[u_{0}]}}$ and assume
$\operatorname{Ind}(u_{0},\pi_{n})\to\operatorname{Ind}(u_{0},\pi_{0})$ as
irreducible representations of $C^{*}(G|_{{[u_{0}]}})$. Since $G|_{{[u_{0}]}}$
is equivalent to $G(u_{0})$, this means $\pi_{n}\to\pi_{0}$ in
$G(u_{0})^{\wedge}$ (by [rw:morita]*Theorem 3.29).

This provides the required lift in the case infinitely many orbits $[u_{n}]$
coincide.

Otherwise, we can pass to a subsequence, relabel, and assume that $n>m\geq 0$
implies $[u_{m}]\not=[u_{n}]$. We can also assume $G(u_{n})\to H\subset
G(u_{0})$.

We let $C=\\{\,u_{n}:n\geq 0\,\\}$ and $F=G\cdot C$. We claim that $F$ is
closed. Suppose that $\gamma_{k}\cdot v_{k}\to v$ with each $v_{k}\in C$. If
$v_{k}=u_{j}$ for infinitely many $k$, then $v\in\overline{[u_{j}]}=[u_{j}]$
and $v\in F$. Otherwise, we can pass to a subsequence, relabel, and assume
$v_{k}=u_{n_{k}}$ with $u_{n_{k}}\to u_{0}$. Then
$\bigl{(}[u_{n_{k}}]\bigr{)}$ converges to both $[u_{0}]$ and $[v]$ in
$G\backslash G^{(0)}$. Since the latter is Hausdorff, this implies
$[v]=[u_{0}]$ and $v\in F$ in this case as well. This proves that $F$ is
closed as claimed.

In view of Lemma 9.1, we get a well-defined function
$\mathbb{H}:F\to\Sigma_{0}$ given by

(9.2) $\mathbb{H}(\gamma\cdot u_{n})=\begin{cases}\gamma\cdot
G(u_{n})&\text{if $n\geq 1$, and}\\\ \gamma\cdot H&\text{if
$n=0$.}\end{cases}$

We claim that $\mathbb{H}$ is continuous. If not, there is a sequence
$\gamma_{k}\cdot v_{k}\to\gamma_{0}\cdot v_{0}$ with each $v_{k}\in C$ and
such that no subsequence of $\bigl{(}\gamma_{k}\cdot\mathbb{H}(v_{k})\bigr{)}$
converges to $\gamma_{0}\cdot\mathbb{H}(v_{0})$. If there are infinitely many
$k$ such that $v_{k}=u_{n}$, then we can pass to subsequence, relabel, and
assume $\gamma_{k}\cdot u_{n}\to\gamma_{0}\cdot v_{0}$. Then $\gamma_{0}\cdot
v_{0}\in\overline{[u_{n}]}=[u_{n}]$. Then we can replace $v_{0}$ with $u_{n}$
and adjust $\gamma_{0}$ so that $\gamma_{k}\cdot u_{n}\to\gamma_{0}\cdot
u_{n}$. Since $G\backslash G^{(0)}$ is Hausdorff, the Mackey-Glimm-Ramsay
Dichotomy [wil:toolkit]*Theorem 2.27 implies $\gamma\mapsto\gamma\cdot u_{n}$
is an open map. Hence we can pass to a subsequence, relabel, and assume that
$\gamma_{k}\to\gamma_{0}$. Then $\mathbb{H}(\gamma_{k}\cdot
u_{n})=\gamma_{k}\cdot\mathbb{H}(u_{n})\to\gamma_{0}\cdot\mathbb{H}(u_{n})=\mathbb{H}(\gamma_{0}\cdot
u_{n})$ which contradicts our choice of $\bigl{(}\gamma_{k}\cdot
v_{k}\bigr{)}$.

Now we assume that no $v_{k}$ is repeated infinitely often. Then we must have
$v_{k}\to u_{0}$. Since $\gamma_{k}\cdot v_{k}\to\gamma_{0}\cdot v_{0}$ and
$G\backslash G^{(0)}$ is Hausdorff, we can adjust $\gamma_{0}$ if need be and
replace $\gamma_{0}\cdot v_{0}$ with $\gamma_{0}\cdot u_{0}$. Since
$\mathbb{H}$ is clearly continuous on $C$, we have
$\mathbb{H}(v_{k})\to\mathbb{H}(u_{0})$. Since $R_{G}$ acts continuously by
assumption,
$\gamma_{k}\cdot\mathbb{H}(v_{k})\to\gamma_{0}\cdot\mathbb{H}(u_{0})$. Again,
this contradicts our assumptions on $\bigl{(}\gamma_{k}\cdot v_{k}\bigr{)}$.
Hence $\mathbb{H}$ is continuous on $F$ as claimed.

