# Orbit Equivalence of actions on Cartan pairs

Massoud Amini  Department of Mathematics, Faculty of Mathematical Sciences,
Tarbiat Modares University, Tehran 14115-134, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Mahdi Moosazadeh

###### Abstract.

We introduce and study the notion of continuous orbit equivalence of actions
of countable discrete groups on Cartan pairs in (twisted) groupoid context. We
characterize orbit equivalence of actions in terms of the corresponding
C∗-algebraic crossed products using Kumjian-Renault theory. We relate our
notion to the classical notion of orbit equivalence of actions on topological
spaces by showing that, under certain conditions, orbit equivalence of actions
on (twisted) groupoids follows from orbit equivalence of restricted actions on
unit spaces. We illustrate our results with concrete examples of continuous
orbit equivalent actions on groupoids coming from odometer transformations.

###### 2020 Mathematics Subject Classification:

Primary 46L55; Secondary 37A20

##  1\. Introduction

Dynamical systems have ever been a fruitful source of examples for operator
algebras. The very early examples in type theory of von Neumann algebras,
given by Murray and von Neumann, used group actions on measure spaces. Later,
W. Krieger showed that for two ergodic non-singular systems, the associated
von Neumann crossed product factors are isomorphic iff the systems are orbit
equivalent [12, 13], building upon an earlier result of H. Dye that all
ergodic p.m.p actions are orbit equivalent. Since then, orbit equivalence is
extensively studied both in the measurable [8] and topological [11], [3]
setting.

Two p.m.p. actions $\Gamma_{1}\curvearrowright X_{1}$ and
$\Gamma_{2}\curvearrowright X_{2}$ are called orbit equivalent if there exists
an isomorphism $\phi:X_{1}\rightarrow X_{2}$ of measure spaces sending orbits
to orbits (a.e.). For discrete countable groups acting essentially free, this
is known to be equivalent to the corresponding measure groupoids being
isomorphic, and to the existence of an isomorphism between the corresponding
von Neumann algebraic crossed products, preserving $L^{\infty}$ masas [21],
[23].

The notion of orbit equivalence in topological setting is studied for
$\mathbb{Z}^{d}$ actions on Cantor set in [10], [11], and in general in [15].
The actions $\Gamma_{1}\curvearrowright X_{1}$ and $\Gamma_{2}\curvearrowright
X_{2}$ of discrete groups on topological spaces are continuous orbit
equivalent if there exist a homeomorphism $\phi:X_{1}\rightarrow X_{2}$ and
continuous maps $a:\Gamma_{1}\times X_{1}\rightarrow\Gamma_{2}$ and
$b:\Gamma_{2}\times X_{2}\rightarrow\Gamma_{1}$ with
$\phi(\gamma_{1}x_{1})=a(\gamma_{1},x_{1})\phi(x_{1})$ and
$\phi^{-1}(\gamma_{2}x_{2})=b(\gamma_{2},x_{2})\phi^{-1}(x_{2})$, for
$\gamma_{i}\in\Gamma_{i}$ and $x_{i}\in X_{i}$, $i=1,2$. The topological
version of the above equivalence is known to hold with $L^{\infty}$ masas
replaced by $C_{0}$ masas [15, Theorem 1.2].

It is natural to ask weather there is a notion of orbit equivalence for
general $C^{*}$-dynamics and this is the main objective of present paper. As
already seen in both measurable and continuous cases, the masas–or more
precisely, the Cartan subalgebras–play a central role in orbit equivalence,
and we rather define orbit equivalence of actions on Cartan pairs, that is,
actions of countable discrete groups on $C^{*}$-algebras, preserving given
Cartan subalgebras. This makes our approach inevitably related to twisted
grroupoids, as any Cartan pair comes from a Weyl twisted groupoid [18], and a
Cartan invariant action on a separable $C^{*}$-algebra induces an action on
the corresponding Weyl twisted groupoid [1, Proposition 3.4].

In this paper, we define continuous orbit equivalence (coe) of actions on
groupoids (Definition 2.6), and give characterizations of continuous orbit
equivalence for topologically principally free actions (Definition 3.1). The
paper is organized as follows. In section 2 we define the main notions of this
paper and briefly review Kumjian-Renault theory. Section 3 is devoted to proof
of the main results. These results are illustrated by concrete examples in the
last section.

The main results of this paper are as follows. All groupoids are locally
compact, Hausdorff, second countable, and étale (unless otherwise specified)
and all groups are countable discrete. Throughout, by an isomorphism of
topological groupoids we mean an algebraic isomorphism which is also a
homeomorphism. We also write $e_{1}$ and $e_{2}$ to denote the identity
elements of groups $\Gamma_{1}$ and $\Gamma_{2}$, respectively.

######  Theorem A.

For topologically free actions $\Gamma_{i}\curvearrowright(G_{i},\Sigma_{i})$
on twisted groupoids, $i=1,2$, consider the following statements:

  1. (1)

$\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$,

  2. (2)

there exists $\mathbb{T}$-equivariant isomorphism
$\psi:\Gamma_{1}\ltimes\Sigma_{1}\rightarrow\Gamma_{2}\ltimes\Sigma_{2}$ such
that $\psi(\Gamma_{1},\Sigma_{1}^{(0)})=(\Gamma_{2},\Sigma_{2}^{(0)})$ and
$\psi(e_{1},\sigma_{1})=(e_{2},\phi(\sigma_{2}))$, where
$(\phi^{\prime},\phi)$ is a twisted groupoid isomorphism between
$(G_{1},\Sigma_{1})$ and $(G_{2},\Sigma_{2})$,

  3. (3)

there exists isomorphism $\theta:C^{*}_{r}(\Gamma_{1}\ltimes
G_{1},\Gamma_{1}\ltimes\Sigma_{1})\cong C^{*}_{r}(\Gamma_{2}\ltimes
G_{2},\Gamma_{2}\ltimes\Sigma_{2})$ such that,

     * •

$\theta(C_{0}(\Sigma_{1}^{(0)}))=C_{0}(\Sigma_{2}^{(0)}),$

     * •

$\theta(\Gamma_{1}\ltimes_{r}C_{0}(G_{1}^{(0)}))=\Gamma_{2}\ltimes_{r}C_{0}(G_{2}^{(0)}),$

     * •

$\theta(C^{*}_{r}(G_{1},\Sigma_{1}))=C^{*}_{r}(G_{2},\Sigma_{2})$.

Then $(1)\Leftrightarrow(2)\Rightarrow(3)$. These are equivalent for
topologically principally free actions.

For the case of trivial twist, we get the following corollary.

######  Corollary B.

For topologically free actions $\Gamma_{i}\curvearrowright G_{i}$ on
groupoids, $i=1,2$, consider the following statements:

  1. (1)

$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}$,

  2. (2)

there exists groupoid isomorphism $\psi:\Gamma_{1}\ltimes
G_{1}\rightarrow\Gamma_{2}\ltimes G_{2}$ such that
$\psi(\Gamma_{1},G_{1}^{(0)})=(\Gamma_{2},G_{2}^{(0)})$ and
$\psi(e_{1},G_{1})=\psi(e_{2},G_{2})$,

  3. (3)

there exists isomorphism $\theta:C^{*}_{r}(\Gamma_{1}\ltimes G_{1})\cong
C^{*}_{r}(\Gamma_{2}\ltimes G_{2})$ such that,

     * •

$\theta(C_{0}(G_{1}^{(0)}))=C_{0}(G_{2}^{(0)}),$

     * •

$\theta(\Gamma_{1}\ltimes_{r}C_{0}(G_{1}^{(0)}))=\Gamma_{2}\ltimes_{r}C_{0}(G_{2}^{(0)}),$

     * •

$\theta(C^{*}_{r}(G_{1}))=C^{*}_{r}(G_{2})$.

Then $(1)\Leftrightarrow(2)\Rightarrow(3)$. These are equivalent for
topologically principally free actions.

We define continuous orbit equivalence of actions on Cartan pairs via the
corresponding Weyl twisted groupoids (Definition 2.11), and get the following
characterization.

######  Corollary C.

Given Cartan pairs $(A_{1},B_{1})$ and $(A_{2},B_{2})$ and Cartan invariant
topologically free actions $\Gamma_{1}\curvearrowright A_{1}$ and
$\Gamma_{2}\curvearrowright A_{2}$, consider the following statements:

  1. (1)

$\Gamma_{1}\curvearrowright(A_{1},B_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(A_{2},B_{2})$,

  2. (2)

there exists isomorphism
$\theta:\Gamma_{1}\ltimes_{r}A_{1}\cong\Gamma_{2}\ltimes_{r}A_{2}$ such that:,

     * •

$\theta(B_{1})=B_{2},$

     * •

$\theta(\Gamma_{1}\ltimes_{r}B_{1})=\Gamma_{2}\ltimes_{r}B_{2},$

     * •

$\theta(A_{1})=A_{2}$.

Then $(1)\Rightarrow(2)$, and these are equivalent for topologically
principally free actions.

We find conditions for rigidity of continuous orbit equivalence of actions on
twisted groupoids, that is conditions under which such equivalence follow from
equivalence of the restricted actions on the corresponding unit spaces.

######  Theorem D.

For Cartan invariant actions $\Gamma_{1}\curvearrowright
C_{r}^{*}(G_{1},\Sigma_{1})$ and $\Gamma_{2}\curvearrowright
C_{r}^{*}(G_{2},\Sigma_{2})$ with isomorphism
$\psi:C_{r}^{*}(G_{1},\Sigma_{1})\rightarrow C_{r}^{*}(G_{2},\Sigma_{2})$
satisfying $\psi(C_{0}(G_{1}^{(0)}))=C_{0}(G_{2}^{(0)})$, and induced actions
$\Gamma_{1}\curvearrowright G_{1}^{(0)}\sim_{\rm
coe}\Gamma_{2}\curvearrowright G_{2}^{(0)}$ with corresponding cocycle maps
$a$ and $b$, if,

(1.1) $\displaystyle a(\gamma_{1},s(g_{1}))=a(\gamma_{1},r(g_{1})),\ \
b(\gamma_{2},s(g_{2}))=b(\gamma_{2},r(g_{2})),$

for $\gamma_{i}\in\Gamma_{i},g_{i}\in G_{i},i=1,2$, then
$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}$.

As an immediate consequence, we get the following further characterization of
continuous orbit equivalence.

######  Corollary E.

In the situation of last theorem when the twists are trivial, for continuous
orbit equivalent topologically free actions $\Gamma_{i}\curvearrowright
G_{i}^{(0)}$, $i=1,2$ with cocycle maps $a$ and $b$, consider the following
statements:

  1. (1)

$a(\gamma_{1},s(g_{1}))=a(\gamma_{1},r(g_{1}))$ and
$b(\gamma_{2},s(g_{2}))=b(\gamma_{2},r(g_{2}))$, for $\gamma_{i}\in\Gamma_{i}$
and $g_{i}\in G_{i}$; $i=1,2$,

  2. (2)

$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}$,

  3. (3)

there exists isomorphism
$\theta:\Gamma_{1}\ltimes_{r}C^{*}_{r}(G_{1})\rightarrow\Gamma_{2}\ltimes_{r}C^{*}_{r}(G_{2})$
such that,

     * •

$\theta(C_{0}(G_{1}^{(0)}))=C_{0}(G_{2}^{(0)}),$

     * •

$\theta(\Gamma_{1}\ltimes_{r}C_{0}(G_{1}^{(0)}))=\Gamma_{2}\ltimes_{r}C_{0}(G_{2}^{(0)}),$

     * •

$\theta(C^{*}_{r}(G_{1}))=C^{*}_{r}(G_{2})$.

Then $(1)\Leftrightarrow(2)\Rightarrow(3)$. These are equivalent for
topologically principally free actions.

##  2\. Continuous Orbit Equivalence

An action $\Gamma\curvearrowright X$ on a topological space is called
topologically free if for each $x\in X$ and non identity element
$\gamma\in\Gamma$, $\\{x\in X|\gamma x\neq x\\}$ is dense in $X$. For a
topological groupoid $G$, Aut$(G)$ is the group of all groupoid isomorphisms
$\phi:G\rightarrow G$. By an action of a group $\Gamma$ on a groupoid $G$ we
mean a group homomorphism $:\Gamma\rightarrow$ Aut$(G)$. We say that
$\Gamma\curvearrowright G$ is (topologically) free if the induced action
$\Gamma\curvearrowright G^{(0)}$ is (topologically) free. The semidirect
product $\Gamma\ltimes G$ is the groupoid $\Gamma\times G$ with the following
operations
$(\gamma,g)(\gamma^{\prime},g^{\prime}):=(\gamma\gamma^{\prime},(\gamma^{\prime-1}g)g^{\prime}),\
(\gamma,g)^{-1}=(\gamma^{-1},\gamma g^{-1})$, for
$\gamma,\gamma^{\prime}\in\Gamma$ and $g,g^{\prime}\in G$.

######  Definition 2.1.

Two actions $\Gamma_{1}\curvearrowright G_{1}$ and $\Gamma_{2}\curvearrowright
G_{2}$ on topological (not necessarily étale) groupoids are said to be
continuous $r$-orbit equivalent if there exist a groupoid isomorphism
$\phi:G_{1}\rightarrow G_{2}$, and continuous maps $a:\Gamma_{1}\times
G_{1}^{(0)}\rightarrow\Gamma_{2}$ and $b:\Gamma_{2}\times
G_{2}^{(0)}\rightarrow\Gamma_{1}$, with
$\phi(\gamma_{1}g_{1})=a(\gamma_{1},r(g_{1}))\phi(g_{1})$ and
$\phi^{-1}(\gamma_{2}g_{2})=b(\gamma_{2},r(g_{2}))\phi^{-1}(g_{2})$, for
$\gamma_{i}\in\Gamma_{i},g_{i}\in G_{i},i=1,2$. The continuous $s$-orbit
equivalence is defined similarly. These actions are called continuous orbit
equivalent, writing $\Gamma_{1}\curvearrowright G_{1}\sim_{\rm
coe}\Gamma_{2}\curvearrowright G_{2}$, if they are both continuous $r$\- and
$s$-orbit equivalent with the same cocycle maps $a$ and $b$. If moreover, the
cocycle maps are independent of their second variable, giving isomorphism of
the underlying groups, the actions are called conjugate, and write
$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm con}\Gamma_{2}\curvearrowright
G_{2}$.

A topological space $X$ could be regarded as a (co-trivial) groupoid with
$G^{(2)}=$ Diag$(X)=:\Delta_{X}$ and trivial inverse and multiplication, and
in this case, the above notion of orbit equivalence is the same as the
classical one.

Note that though we do not assume a priory algebraic conditions on maps $a$
and $b$, they are indeed not only cocycle maps in the usual sense of classical
dynamics (compare with [15, Lemma 2.8]), but also in the sense of actions on
groupoids. More precisely, we have the following result.

######  Lemma 2.2.

Let $\Gamma_{1}\curvearrowright G_{1}$ and $\Gamma_{2}\curvearrowright G_{2}$
be two continuous $r$-orbit equivalent topologically free actions. Let the
maps $\phi,a,b$ be as they defined in Definition 2.1. For
$\gamma_{i},\gamma_{i}^{\prime}\in\Gamma_{i}$ and $g_{i},g_{i}^{\prime}\in
G_{i}$, $i=1,2$,

  1. (1)

$a(\gamma_{1},s(g_{1}))=a(\gamma_{1},r(g_{1}))$ and
$b(\gamma_{2},s(g_{2}))=b(\gamma_{2},r(g_{2}))$,

  2. (2)

when products are defined,

     * •

$a(\gamma_{1}^{\prime},r(g_{1}^{\prime}))a(\gamma_{1},r(g_{1}))=a(\gamma_{1}^{\prime}\gamma_{1},r((\gamma_{1}^{-1}g_{1}^{\prime})g_{1})),$

     * •

$b(\gamma_{2}^{\prime},r(g_{2}^{\prime}))b(\gamma_{2},r(g_{2}))=b(\gamma_{2}^{\prime}\gamma_{2},r((\gamma_{2}^{-1}g_{2}^{\prime})g_{2})),$

  3. (3)

$a(\gamma_{1},r(g_{1}))^{-1}=a(\gamma_{1}^{-1},r(\gamma_{1}g_{1}^{-1}))$,
$b(\gamma_{2},r(g_{2}))^{-1}=b(\gamma_{2}^{-1},r(\gamma_{2}g_{2}^{-1})),$

  4. (4)

$b(a(\gamma_{1},r(g_{1})),\phi(g_{1}))=\gamma_{1}$,
$a(b(\gamma_{2},r(g_{2})),\phi(g_{2}))=\gamma_{2}$.

