# The C*-algebra of a twisted groupoid extension

Jean N. Renault  Institut Denis Poisson (UMR 7013)
Université d’Orléans et CNRS
45067 Orléans Cedex 2, FRANCE<EMAIL_ADDRESS>
###### Abstract.

This written version of a talk given in July 2020 at the Western Sydney Abend
seminar and based on the joint work [6] gives a decomposition of the
C*-algebra of a locally compact groupoid with Haar system, possibly endowed
with a twist, in presence of a normal subgroupoid. The natural expression of
this result uses Fell bundles over groupoids. When the normal subgroupoid and
the twist over it are abelian, one obtains another twisted groupoid
C*-algebra.

###### Key words and phrases:

groupoid extension, twist, Mackey machine, Fell bundle.

##  1\. Introduction.

The Mackey normal subgroup analysis (also called the Mackey machine) describes
the representations of a group $G$ in terms of a normal subgroup $S$ and the
quotient $H=G/S$. A semidirect product of groups $G=S\rtimes H$ such as the
group of rigid motions or the Poincaré group is the simplest example. From the
C*-algebraic perspective, it gives a description of $C^{*}(G)$ as a crossed
product. In the simple case of a semidirect product, we easily have

$C^{*}(S\rtimes H)=C^{*}(S)\rtimes H=C^{*}(H,C^{*}(S))$

A semidirect product is a trivial extension. When the extension is not
trivial, a twist appears:

######  Theorem 1.1 (Green [5]).

$C^{*}(G)=C^{*}(G,C^{*}(S),\tau_{S})$

where the right handside is a twisted crossed product.

When $S$ is abelian, one can go one step further, namely use the Gelfand
transform

$C^{*}(S\rtimes
H)=C^{*}(H,C^{*}(S))=C^{*}(H,C_{0}(\hat{S}))=C^{*}(\hat{S}\rtimes H)$

The last term is a groupoid C*-algebra, where the groupoid $\hat{S}\rtimes H$
has less isotropy than the initial group $S\rtimes H$. One may want to iterate
the process. It is then necessary to extend the Mackey machine to a groupoid
$G$ rather than a group. The original motivation was the analysis of nilpotent
group C*-algebras. It is limited here to the example of the Heisenberg group,
which is presented in the last section.

This article is a written version of a talk I gave at the Western Sydney Abend
seminar in July 2020. It is based on a joint work [6] with M. Ionescu, A.
Kumjian, A. Sims and D. Williams, whom I thank for a stimulating and enjoyable
collaboration. The situation considered in the present version is more general
than that of [6], since the initial groupoid may be twisted. While most proofs
are the same as in [6], it seems preferable to give a separate presentation of
the general result because it requires some changes all along. Note however
the case of a twisted groupoid extension considered here can be deduced
directly from the untwisted case considered in [6]. This is done in [7,
Proposition 3.5]. There is an attempt to make this article self-contained but
the reader is directed to [6] (and also [7]) for some proofs. We also refer to
[6] for unexplained notation. Here is a notation which is frequently used:
given two maps $p:X\to T$ and $q:Y\to T$ with the same range, their fibre
product over $T$ is denoted by $X*Y$ when there is no ambiguity about the
maps.

##  2\. Fell bundles and groupoid C*-dynamical systems

######  Definition 2.1.

A groupoid extension is a short exact sequence of groupoids

$S\rightarrowtail G\twoheadrightarrow H$

with common unit space $G^{(0)}$. Equivalently, an extension of the groupoid
$H$ is a surjective homomorphism $\pi:G\twoheadrightarrow H$ such that
$\pi^{(0)}:G^{(0)}\to H^{(0)}$ is a bijection (we shall assume that
$G^{(0)}=H^{(0)}$ and that $\pi^{(0)}$ is the identity map). We shall write
$\dot{\gamma}=\pi(\gamma)$ when there is no ambiguity about the projection
map.