Therefore $\mathbb{H}$ is an equivariant subgroup section for $G|_{{F}}$.
Since $F$ is closed and $G$-invariant, we can replace $G$ by $G|_{{F}}$. Let
$P^{*}:\mathcal{I}(C^{*}(G|_{{F}}))\to\mathcal{I}(C_{0}(\widehat{\Sigma}_{F})$
be the corresponding map. Since
$\kappa(u_{k},\pi_{k})\to\kappa(u_{0},\pi_{0})$ in
$\operatorname{Prim}\bigl{(}C^{*}(G|_{{F}})\bigr{)}$, we have
$P^{*}(\kappa(u_{k},\pi_{k}))\to P^{*}(\kappa(u_{0},\pi_{0}))$ in
$\mathcal{I}(C_{0}(\widehat{\Sigma}_{F})$. By Proposition 6.6, if $k\geq 1$,
then $P^{*}(\kappa(u_{k},\pi_{k})$ is the ideal of functions in
$C_{0}(\widehat{\Sigma}_{F})$ vanishing on
$G|_{{F}}\cdot(G(u_{k}),\pi_{k})=G\cdot(G(u_{k}),\pi_{k})$. Similarly,
$P^{*}(\kappa(u_{0},\pi_{0}))$ is the ideal vanishing on
$H\cdot(H,\pi_{0}|_{{H}})$. We can use Lemma 2.3 to pass to a subsequence,
relabel, so that there are $\gamma_{k}\in G$ such that
$\gamma_{k}\cdot(G(u_{k}),\pi_{k})\to(H,\pi_{0}|_{{H}})$ in
$\widehat{\Sigma}$. But then $(u_{k},\pi_{k})\to(u_{0},\pi_{0})$ in
$\operatorname{Stab}(G)$ (by Proposition 3.4). Since
$\kappa(\gamma_{k}\cdot(u_{k},\pi_{k}))=\kappa(u_{k},\pi_{k})$, this completes
the proof. ∎

Before closing out this section, we note that there are natural circumstances
where $R_{G}$ acts continuously without having to assume either that $G$ is
proper or invoking Example 9.5.

######  Lemma 9.9.

Let $(\mathcal{G},X)$ be a locally compact transformation group with
$\mathcal{G}$ abelian, and let $G=G(\mathcal{G},X)$ be the corresponding
action groupoid as in Section 8. Then $R_{G}$ acts continuously on
$\Sigma_{0}$.

######  Remark 9.10.

It would suffice for all the $S_{x}$ to be central in $G$.

###### Proof.

As usual, we identify $G^{(0)}=\\{\,(x,e,x)\in G:x\in X\,\\}$ with $X$. Then
the isotropy group $G(x)$ is $\\{\,(x,h,x):h\in S_{x}\,\\}$ where
$S_{x}=\\{\,h\in\mathcal{G}:h\cdot x=x\,\\}$ is the stability group at $x$.
Let $c:G\to\mathcal{G}$ be the cocycle $c(y,g,x)=g$. Note that $c$ restricts
to an isomorphism of $G(x)=\\{\,(x,g,x):x\in S_{x}\,\\}$ with $S_{x}$. More
generally, if $H\in\Sigma_{0}$ and $H\subset G(x)$, then $c(H)$ is a subgroup
of $\mathcal{G}(x)$. Using [wil:toolkit]*Lemma 3.22 for example, we have
$H_{n}\subset G(x_{n})$ converging to $H\subset G(x)$ if and only if $x_{n}\to
x$ and $c(H_{n})\to c(H)$ as subgroups of $\mathcal{G}$. Since

(9.3) $(y,g^{-1},x)(x,h,x)(x,g,y)=(y,g^{-1}hg,y)=(y,h,y)$

it follows that $R_{G}$ acts continuously on $\Sigma_{0}$. ∎

Recall that a continuous surjection $f:X\to Y$ has _local sections_ if for
each $y\in Y$ there is a neighborhood $V\subset Y$ and a continuous function
$c:V\to X$ such that $f(c(z))=z$ for all $z\in V$.