###### Proof.

By topological freeness, there is a dense subset $D_{i}\subset G_{i}^{(0)}$
such that $\gamma d=d$ forces $\gamma$ to be the identity of ${\Gamma_{i}}$,
for $\gamma\in\Gamma_{i}$, $d\in D_{i}$, $i=1,2$. Since groupoids are étale,
for an arbitrary open subset $U_{i}\subset G_{i}$, $r(U_{i})$ is open and
$r(U_{i})\cap D_{i}\neq\emptyset$. Thus, $r^{-1}(D_{i})$ is dense in $G_{i}$,
$i=1,2$. For $\gamma_{i},\gamma_{i}^{\prime}\in\Gamma_{i}$ and
$g_{i},g_{i}^{\prime}\in G_{i}$, $i=1,2$, observe that
$\phi(\gamma_{1}g_{1})^{-1}=(a(\gamma_{1},r(g_{1}))\phi(g_{1}))^{-1}$ is
equivalent to
$a(\gamma_{1},r(g_{1}^{-1}))\phi(g_{1}^{-1})=a(\gamma_{1},r(g_{1}))\phi(g_{1}^{-1})$,
and so by topological freeness and continuity of $a$, we get (1). By a similar
argument, we also have $b(\gamma_{2},s(g_{2}))=b(\gamma_{2},r(g_{2}))$. For
(2), observe that,

$\displaystyle\phi((\gamma_{1}^{\prime}\gamma_{1})((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))$
$\displaystyle=\phi(\gamma_{1}^{\prime}(g_{1}^{\prime}(\gamma_{1}g_{1})))=a(\gamma_{1}^{\prime},r(g_{1}^{\prime}(\gamma_{1}g_{1})))\phi(\gamma_{1}((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))$
$\displaystyle=a(\gamma_{1}^{\prime},r(g_{1}^{\prime}(\gamma_{1}g_{1})))a(\gamma_{1},r((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))\phi((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}),$

and
$\phi((\gamma_{1}^{\prime}\gamma_{1})((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))=a(\gamma_{1}^{\prime}\gamma_{1},r((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))\phi((\gamma_{1}^{-1}g_{1}^{\prime})g_{1})$,
and again by topological freeness,

$\displaystyle
a(\gamma_{1}^{\prime}\gamma_{1},r((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))$
$\displaystyle=a(\gamma_{1}^{\prime},r(g_{1}^{\prime}(\gamma_{1}g_{1})))a(\gamma_{1},r((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))$
$\displaystyle{=}a(\gamma_{1}^{\prime},r(g_{1}^{\prime}))a(\gamma_{1},s((\gamma_{1}^{-1}g_{1}^{\prime})g_{1}))=a(\gamma_{1}^{\prime},r(g_{1}^{\prime}))a(\gamma_{1},s(g_{1}))$
$\displaystyle{=}a(\gamma_{1}^{\prime},r(g_{1}^{\prime}))a(\gamma_{1},r(g_{1})),$

and the same for $b$. Now (3) follows from (2) and (4) is proved similarly. ∎

Recall that the identity elements of $\Gamma_{1}$ and $\Gamma_{2}$ are denoted
by $e_{1}$ and $e_{2}$.

######  Theorem 2.3.

Let $\Gamma_{1}\curvearrowright G_{1}$, $\Gamma_{2}\curvearrowright G_{2}$ be
two topologically free actions. The fallowing are equivalent:

  1. (1)

$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2},$

  2. (2)

there exists groupoid isomorphism $\psi:\Gamma_{1}\ltimes
G_{1}\rightarrow\Gamma_{2}\ltimes G_{2}$ such that
$\psi(\Gamma_{1},G_{1}^{(0)})=(\Gamma_{2},G_{2}^{(0)})$ and
$\psi(e_{1},G_{1})=(e_{2},G_{2}).$

###### Proof.

$(1)\Rightarrow(2)$: Let $\phi:G_{1}\rightarrow G_{2}$ be the isomorphism
given by orbit equivalence. We claim that the maps $\psi:\Gamma_{1}\ltimes
G_{1}\rightarrow\Gamma_{2}\ltimes G_{2};\
(\gamma_{1},g_{1})\mapsto(a(\gamma_{1},r(g_{1})),\phi(g_{1}))$, and
$\chi:\Gamma_{2}\ltimes G_{2}\rightarrow\Gamma_{1}\ltimes G_{1};\
(\eta_{2},h_{2})\mapsto(b(\eta_{2},r(h_{2})),\phi^{-1}(h_{2}))$, are groupoid
homomorphisms. Let
$((\gamma_{1},g_{1}),(\gamma_{2},g_{2}))\in(\Gamma_{1}\ltimes G_{1})^{(2)}$,
then $(\gamma_{2}^{-1}g_{1},g_{2})\in G_{1}^{(2)}$,
$r(g_{2})=s(\gamma_{2}^{-1}g_{1})=\gamma_{2}^{-1}s(g_{1})$, and
$\gamma_{2}r(g_{2})=s(g_{1})$. For $\gamma\in\Gamma_{1}$,

$a(\gamma,\gamma_{2}r(g_{2}^{-1}))=a(\gamma,\gamma_{2}s(g_{2}))=a(\gamma,\gamma_{2}r(g_{2}))=a(\gamma,s(g_{1}))=a(\gamma,r(g_{1})).$

Using this and Lemma 2.2, we get,

$\displaystyle\psi(\gamma_{1},g_{1})\psi(\gamma_{2},g_{2})$
$\displaystyle=(a(\gamma_{1},r(g_{1})),\phi(g_{1}))(a(\gamma_{2},r(g_{2})),\phi(g_{2}))$
$\displaystyle=(a(\gamma_{1},r(g_{1}))a(\gamma_{2},r(g_{2})),(a(\gamma_{2},r(g_{2}))^{-1}\phi(g_{1}))\phi(g_{2}))$
$\displaystyle=(a(\gamma_{1}\gamma_{2},r((\gamma_{2}^{-1}g_{1})g_{2})),(a(\gamma_{2}^{-1},r(\gamma_{2}g_{2}^{-1}))\phi(g_{1}))\phi(g_{2}))$
$\displaystyle=(a(\gamma_{1}\gamma_{2},r((\gamma_{2}^{-1}g_{1})g_{2})),(\phi(\gamma_{2}^{-1}g_{1}))\phi(g_{2}))$
$\displaystyle=\psi(\gamma_{1}\gamma_{2},(\gamma_{2}^{-1}g_{1})g_{2})=\psi((\gamma_{1},g_{1})(\gamma_{2},g_{2})),$

for $\gamma_{1},\gamma_{2}\in\Gamma_{1},g_{1},g_{2}\in G_{1}$, thus, $\psi$ is
a groupoid homomorphism. The proof for $\chi$ is similar. By Lemma 2.2, these
maps are inverse of each other, and $\psi$ satisfies the required conditions.

$(2)\Rightarrow(1)$: Let $\psi:\Gamma_{1}\ltimes
G_{1}\rightarrow\Gamma_{2}\ltimes G_{2}$ and $\phi:G_{1}\rightarrow G_{2}$ be
isomorphisms such that $\psi(\Gamma_{1},G_{1}^{(0)})=(\Gamma_{2},G_{2}^{(0)})$
and $\psi(e_{1},g_{1})=(e_{2},\phi(g_{1}))$, for $g_{1}\in G_{1}$. Let
$a:\Gamma_{1}\times G_{1}^{(0)}\rightarrow\Gamma_{2}$ map
$(\gamma_{1},r(g_{1}))$ to the first component of $\psi(\gamma_{1},r(g_{1}))$
and $b:\Gamma_{2}\times G_{2}^{(0)}\rightarrow\Gamma_{1}$ map
$(\gamma_{2},r(g_{2}))$ to the first component of
$\psi^{-1}(\gamma_{2},r(g_{2}))$. By assumption, if
$\psi(\gamma_{1},r(g_{1}))=(a(\gamma_{1},r(g_{1})),g_{2})$, then $g_{2}\in
G_{2}^{(0)}$. Also,
$\psi(s(\gamma_{1},r(g_{1})))=s(a(\gamma_{1},r(g_{1})),g_{2})$ implies
$\psi(e_{1},r(g_{1}))=(e_{2},g_{2})$, and $g_{2}=\phi(r(g_{1}))$. This means
that, $\psi(\gamma_{1},r(g_{1}))=(a(\gamma_{1},r(g_{1})),\phi(r(g_{1}))).$
Furthermore,

$\displaystyle\psi(\gamma_{1},g_{1})$
$\displaystyle=\psi(\gamma_{1},r(g_{1}))\psi(e_{1},g_{1})=(a(\gamma_{1},r(g_{1})),\phi(r(g_{1})))(e_{1},\phi(g_{1}))$
$\displaystyle=(a(\gamma_{1},r(g_{1})),\phi(g_{1})),$

and

$\displaystyle(a(\gamma_{1}^{-1},\gamma_{1}r(g_{1})),\phi(\gamma_{1}g_{1}))$
$\displaystyle=\psi(\gamma_{1},g_{1}^{-1})^{-1}=(a(\gamma_{1},r(g_{1}^{-1})),\phi(g_{1}^{-1}))^{-1}$
$\displaystyle=(a(\gamma_{1},r(g_{1}^{-1}))^{-1},a(\gamma_{1},r(g_{1}^{-1}))\phi(g_{1})).$

Thus,
$a(\gamma_{1},r(g_{1}))\phi(g_{1})=a(\gamma_{1},s(g_{1}))\phi(g_{1})=\phi(\gamma_{1}g_{1}),$
for $\gamma_{1}\in\Gamma_{1}$ and $g_{1}\in G_{1}$. Similarly,
$b(\gamma_{2},r(g_{2}))\phi^{-1}(g_{2})=\phi^{-1}(\gamma_{2}g_{2}),$ for
$\gamma_{2}\in\Gamma_{2}$ and $g_{2}\in G_{2}$. ∎

######  Corollary 2.4.

Let $\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}$ be topologically free actions then $C^{*}_{r}(\Gamma_{1}\ltimes
G_{1})\cong C^{*}_{r}(\Gamma_{2}\ltimes G_{2})$.

Let $G$ be a groupoid and for $u,v\in G^{(0)}$, put $G_{u}:=s^{-1}(u)$,
$G^{u}:=r^{-1}(u)$ and $G_{u}^{v}:=G_{u}\cap G^{v}$. Then $G$ is called
principal if for each $u\in G^{(0)}$, $G_{u}^{u}:=\\{u\\}$. When $G$ is a
topological groupoid, it is called topologically principal if the set of units
$u\in G^{(0)}$ with $G_{u}^{u}=\\{u\\}$ is dense in $G^{(0)}$. By a bisection
we mean a subgroupoid $S$ of groupoid $G$ with restrictions $r|_{S}$ and
$s|_{S}$ being injective.

Twist over groupoids are first introduced by Alex Kumjian [14]. Here we recall
the definition of a twisted groupoid and thier reduced C∗-algebras.

######  Definition 2.5.

([20], Definition 11.1.1) Let $G$ be a locally compact, Hausdorff, second
countable, étale groupoid, and $\Sigma$ be a locally compact, Hausdorff
groupoid and $\mathbb{T}\times G^{(0)}$ be a trivial group bundle with fibers
$\mathbb{T}$. Then $(G,\Sigma)$ is a twisted groupoid if there is a sequence

$\mathbb{T}\times
G^{(0)}\overset{\iota}{\rightarrowtail}\Sigma\overset{\pi}{\twoheadrightarrow}G$

such that the following conditions are satisfied:

  1. (1)

$\iota$ and $\pi$ are groupoid morphisms with $\iota$ one-to-one and $q$ onto,

  2. (2)

$\iota$ and $\pi$ by reduction respectively define homeomorphisms
$\\{1\\}\times G^{(0)}\cong\Sigma^{(0)}$ and $\Sigma^{(0)}\cong G^{(0)}$,

  3. (3)

For $\sigma\in\Sigma$ and $t\in\mathbb{T}$,
$\iota(t,r(\sigma))\sigma=\sigma\iota(t,s(\sigma))$,

  4. (4)

$\pi^{-1}(u)=\iota(\mathbb{T}\times\\{u\\})$ for $u\in G^{(0)}$.

The twist is said to be locally trivial if in addition,

  1. (5)

For each $g\in G$ there is an open bisection $U_{g}$ containing $g$ and a
continuous map $\phi_{g}:U_{g}\rightarrow\Sigma$ with
$\pi\circ\phi_{g}=id|_{U_{g}}$, making $(t,u)\mapsto\iota(t,r(u))\phi_{g}(u)$
a homeomorphism between $\mathbb{T}\times U$ and $\pi^{-1}(U)$.

The particular case $\mathbb{T}\times G^{(0)}\rightarrowtail\mathbb{T}\times
G\twoheadrightarrow G$, gives the trivial twist. A twisted groupoid is called
topologically principal if $G$ is so. In this case, there exist an action of
the unit circle $\mathbb{T}$ on $\Sigma$ such that when
$\pi(\sigma_{1})=\pi(\sigma_{2})$, there is a unique $t\in\mathbb{T}$ with
$t\sigma_{1}=\sigma_{2}$ [20, 11.1.2 and 11.1.3].

Let $G$ be an étale groupoid with Haar system of counting measures and
$(G,\Sigma)$ be a twisted groupoid with corresponding maps $\iota$ and $\pi$.
We have the $*$-algebra

$C_{c}(G,\Sigma):=\\{f\in
C_{c}(\Sigma,\mathbb{C}):f(t\sigma)=\bar{t}f(\sigma),\text{ for all
}\sigma\in\Sigma,t\in\mathbb{T}\\}$

under the operations $f*h(\sigma)=\sum_{\dot{\tau}\in
G_{s(\sigma)}}f(\sigma\tau^{-1})h(\tau),\ \
f^{*}(\sigma)=\overline{f(\sigma^{-1})},$ For $u\in G^{(0)}$, on the Hilbert
space

$\mathscr{H}_{u}:=\\{\xi:\Sigma_{u}\rightarrow\mathbb{C}:\xi(t\sigma)=\bar{t}\xi(\sigma),\sum_{\dot{\sigma}\in
G_{u}}|\xi(\sigma)|^{2}<\infty\\}$

we define $*$-representation ${\rm Ind}\delta_{u}$ of $C_{c}(G,\Sigma)$ on
$\mathscr{H}_{u}$ by

${\rm
Ind}\delta_{u}(f)\xi(\sigma):=\sum_{\tau\in\Sigma_{u}}f(\sigma\tau^{-1})\xi(\tau),\
\ (\sigma\in\Sigma_{u},\xi\in\mathscr{H}_{u},f\in C_{c}(G,\Sigma)),$

and define the reduced twisted groupoid $C^{*}$-algebra $C^{*}_{r}(G,\Sigma)$
as the completion of $C_{c}(G,\Sigma)$ in the reduced norm
$\|f\|_{r}:=\text{sup}_{u\in G^{(0)}}\|{\rm Ind}\delta_{u}(f)\|$. Replacing
sums with integrals with respect to a Haar system and a quasi-invariant
measure on $G^{(0)}$, the same construction works when $G$ is not étale [18].