Then $S=\text{Ker}(\pi)$ is a subgroup bundle of the isotropy group bundle
$G^{\prime}$. Moreover, it is normal in the sense that for all compatible pair
$(\gamma,s)\in G*S$, $\gamma s\gamma^{-1}$ belongs to $S$. Note that
subgroupoids which are normal in this sense are necessarily subgroup bundles
of the isotropy group bundle. Note also that $H$ is naturally isomorphic to
the quotient groupoid $G/S$. Since the normal subgroupoid $S$ of $G$
determines the extension, we shall often use the terminology of normal
subgroupoid rather than extension. Then $H$ is the quotient groupoid. We
assume that $H$ and $G$ are locally compact Hausdorff groupoids and that $\pi$
is continuous and open. In particular $S$ is a closed normal subgroupoid of
$G$. We also assume that $H$ has a Haar system $\alpha$ and that $S$ has a
Haar system $\beta$. There is a homomorphism $\delta:G\to{\mathbf{R}}_{+}^{*}$
such that for all $\gamma\in G$,
$\gamma\beta^{s(\gamma)}\gamma^{-1}=\delta(\gamma)\beta^{r(\gamma)}$. This
homomorphism is called the modular cocycle of the extension. Its cohomology
class does not depend on the choice of $\beta$. For all $x\in G^{(0)}$, its
restriction to the group $S_{x}$ is the modular function of $S_{x}$. It is
continuous (see [6, Lemma 2.4]). Given $\alpha$ and $\beta$, we define the
Haar system $\lambda$ for $G$ by the formula:

$\int f(\gamma)d\lambda^{x}(\gamma)=\int_{H}\int_{S}f(\gamma
t)d\beta^{s(\gamma)}(t)d\alpha^{x}(\dot{\gamma})$

######  Definition 2.2.

Under these assumptions, we say that $S\rightarrowtail G\twoheadrightarrow H$
is a locally compact groupoid extension with Haar systems.

It will be convenient in the sequel to define an extension with Haar system as
a pair $(G,S)$ where $S$ is a closed normal subgroupoid of the locally compact
groupoid $G$ which admits a Haar system and such that the quotient groupoid
$H=G/S$ admits a Haar system.

######  Definition 2.3.

[16] A groupoid C*-dynamical system (or dynamical system for short) is a
triple $(G,S,{\mathcal{A}})$ where $G$ is a locally compact groupoid, $S$ is a
closed normal subgroupoid, and $\mathcal{A}$ is an upper semi-continuous
bundle of C*-algebras over $G^{(0)}$ endowed with a continuous action
$G*{\mathcal{A}}\to{\mathcal{A}}$ such that $S$ is unitarily implemented in
the multiplier algebra bundle $M({\mathcal{A}})$, meaning the existence of a
bundle homomorphism $\chi$ from $S$ to the unitary bundle of
$M({\mathcal{A}})$, such that

  1. (i)

the map $S*{\mathcal{A}}\to{\mathcal{A}}$ sending $(s,a)$ to $\chi(s)a$ is
continuous;

  2. (ii)

$s.a=\chi(s)a\chi(s)^{-1}$ for all $(s,a)\in S*{\mathcal{A}}$;

  3. (iii)

$\chi(\gamma s\gamma^{-1})=\gamma.\chi(s)$ for all $(\gamma,s)\in G*S$.

In [16, Section 3], it was assumed that the kernel $S$ was abelian, in the
sense that it was a bundle of abelian groups. However, as shown in [6], this
assumption is not necessary and most results of [16] remain valid.
To distinguish bundles of algebras (or of linear spaces) from algebras,
algebra bundles will usually be denoted by calligraphic letters such as
$\mathcal{A}$ while algebras will be denoted by Roman letters such as $A$. For
example, if $p:G\twoheadrightarrow X$ is a bundle of groupoids with Haar
systems, ${\mathcal{C}}_{c}(G)$ and ${\mathcal{C}}^{*}(G)$ denote the bundles
with respective fibers $C_{c}(G(x))$ and $C^{*}(G(x))$. On the other hand,
$C_{c}(G)$ and $C^{*}(G)$ denote the usual $*$-algebras of the groupoid $G$,
which are the sectional algebras of the above bundle. Groupoid dynamical
systems fit into the more general framework of Fell bundles over groupoids.