######  Lemma 9.11.

Suppose that $\pi:G\to R_{G}$ has local sections. Then $R_{G}$ acts
continuously on $\Sigma_{0}$.

###### Proof.

Suppose that $H_{n}\to H_{0}$ with $p_{0}(H_{n})=u_{n}$. Suppose also that
$\pi(\gamma_{n})=(v_{n},u_{n})\to\pi(\gamma_{0})=(u_{0},v_{0})$. Let $U$ be a
neighborhood of $(v_{0},u_{0})$ and $c:U\to G$ a local section for $\pi$. We
can assume that $\bigl{(}(v_{n},u_{n})\bigr{)}\subset U$ and let
$\eta_{n}=c(v_{n},u_{n})$. Then $\eta_{n}\to\eta_{0}$ and
$\pi(\eta_{n})=\pi(\gamma_{n})$ for all $n\geq 0$. Then

(9.4) $\displaystyle\pi(\gamma_{n})\cdot H_{n}$ $\displaystyle=\eta_{n}\cdot
H_{n}\to\eta_{0}\cdot H_{0}=\pi(\gamma_{0})\cdot H_{0}.\qed$

##  10\. Examples

In order to see that Theorem 7.3 has new interesting applications, we want to
exhibit some elementary examples where the isotropy is continuous except for
jump discontinuities as in Definition 7.1. For ease of exposition, all of the
constructions here come from directed graphs, where the primitive ideal space
of the corresponding groupoid $C^{*}$-algebra can be computed via
[honszy:jmsj04]. Nevertheless, these examples show the phenomena is reasonably
common.

The groupoids we will construct will be étale groupoids and are now usually
called Deaconu–Renault groupoids. The construction originated from directed
graphs in [kprr:jfa97] and emerged on its own in [ren:otm00]. For a more
general construction, one can consult [renwil:tams16]*§5. We review the basics
for the constructions we need here.

We let $X$ be a locally compact space and let $T:\operatorname{dom}(T)\subset
X\to X$ be a local homeomorphism with $\operatorname{dom}(T)$ and
$\operatorname{ran}(T)=T(\operatorname{dom}(T))$ open subsets. Let
$\mathbf{N}_{0}=\\{\,0,1,2,\dots\,\\}$. Then

(10.1) $G_{T}=\\{\,(x,m-n,y)\in X\times\mathbf{Z}\times
X:\text{$T^{m}x=T^{n}y$ with $m,n\in\mathbf{N}_{0}$}\,\\}$

is a groupoid with respect to the natural operations. The sets

(10.2) $Z(U,m,n,V)=\\{\,(x,m-n,y)\in G_{T}:T^{m}x=T^{n}y\,\\}$

with $U$ and $V$ open in $X$ and $m,n\in\mathbf{N}_{0}$ form a basis for a
locally compact Hausdorff topology on $G_{T}$ with respect to which $G_{T}$ is
an étale groupoid, and such that the map $c:G_{\mathbf{T}}\to\mathbf{Z}$ given
by $c(x,k,y)=k$, is a continuous cocycle on $G_{T}$. Furthermore, the groupoid
$G_{T}$ is always amenable (see [ren:otm00]*Proposition 2.9 or
[renwil:tams16]*Theorem 5.13) so that our results apply. This construction of
$G_{T}$ from $X$ and $T$ is an example of a Deaconu–Renault groupoid.

Given a general directed graph $E$, there is now a well understood way of
associating to it a Deaconu–Renault groupoid $G_{T}$ as above. We let $X$ be
the boundary path space $\partial E$ and $T$ the edge shift. Although we don’t
actually require them here, the details of the construction can be found in
[bcw:etds17] or [web:pams14].