By an isomorphism $(\phi_{0},\phi_{1},\phi_{2})$ between twisted groupoids
$(G_{1},\Sigma_{1})$ and $(G_{2},\Sigma_{2})$ we mean a commutative diagram,

$\textstyle{\mathbb{T}\times
G^{(0)}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\textstyle{\Sigma_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathbb{T}\times
G^{(0)}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{2}}$

where $\phi_{0}$, $\phi_{1}$, and $\phi_{2}$ are isomorphisms. We identify
$\mathbb{T}\times G_{1}^{(0)}$ with $\mathbb{T}\times G_{2}^{(0)}$ and write
$(\phi_{2},\phi_{1}):(G_{1},\Sigma_{1})\rightarrow(G_{2},\Sigma_{2})$ for this
isomorphism. In this case, we simply say that $\phi_{1}$ is
$\mathbb{T}$-equivariant.

By an action of a group $\Gamma$ on a twisted groupoid $(G,\Sigma)$ we mean
two actions $\Gamma\curvearrowright\Sigma$ and $\Gamma\curvearrowright G$ with
the following equivariance conditions,

$\displaystyle\iota(t,\gamma u)=\gamma\iota(t,u),\ \
\pi(\gamma\sigma)=\gamma\pi(\sigma),\ \
(\gamma\in\Gamma,t\in\mathbb{T},\sigma\in\Sigma,u\in G^{(0)}).$

An action of $\Gamma$ on $(G,\Sigma)$ is called topologically free if the
corresponding restricted action $\Gamma\curvearrowright G^{(0)}$ is
topologically free in the classical sense.

######  Definition 2.6.

Two actions $\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})$ and
$\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$ are called $r$-continuous
($s$-continuous) orbit equivalent if there exists a $\mathbb{T}$-equivariant
isomorphism $\phi:\Sigma_{1}\rightarrow\Sigma_{2}$ such that the induced
actions $\Gamma_{1}\curvearrowright\Sigma_{1}$ and
$\Gamma_{2}\curvearrowright\Sigma_{2}$ are $r$-continuous ($s$-continuous)
orbit equivalent with respect to $\phi$. When the actions are both
$r$-continuous and $s$-continuous orbit equivalent, with the same isomorphism
and cocyle maps, the actions are called continuous orbit equivalent, writing
$\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$. Finally, when
$\Gamma_{1}\curvearrowright\Sigma_{1}\sim_{\rm
con}\Gamma_{2}\curvearrowright\Sigma_{2}$ with respect to a
$\mathbb{T}$-equivariant isomorphism $\phi$, we say that the actions are
conjugate.

######  Proposition 2.7.

Given topologically free actions
$\Gamma_{i}\curvearrowright(G_{i},\Sigma_{i})$, $i=1,2$, such that there
exists an isomorphism
$(\phi^{\prime},\phi):(G_{1},\Sigma_{1})\rightarrow(G_{2},\Sigma_{2})$,
consider following statements:

  1. (1)

$\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$ with respect to $\phi$,

  2. (2)

$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}.$

Then $(1)$ implies $(2)$.

###### Proof.

For $i=1,2$, let $(\iota_{i},\pi_{i})$ be the twist map of
$(G_{i},\Sigma_{i})$, and consider the twist $\mathbb{T}\times
G_{i}^{(0)}\overset{(c_{\Gamma_{i}},\iota_{i})}{\longrightarrow}\Gamma_{i}\ltimes\Sigma_{i}\overset{(id_{i},\pi_{i})}{\longrightarrow}\Gamma_{i}\ltimes
G_{i}$, where $c_{\Gamma_{i}}$ is the constant map to the identity element of
$\Gamma_{i}$. By the definition of twisted isomorphism $(\phi^{\prime},\phi)$
and equality
$(\gamma_{1},\sigma_{1})=(\gamma_{1},r(\sigma_{1}))(e_{1},\sigma_{1})$, we get
the equivariance condition
$\psi(\gamma_{1},t\sigma_{1})=t\psi(\gamma_{1},\sigma_{1})$. Thus, $\psi$
induces $\psi^{\prime}:\Gamma_{1}\ltimes G_{1}\rightarrow\Gamma_{2}\ltimes
G_{2}$ with $(\psi^{\prime},\psi)$ an isomorphism of twisted groupoids. For
$g_{1}\in G_{1}$, $\sigma_{1}\in\Sigma_{1}$ with $\pi_{1}(\sigma_{1})=g_{1}$,
$\psi^{\prime}(e_{1},g_{1})=\psi(e_{1},\pi_{2}(\phi(\sigma_{1})))$, that is,
$\psi^{\prime}(e_{1},G_{1})\subseteq(e_{2},G_{2})$, and by the same argument,
$\psi^{\prime}(e_{1},G_{1})=(e_{2},G_{2})$. Also, for $u\in G_{1}^{(0)},$

$\displaystyle\psi^{\prime}(\gamma_{1},u)$
$\displaystyle\in\psi^{\prime}\circ(id_{1},\pi_{1})(\gamma_{1},\pi_{1}^{-1}(u))=(id_{2},\pi_{2})\circ\psi(\gamma_{1},\pi_{1}^{-1}(u))$
$\displaystyle\subseteq(id_{2},\pi_{2})(\Gamma_{2},\mathbb{T}\times\Sigma_{2}^{(0)})=(\Gamma_{2},\Sigma_{2}^{(0)}),$

thus
$\psi^{\prime}(\Gamma_{1},\Sigma_{1}^{(0)})\subset(\Gamma_{2},\Sigma_{2}^{(0)}),$
and by the same argument,
$\psi^{\prime}(\Gamma_{1},\Sigma_{1}^{(0)})=(\Gamma_{2},\Sigma_{2}^{(0)})$,
which completes the proof by Theorem 2.3 ∎

In the notation of the above proposition,
$\Gamma_{1}\curvearrowright\Sigma_{1}\sim_{\rm
coe}\Gamma_{2}\curvearrowright\Sigma_{2}$ with respect to $\phi$ implies
$\Gamma_{1}\curvearrowright G_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
G_{2}$ with respect to $\phi^{\prime}$.

A sub-C∗-algebra $B$ of C∗-algebra $A$ is called a Cartan subalgebra if,

  1. (1)

$B$ is a maximal abelian subalgebra of $A$,

  2. (2)

$B$ is regular, i.e., $N_{B}(A):=\\{n\in A:nBn^{*}\subseteq B\text{ and
}n^{*}Bn\subseteq B\\}$ generates $A$ as a C∗-algebra,

  3. (3)

there exists a faithful conditional expectation $P:A\rightarrow B$.

The pair $(A,B)$ is called a Cartan pair. In this case, the existence of an
approximate identity of $A$ in $B$ is automatic [16, Theorem 2.6].

We say that two Cartan pairs $(A_{1},B_{1}),(A_{2},B_{2})$ are said to be
isomorphic, writing $(A_{1},B_{1})\simeq(A_{2},B_{2})$, if there is a
$C^{*}$-algebra isomorphism from $A_{1}$ from $A_{2}$ mapping $B_{1}$ onto
$B_{2}$. There is a correspondence between separable C∗-algebras containing a
Cartan subalgebra and twisted groupoid C∗-algebras, given by Renault in the
following result [18, Theorem 5.9].

######  Proposition 2.8.

For a Cartan pair $(A,B)$ with $A$ separable, there exists a twisted
Hausdorff, locally compact, second countable, topologically principal, étale
groupoid $(G,\Sigma)$ with $(A,B)\cong(C^{*}_{r}(G,\Sigma),C_{0}(G^{(0)}))$.

The twisted groupoid $(G,\Sigma)$ in the above proposition is called the Weyl
twisted groupoid associated to the Cartan pair $(A,B)$. Let us recall the
construction of the Weyl twisted groupoid (for details, see [18] and [1,
Remark 2.8]). Let $X$ be the spectrum of the abelian C∗-algebra $B$. For each
normalize $n\in N_{A}(B)$, by [18, Lemma 4.6], $n^{*}n$ and $nn^{*}$ are in
$B$. Put, $\text{dom}(n):=\\{x\in X;n^{*}n(x)>0\\}$ and
$\text{ran}(n):=\\{x\in X;nn^{*}(x)>0\\}$. By [18, Proposition 4.7], there is
a partial homeomorphism $\alpha_{n}:\text{dom}(n)\rightarrow\text{ran}(n)$
such that $n^{*}bn(x)=b(\alpha_{n}(x))n^{*}n(x)$, for $x\in\text{dom}(n)$ and
$b\in B$. Now the groupoid $G=G(B)$ is defined by,

$G:=\\{(x,\alpha_{n},y):n\in
N_{A}(B),y\in\text{dom}(n),\alpha_{n}(y)=x\\}/\sim,$

where $(x,\alpha_{n},y)\sim(x^{\prime},\alpha_{m^{\prime}},y^{\prime})$ iff
$y=y^{\prime}$ and there exist neighbourhood $U\subseteq X$ of $y$ such that
$\alpha_{n}|_{U}=\alpha_{n^{\prime}}|_{U}$. We denote the equivalence class of
$(x,\alpha_{b},y)$ by $[x,\alpha_{n},y]$. The unit space $G^{(0)}$ could be
identified with $\\{[x,\alpha_{b},x];b\in B,x\in$ supp$(b)\\}$. The twist
$\Sigma=\Sigma(B)$ is defined by,

$\Sigma:=\\{(x,n,y)\in\text{ran}(n)\times
N_{A}(B)\times\text{dom}(n):\alpha_{n}(y)=x\\}/\approx,$

where $(x,n,y)\approx(x^{\prime},n^{\prime},y^{\prime})$ iff $y=y^{\prime}$
and there exist $b,b^{\prime}\in B$ with $b(y),b^{\prime}(y)>0$ and
$nb=n^{\prime}b^{\prime}$. We use the same notation for the equivalence
classes. The unit space could be identified with $\\{[x,b,x]:b\in B,x\in{\rm
dom}(b)\\}$. The map $[x,n,y]\mapsto[x,\alpha_{n},y]$ yields a central
extension, making $(G,\Sigma)$ a twisted groupoid with
$(A,B)\cong(C^{*}_{r}(G,\Sigma),C_{0}(G^{(0)}))$. This extension uses the
identification of $\mathbb{T}\times X$ with $\mathscr{B}:=\\{[x,b,x]:b\in B,\
b(x)\neq 0\\}\subseteq\Sigma(B)$ given by, $[x,b,x]\mapsto(b(x)/|b(x)|,x).$

The next result ensures the uniqueness of twisted groupoid in Proposition 2.8.

######  Proposition 2.9 (Proposition 4.15 of [18]).

Let $(G,\Sigma)$ be a twisted Hausdorff, locally compact, second countable,
topologically principal étale groupoid. Let $A:=C^{*}_{r}(G,\Sigma)$ and
$B:=C_{0}(G^{(0)})$. Then there is an isomorphism of twisted groupoids,

$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\textstyle{\Sigma(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{G(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\textstyle{\mathbb{T}\times
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$

As mentioned in [18], using the canonical action
$\mathbb{T}\curvearrowright\Sigma$ in the twisted groupoid $(G,\Sigma)$, one
can define a complex line bundle $L:=\frac{\mathbb{C}\times\Sigma}{\sim}$,
where $(c,\sigma)\sim(c^{\prime},\sigma^{\prime})$ iff there exists
$t\in\mathbb{T}$ such that $(c,\sigma)=(\bar{t}c,t\sigma)$. Let $[c,\sigma]$
be the corresponding equivalence class, and let
$\omega:L\rightarrow\mathbb{C};[c,\sigma]\mapsto c$. Then each section of $L$
after composing with $\omega$ could be viewed as a map
$f:G\rightarrow\mathbb{C}$. For each section $f$, the open support of $f$ is
defined to be ${\rm supp}^{\prime}(f)=\\{g\in G;f(g)\neq 0\\}.$ By [17,
Proposition 2.4.2], each element of $C^{*}_{r}(G,\Sigma)$ could be viewed as a
continuous section of $L$, whose open support is the image of the open support
of $f\in C_{c}(G,\Sigma)$ by the twist map $\pi$ (see, section 2.2 in [5]).
With this convention, we have the identification, $C_{0}(G^{(0)})=\\{f\in
C^{*}_{r}(G,\Sigma):{\rm supp}^{\prime}(f)\subset G^{(0)}\\},$ where $h\in
C_{0}(G^{(0)})$ is identified with $f(\sigma)=\bar{t}h(x)$, if
$\sigma\in\mathbb{T}\times G^{(0)}$, and $f(\sigma)=0$, otherwise. This could
be used to explicitly write the twisted groupoid isomorphism of Proposition
2.9 (compare with [18, Proposition 4.7]): Let $n\in N_{A}(B)$, since $G$ is
topologically principal, $S:={\rm supp}^{\prime}(n)$ is a bisection of $G$
[18, Proposition 4.7]. This gives groupoid isomorphisms
$\phi_{1}:\Sigma(B)\rightarrow\Sigma;\
(y,n,x)\mapsto(n(Sy)/\sqrt{n^{*}n(y)},Sx)$ and $\phi_{2}:G(B)\rightarrow G;\
[y,\alpha_{n},x]\mapsto Sx$.

######  Proposition 2.10 (Proposition 3.4 of [1]).

Let $A$ be a separable C∗-algebra admitting a Cartan subalgebra $B\subseteq
A$. For a Cartan invariant action of a countable group $\Gamma$ on $(A,B)$,
then there is an action of $\Gamma$ on $(G(B),\Sigma(B))$ such that,

$\Gamma\ltimes_{r}A\cong\Gamma\ltimes_{r}C^{*}_{r}(G(B),\Sigma(B))\cong
C^{*}_{r}(\Gamma\ltimes G(B),\Gamma\ltimes\Sigma(B)).$

Since $\alpha:\Gamma\curvearrowright A$ is Cartan invariant, there is a
restricted action of $\Gamma$ on $B\cong C_{0}(G^{(0)})$, or equivalently, on
$G^{(0)}$. This guarantees that $N_{B}(A)$ is invariant under the action. A
simple calculation shows that $\alpha_{\gamma n}(\gamma
y)=\gamma\alpha_{n}(y).$ We have the actions, $\Gamma\curvearrowright G$ and
$\Gamma\curvearrowright\Sigma$ given by,

(2.1) $\displaystyle\gamma[x,\alpha_{n},y]=[\gamma x,\alpha_{\gamma n},\gamma
y],\ \ \gamma[x,n,y]=[\gamma x,\gamma n,\gamma y],$

which yields an action of $\Gamma$ on the twisted groupoid $(G(B),\Sigma(B))$.