######  Definition 2.4.

[22, 10] A Fell bundle over a groupoid is a bundle
${\mathcal{B}}\twoheadrightarrow H$ where $H$ is a locally compact groupoid
and $\mathcal{B}$ is an upper semi-continuous bundle of Banach spaces over $H$
endowed with a continuous multiplication
${\mathcal{B}}*{\mathcal{B}}\to{\mathcal{B}}$ and a continuous involution
${\mathcal{B}}\to{\mathcal{B}}$ satisfying the C*-algebra axioms whenever they
make sense.

This definition implies that the fibers $B_{x}$ over $x\in H^{(0)}$ become
C*-algebras, the fibres $B_{h}$ over $h\in H$ become
$(B_{r(h)},B_{s(h)})$-C*-bimodules and $h\mapsto B_{h}$ is functorial. One
says that the Fell bundle is saturated if the $B_{h}$’s are equivalence
C*-bimodules.

A groupoid C*-dynamical system $(G,S,{\mathcal{A}})$ as above gives rise to a
Fell bundle ${\mathcal{B}}$ over $H=G/S$. This is [12, Example 7.3] which we
recall now. We form

${\mathcal{A}}*G=\\{(a,\gamma)\in{\mathcal{A}}\times G:a\in A_{r(\gamma)}\\}.$

We let $S$ act on it by $s(a,\gamma)=(a\chi(s^{-1}),s\gamma)$ and consider the
quotient ${\mathcal{B}}=({\mathcal{A}}*G)/S$. The image of $(a,\gamma)$ in
${\mathcal{B}}$ is denoted by $[a,\gamma]$. The bundle map
$p:{\mathcal{B}}\twoheadrightarrow H$ sends $[a,\gamma]$ to $\pi(\gamma)$
where $\pi:G\to G/S$ is the quotient map. A choice of $\gamma$ in
$\pi^{-1}(h)$ gives a Banach space isomorphism $[a,\gamma]\mapsto a$ from
$B_{h}$ to $A_{r(h)}$. The multiplication in ${\mathcal{B}}$ is given by

$[a,\gamma][b,\gamma^{\prime}]=[a(\gamma.b),\gamma\gamma^{\prime}]$

and the involution by

$[a,\gamma]^{*}=[\gamma^{-1}.a^{*},\gamma^{-1}]$

######  Lemma 2.1.

[6, Lemma 1.5] The above bundle ${\mathcal{B}}\twoheadrightarrow H$ is a
saturated Fell bundle.

######  Definition 2.5.

This bundle ${\mathcal{B}}\twoheadrightarrow G/S$ is called the Fell bundle of
the groupoid C*-dynamical system $(G,S,{\mathcal{A}})$.

Recall the construction of the crossed products (see [16, Section 3] and
[22]). Let $(G,S,{\mathcal{A}})$ be a groupoid C*-dynamical system. We assume
that $H=G/S$ has a Haar system $(\alpha^{x})_{x\in G^{(0)}}$ and that $S$ has
a Haar system $(\beta^{x})_{x\in G^{(0)}}$. One first form the $*$-algebra
$C_{c}(G,S,{\mathcal{A}})$. Its elements are continuous functions
$f:G\to{\mathcal{A}}$ such that

  1. (i)

$f(\gamma)$ belongs to $A_{r(\gamma)}$ for all $\gamma\in G$;

  2. (ii)

$f(s\gamma)=f(\gamma)\chi(s^{-1})$ for all $(s,\gamma)\in S*G$;

  3. (iii)

$f$ has compact support modulo $S$.