$v$$e$$g$$w$$f$ Figure 1. A Simple Graph

To start, let $E$ be the graph in Figure 1 which comes from
[simwil:art15]*Example 3.4. In this case, the boundary paths are all infinite
paths and consist of $e^{\infty}$, $g^{\infty}$, and $g^{n-1}fe^{\infty}$ for
$n\geq 1$. Topologically,
$\\{\,e^{\infty}\,\\}\cup\\{\,g^{n-1}fe^{\infty}\,\\}$ is discrete while every
neighborhood of $g^{\infty}$ contains $\\{\,g^{n-1}fe^{\infty}\,\\}$ for
$n\geq N$ for some $N$. In particular,
$\lim_{n\to\infty}g^{n-1}fe^{\infty}=g^{\infty}$. At this point, we can forget
about the graph and reduce to simply specifying a space $X$ and local
homeomorphism $T$. Then we can construct $G_{T}$ as above.

######  Example 10.1 ([simwil:art15]*Example 3.4).

We let $X=\partial E$ and think of $x_{n}=g^{n-1}fe^{\infty}$ as a point in
$X$ for $n\geq 1$. Then $X$ is the space
$\\{\,x_{0},x_{1},x_{2},\dots,x_{\infty}\,\\}$ where
$X\setminus\\{x_{\infty}\\}$ is discrete and the open neighborhoods of
$x_{\infty}$ are all of the form $\\{\,x_{k}:k\geq N\,\\}$ for some integer
$N\geq 0$. The map $T:X\to X$ is given by

(10.3) $T(x_{k})=\begin{cases}x_{0}&\text{if $k=0$,}\\\ x_{k-1}&\text{if
$1\leq k<\infty$, and}\\\ x_{\infty}&\text{if $k=\infty$.}\end{cases}$

Since $T^{k}x_{k}=x_{0}$ for all $0\leq k<\infty$, we have
$(x_{0},-k,x_{k})\in G_{T}$ and $[x_{0}]=\\{\,x_{k}:0\leq k<\infty\,\\}$. On
the other hand, $[x_{\infty}]=\\{x_{\infty}\\}$. Note that $[x_{0}]$ is dense
and the orbit space $G_{T}\backslash X$ is $T_{0}$ consisting of the dense
point $[x_{0}]$ and the closed point $[x_{\infty}]$.

If $0\leq k<\infty$, then $T^{n}x_{k}=x_{0}$ for all $n\geq k$. Hence
$(x_{k},m+k-(n+k),x_{k})\in G_{T}$ for all $m,n\in\mathbf{N}$. Thus
$G_{T}(x_{k})=\\{\,(x_{k},l,x_{k}):l\in\mathbf{Z}\,\\}$ for all $0\leq
k<\infty$. This clearly holds for $x_{\infty}$ as well.

Now let $(x_{\infty},m-n,x_{\infty})\in G_{T}(x_{\infty})$ and let
$Z(U,m,n,V)$ be a basic open neighborhood of $(x_{\infty},m-n,x_{\infty})$.
Let $k>\max\\{\,m,n\,\\}$. Then $T^{m}x_{k}=T^{n}x_{k}$ only if $m=n$. Thus if
$m\not=n$, then $G_{T}(x_{k})\cap Z(U,m,n,V)=\emptyset$ if
$k>\max\\{\,m,n\,\\}$. It follows that
$G_{T}(x_{k})\to\\{(x_{\infty},0,x_{\infty})\\}$.

Since $\\{\,x_{k}:0\leq k<\infty\,\\}$ is discrete, we have established that
$G_{T}$ has isotropy which is continuous except for a jump discontinuity at
$x_{\infty}$.

Now we can identify $\operatorname{Stab}(G_{T})$ with $X\times\mathbf{T}$ and
the sets $\\{x_{k}\\}\times\mathbf{T}$ are each homeomorphic to $\mathbf{T}$
for all $k$ and open if $k<\infty$. Consider a sequence
$\bigl{(}(x_{k},z_{k})\bigr{)}$. Recall that $\bigl{(}G_{T}(x_{k})\bigr{)}$
converges to the trivial subgroup of $G_{T}(x_{\infty})$. Thus if $a_{k}\in
G_{T}(x_{k})$ and $a_{k}$ converges, it must converge to the identity in
$G_{T}(x_{\infty})$. Since the cocycle $c:G_{T}\to\mathbf{Z}$ is continuous,
$a_{k}$ must eventually be the identity of $G_{T}(x_{k})$. It follows that
$\bigl{(}(x_{k},z_{k})\bigr{)}$ converges to $(x_{\infty},w)$ for all
$w\in\mathbf{T}$ by Proposition 3.4. ∎