Let $(A_{1},B_{1})$ and $(A_{2},B_{2})$ be Cartan pairs, and
$\psi:A_{1}\rightarrow A_{2}$ be a Cartan invariant isomorphism. By
Proposition 2.8,
$(A_{i},B_{i})\cong(C^{*}_{r}(G_{i},\Sigma_{i}),C_{0}(G^{(0)}))$, for $i=1,2$.
The Cartan invariance gives an isomorphism $G_{1}^{(0)}\cong G_{2}^{(0)}$,
again denoted by $\psi$. For $n\in N_{A_{1}}(B_{1})$, $x\in G_{1}^{(0)}$, and
$b^{\prime}\in B_{2}$, say, $b^{\prime}=\psi(b)$, for some $b\in B_{1}$, we
have,

$\displaystyle\psi(n^{*})\psi(b)\psi(n)(\psi(x))$
$\displaystyle=\psi(b)(\alpha_{\psi(n)}(\psi(x)))\psi(n^{*}n)(\psi(x)),$
$\displaystyle n^{*}bn(x)$
$\displaystyle=b(\psi^{-1}(\alpha_{\psi(n)}(\psi(x))))n^{*}n(x),$
$\displaystyle b(\alpha_{n}(x))n^{*}n(x)$
$\displaystyle=b(\psi^{-1}(\alpha_{\psi(n)}(\psi(x))))n^{*}n(x).$

Since $x\in$ dom$(n)$, $n^{*}n(x)>0$, and the above equalities hold for all
$b\in B_{1}$, we get, $\alpha_{\psi(n)}(\psi(x))=\psi(\alpha_{n}(x))$. Thus,
there exists a twisted groupoid isomorphism
$(\theta,\theta^{\prime}):(G_{1},\Sigma_{1})\rightarrow(G_{2},\Sigma_{2})$,
given by,

(2.2)
$\displaystyle\theta([\alpha_{n_{1}}(x_{1}),\alpha_{n_{1}},x_{1}])=[\alpha_{\psi(n_{1})}(\psi(x_{1})),\alpha_{\psi(n_{1})},\psi(x_{1})],$
$\displaystyle\theta^{\prime}[\alpha_{n_{1}}(x_{1}),n_{1},x_{1}]=[\alpha_{\psi(n_{1})}(\psi(x_{1})),\psi(n_{1}),\psi(x_{1})].$

######  Definition 2.11.

Let $(A_{1},B_{1})$ and $(A_{2},B_{2})$ be Cartan pairs. Two Cartan invarinat
actions $\Gamma_{1}\curvearrowright A_{1}$ and $\Gamma_{2}\curvearrowright
A_{2}$ are said to be continuous orbit equivalent if the induced action on the
corresponding Weyl twisted groupoids are continuous orbit equivalent. In this
case, we write $\Gamma_{1}\curvearrowright(A_{1},B_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(A_{2},B_{2})$, or simply
$\Gamma_{1}\curvearrowright A_{1}\sim_{\rm coe}\Gamma_{2}\curvearrowright
A_{2}$, if the Cartan subalgebras are known from the context. Two Cartan
invarinat actions $\Gamma_{1}\curvearrowright(A_{1},B_{1})$ and
$\Gamma_{2}\curvearrowright(A_{2},B_{2})$ are said to be conjugate if there is
a covariant isomorphism $\phi:A_{1}\to A_{2}$ with $\phi(B_{1})=B_{2}$. In
this case, we write $\Gamma_{1}\curvearrowright(A_{1},B_{1})\sim_{\rm
con}\Gamma_{2}\curvearrowright(A_{2},B_{2})$.

We usually start with given actions on (twisted) groupoids and then lift the
actions to the corresponding reduced groupoid $C^{*}$-algebras. If these
groupoid $C^{*}$-algebras happen to have Cartan subalgebras and the induced
actions are Cartan invariant, then we get actions on the corresponding Weyl
twisted groupoids. The next lemma follows from Corollary C.

######  Lemma 2.12.

Topologically principally free conjugate actions on Cartan pairs are
continuous orbit equivalent.

In practice, the original actions might be on non étale groupoids, and so the
actions could not be orbit equivalent in the level of groupoids, but they are
equivalent (and indeed conjugate) in the level of $C^{*}$-algebras. To clarify
this rather vague description of the situation, we need some definitions.

######  Definition 2.13.

Let $(G,\Sigma)$ be a (not necessarily étale) twisted groupoid such that
$C^{*}_{r}(G,\Sigma)$ has a Cartan subalgebra $C$. Let
$\Gamma\curvearrowright(G,\Sigma)$ be an action such that the lifted action on
$C^{*}_{r}(G,\Sigma)$ leaves $C$ invariant. Then the unique Weyl twisted
(étale) groupoid $(\tilde{G},\tilde{\Sigma})$ satisfying
$\big{(}C^{*}_{r}(G,\Sigma),C\big{)}\simeq\big{(}C^{*}_{r}(\tilde{G},\tilde{\Sigma}),C_{0}(\tilde{G}^{(0)})\big{)}$
is called an étale realization of $(G,\Sigma)$, or more precisely, the étale
realization of $(G,\Sigma)$ with respect to $C$. If moreover,
$\Gamma\curvearrowright\big{(}C^{*}_{r}(G,\Sigma),C\big{)}\sim_{\rm
con}\Gamma\curvearrowright\big{(}C^{*}_{r}(\tilde{G},\tilde{\Sigma}),C_{0}(\tilde{G}^{(0)})\big{)}$,
then $(\tilde{G},\tilde{\Sigma})$ is called an equivariant étale realization
of $(G,\Sigma)$, or more precisely, the equivariant étale realization of
$(G,\Sigma)$ with respect to $C$.

The next lemma is just a restatement of transitivity of continuous orbit
equivalence of actions on (étale) twisted groupoids.

######  Lemma 2.14.

Given Cartan invariant actions $\Gamma_{i}\curvearrowright(A_{i},B_{i})$ and
$\Lambda_{i}\curvearrowright(D_{i},C_{i})$ with
$\Gamma_{i}\curvearrowright(A_{i},B_{i})\sim_{\rm
coe}\Lambda_{i}\curvearrowright(D_{i},C_{i})$, for $i=1,2$, we have
$\Gamma_{1}\curvearrowright(A_{1},B_{1})\sim_{\rm
coe}\Gamma_{1}\curvearrowright(A_{1},B_{1})$ iff
$\Lambda_{1}\curvearrowright(D_{1},C_{1})\sim_{\rm
coe}\Lambda_{1}\curvearrowright(D_{1},C_{1})$.

The next corollary is an immediate consequence of the above lemma, but despite
its easy proof, it is crucial in handling orbit equivalence of actions on non
étale groupoids.

######  Corollary 2.15.

For $i=1,2$, let $(G_{i},\Sigma_{i})$ be (not necessarily étale) twisted
groupoids such that $C^{*}_{r}(G_{i},\Sigma_{i})$ has a Cartan subalgebra
$C_{i}$. Let $\Gamma_{i}\curvearrowright(G_{i},\Sigma_{i})$ be an action such
that the lifted action on $C^{*}_{r}(G_{i},\Sigma_{i})$ leaves $C_{i}$
invariant, and let $(G_{i},\Sigma_{i})$ has an equivariant étale realization
$(\tilde{G}_{i},\tilde{\Sigma}_{i})$ with respect to $C_{i}$. Then,

$(i)$ $\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$ implies
$\Gamma_{1}\curvearrowright(\tilde{G}_{1},\tilde{\Sigma}_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(\tilde{G}_{2},\tilde{\Sigma}_{2})$.

$(ii)$ $\Gamma_{1}\curvearrowright(G_{1},\Sigma_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright(G_{2},\Sigma_{2})$ implies
$\Gamma_{1}\curvearrowright(C^{*}_{r}(G_{1},\Sigma_{1}),C_{1})\sim_{\rm
coe}\Gamma_{2}\curvearrowright C^{*}_{r}(G_{2},\Sigma_{2}),C_{2}).$

We later use this to build concrete examples of orbit equivalent actions on
Cartan pairs (Example 4.4). We warn the reader that the converse implication
in part $(i)$ of the above Corollary may fail in general. With the same token,
the implication in part $(ii)$ may fail if the twisted groupoids
$(G_{i},\Sigma_{i})$ happen not to have an equivariant étale realization.

##  3\. Proof of The Main Results

In this section we prove the main results of the paper. One of the main steps
in proving our first main result is to use Proposition 2.9 for
$C^{*}_{r}(\Gamma\ltimes G,\Gamma\ltimes\Sigma)$. In order to do this, we need
a condition to ensure topological principality of $(\Gamma\ltimes
G,\Gamma\ltimes\Sigma)$.

######  Definition 3.1.

An action $\Gamma\curvearrowright G$ is called principally free if $\gamma
r(g)=s(g)$ forces $\gamma$ to be the identity of $\Gamma$ and $g$ to be a unit
element in $G^{(0)}.$ It is called topologically principally free if there is
a dense subset $U\subseteq G^{(0)}$ such that the above condition holds for
each $g\in G_{u}$ and each $u\in U$.

The (topologically) principally free actions are (topologically) free. An
action $\Gamma\curvearrowright G$ is principally free iff $\Gamma\ltimes G$ is
(topologically) principal. Note that, this notion is defined only for actions
on (topologically) principal groupoids.

Let $(G,\Sigma)$ and $(H,E)$ be twisted groupoids. By an open embedding of
$(H,E)$ in $(G,\Sigma)$, written as $(H,E)\subset^{\circ}(G,\Sigma)$, we mean
open inclusions $G^{(0)}\subseteq H\subseteq G$ and $\Sigma^{(0)}\subset
E\subset\Sigma$, such that the following diagram commutes,

Let $(G,\Sigma)$ be a twisted groupoid with twist maps $\iota_{0}$ and
$\pi_{0}$, and let $\Gamma$ act on $(G,\Sigma)$. Then

$\mathbb{T}\times
G^{(0)}\overset{\iota}{\rightarrowtail}\Gamma\ltimes\Sigma\overset{\pi}{\twoheadrightarrow}\Gamma\ltimes
G$

is a twist, where $\iota(t,v):=(e,\iota_{0}(t,v))$, for the identity element
$e$ of $\Gamma$, and $\pi(\gamma,\sigma):=(\gamma,\pi_{0}(\sigma))$.

  1. (1)

There exist an open embedding which embeds $(G,\Sigma)$ in $(\Gamma\ltimes
G,\Gamma\ltimes\Sigma)$,

$\textstyle{\mathbb{T}\times
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\Gamma\ltimes\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Gamma\ltimes
G}$$\textstyle{\mathbb{T}\times
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{0}}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\pi_{0}}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$

where $\eta(\sigma):=(e,\sigma)$ and $\eta^{\prime}(g):=(e,g)$, for
$\sigma\in\Sigma,g\in G$. By [2, Lemma 3.4], there is a canonical embedding
$C^{*}_{r}(G,\Sigma)\hookrightarrow C^{*}_{r}(\Gamma\ltimes
G,\Gamma\ltimes\Sigma)$.

  2. (2)

There is an open embedding of $\Gamma\ltimes G^{(0)}$ with trivial twist in
$(\Gamma\ltimes G,\Gamma\ltimes\Sigma)$,

$\textstyle{\mathbb{T}\times
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\Gamma\ltimes\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\Gamma\ltimes
G}$$\textstyle{\mathbb{T}\times
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(id_{\mathbb{T}},id_{G})}$$\textstyle{\mathbb{T}\times(\Gamma\ltimes
G^{(0)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\Gamma\ltimes
G^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(id_{\mathbb{T}},id_{G})}$

where $\theta(t,(\gamma,v))=(\gamma,\iota_{0}(t,v))$. By the definition of
twisted groupoid, $\pi_{0}^{-1}(u)=\iota_{0}(\mathbb{T}\times\\{u\\})$, for
$u\in G^{(0)}$, and $(\Gamma,\iota_{0}(\mathbb{T}\times
G^{(0)}))=(\Gamma,\pi^{-1}(G^{(0)}))$ is open, since $\Gamma$ is discrete and
$G^{(0)}$ is open. We use the notation $\Gamma\ltimes
G^{(0)}\subset^{\circ}(G,\Sigma)$ for this open inclusion. There is a natural
isomorphism between $\Gamma\ltimes_{r}C_{0}(G^{(0)})$ and
$C^{*}_{r}(\Gamma\ltimes G^{(0)})$, and again by [2, Lemma 3.4], there is a
canonical embedding $\Gamma\ltimes_{r}C_{0}(G^{(0)})\hookrightarrow
C^{*}_{r}(\Gamma\ltimes G,\Gamma\ltimes\Sigma)$.

######  Lemma 3.2.

Let $(G,\Sigma)$ and $(H,E)$ be twisted groupoids with
$(G,\Sigma)\subset^{\circ}(H,E)$. Then,

  1. (1)

$C^{*}_{r}(H,E)\overset{\iota}{\hookrightarrow}C^{*}_{r}(G,\Sigma)$ and the
restriction of $\iota$ to $C_{0}(H^{(0)})$ is an isomorphism onto
$C_{0}(G^{(0)})$.

If moreover the twisted groupoids are topologically principal,

  1. (2)

$H(C_{0}(H^{(0)}))\overset{\iota_{2}}{\hookrightarrow}G(C_{0}(G^{(0)}))$ and
$E(C_{0}(H^{(0)}))\overset{\iota_{1}}{\hookrightarrow}\Sigma(C_{0}(H^{(0)}))$,
where $\iota_{1}[\alpha_{n}(x),n,x]:=[\alpha_{\iota(n)}(x),\iota(n),x]$
$\iota_{2}[\alpha_{n}(x),\alpha_{n},x]:=[\alpha_{\iota(n)}(x),\alpha_{\iota(n)},x]$,

  2. (3)

for isomorphisms
$(\phi_{2},\phi_{1}):\big{(}G(C_{0}(G^{(0)})),\Sigma(C_{0}(G^{(0)}))\big{)}\rightarrow(G,\Sigma)$
and
$(\phi^{\prime}_{2},\phi^{\prime}_{1}):\big{(}H(C_{0}(H^{(0)})),E(C_{0}(H^{(0)}))\big{)}\rightarrow(H,E)$,
defined in Proposition 2.9, the following diagrams commute,

###### Proof.

Since $(H,E)\subset^{\circ}(G,\Sigma)$, the natural map $C_{c}(H,E)\rightarrow
C_{c}(G,\Sigma)$ extends to an isometric homomorphism
$:C^{*}_{r}(H,E)\hookrightarrow C^{*}_{r}(G,\Sigma)$, by [2, Lemma 3.4]. Since
the open support of an element in $C_{c}(H,E)$ is a subset of the open support
of that element in $C_{c}(G,\Sigma)$, and after identification,
$\iota|_{C_{0}(H^{(0)})}:C_{0}(H^{(0)})\cong C_{0}(G^{(0)})$, which proves
(1). Put $A:=C^{*}_{r}(G,\Sigma)$, $B:=C^{*}_{r}(H,E)$, and
$C:=C_{0}(G^{(0)})\cong C_{0}(H^{(0)})$. For (2), let
$[\alpha_{n}(x),\alpha_{n},x]\in H(C_{0}(H^{(0)}))$, i.e., $n\in N_{B}(C)$,
$x\in$ dom$(n)$. By definition of normalizers,
$\iota(N_{B}(C))=N_{A}(C)\cap\iota(B)$. On the other hand, since $\iota$ maps
$C_{0}(H^{(0)})$ into $C_{0}(G^{(0)})$, dom$(\iota(n))=$ dom$(n)$, thus
$[\alpha_{\iota(n)}(x),\alpha_{\iota(n)},x]\in G(C_{0}(G^{(0)}))$. It is easy
to check that $\iota_{2}$ is a groupoid embedding, and the same for
$\iota_{1}$. To prove (3), let $n\in N_{B}(C)$ and put $S_{1}:={\rm
supp}^{\prime}(\iota(n))\subseteq G$ and $S_{2}={\rm
supp}^{\prime}(n)\subseteq H$, then
$\phi^{\prime}_{2}([\alpha_{n}(x),\alpha_{n},x])=S_{1}x$ and
$\phi_{2}([\alpha_{\iota(n)}(x),\alpha_{\iota(n)},x])=S_{2}x$. By the proof of
part (1), $S_{1}\subset S_{2}$, and [18, Proposition 4.7] implies that $S_{1}$
and $S_{2}$ are bisections with $S_{1}x=S_{2}x$, for each $x$. Finally,

$\displaystyle\phi^{\prime}_{1}[y,n,x]$
$\displaystyle=(n(S_{1}x)/\sqrt{n^{*}n(x)},S_{1}x)=(\iota(n)(S_{2}x)/\sqrt{\iota(n^{*}n)(x)},S_{2}x)$
$\displaystyle=\phi_{1}[y,\iota(n),x],$

as required. ∎

######  Lemma 3.3.

Let $B_{i}:=C^{*}_{r}(H_{i},E_{i})\subset C^{*}_{r}(G_{i},\Sigma_{i}):=A_{i}$,
$i=1,2$, where $(G_{i},\Sigma_{i})$ and $(H_{i},E_{i})$ are twisted
topologically principal groupoids with
$(H_{i},E_{i})\subset^{\circ}(G_{i},\Sigma_{i})$. Assume that there exists a
Cartan invariant isomorphism $\psi:A_{1}\rightarrow A_{2}$, mapping $B_{1}$
onto $B_{2}$. Then the isomorphism
$(\phi^{\prime},\phi):(G_{1},\Sigma_{1})\rightarrow(G_{2},\Sigma_{2})$ induced
by the Renault’s canonical isomorphism (2.2) maps $E_{1}$ and $H_{1}$ to
$E_{2}$ and $H_{2}$, respectively, yielding a twisted groupoid isomorphism
$(H_{1},E_{1})\cong(H_{2},E_{2})$.