The product and the involution are respectively given by

$f*g(\gamma)=\int
f(\tau)[\tau.g(\tau^{-1}\gamma)]d\alpha^{r(\gamma)}(\dot{\tau})$

and

$f^{*}(\gamma)=\gamma.(f(\gamma^{-1}))^{*}$

The crossed product C*-algebra $C^{*}(G,S,{\mathcal{A}})$ is the completion of
$C_{c}(G,S,{\mathcal{A}})$ for the full C*-norm.

On the other hand, the sectional C*-algebra of a Fell bundle ${\mathcal{B}}$
over a locally groupoid $H$ endowed with a Haar system $\alpha$ is constructed
from the $*$-algebra $C_{c}(H,{\mathcal{B}})$ whose elements are continuous
compactly supported sections $F:H\to{\mathcal{B}}$. The product and the
involution are respectively given by

$F*G(h)=\int F(\eta)G(\eta^{-1}h)d\alpha^{r(h)}(\eta)$

and

$F^{*}(h)=F(h^{-1})^{*}$

Again, the C*-algebra $C^{*}(H,{\mathcal{B}})$ is obtained as the
C*-completion for the full C*-norm.

######  Proposition 2.2.

[6, Section 1.4] Let $(G,S,{\mathcal{A}})$ be a groupoid C*-dynamical system
where $S$ and $H=G/S$ have a Haar system and let
${\mathcal{B}}\twoheadrightarrow H$ be its Fell bundle. Then the C*-algebras
$C^{*}(G,S,{\mathcal{A}})$ and $C^{*}(H,{\mathcal{B}})$ are canonically
isomorphic.

###### Proof.

This is just a sketch of the proof. We refer the reader to [6] and the
references given there. Let $f\in C_{c}(G,S,{\mathcal{A}})$. Note that
$[f(\gamma),\gamma]\in B_{\pi(\gamma)}$ depends on $h=\pi(\gamma)$ only. Call
this element $F(h)$. Then check that

  * •

$F$ belongs to $C_{c}(H,{\mathcal{B}})$;

  * •

$f\mapsto F$ is a $*$-homomorphism;

  * •

this $*$-homomorphism extends to an isomorphism $C^{*}(G,S,{\mathcal{A}})\to
C^{*}(H,{\mathcal{B}})$.

∎

######  Remark 2.1.

The notion of groupoid Fell bundle $(H,{\mathcal{B}})$ generalizes that of
groupoid C*-dynamical system $(G,S,{\mathcal{A}})$, where $H=G/S$ as above. A
Fell bundle over a groupoid $H$ is sometimes called an action by
C*-correspondences. The sectional C*-algebra $C^{*}(H,{\mathcal{B}})$ is then
called its crossed-product C*-algebra (see for example [1]).

##  3\. Mackey analysis of a twisted groupoid C*-algebra.

###  3.1. Twists

We have given earlier the general notion of an extension. The following
special case has been introduced by Kumjian in [9] in the framework of
groupoids.

######  Definition 3.1.

A central groupoid extension

$G^{(0)}\times{\mathbf{T}}\rightarrowtail\Sigma\twoheadrightarrow G$

where ${\mathbf{T}}$ is the group of complex numbers of module 1, is called a
twist. Then, we say that $(G,\Sigma)$ is a twisted groupoid.

We need to distinguish arbitrary extensions as above and twists, because they
do not play the same role in this study. While a twisted groupoid is denoted
by $(G,\Sigma)$ where $\Sigma$ is the middle term, an arbitrary extension will
be determined by a closed normal subgroupoid and denoted for example by
$(G,S)$ where $S$ is the kernel of the extension.