The construction of the boundary path space is a bit more complicated if we
allow our graph to have sinks. For example, let $E$ be the graph in Figure 2.
The boundary path space for the graph is

(10.4) $\partial
E=\\{\,(e_{1}e_{2})^{n}h,e_{2}(e_{1}e_{2})^{n}h,(e_{1}e_{2})^{\infty},(e_{2}e_{1})^{\infty}:n\in\mathbf{N}_{0}\,\\}.$

As in the previous example, the construction of $G_{T}$ becomes a bit more
transparent if we just describe the boundary path space as a space without
reference to the graph.

$e_{1}$$e_{2}$$h$$u$$v$$z$ Figure 2. A Graph With a Sink

######  Example 10.2.

We let $x_{n}=(e_{1}e_{2})^{n}h$ for $n\in\mathbf{N}_{0}$,
$x_{\infty}=(e_{1}e_{2})^{\infty}$, $y_{n}=e_{2}(e_{1}e_{2})^{n}h$ for
$n\in\mathbf{N}_{0}$, and $y_{\infty}=(e_{2}e_{1})^{\infty}$. Then

(10.5)
$X=\\{\,x_{0},x_{1},\dots,x_{\infty}\,\\}\cup\\{\,y_{0},y_{1},\dots,y_{\infty}\,\\}$

where $X\setminus\\{\,x_{\infty},y_{\infty}\,\\}$ is discrete and
$\\{\,x_{k}:k\geq N\,\\}$ and $\\{\,y_{k}:k\geq N\,\\}$ are neighborhoods of
$x_{\infty}$ and $y_{\infty}$, respectively, for all $N\in\mathbf{N}$. If we
let $\operatorname{dom}(T)=X\setminus\\{x_{0}$}, then we define a local
homeomorphism $T:\operatorname{dom}(T)\subset X\to X$ by

(10.6) $T(x)=\begin{cases}y_{k-1}&\text{if $x=x_{k}$ with $1\leq
k<\infty$},\\\ y_{\infty}&\text{if $x=x_{\infty}$},\\\ x_{k}&\text{if
$x=y_{k}$, with $0\leq k<\infty$, and}\\\ x_{\infty}&\text{if
$x=y_{\infty}$.}\end{cases}$

Thus there are only two orbits in $G_{T}^{(0)}$; namely, the dense orbit
$[x_{0}]=\\{\,x_{k},y_{k}:0\leq k<\infty\,\\}$ and the closed orbit
$[x_{\infty}]=\\{\,x_{\infty},y_{\infty}\,\\}$. Clearly, the orbit space is
$T_{0}$.

If $k<\infty$, then $T^{n}x_{k}$ is only defined if $n\leq 2k$. Then it is not
hard to see that

(10.7) $G_{T}(x_{k})=\\{\,(x_{k},0,x_{k})\,\\}\quad\text{for all $0\leq
k<\infty$.}$

Similarly, $G_{T}(y_{k})$ is trivial for all $0\leq k<\infty$. On the other
hand,

(10.8)
$G_{T}(x_{\infty})=\\{\,(x_{\infty},2k,x_{\infty}):k\in\mathbf{N}_{0}\,\\}$

and a similar formula holds for $G_{T}(y_{\infty})$. Hence is isotropy is
trivial except at $D=\\{\,x_{\infty},y_{\infty}\,\\}$ and $G_{T}$ has
continuous isotropy except for jump discontinuities as in Remark 7.2.

Furthermore, the sequence $\bigl{(}(x_{k},1)\bigr{)}$ converges to
$(x_{\infty},w)$ in $\operatorname{Stab}(G_{T})$ for all $w\in\mathbf{T}$.
Similarly, $\bigl{(}(y_{k},1)\bigr{)}$ converges to $(y_{\infty},w)$ for all
$w\in\mathbf{T}$. ∎

We can now combine the phenomena exhibited by the previous two examples by
starting with a loop and then attaching exits to sinks or additional loops.
Consider the graph in Figure 3. Now

(10.9) $\partial
E=\\{\,g^{n-1}f(e_{2}e_{1})^{\infty},g^{\infty},(e_{2}e_{1})^{\infty},e_{2}(e_{1}e_{2})^{n-1}h,(e_{1}e_{2})^{\infty},(e_{1}e_{2})^{n}h\,\\}.$

Now let $x_{\infty}=g^{\infty}$, $x_{n}=g^{n-1}f(e_{2}e_{1})^{\infty}$,
$x_{0}=(e_{2}e_{1})^{\infty}$, $z_{n}=e_{2}(e_{1}e_{2})^{n-1}h$,
$y_{\infty}=(e_{1}e_{2})^{\infty}$, and $y_{n}=(e_{1}e_{2})^{n}h$.