###### Proof.

Let us identify $C_{i}:=C_{0}(G_{i}^{(0)})$ with $C_{0}(H_{i}^{(0)})$, for
$i=1,2$. Let
$(\phi_{i},\phi_{i}^{\prime}):(G_{i}(C_{i}),\Sigma_{i}(C_{i}))\rightarrow(G_{i},\Sigma_{i})$
be the isomorphism defined in Proposition 2.9, and
$(\omega^{\prime},\omega):(G_{1}(C_{1}),\Sigma_{1}(C_{1}))\rightarrow(G_{2}(C_{2}),\Sigma_{2}(C_{2}))$
be as in (2.2). We have commutative diagrams,

By Lemma 3.2, we also have the following commutative diagrams,

It is enough to observe that $\omega(E_{1}(C_{1}))=E_{2}(C_{2})$ and
$\omega^{\prime}(H_{1}(C_{1}))=H_{2}(C_{2})$. Let $n\in
N_{B_{1}}(C_{1})=N_{A}(C_{1})\cap B_{1}$ and $x\in H_{1}^{(0)}=G_{1}^{(0)}$
with $x\in$ dom$(n)$, then $[\alpha_{n}(x),n,x]\in E_{1}(C_{1})$, and by the
paragraph before Definition 2.11, we have,
$\omega([\alpha_{n}(x),n,x])=[\psi(\alpha_{n}(x)),\psi(n),\psi(x)]=[\alpha_{\psi(n)}(\psi(x)),\psi(n),\psi(x)].$
Since $\psi$ is Cartan invariant, $\psi(n)\in N_{B_{2}}(C_{2})$ and
$\psi(x)\in G_{2}^{(0)}$, and in particular,
$[\alpha_{\psi(n)}(\psi(x)),\psi(n),\psi(x)]\in E_{2}(C_{2})$. Similarly,
$\omega^{\prime}(H_{1}(C_{1}))=H_{2}(C_{2})$. As the following diagram
commutes,

$\textstyle{\mathbb{T}\times
H^{(0)}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}\times
G^{(0)}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}\times
G^{(0)}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}\times
H^{(0)}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\Sigma_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{G_{2}}$$\textstyle{H_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the restrictions $\phi|_{E_{1}}$ and $\phi^{\prime}|_{H_{1}}$ induce an
isomorphism: $(H_{1},E_{1})\rightarrow(H_{2},E_{2})$. ∎

######  Lemma 3.4.

Let $(A,B)$ be a Cartan pair, $\Gamma\curvearrowright A$ be a Cartan invariant
action, and $\Gamma\curvearrowright(G(B),\Sigma(B))$ be as in (2.1). Then, for
$X:=\hat{B}$, the following are equivalent:

  1. (1)

$\Gamma\ltimes G(B)$ is topologically principal,

  2. (2)

$\beta:\Gamma\curvearrowright G(B)$ is topologically principally free,

  3. (3)

the elements $x\in X$ with the following property form a dense subset of $X$:
$\alpha_{n}(x)=\gamma x$ with $n\in N_{A}(B)$ and $x\in{\rm dom}(n)$ holds
only when $\gamma$ is the identity of $\Gamma$ and $n\in B$,

  4. (4)

The elements $x\in X$ with the following property form a dense subset of $X$:
$n^{*}(\gamma b)n(x)=b(x)n^{*}n(x)$ with $n\in N_{A}(B)$ and $x\in{\rm
dom}(n)$, for all $b\in B$, holds only when $\gamma$ is the identity of
$\Gamma$ and $n\in B$.

###### Proof.

$(1)$ and $(2)$ are trivially equivalent. For
$g=[\alpha_{n}(x),\alpha_{n},x]\in G(B)$, $\gamma r(g)=s(g)$ is equivalent to
$\gamma\alpha_{n}(x)=x$. By the definition of $G(B)$, $g$ is in the unit space
if $n\in B$. Thus $x\in U_{\beta}$ simply means that
$\alpha_{n}(x)=\gamma^{-1}x$ holds only when $\gamma$ is the identity of
$\Gamma$ and $n\in B$, for $n\in N_{A}(B)$ with $x\in$ dom$(n)$. Thus $(2)$
and $(3)$ are equivalent. To show the equivalence of $(3)$ and $(4)$, it is
enough to observe that $n^{*}(\gamma b)n(x)=b(x)n^{*}n(x)$, for $b\in B$, if
and only if $\alpha_{n}(x)=\gamma^{-1}x$, for $n\in N_{A}(B)$ with $x\in$
dom$(n)$. For this, let $n\in N_{A}(B)$, $x\in$ dom$(n)$ and $\gamma\in\Gamma$
such that $n^{*}(\gamma b)n(x)=b(x)n^{*}n(x)$, for $b\in B$, then,

$b(\gamma^{-1}\alpha_{n}(x))n^{*}n(x)=(\gamma
b)(\alpha_{n}(x))n^{*}n(x)=n^{*}(\gamma b)n(x)=b(x)n^{*}n(x),$

for $b\in B$. Since $x\in$ dom$(n)$ and $b$ is arbitrary,
$\gamma^{-1}\alpha_{n}(x)=x$. Conversely, if $\gamma^{-1}\alpha_{n}(x)=x$,
then $b(x)n^{*}n(x)=b(\gamma^{-1}\alpha_{n}(x))n^{*}n(x)=n^{*}(\gamma b)n(x)$.
∎

######  Definition 3.5.

A Cartan invariant action $\Gamma\curvearrowright A$ is called topologically
principally free if the condition (4) above holds.

###### Proof of Theorem A.

$(1)\Leftrightarrow(2)$: This follows directly from Definition 2.6 and Theorem
2.3.

$(2)\Rightarrow(3)$: The map $\psi$ lifts to
$\theta^{\prime}:C_{c}(\Gamma_{1}\ltimes
G_{1},\Gamma_{1}\ltimes\Sigma_{1})\rightarrow C_{c}(\Gamma_{2}\ltimes
G_{2},\Gamma_{2}\ltimes\Sigma_{2})$, defined by
$\theta^{\prime}(f)(\gamma_{2},\sigma_{2})=f(\psi^{-1}(\gamma_{2},\sigma_{2}))$.
Using open inclusions of the paragraph after Definition 3.1, one can see that
$\theta^{\prime}$ maps $C_{c}(G_{1}^{(0)},\mathbb{T}\times G_{1}^{(0)})$ and
$C_{c}(G_{1},\Sigma_{1})$ onto $C_{c}(G_{2}^{(0)},\mathbb{T}\times
G_{2}^{(0)})$ and $C_{c}(G_{2},\Sigma_{2})$, and $C_{c}(\Gamma_{1}\ltimes
G_{1}^{(0)},\mathbb{T}\times(\Gamma_{1}\ltimes G_{1}^{(0)}))$ onto
$C_{c}(\Gamma_{2}\ltimes G_{2}^{(0)},\mathbb{T}\times(\Gamma_{2}\ltimes
G_{2}^{(0)}))$, respectively. This induces an isomorphism

$\theta:C^{*}_{r}(\Gamma_{1}\ltimes
G_{1},\Gamma_{1}\ltimes\Sigma_{1})\rightarrow C^{*}_{r}(\Gamma_{2}\ltimes
G_{2},\Gamma_{2}\ltimes\Sigma_{2}).$

The embedding of trivial twist of $\Gamma_{i}\ltimes G_{i}^{(0)}$ in
$(\Gamma_{i}\ltimes G_{i},\Gamma_{i}\ltimes\Sigma_{i})$ is an open embedding,
$(G_{i},\Sigma_{i})\subset^{\circ}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})$, and
$(e_{i},G_{i}^{(0)})\subset^{\circ}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})$, for $i=1,2$. By the proof of [2, Lemma
3.4], given $f_{i}\in C_{c}(G_{i}^{(0)},\mathbb{T}\times G_{i}^{(0)})$,
$f^{\prime}_{i}\in C_{c}(\Gamma_{i}\ltimes
G_{i}^{(0)},\mathbb{T}\times\Gamma_{i}\ltimes G_{i}^{(0)})$, and
$f^{\prime\prime}_{i}\in C_{c}(G_{i},\Sigma_{i})$, we have,
$||f_{i}||_{C_{0}(G_{i}^{(0)})}=||f_{i}||_{C^{*}_{r}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})}$,
$||f^{\prime}_{i}||_{C^{*}_{r}(\Gamma_{i}\ltimes
G_{i}^{(0)})}=||f^{\prime}_{i}||_{C^{*}_{r}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})}$, and
$||f^{\prime\prime}_{i}||_{C^{*}_{r}(G_{i},\Sigma_{i})}=||f^{\prime\prime}_{i}||_{C^{*}_{r}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})}$. Therefore, $\theta$ maps
$C_{0}(G_{1}^{(0)})$,
$\Gamma_{1}\ltimes_{r}C^{*}_{r}(G_{1}^{(0)})=C^{*}_{r}(\Gamma_{1}\ltimes
G_{1}^{(0)})$, and $C^{*}_{r}(G_{1},\Sigma_{1})$ onto $C_{0}(G_{2}^{(0)})$,
$\Gamma_{2}\ltimes_{r}C_{0}(G_{2}^{(0)})$, and $C^{*}_{r}(G_{2},\Sigma_{2})$,
respectively.

$(3)\Rightarrow(2)$: Put $A_{i}:=C^{*}_{r}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})$, $B_{i}:=C^{*}(G_{i},\Sigma_{i})$,
$C_{i}:=C_{0}(G_{i}^{(0)})$, and $\Gamma_{i}\ltimes_{r}C_{0}(G_{i}^{(0)})\cong
C^{*}_{r}(\Gamma_{i}\ltimes G_{i}^{(0)})=:D_{i}$, for $i=1,2$. Then, there
exists an isomorphism $A_{1}\cong A_{2}$, mapping $B_{1},C_{1}$, and $D_{1}$
onto $B_{2},C_{2}$, and $D_{2}$, respectively. The additional condition
ensures that the groupoids $\Gamma_{1}\ltimes G_{1}$ and $\Gamma_{1}\ltimes
G_{1}$ are topologically principal, and by Proposition 2.9, there exists a
twisted groupoid isomorphism $(\psi^{\prime},\psi):(\Gamma_{1}\ltimes
G_{1},\Gamma_{1}\ltimes\Sigma_{1})\rightarrow(\Gamma_{2}\ltimes
G_{2},\Gamma_{2}\ltimes\Sigma_{2})$. Since
$(G_{i},\Sigma_{i})\subset^{\circ}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})$, by Lemma 3.3,
$\psi^{\prime}(e_{1},G_{1})=(e_{2},G_{2})$, and
$\psi(e_{1},\Sigma_{1})=(e_{2},\Sigma_{2})$, yielding a twisted groupoid
isomorphism $(G_{1},\Sigma_{1})\cong(G_{2},\Sigma_{2})$. Define
$\phi^{\prime}:=\psi^{\prime}|_{(e_{1},G_{1})}$ and
$\phi:=\psi|_{(id,\Sigma_{1})}$, and we have twisted groupoid ismomorphism
$(\phi^{\prime},\phi):(G_{1},\Sigma_{1})\rightarrow(G_{2},\Sigma_{2})$.
Finally, since $\Gamma_{i}\ltimes G_{i}^{(0)}\subset^{\circ}(\Gamma_{i}\ltimes
G_{i},\Gamma_{i}\ltimes\Sigma_{i})$, $\psi$ maps $(\Gamma_{1},G_{1}^{(0)})$
onto $(\Gamma_{2},G_{2}^{(0)})$. ∎

Given a Cartan pair $(A,B)=:(A,C_{0}(X))$, a Cartan invariant action
$\Gamma\curvearrowright(A,B)$ is called topologically free if the induced
action $\Gamma\curvearrowright X$ is topologically free in the classical
sense. Now to see that Corollary C follows from Proposition 2.8 and Theorem A,
simply observe that by Proposition 2.10, statement (3) of Theorem A could be
rephrased as the requirement that there exists an isomorphism
$\theta:\Gamma_{1}\ltimes_{r}C^{*}_{r}(G_{1},\Sigma_{1})\cong\Gamma_{2}\ltimes_{r}C^{*}_{r}(G_{2},\Sigma_{2})$
with,

  * •

$\theta(C_{0}(G_{1}^{(0)}))=C_{0}(G_{2}^{(0)}),$

  * •

$\theta(\Gamma_{1}\ltimes_{r}C_{0}(G_{1}^{(0)}))=\Gamma_{2}\ltimes_{r}C_{0}(G_{2}^{(0)}),$

  * •

$\theta(C^{*}_{r}(G_{1},\Sigma_{1}))=C^{*}_{r}(G_{2},\Sigma_{2}).$

###### Proof of Theorem D.

Put $C_{i}:=C_{0}(G_{i}^{(0)})$, $i=1,2,$ and let,

$(\theta,\theta^{\prime}):(G_{1}(C_{1}),\Sigma(C_{1}))\rightarrow(G_{2}(C_{2}),\Sigma_{2}(C_{2}))$

be as in (2.2). Since the actions are Cartan invariant, $\Gamma_{i}$ acts on
$(G_{i}(C_{i}),\Sigma_{i}(C_{i}))$ by (2.1). Let $a:\Gamma_{1}\times
G_{1}^{(0)}\rightarrow\Gamma_{2}$ and $b:\Gamma_{2}\times
G_{2}^{(0)}\rightarrow\Gamma_{1}$ be continuous maps yielding
$\Gamma_{1}\curvearrowright G_{1}^{(0)}\sim_{\rm
coe}\Gamma_{2}\curvearrowright G_{2}^{(0)}$. Then,
$\theta(\gamma[y,\alpha_{n},x])=a(\gamma,x)\theta([y,\alpha_{n},x]),$ for
$\gamma\in\Gamma_{1}$, $\gamma^{\prime}\in\Gamma_{2}$, $[y,\alpha_{n},x]\in
G_{1}(C_{1})$, and $[y^{\prime},\alpha_{n^{\prime}},x^{\prime}]\in
G_{2}(C_{2})$. By continuity of $a$, there exists an open neighborhood
$U\subset G_{1}^{(0)}$ of $x$ such that $a(\gamma,u)=a(\gamma,x)$, for $u\in
U$. By assumption,

$\displaystyle\begin{split}a(\gamma,x)&=a(\gamma,u)=a(\gamma,s([\alpha_{n}(u),\alpha_{n},u]))=a(\gamma,r([\alpha_{n}(u),\alpha_{n},u]))=a(\gamma,\alpha_{n}(u)),\end{split}$

and since the actions preserve Cartan subalgebras, $\gamma_{1}n_{1}$ and
$\gamma_{2}n_{2}$ are normalizers in $C^{*}_{r}(G_{1},\Sigma_{1})$ and
$(G_{2},\Sigma_{2})$, respectively, for $\gamma_{i}\in\Gamma_{i}$ and
normalizers $n_{i}$ in $C^{*}_{r}(G_{i},\Sigma_{i})$, $i=1,2$. For $x\in
G_{1}^{(0)}$,

$(a(\gamma,x)\psi(n^{*}n))(\psi(\gamma x))=\psi(n^{*}n)(\psi(x))=n^{*}n(x).$

Thus, dom$(a(\gamma,x)\psi(n))=\psi($dom$(n))$. Therefore,

$\displaystyle\alpha_{a(\gamma,x)\psi(n)}(\psi(\gamma u))$
$\displaystyle=\alpha_{a(\gamma,x)\psi(n)}(a(\gamma,x)\psi(u))=a(\gamma,x)\alpha_{\psi(n)}(\psi(u))$
$\displaystyle=a(\gamma,\alpha_{n}(u))\psi(\alpha_{n}(u))=\psi(\gamma\alpha_{n}(u))$
$\displaystyle=\psi(\alpha_{\gamma n}(\gamma u))=\alpha_{\psi(\gamma
n)}(\psi(\gamma u)),$

i.e., $\alpha_{a(\gamma,x)\psi(n)}|_{\psi(\gamma U)}=\alpha_{\psi(\gamma
n)}|_{\psi(\gamma U)}$. Finally, by the paragraph before Definition 2.11,
$\theta(\gamma[y,\alpha_{n},x])=[\psi(\gamma y),\alpha_{\psi(\gamma
n)},a(\gamma,x)\psi(x)]=a(\gamma,x)\theta([y,\alpha_{n},x]).$ Similar equality
holds for $\theta^{-1}$ and $b$. ∎

Finally, by Lemma 2.2 and Theorem D, we get Corollary E.