###  3.2. Twisted extensions

Let $(G,\Sigma)$ be a twisted groupoid. Then
$(\Sigma,G^{(0)}\times{\mathbf{T}},G^{(0)}\times{\mathbf{C}})$ is a groupoid
dynamical system with the action $\sigma(s(\sigma),a)=(r(\sigma),a)$ for
$(\sigma,a)\in\Sigma\times{\mathbf{C}}$ and $\chi(x,\theta)=\theta$ for
$(x,\theta)\in G^{(0)}\times{\mathbf{T}}$. If $G$ is a locally compact
groupoid with Haar system, we can construct the crossed product C*-algebra,
which we denote by $C^{*}(G,\Sigma)$ rather than
$C^{*}(\Sigma,G^{(0)}\times{\mathbf{T}},G^{(0)}\times{\mathbf{C}})$ and which
we call the twisted groupoid C*-algebra. The principle of our version of
Mackey analysis is to decompose this C*-algebra when $G$ possesses a closed
normal subgroupoid $S$ endowed with a Haar system. This is a strong condition,
which is often not satisfied by the isotropy bundle itself.

######  Definition 3.2.

We call $(G,\Sigma,S)$ a twisted extension with Haar systems when $(G,\Sigma)$
is a twisted groupoid, $S$ is a closed normal subgroupoid of $G$ with Haar
system and $G/S$ has a Haar system.

In the sequel, we shall denote $H=G/S$ the quotient groupoid and $\pi:G\to H$
the quotient map. We shall denote by $(\alpha^{x})_{x\in G^{(0)}}$ [resp.
$(\beta^{x})_{x\in G^{(0)}}$] the Haar system of $H$ [resp. $S$ ]. We denote
by $(\lambda^{x})_{x\in G^{(0)}}$ the Haar system of $G$ described earlier.
The following diagram summarizes the situation:

$\textstyle{G^{(0)}\times{\mathbf{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{(0)}\times\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\mathbf{T}}}$$\textstyle{\Sigma_{|S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\pi_{\Sigma}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{H}$

###  3.3. The tautological Fell bundle

There is a Fell bundle ${\mathcal{C}}\twoheadrightarrow H$ naturally
associated to a twisted extension $(G,\Sigma,S)$. The construction below is a
particular case of the construction of Section 6 of [1], where the authors
consider the more general framework of fibrations. The idea is very clear: the
twisted groupoid $(G,\Sigma)$ defines a C*-category over the groupoid $H$. We
first note that $\Sigma$ defines a Fell line bundle
${\mathcal{L}}=\Sigma\otimes_{\mathbf{T}}{\mathbf{C}}$ over $G$, with
multiplication
$(\sigma_{1}\otimes\lambda_{1})(\sigma_{2}\otimes\lambda_{2})=\sigma_{1}\sigma_{2}\otimes\lambda_{1}\lambda_{2}$
and involution
$(\sigma\otimes\lambda)^{*}=\sigma^{-1}\otimes\overline{\lambda}$. In the next
section, we shall view a section of ${\mathcal{L}}\twoheadrightarrow G$ as a
function $f:\Sigma\to{\mathbf{C}}$ such that
$f(\theta\tau)=f(\tau){\overline{\theta}}$ for
$(\theta,\tau)\in{\mathbf{T}}\times\Sigma$, but here we use the line bundle
framework. We associate to the unit $x\in H^{(0)}$ the C*-algebra
$C_{x}=C^{*}(S_{x},\Sigma_{|S_{x}})$. We want to associate to the arrow $h\in
H$ a suitable completion $C_{h}$ of $C_{c}(G(h),{\mathcal{L}}(h))$ where
$G(h)=\pi^{-1}(h)$ and ${\mathcal{L}}(h)$ is the restriction to $G(h)$ of the
line bundle ${\mathcal{L}}$. Let us write $\Sigma(h)=\pi_{\Sigma}^{-1}(h)$.
Since $\Sigma$ is a right principal $\Sigma_{|S}$-space, a choice of
$\tau\in\Sigma(h)$ defines a homeomorphism $L(\tau):\Sigma(s(h))\to\Sigma(h)$
sending $\sigma$ to $\tau\sigma$. In fact, this defines a line bundle
isomorphism, still denoted by $L(\tau)$, from the line bundle
${\mathcal{L}}(s(h))\twoheadrightarrow S_{s(h)}$ to the line bundle
${\mathcal{L}}(h)\twoheadrightarrow G(h)$, hence an isomorphism
$L(\tau)^{*}:C_{c}(G(h),{\mathcal{L}}(h))\to
C_{c}(S_{s(h)},{\mathcal{L}}(s(h)))\subset
C^{*}(S_{s(h)},\Sigma_{|S_{s(h)}})$. The norm $\|L(\tau)^{*}(f)\|$ of $f\in
C_{c}(G(h),{\mathcal{L}}(h))$ depends on $h$ only and we write it $\|f\|_{h}$.
We let $C_{h}$ be the completion of $C_{c}(G(h),{\mathcal{L}}(h))$ with
respect to this norm. We define ${\mathcal{C}}$ as the disjoint union
${\mathcal{C}}=\bigsqcup_{h\in H}C_{h}$. We take
$\Gamma=C_{c}(G,{\mathcal{L}})$ as a fundamental family of continuous sections
to define its topology. One checks that the following conditions are
satisfied.