$e_{1}$$e_{2}$$h$$u$$v$$z$$f$$w$$g$ Figure 3. A Combination

######  Example 10.3.

We let

(10.10)
$X=\\{\,x_{0},x_{1},\dots,x_{\infty}\,\\}\cup\\{\,z_{0},z_{1},\dots\,\\}\cup\\{\,y_{0},y_{1},\dots,y_{\infty}\,\\}$

where $X\setminus\\{\,x_{\infty},x_{0},y_{\infty}\,\\}$ is discrete and
$\\{\,x_{k}:k\geq N\,\\}$, $\\{\,z_{k},k\geq N\,\\}$ and $\\{\,y_{k}:k\geq
N\,\\}$ are neighborhoods of $x_{\infty}$, $x_{0}$, and $y_{\infty}$,
respectively, for all $N\in\mathbf{N}$. Here
$\operatorname{dom}(T)=X\setminus\\{y_{0}\\}$ and

(10.11) $T(x)=\begin{cases}x_{\infty}&\text{if $x=x_{\infty}$,}\\\
x_{k-1}&\text{if $x=x_{k}$ for $1\leq k<\infty$,}\\\ y_{\infty}&\text{if
$x=x_{0}$,}\\\ y_{k-1}&\text{if $x=z_{k}$ for $1\leq k<\infty$,}\\\
y_{0}&\text{if $x=z_{0}$,}\\\ x_{0}&\text{if $x=y_{\infty}$, and}\\\
z_{k}&\text{if $x=y_{k}$ for $1\leq k<\infty$.}\end{cases}$

The orbits consist of $[x_{\infty}]=\\{x_{\infty}\\}$,
$[x_{0}]=\\{\,x_{k}:0\leq k<\infty\,\\}\cup\\{y_{\infty}\\}$, and
$[z_{0}]=\\{\,z_{k},y_{k}:0\leq k<\infty\,\\}$. As in the previous examples,
it is not hard to see that $G_{T}(z_{k})$ and $G_{T}(y_{k})$ are trivial if
$1\leq k<\infty$ while $G_{T}(y_{\infty})$ and $G_{T}(x_{k})$, for $0\leq
k\leq\infty$ are all isomorphic to $\mathbf{Z}$. However
$G_{T}(x_{k})\to\\{\,(x_{\infty},0,x_{\infty})\,\\}$.

To see that $G_{T}$ has continuous isotropy except for jump discontinuities,
we define $\mathbb{H}:G_{T}^{(0)}\to\Sigma_{0}$ by

(10.12) $\mathbb{H}(x)=\begin{cases}\\{(x_{\infty},0,x_{\infty})\\}&\text{if
$x=x_{\infty}$,}\\\ \\{(y_{\infty},0,y_{\infty})\\}&\text{if $x=y_{\infty}$,
and}\\\ G_{T}(x)&\text{otherwise.}\end{cases}$

It follows that any sequence of the form $\bigl{(}(x_{k},z_{k})\bigr{)}$
converges to $(x_{\infty},w)$ in $\operatorname{Stab}(G_{T})$ for any
$w\in\mathbf{T}$. On the other hand, $\bigl{(}(z_{k},1)\bigr{)}$ converges to
$(x_{0},w)$ in $\operatorname{Stab}(G_{T})$ for all $w\in\mathbf{T}$.
Similarly, $\bigl{(}(y_{k},1)\bigr{)}$ converges to $(y_{\infty},w)$ in
$\operatorname{Stab}(G_{T})$ for all $w\in\mathbf{T}$. ∎

These examples make it clear that more complicated examples of groupoids with
continuous isotropy except for jump discontinuities can be built by starting
with a loop and then attaching sinks or additional loops.

## References