##  4\. Examples

In this section, we give examples of orbit equivalence in our sense which give
rise to non classical orbit equivalent actions. Let us first see what we mean
by a classical action. For a continuous action of a discrete group $G$ on a
locally compact, Hausdorff space $X$, we might regard $X$ as a groupoid
identified with the diagonal $\Delta_{X}\subseteq X\times X$, then
$C^{*}_{r}(\Delta_{X})=C_{0}(X)$ with Cartan subalgebra being $C_{0}(X)$
itself. Conversely, a Cartan invariant action $\Gamma\curvearrowright(A,B)$ is
classical if $A=B$, in which case $X$ would be the spectrum of the commutative
$C^{*}$-algebras $A$. In this sense, an action
$\Gamma\curvearrowright(G,\Sigma)$ is non classical whenever
$C^{*}_{r}(G,\Sigma)$ is non abelian.

Our strategy to build non classical examples is to start with a classical
instance of continuous orbit equivalence with cocycle maps $a:\Gamma\times
X\rightarrow\Lambda$, $b:\Lambda\times Y\rightarrow\Gamma$, and given
$\gamma\in\Gamma$, choose a (the largest) open neighborhood (depending on
$\gamma$) of given point $x\in X$ on which $a(\gamma,)$ is constant, and the
same for $b$ and points in $Y$, and use these to build equivalence relations
on $X$ and $Y$. We then find canonical actions on these equivalence relations,
which are topologically free and orbit equivalent. The main challenge is that
though these equivalence relations are topologically principal (and usually
amenable), they are rarely étale. The good news is that since the actions are
Cartan invariant in the level of $C^{*}$-algebras, by Kumjian-Renault theory,
there are topologically principal, étale (twisted) groupoids giving the same
Cartan pairs. Therefore, in concrete examples the real challenge is to find
these étale realizations and find their isomorphic copies inside the groupoid
$C^{*}$-algebras involved. We illustrate how to handle this in some concrete
examples, by explicitly calculating the corresponding groupoid
$C^{*}$-algebras and étale realizations of the involved equivalence relations.
The next lemma gives a sufficient condition for conjugacy of actions on
groupoids.

######  Lemma 4.1.

Each continuous action $\Gamma\curvearrowright G$ on a (not necessarily étale)
groupoid induces an action $\Gamma\curvearrowright C^{*}_{r}(G)$, which is
also Cartan invariant if $G$ is topologically principle and étale. In the
latter case, for any invariant Cartan subalgebra $B$ of $C^{*}_{r}(G)$, the
induced action $\Gamma\curvearrowright G(B)$ is conjugate to
$\Gamma\curvearrowright G$.

###### Proof.

Let us define $\gamma f(g):=f(\gamma^{-1}g)$, for $\gamma\in\Gamma,g\in G$,
and $f\in C_{c}(G)$. For $f,h\in C_{c}(G)$ observe that,

$\displaystyle(\gamma f)*(\gamma h)(g)$ $\displaystyle=\int_{G}(\gamma
f)(\eta)(\gamma h)(\eta^{-1}g)d\lambda^{r(g)}(\eta)$
$\displaystyle=\int_{G}f(\gamma^{-1}\eta)h((\gamma^{-1}\eta^{-1})(\gamma^{-1}g))d\lambda^{r(g)}(\eta)$
$\displaystyle=\int_{G}f(\eta)h(\eta^{-1}(\gamma^{-1}g))d\lambda^{r(\gamma^{-1}g)}(\eta)$
$\displaystyle=f*h(\gamma^{-1}g)=\gamma(f*h)(g),$

for $g\in G$. Next let us observe that the above action is bounded (indeed an
isometry) in the reduced norm, defined by $\|f\|_{r}:=\sup_{u\in
G^{(0)}}\|{\rm Ind}\delta_{u}(f)\|$, where

${\rm
Ind}\delta_{u}(f)\xi(g):=\int_{G}f(\eta)\xi(\eta^{-1}g)d\lambda^{r(g)}(\eta),$

for $g\in G_{u}$ and $\xi\in C_{c}(G_{u})$. By a change of variable similar to
the above calculation,

$\displaystyle{\rm Ind}\delta_{u}(\gamma f)\xi(g)={\rm
Ind}\delta_{u}(f)(\gamma^{-1}\xi)(\gamma^{-1}g)=\gamma{\rm
Ind}\delta_{u}(f)(\gamma^{-1}\xi)(g),$

thus regarding $\xi$ as an element in $L^{2}(G^{u},\lambda^{u})$,

$\|{\rm Ind}\delta_{u}(\gamma f)\xi\|_{2}=\|\gamma{\rm
Ind}\delta_{u}(f)(\gamma^{-1}\xi)\|_{2}=\|{\rm
Ind}\delta_{u}(f)(\gamma^{-1}\xi)\|_{2}\leq\|{\rm
Ind}\delta_{u}(f)\|\|\xi\|_{2},$

which by a density argument gives $\|{\rm Ind}\delta_{u}(\gamma f)\|\leq\|{\rm
Ind}\delta_{u}(f)\|$, leading to an equality by the same argument for $\gamma$
replaced by $\gamma^{-1}$. We may thus extend this to an action on the
$C^{*}$-algebra $C^{*}_{r}(G)$. When the groupoid is moreover topologically
principle and étale, this latter action is also Cartan invariant, since the
action $\Gamma\curvearrowright G$ keeps $G^{(0)}$ invariant and the Cartan
subalgebra is nothing but $C_{0}(G^{(0)})$. More generally, for any Cartan
subalgebra $B$ of $C^{*}_{r}(G,)$, let $G(B)$ be the corresponding Weyl
groupoid, $\phi_{2}$ be as it defined in Proposition 2.9 then for
$[x,\alpha_{n},y]\in G(B)$,

$\displaystyle\phi_{2}(\gamma[x,\alpha_{n},y])=\phi_{2}[\gamma
x,\alpha_{\gamma n},\gamma y]=S_{\gamma n}(\gamma
x)=\gamma\phi_{2}[x,\alpha_{n},y].$

This simply means that the induced action on Renault-Kumjian groupoid is
conjugate to the original action, as claimed. ∎

Let $\Gamma\curvearrowright X$ and $\Lambda\curvearrowright Y$ be
topologically free actions of discrete groups, with respect to maps
$a:\Gamma\times X\rightarrow\Lambda$, $b:\Lambda\times Y\rightarrow\Gamma$ and
homeomorphism $\phi:X\rightarrow Y$. For $\gamma\in\Gamma$ ,
$\lambda\in\Lambda$, and $x\in X$ (resp., $y\in Y$), define $U_{x}^{\gamma}$
(resp., $V_{y}^{\lambda}$) to be the open neighbourhood of $x$ (resp., $y$)
that $a(\gamma,U_{x}^{\gamma})=a(\gamma,x)$ (resp.,
$b(\lambda,V_{y}^{\lambda})=b(\lambda,y)$). In another words,
$U_{x}^{\gamma}=a^{-1}(a(\gamma,x))\cap(\gamma,X)$ (resp.,
$V_{y}^{\lambda}=b^{-1}(b(\lambda,y))\cap(\lambda,Y)$). Also, define
$U_{x}:=\bigcap_{\gamma\in\Gamma}U_{x}^{\gamma}$ and
$\hat{U}_{x}=U_{x}\cap\phi^{-1}(V_{\phi(x)})$ (resp.,
$V_{y}:=\bigcap_{\lambda\in\Lambda}V_{y}^{\lambda}$ and
$\hat{V}_{y}=V_{y}\cap\phi(U_{\phi^{-1}(y)})$).

######  Lemma 4.2.

In the above notation,

$(i)$ $\gamma U_{x}=U_{\gamma x}$ and $\lambda V_{y}=V_{\lambda y}$,

$(ii)$ $\gamma\hat{U}_{x}=\hat{U}_{\gamma x}$ and
$\lambda\hat{V}_{y}=\hat{V}_{\lambda y}$,

for each $\gamma\in\Gamma$ , $\lambda\in\Lambda$, and $x\in X,y\in Y$.

###### Proof.

$(i)$ Since actions are topologically free,
$a(\gamma^{\prime}\gamma,x)=a(\gamma^{\prime},\gamma x)a(\gamma,x)$ by Lemma
2.2 . Given $u\in U_{x}$,

$a(\gamma^{\prime},\gamma
x)=a(\gamma^{\prime}\gamma,x)a(\gamma,x)^{-1}=a(\gamma^{\prime}\gamma,u)a(\gamma,u)^{-1}=a(\gamma^{\prime},\gamma
u),$

for each $\gamma^{\prime}\in\Gamma$. This means that, $\gamma U_{x}=U_{\gamma
x}$, and by the same argument, we have $\lambda V_{y}=V_{\lambda y}$.

$(ii)$ Assume further that $u\in\hat{U}_{x}$. Let us observe that $\gamma
u\in\phi^{-1}(V_{\phi(\gamma x)})$. Given $\lambda\in\Lambda$,

$b(\lambda,\phi(\gamma
u))=b(\lambda,a(\gamma,u)\phi(u)){=}b(\lambda,a(\gamma,u)\phi(x))=b(\lambda,a(\gamma,x)\phi(x))=b(\lambda,\phi(\gamma
x)),$

thus, $\phi(\gamma u)\in V_{\phi(\gamma x)}$, as claimed. Similarly,
$\lambda\hat{V}_{y}=\hat{V}_{\lambda y}$. ∎

The next proposition is an immediate consequence of Lemma 4.2.

######  Proposition 4.3.

Let $\Gamma\curvearrowright X\sim_{\rm coe}\Lambda\curvearrowright Y$ with
respect to homeomorphism $\phi:X\rightarrow Y$ and let $\hat{U}_{x}$ and
$\hat{V}_{y}$ be defined as above. Then,

$\displaystyle G_{X}:=\\{(x,u);x\in X,u\in\hat{U}_{x}\\},\ \
G_{Y}:=\\{(y,v);y\in Y,v\in\hat{V}_{y}\\}$

are groupoids with respect to operations $(l,m)(m,n)=(l,n)$ and
$(l,m)^{-1}=(m,l)$. The actions $\Gamma\curvearrowright G_{X}$ and
$\Lambda\curvearrowright G_{Y}$, respectively defined by $\gamma(x,u)=(\gamma
x,\gamma u)$ and $\lambda(y,v)=(\lambda y,\lambda v)$, are topologically free,
and $\Gamma\curvearrowright G_{X}\sim_{\rm coe}\Lambda\curvearrowright G_{Y}$,
with respect to maps $a$, $b$ and groupoid isomorphism mapping $(x,u)$ on
$(\phi(x),\phi(u))$.

To illustrate the above typical construction in a concrete example, let us use
the example of odometer transformations in [15, Theorem 3.3] (with the warning
that there is a typo in the definition of cocycles in the proof of the above
cited result). Consider supernatural number $M=\Pi_{p\in P}p^{v_{p}(M)}$,
where $P$ is the set of all prime numbers and $\sum_{p}=v_{p}(M)=\infty$. For
a natural number $m$, let $\mathbb{Z}_{m}$ be the finite cyclic group of
integers mod $m$. Fix a sequence $(m_{k})_{k}$ of natural numbers such that
for all prime number $p$, $v_{p}(m_{k})\nearrow v_{p}(M)$, as
$k\rightarrow\infty$. Now define $\mathbb{Z}_{M}$ to be the inverse limit of
$\mathbb{Z}_{m_{k}}$. Natural embedding of $\mathbb{Z}$ in $\mathbb{Z}_{M}$
yields an action $\mathbb{Z}\curvearrowright\mathbb{Z}_{M}$ which is called
odometer transformation for $M$. We denote the action
$\mathbb{Z}\curvearrowright\mathbb{Z}_{M}$ and its corresponding diagonal
action $\mathbb{Z}\curvearrowright\mathbb{Z}_{M}\times\mathbb{Z}_{M}$ by
$\lambda_{M}$ and $\lambda^{d}_{M}$, respectively.

######  Example 4.4 (Odometer transformation).

For $l\in\mathbb{N}$ and supernatural number $L$, define
$X=X_{lL}=\mathbb{Z}_{lL}$, $\tilde{X}=\tilde{X}_{lL}=l.(\mathbb{Z}_{lL})$ and
$Y=\mathbb{Z}_{l}\times\tilde{X}$. Let $\alpha:\mathbb{Z}\curvearrowright X$
be the odometer transformation,
$\tilde{\alpha}:l\mathbb{Z}\curvearrowright\tilde{X}$ be the corresponding
restricted action, and
$\lambda_{l}:\mathbb{Z}_{l}\curvearrowright\mathbb{Z}_{l}$ be given by
translation. Then $\alpha\sim_{\rm coe}\lambda_{l}\times\tilde{\alpha}$ (c.f.,
the proof of [15, Theorem 3.3]). Indeed, for homeomorphism,

$\displaystyle\phi:X=\bigsqcup_{k=0}^{l-1}k+\tilde{X}\rightarrow\mathbb{Z}_{l}\times\tilde{X};\
\phi(k+x)=(k,x),\ \ (x\in\tilde{X}),$

with inverse $\psi:\mathbb{Z}_{l}\times\tilde{X}\rightarrow X$, given by
$\psi(k,x)=k+x$, for $x\in\tilde{X}$, and the cocycle maps,

$\displaystyle a:\mathbb{Z}\times
X=\bigsqcup_{k=0}^{l-1}k+\mathbb{Z}_{l}\times\bigsqcup_{k=0}^{l-1}k+\tilde{X}\rightarrow\mathbb{Z}_{l}\times
l\mathbb{Z};$
$\displaystyle\big{(}(j+h),(k+x)\big{)}\mapsto\begin{cases}(j,h)&j+k<l\\\
(j,h+l)&j+k\geq l,\end{cases}$

and,

$\displaystyle b:(\mathbb{Z}_{l}\times
l\mathbb{Z})\times(\mathbb{Z}_{l}\times\tilde{X})\rightarrow\mathbb{Z};\ \
\big{(}(j,h),(k,x)\big{)}\mapsto\begin{cases}j+h&j+k<l\\\ j+h-l&j+k\geq
l\end{cases}$

and element $t=\tilde{t}l+k_{0}$ in $\mathbb{Z}$,

$\displaystyle\begin{split}\phi(t(k+x))=\phi((k_{0}+\tilde{t}l)(k+x))&=\begin{cases}((k+k_{0}),(\tilde{t}l+x))&k_{0}+k<l\\\
((k+k_{0}-l),(\tilde{t}l+l+x))&k_{0}+k\geq l\end{cases}\\\
&=\begin{cases}(k_{0},\tilde{t}l)(k,x)&k_{0}+k<l\\\
(k_{0},\tilde{t}l+l)(k,x)&k_{0}+k\geq l\end{cases}\\\
&=a((k_{0}+\tilde{t}l),(k+x))\phi(k+x),\end{split}$

and,

$\displaystyle\begin{split}\psi((k_{0},\tilde{t})(k,x))&=\begin{cases}(k+k_{0})+(\tilde{t}l+x)&k_{0}+k<l\\\
(k+k_{0}-l)+(\tilde{t}l+x)&k_{0}+k\geq l\end{cases}\\\
&=\begin{cases}(k_{0}+\tilde{t}l)(k+x)&k_{0}+k<l\\\
(k_{0}+\tilde{t}l-l)(k+x)&k_{0}+k\geq l\end{cases}\\\
&=b((k_{0},\tilde{t}l),(k,x))\phi^{-1}(k,x).\end{split}$

Next, for $x\in\tilde{X}$ and $k\in\\{0,...,l-1\\}$, by the definition of
$U_{k+x}^{t}$,

$\displaystyle
U_{k+x}^{k_{0}+\tilde{t}l}=\begin{cases}\\{k<l-k_{0}\\}+\tilde{X}&k_{0}+k<l\\\
\\{k\geq l-k_{0}\\}+\tilde{X}&k_{0}+k\geq l,\end{cases}$

and similarly,

$\displaystyle
V_{(k,x)}^{(k_{0},\tilde{t}l)}=\begin{cases}\\{k<l-k_{0}\\}\times\tilde{X}&k_{0}+k<l\\\
\\{k\geq l-k_{0}\\}\times\tilde{X}&k_{0}+k\geq l.\end{cases}$

Thus, $U_{k+x}=\bigcap_{t\in\mathbb{Z}}U_{k+x}^{t}=k+\tilde{X}$ and
$V_{(k,x)}=\bigcap_{t\in\mathbb{Z}_{l}\times
l\mathbb{Z}}V_{(k,x)}^{t}=\\{k\\}\times\tilde{X}$. Similarly,
$\hat{U}_{k+x}=U_{k+x}$ and $\hat{V}_{(k,x)}=V_{(k,x)}$. Consider the
groupoids $G_{L}^{l}$ and $H_{L}^{l}$ given by,

$\displaystyle G_{L}^{l}=\\{(k+x,k+u);k=0,...,l-1,x,u\in\tilde{X}\\},$
$\displaystyle H_{L}^{l}=\\{((k,x),(k,u));k=0,...,l-1,x,u\in\tilde{X}\\}.$

The actions $\mathbb{Z}\curvearrowright G_{L}^{l}$ and $\mathbb{Z}_{l}\times
l\mathbb{Z}\curvearrowright H_{L}^{l}$ are defined by,

$\displaystyle z(k+x,k+u)=(z+k+x,z+k+u)$ $\displaystyle
z((k,x),(k,u))=\begin{cases}((k+k_{0},l\tilde{z}+x),(k+k_{0},l\tilde{z}+u))&k_{0}+k<l\\\
((k+k_{0}-l,l\tilde{z}+x),(k+k_{0}-l,l\tilde{z}+u))&k_{0}+k\geq l\end{cases}$

where $z=\tilde{z}l+k_{0}$. The above groupoids are second-countable,
Hausdorff, amenable, and principal. By Proposition 4.3, the actions are
continuous orbit equivalent and by Lemma 4.1 these action could be lift to
their C∗-algebra. In order to calculate the groupoid C∗-algebra, first observe
$H_{L}^{l}\cong\mathbb{Z}_{l}\times(\tilde{X}\times\tilde{X})$, where
$\mathbb{Z}_{l}$ is regarded as co-trivial groupoid and
$(\tilde{X}\times\tilde{X})$ is the groupoid of the full equivalence relation.
For any Radon measure $\mu$ of full support on $\tilde{X}$,
$C^{*}_{r}(G_{L}^{l})\cong C^{*}_{r}(H_{L}^{l})$, both being isomorphic to
$C^{*}_{r}(\mathbb{Z}_{l}\times(\tilde{X}\times\tilde{X}))\cong\mathbb{C}^{l}\otimes
C^{*}_{r}(\tilde{X}\times\tilde{X})\cong\bigoplus_{1}^{l}\mathscr{K}(L^{2}(\tilde{X},\mu)),$
where the last isomorphism is well known (c.f., [24, Example 1.54]), and the
first holds by the fact that $C^{*}_{r}(G_{1}\times G_{2})\cong
C^{*}_{r}(G_{1})\otimes C^{*}_{r}(G_{2})$, for locally compact, Hausdorff
groupoids $G_{1},G_{2}$ (see for instance, [4, Lemma 5.6], where the same fact
is stated in the amenable étale case, but these conditions are not needed).

In order to identify the étale realizations of the groupoids in the above
examples via the Kumjian-Renault theory, we need the following essentially
known lemma, in which we restrict ourselves to the simpler case of full
equivalence relations. The first two statements are well known (see, [24,
Example 1.54], with a warning that the cited example has some typos), but we
sketch the proof, for the sake of completeness.

######  Lemma 4.5.

Let $X$ be a locally compact Hausdorff space and $\mu$ be a full support Radon
measure on $X$, then for each $x\in X$, the induced representation
$L^{x}:={\rm Ind}\delta_{x}$ extends to an isomorphism of $C^{*}_{r}(X\times
X)$ onto the algebra $\mathscr{K}(L^{2}(X,\mu))$ of compact operators, with
Cartan subalgebra $\mathscr{D}_{X}$, consisting of diagonal operators
$D_{\lambda}:={\rm diag(\lambda)}$, for $\lambda=(\lambda_{k})\in
c_{0}(\mathbb{Z})$. The isomorphisms $L^{x}$ is independent of $x$ up to
equivalence, and given $\lambda=(\lambda_{k})\in c_{0}(\mathbb{Z})$ and a
countable ONB $(\xi_{n})_{n\in\mathbb{Z}}$ of $L^{2}(X,\mu)$ inside $C(X)$, it
maps the elements $f_{\lambda}\in C_{0}(X\times X)$ defined by,

$f_{\lambda}(x,y):=\sum_{k\in\mathbb{Z}}\lambda_{k}\xi_{k}(x)\bar{\xi}_{k}(y),\
\ (x,y\in X),$

to the diagonal operator $D_{\lambda}$. In particular, the $C^{*}$-subalgebra
$\mathscr{C}_{X}$ generated by these continuous functions in
$C^{*}_{r}(X\times X)$ is the Cartan subalgebra of $C^{*}_{r}(X\times X)$,
mapped by $L^{x}$ isometrically onto $\mathscr{D}_{X}$.

###### Proof.

It is known that, for $x\in X$, the maps $L^{x}$ are all equivalent [24,
Exercise 1.4.8], and give the reduced $C^{*}$-norm. Identifying
$L^{2}(\\{x\\}\times X,\delta_{x}\times\mu)$ with $L^{2}(X,\mu)$, one could
write $L^{x}$ as,

$L^{x}(f)\xi(y)=\int_{X}f(y,z)\xi(z)d\mu(z),\ \ \big{(}y\in X,f\in
C_{c}(X\times X),\xi\in L^{2}(X,\mu)\big{)},$

which is a Hilbert-Schmidt operator with kernel $f$. In particular, $L^{x}$ is
surjective. Since $X\times X$ is equivalent to the trivial group,
$C^{*}_{r}(X\times X)$ is Morita equivalent to $\mathbb{C}$, by the celebrated
Renault’s equivalence theorem for reduced algebras [24, Theorem 2.71], and so
simple. This forced $L^{x}$ to be one-one, which is the only non trivial part
of showing that it is a $C^{*}$-isomorphism.

The fact that $\mathscr{D}$ is the Cartan subalgebra of
$\mathscr{K}(L^{2}(X,\mu))$ is well known. Fix a countable ONB
$(\xi_{n})_{n\in\mathbb{Z}}$ of $L^{2}(X,\mu))$ inside $C(X)$, which exists by
standard Fourier theory. Next, for each $n\in\mathbb{Z}$ and
$\lambda=(\lambda_{k})\in c_{0}(\mathbb{Z})$,

$\displaystyle L^{x}(f_{\lambda})\xi_{n}(y)$
$\displaystyle=\int_{X}f_{\lambda}(y,z)\xi(z)d\mu(z)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda_{k}\int_{X}\xi_{k}(y)\bar{\xi}_{k}(z)\xi_{n}(z)d\mu(z)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda_{k}\langle\xi_{n},\xi_{k}\rangle\xi_{k}(y)=\lambda_{n}\xi_{n}(y),$

that is, $L^{x}(f_{\lambda})=D_{\lambda}$. The last statement now follows, as
$L^{x}$ is an isomorphism. ∎

######  Lemma 4.6.

In the notations of Example 4.4,

$(i)$ there is a full support Radon measure $\mu$ on $X$ and a
$\mathbb{Z}$-equivariant isomorphism
$\theta_{X}:L^{2}(X,\mu)\to\ell^{2}(\mathbb{Z})$, lifting to a Cartan
preserving $\mathbb{Z}$-equivariant isomorphism
$\tilde{\theta}_{X}:C^{*}_{r}(X\times X)\to
C^{*}_{r}(\mathbb{Z}\times\mathbb{Z})$.

$(ii)$ there is a full support Radon measure $\nu$ on $Y$ and a
$\mathbb{(}Z_{l}\times l\mathbb{Z})$-equivariant isomorphism
$\theta_{Y}:L^{2}(Y,\nu)\to\ell^{2}(\mathbb{Z}_{l}\times l\mathbb{Z})$,
lifting to a Cartan preserving $(Z_{l}\times l\mathbb{Z})$-equivariant
isomorphism $\tilde{\theta}_{Y}:C^{*}_{r}(Y\times Y)\to
C^{*}_{r}\big{(}(\mathbb{Z}_{l}\times l\mathbb{Z})\times(\mathbb{Z}_{l}\times
l\mathbb{Z})\big{)}.$

###### Proof.

$(i)$ First note that $X=\varprojlim\mathbb{Z}_{m_{k}}$ is a procyclic
(compact abelian) group with (discrete) Pontryagin dual
$\hat{X}=\varinjlim\mathbb{Z}_{m_{k}}$ [19, Lemma 2.9.3], and the latter is an
ONB for $L^{2}(X,\mu)$, for any normalized left Haar measure $\mu$ on $X$ [9,
Corollary 4.27]. Each element of $\hat{X}=\bigcup\mathbb{Z}_{m_{k}}$ is a
positive integer $\ell\ ({\rm mod}\ m_{k})$, which corresponds to a character

$\xi_{\ell}^{k}\big{(}(n_{j})_{j=1}^{\infty}\big{)}:={\rm exp}\big{(}-2\pi
i(\ell+n_{k})/m_{k}\big{)};\ \ \big{(}(n_{j})_{j=1}^{\infty}\in X).$

For simplicity, let us use the notation $[n]_{k}:=n\ ({\rm mod}\ m_{k})$, for
$n\in\mathbb{Z}$. Take a bijection between
$\hat{X}=\\{[n]_{k}:n\in\mathbb{Z}\\}$ and $\mathbb{Z},$ and let $n(\ell,k)$
be the image of $[\ell]_{k}$ under this bijection, for $\ell=0,s,m_{k}-1$. Let
$\mathbb{Z}$ act on itself by translation, then for $t\in\mathbb{Z}$,
$n(\ell,k)+t$ is the image of $[\ell]_{k}+t=[\ell+t]_{k}$, thus
$n(\ell,k)+t=n(\ell+t,k)$. Let $(e_{n})_{n\in\mathbb{Z}}$ be the canonical ONB
of $\ell^{2}(\mathbb{Z})$ and $\theta_{X}:L^{2}(X,\mu)\to\ell^{2}(\mathbb{Z})$
be the isomorphism mapping $\xi_{\ell}^{k}$ to $e_{n(\ell,k)}$, for $k\geq 1$,
$\ell=0,s,m_{k}-1$. Then,

$\displaystyle t\xi_{\ell}^{k}\big{(}(n_{j})\big{)}$
$\displaystyle=\xi_{\ell}^{k}\big{(}([n_{j}-t]_{j})\big{)}$
$\displaystyle={\rm exp}\big{(}-2\pi i(\ell+[n_{k}-t]_{k})/m_{k}\big{)}$
$\displaystyle={\rm exp}\big{(}-2\pi i(\ell+n_{k}-t)/m_{k}\big{)}$
$\displaystyle={\rm exp}\big{(}-2\pi
i([\ell-t]_{k}+n_{k})/m_{k}\big{)}=\xi_{[\ell-t]_{k}}^{k}\big{(}(n_{j})\big{)}$

for $k\geq 1$, $\ell=0,s,m_{k}-1$, and $(n_{j})_{j=1}^{\infty}\in X.$ On the
other hand,

$te_{n(\ell,k)}=e_{n(\ell,k)-t}=e_{n(\ell-t,k)},$

thus $\theta_{X}$ is equivariant. The induced isomorphism
$\tilde{\theta}_{X}$, after identification of Lemma 4.5, is given by

$\langle\tilde{\theta}_{X}(T),\eta\rangle:=\langle
T,\theta_{X}^{-1}(\eta)\rangle;\ \
\big{(}\eta\in\ell^{2}(\mathbb{Z}),T\in\mathscr{K}(L^{2}(X,\mu))\big{)},$

and the $\mathbb{Z}$-action on $\mathscr{K}(L^{2}(X,\mu))$ is given by
$tT(\xi)(x):=T(-t\xi)(-tx),$ for $t\in\mathbb{Z},x\in X$, $\xi\in
L^{2}(X,\mu)$, and $T\in\mathscr{K}(L^{2}(X,\mu))$, where $t\xi(x):=\xi(-tx)$,
and likewise for the $\mathbb{Z}$-action on
$\mathscr{K}(\ell^{2}(\mathbb{Z}))$. Now the fact that $\tilde{\theta}_{X}$ is
equivariant follows directly from the same fact for $\theta_{X}$. Next, let us
observe that $\tilde{\theta}_{X}$ is Cartan preserving. Indeed, we show the
stronger assertion that the $\mathbb{Z}$-action fixes each point of the Cartan
subalgebra $\mathscr{D}_{X}$. Again by Lemma 4.5, it is enough to show that
each function $f_{\lambda}$ is fixed by the $\mathbb{Z}$-action. For this, let
$t\in\mathbb{Z}$ and $\lambda=(\lambda_{n{(\ell,k)}})\in c_{0}(\mathbb{Z})$,
and observe that,

$\displaystyle f_{\lambda}((x_{j}),(y_{j}))$ $\displaystyle=\sum_{k\geq
1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}\xi_{\ell}^{k}((x_{j}))\bar{\xi}_{\ell}^{k}((y_{j}))$
$\displaystyle=\sum_{k\geq 1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}{\rm
exp}\big{(}-2\pi i(\ell+x_{k})/m_{k}\big{)}{\rm exp}\big{(}2\pi
i(\ell+y_{k})/m_{k}\big{)}$ $\displaystyle=\sum_{k\geq
1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}{\rm exp}\big{(}2\pi
i(y_{k}-x_{k})/m_{k}\big{)},$

hence,

$\displaystyle tf_{\lambda}((x_{j}),(y_{j}))$
$\displaystyle=f_{\lambda}(([x_{j}-t]_{k}),([y_{j}-t]_{k}))$
$\displaystyle=\sum_{k\geq 1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}{\rm
exp}\big{(}2\pi i([y_{j}-t]_{k}-[x_{j}-t]_{k})/m_{k}\big{)}$
$\displaystyle=\sum_{k\geq 1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}{\rm
exp}\big{(}2\pi i([y_{j}-x_{j}]_{k})/m_{k}\big{)}$ $\displaystyle=\sum_{k\geq
1}\sum_{\ell=0}^{m_{k}}\lambda_{n(\ell,k)}{\rm exp}\big{(}2\pi
i(y_{j}-x_{j})/m_{k}\big{)}=f_{\lambda}((x_{j}),(y_{j})),$

for each $x=(x_{j}),y=(y_{j})\in X$, establishing the claim. On the other
hand, $t\lambda=t(\lambda_{n{(\ell,k)}})=(\lambda_{n{([\ell-t]_{k},k)}})$,
which is again inside $c_{0}(\mathbb{Z})$, since for $k^{{}^{\prime}}$ large
enough, $m_{k^{{}^{\prime}}}\geq m_{k}+|t|$, and for $0\leq\ell\leq m_{k}-1,$
we have, $0\leq\ell-t\leq m_{k}+|t|-1\leq m_{k^{\prime}}-1$, thus
$n{([\ell-t]_{k},k)}=n([\ell-t]_{k^{{}^{\prime}}},k^{{}^{\prime}})=n(\ell-t,k^{{}^{\prime}})$.
Therefore, though the $\mathbb{Z}$-action does not fix the diagonal operators
$D_{\lambda}$, it keeps the Cartan subalgebra $\mathscr{C}_{X}$ invariant.

$(ii)$ For $Y=\mathbb{Z}_{l}\times lX$, we have
$L^{2}(Y,\mu_{l}\times\nu)=\ell^{2}(\mathbb{Z}_{l})\otimes L^{2}(lX,\nu)$,
where $\mu_{l}$ is the counting measure on $\mathbb{Z}_{l}$, and the
normalized left Haar measure $\nu$ of the clopen subgroup $lX$ of $X$ is
simply the restriction of $\mu$. Now $\theta_{X}$ maps the subspace
$L^{2}(lX,\nu)$ of $L^{2}(X,\mu)$ onto the subspace $\ell^{2}(l\mathbb{Z})$.
Therefore, $\theta_{Y}={\rm id}\otimes\theta_{X}$, where the first leg is the
identity operator on $\ell^{2}(\mathbb{Z}_{l})$. Likewise, the action of
$\mathbb{Z}_{l}\times l\mathbb{Z}$ on $L^{2}(Y,\mu_{l}\times\nu)$ breaks into
the translation action on $\mathbb{Z}_{l}$ and the restriction action of the
subgroup $l\mathbb{Z}$ of $\mathbb{Z}$ on the invariant subspace
$L^{2}(lX,\nu)$, thus all the assertions follow from part $(i)$. ∎

Example 4.4 (continued). For $l\in\mathbb{Z}$, consider the translation
actions $\mathbb{Z}_{l}\times l\mathbb{Z}\curvearrowright\mathbb{Z}_{l}\times
l\mathbb{Z}$ and $\mathbb{Z}\curvearrowright\mathbb{Z}$. Consider the
bijection $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_{l}\times l\mathbb{Z};\ \
(k+tl)\mapsto(k,tl).$ Then
$\mathbb{Z}\curvearrowright\mathbb{Z}\sim_{coe}\mathbb{Z}_{l}\times
l\mathbb{Z}\curvearrowright\mathbb{Z}_{l}\times l\mathbb{Z}$, with respect to
$\phi$ and the following continuous maps,

$\displaystyle a:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}_{l}\times
l\mathbb{Z};\ \
\big{(}(k_{1}+t_{1}l),(k_{2}+t_{2}l)\big{)}\mapsto\begin{cases}(k_{1},t_{1}l)&k_{1}+k_{2}<l\\\
(k_{1},t_{1}l+l)&k_{1}+k_{2}\geq l,\end{cases}$ $\displaystyle
b:(\mathbb{Z}_{l}\times
l\mathbb{Z})\times(\mathbb{Z}_{l}\times\mathbb{Z}_{l})\rightarrow\mathbb{Z};\
\
\big{(}(k_{1},t_{1}l),(k_{2},t_{2}l)\big{)}\mapsto\begin{cases}k_{1}+t_{1}l&k_{1}+k_{2}<l\\\
k_{1}+t_{1}l-l&k_{1}+k_{2}\geq l.\end{cases}$

Applying the Proposition 4.3 again, the corresponding actions of $\mathbb{Z}$
and $\mathbb{Z}_{l}\times l\mathbb{Z}$ on
$G_{l}:=\\{(lm+k,ln+k):k=0,...,l-1,m,n\in\mathbb{Z}\\}$ and
$H_{l}:=\\{((lm,k),(ln,k)):k=0,...,l-1,m,n\in\mathbb{Z}\\}$ are continuous
orbit equivalent. These are discrete groupoids with $C^{*}_{r}(G_{l})\cong
C^{*}_{r}(G_{l})\cong\bigoplus_{0}^{l-1}\mathscr{K}(\ell^{2}(\mathbb{Z}))$.
Now Lemma 4.6 guarantees this to give equivariant étale realization of the
actions discussed in the beginning of this example.

Next, we give another non classical example of continuous orbit equivalence.
First we need a lemma, which is implicit in [15].

######  Lemma 4.7.

In the notation of Example 4.4, we have,

$(i)$ $\lambda_{n}\times\lambda_{m}\sim_{\rm coe}\lambda_{mn}$,

$(ii)$ for supernatural numbers $M,N$ and natural numbers $m,n$,

$((\mathbb{Z}_{m}\times m\mathbb{Z})\curvearrowright
H_{M}^{m})\times((\mathbb{Z}_{n}\times n\mathbb{Z})\curvearrowright
H_{N}^{n})\sim_{\rm coe}((\mathbb{Z}_{mn}\times mn\mathbb{Z})\curvearrowright
H_{M}^{mn})\times(\mathbb{Z}\curvearrowright H_{N}^{1}).$

###### Proof.

Consider the bijection
$\phi:\mathbb{Z}_{n}\times\mathbb{Z}_{m}\rightarrow\mathbb{Z}_{nm};\ \
(t_{n},t_{m})\mapsto t_{m}n+t_{n}$, and define the cocycle maps
$a:(\mathbb{Z}_{n}\times\mathbb{Z}_{m})\times(\mathbb{Z}_{n}\times\mathbb{Z}_{m})\rightarrow\mathbb{Z}_{nm}$
and
$b:\mathbb{Z}_{nm}\times\mathbb{Z}_{nm}\rightarrow(\mathbb{Z}_{n}\times\mathbb{Z}_{m})$,
defined by,

$a((\gamma_{m},\gamma_{n}),(t_{m},t_{n}))=\begin{cases}(\gamma_{m}-m)n+(\gamma_{n}-n)&t_{m}+\gamma_{m}\geq
m,\,t_{n}+\gamma_{n}\geq n\\\
\gamma_{m}n+(\gamma_{n}-n)&t_{m}+\gamma_{m}<m,\,t_{n}+\gamma_{n}\geq n\\\
(\gamma_{m}-m)n+\gamma_{n}&t_{m}+\gamma_{m}\geq m,\,t_{n}+\gamma_{n}<n\\\
\gamma_{m}n+\gamma_{n}&t_{m}+\gamma_{m}<m,\,t_{n}+\gamma_{n}<n\end{cases}$

and,

$\displaystyle
b(\gamma,t)=\begin{cases}\gamma_{m}-m+1,\gamma_{n}-n&\gamma+t\geq
mn,\,\gamma_{n}+t_{n}\geq n\\\ (\gamma_{m}-m,\gamma_{n})&\gamma+t\geq
mn,\,\gamma_{n}+t_{n}<n\\\
(\gamma_{m}+1,\gamma_{n}-n)&\gamma+t<mn,\,\gamma_{n}+t_{n}\geq n\\\
(\gamma_{m},\gamma_{n})&\gamma+t<mn,\,\gamma_{n}+t_{n}<n\end{cases}$

for $\gamma=\gamma_{m}n+\gamma_{n}$ and $t=t_{m}n+t_{n}$ in $\mathbb{Z}_{nm}$
and observe that,

$\displaystyle\phi((\gamma_{m},\gamma_{n})(t_{m},t_{n}))$
$\displaystyle=t_{m}n+t_{n}+(\gamma_{m}-m)n+(\gamma_{n}-n)$
$\displaystyle=a((\gamma_{m},\gamma_{n}),(t_{m},t_{n}))\phi(t_{m},t_{n}),$

when $t_{m}+\gamma_{m}\geq m,t_{n}+\gamma_{n}\geq n$, and the same for the
other three cases. Similarly, $\phi^{-1}(\gamma.t)=b(\gamma,t)\phi^{-1}(t)$,
showing the first statement. To see the second one, we use the identification
$H_{L}^{l}$ and $\mathbb{Z}_{L}\times\tilde{X}_{lL}$ mentioned earlier. Let us
identify actions
$l\mathbb{Z}\curvearrowright\tilde{X}_{lL}=l\mathbb{Z}\curvearrowright
l.(\mathbb{Z}_{lL})$ with $\mathbb{Z}\curvearrowright\mathbb{Z}_{L}$ and
observe that, for the diagonal action
$\lambda^{d}_{M}:\mathbb{Z}\curvearrowright X_{M}\times X_{M}$,

$\displaystyle((\mathbb{Z}_{m}\times m\mathbb{Z})\curvearrowright
H_{M}^{m})\times((\mathbb{Z}_{n}\times$ $\displaystyle
n\mathbb{Z})\curvearrowright
H_{N}^{n})=\lambda_{m}\times\lambda_{n}\times\lambda^{d}_{M}\times\lambda^{d}_{N}$
$\displaystyle\sim_{\rm
coe}\lambda_{mn}\times\lambda^{d}_{M}\times\lambda^{d}_{N}$
$\displaystyle=((\mathbb{Z}_{mn}\times
mn\mathbb{Z})\curvearrowright\mathbb{Z}_{mn}\times(\tilde{X}_{mn}\times\tilde{X}_{mn}))\times\lambda^{d}_{N}$
$\displaystyle=((\mathbb{Z}_{mn}\times mn\mathbb{Z})\curvearrowright
H_{M}^{mn})\times(\mathbb{Z}\curvearrowright H_{N}^{1}).$

∎

######  Example 4.8 (Product odometer transformation).

Let $n_{i},m_{i}\in\mathbb{N}$, and let $L_{i}$ be a supernatural number, for
$i\in I:=\\{1,2,\cdots,r\\}$, and put $n:=\prod_{I}n_{i}=\prod_{I}m_{i}=:m$.
Put $X:=\prod_{I}\mathbb{Z}_{m_{i}L_{i}}$ and
$Y:=\prod_{I}\mathbb{Z}_{n_{i}L_{i}}$. The product odometer transformation
actions $\mathbb{Z}^{r}\curvearrowright X$ and $\mathbb{Z}^{r}\curvearrowright
Y$ are continuous orbit equivalent by Lemma 4.7. Therefore,

$\displaystyle\prod_{I}(\mathbb{Z}_{m_{i}}\times
m_{i}\mathbb{Z})\curvearrowright H^{m_{i}}_{L_{i}}$ $\displaystyle\sim_{\rm
coe}(\mathbb{Z}_{m}\times\prod_{I}{m_{i}}\mathbb{Z})\curvearrowright
H_{L_{1}}^{m}\times\prod_{I\backslash\\{1\\}}\mathbb{Z}\curvearrowright
H_{L_{i}}^{1}$
$\displaystyle=(\mathbb{Z}_{n}\times\prod_{I}{n_{i}}\mathbb{Z})\curvearrowright
H_{L_{1}}^{n}\times\prod_{I\backslash\\{1\\}}\mathbb{Z}\curvearrowright
H_{L_{i}}^{1}$ $\displaystyle\sim_{\rm coe}\prod_{I}(\mathbb{Z}_{n_{i}}\times
n_{i}\mathbb{Z})\curvearrowright H^{n_{i}}_{L_{i}}.$

Thus, $\mathbb{Z}^{r}\curvearrowright\prod_{I}G^{m_{i}}_{L_{i}}\sim_{\rm
coe}\mathbb{Z}^{r}\curvearrowright\prod_{I}G^{n_{i}}_{L_{i}}$. By a similar
argument as in Example 4.4,

$C^{*}_{r}(G^{m_{i}}_{L_{i}})\cong\bigotimes_{i=1}^{r}\bigoplus_{1}^{m_{i}}\mathscr{K}(L^{2}(\tilde{X}_{m_{i}L_{i}},\mu_{i})),\
C^{*}_{r}(G^{n_{i}}_{L_{i}})\cong\bigotimes_{i=1}^{r}\bigoplus_{1}^{n_{i}}\mathscr{K}(L^{2}(\tilde{X}_{n_{i}L_{i}},\nu_{i})),$

and
$\mathbb{Z}\curvearrowright\bigotimes_{i=1}^{r}\bigoplus_{1}^{n_{i}}\mathscr{K}(L^{2}(\tilde{X}_{n_{i}L_{i}},\nu_{i}))\sim_{\rm
coe}\mathbb{Z}\curvearrowright\bigotimes_{i=1}^{r}\bigoplus_{1}^{m_{i}}\mathscr{K}(L^{2}(\tilde{X}_{m_{i}L_{i}},\mu_{i}))$.

Finally, let us give an example, as promised, of non orbit equivalent actions
with orbit equivalent (even conjugate) induced actions on étale realizations.

######  Example 4.9.

Consider the translation action $\mathbb{Z}\curvearrowright\mathbb{Z}$, and
the action $\mathbb{Z}\curvearrowright\mathbb{T}$, defined by
$ne^{i\theta}:=e^{i(\theta+n)},$ for $n\in\mathbb{Z},\theta\in\mathbb{R}$.
Then the actions $\mathbb{Z}\curvearrowright\mathbb{Z}\times\mathbb{Z}$ and
$\mathbb{Z}\curvearrowright\mathbb{T}\times\mathbb{T}$ could not be continuous
orbit equivalent (as these equivalence relations are not topologically
isomorphic), but they give rise to conjugate actions on their common étale
realization $\mathbb{Z}\times\mathbb{Z}$. We proceed as in Lemma 4.6 by
constructing a $\mathbb{Z}$-equivariant isomorphism in the level of Hilbert
spaces. Indeed, $\theta_{\mathbb{T}}:L^{2}(\mathbb{T})\to L^{2}(\mathbb{Z})$
is noting but the Fourier-Plancherel transform $f\mapsto\hat{f}$,

$\hat{f}(n):=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})e^{-in\theta}d\theta,\
\ (n\in\mathbb{Z},f\in L^{2}(\mathbb{T})).$

The orthogonal basis of $L^{2}(\mathbb{T})$ is $\hat{\mathbb{T}}=\mathbb{Z}$,
given by $\xi_{k}(e^{i\theta})=e^{in\theta}$, for
$k,n\in\mathbb{Z},\theta\in\mathbb{R}$. Now it is easy to see that
$\theta_{\mathbb{T}}$ is $\mathbb{Z}$-equivariant,

$\displaystyle n\hat{\xi}_{k}(m)$
$\displaystyle=\hat{\xi}_{k}(m-n)=\frac{1}{2\pi}\int_{0}^{2\pi}\xi_{k}(e^{i\theta})e^{-i(m-n)\theta}d\theta$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-i(m-n-k)\theta}d\theta=\frac{1}{2\pi}\int_{0}^{2\pi}\xi_{n+k}(e^{i\theta})e^{-im\theta}d\theta=\hat{\xi}_{n+k}(m),$

for $m,n,k\in\mathbb{Z},$ that is,
$n\theta_{\mathbb{T}}(\xi_{k})=ne_{k}=e_{n+k}=\theta_{\mathbb{T}}(\xi_{k}),$
as claimed. Let
$\tilde{\theta}_{\mathbb{T}}:\mathscr{K}(L^{2}(\mathbb{T}))\to\mathscr{K}(\ell^{2}(\mathbb{Z}))$
be the $\mathbb{Z}$-equivariant lift on compact operators and for functions
$f_{\lambda}$ defined by,

$f_{\lambda}(e^{i\theta_{1}},e^{i\theta_{2}}):=\sum_{k\in\mathbb{Z}}\lambda_{k}\xi_{k}(e^{i\theta_{1}})\bar{\xi}_{k}(e^{i\theta_{2}})=\sum_{k\in\mathbb{Z}}\lambda_{k}e^{i(\theta_{1}-\theta_{2})}),\
\ (\lambda=(\lambda_{k})\in c_{0}(\mathbb{Z})),$

and for $n\in\mathbb{Z}$ and $\theta_{1},\theta_{2}\in\mathbb{R}$, observe
that,

$nf_{\lambda}(e^{i\theta_{1}},e^{i\theta_{2}})=f_{\lambda}(e^{i(\theta_{1}-n)},e^{i(\theta_{2}-n)})=f_{\lambda}(e^{i\theta_{1}},e^{i\theta_{2}}),$

thus, $\mathbb{Z}$ fixes points of $\mathscr{D}_{\mathbb{T}}$, and it clearly
also leaves $\mathscr{C}_{\mathbb{Z}}=c_{0}(\mathbb{Z})$ invariant, showing
that the induced actions
$\mathbb{Z}\curvearrowright\mathbb{Z}\times\mathbb{Z}$ on the corresponding
étale realizations are conjugate.

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