  1. (i)

$\Gamma$ is a linear subspace of the complex linear space $\Pi_{h\in H}C_{h}$;

  2. (ii)

for all $h\in H$, the evaluation map $ev_{h}:\Gamma\to C_{h}$ has a dense
image;

  3. (iii)

for all $\xi\in\Gamma$, the norm map $h\mapsto\|\xi(h)\|_{h}$ is upper
semicontinuous;

  4. (iv)

if $\eta\in\Pi_{h\in H}C_{h}$ satisfies: for all $h\in H$ and all
$\epsilon>0$, there exists $\xi\in\Gamma$ and a neighborhood $U$ of $h$ in $H$
such that $\|\eta(h^{\prime})-\xi(h^{\prime})\|_{h^{\prime}}\leq\epsilon$ for
all $h^{\prime}\in U$, then $\eta$ belongs to $\Gamma$.

Before describing the product and the involution of ${\mathcal{C}}$, it is
useful to define a system of measures for the map $\pi:G\twoheadrightarrow H$.
Such a system of measures already appears in [21] in the transitive case.
Recall that, by assumption, the group bundle $S$ is equipped with a Haar
system $\beta=(\beta^{x})_{x\in G^{(0)}}$. We extend it to a $\pi$-system by
left invariance: for $h\in H$, we set $\beta^{h}=\gamma\beta^{s(h)}$, where
$\gamma$ is an arbitrary element of $G(h)$; it depends on $h$ only. It is a
continuous $\pi$-system of measures. It is left-invariant: for
$(h,h^{\prime})\in H^{(2)}$ and $\gamma\in G(h)$,
$\gamma\beta^{h^{\prime}}=\beta^{hh^{\prime}}$. Note also the relation
$(\delta(\gamma)\beta^{h})^{-1}=\beta^{h^{-1}}$. Given $(h_{1},h_{2})\in
H^{(2)}$, $f_{1}\in C_{c}(G(h_{1}),{\mathcal{L}}(h_{1}))$, $f_{2}\in
C_{c}(G(h_{2}),{\mathcal{L}}(h_{2}))$ and $\gamma\in G(h_{1}h_{2})$, we define

$f_{1}*_{\beta}f_{2}(\gamma)=\int
f_{1}(\gamma^{\prime})f_{2}(\gamma^{\prime-1}\gamma)d\beta^{h_{1}}(\gamma^{\prime})$

and for $\gamma\in G(h^{-1})$,

$f^{*}(\gamma)=f(\gamma^{-1})^{*}$

One checks that these operations extend to ${\mathcal{C}}$ and turn it into a
Fell bundle over $H$. We can also use the fact proved below that
${\mathcal{C}}$ is isomorphic to the Fell bundle of a groupoid dynamical
system to complete the proof.

######  Definition 3.3.

The Fell bundle ${\mathcal{C}}$ constructed above is called the tautological
Fell bundle of the twisted extension $(G,\Sigma,S)$.

The following result is a particular case of [1, Theorem 6.2].

######  Theorem 3.1 (Buss-Meyer [1]).

Let $(G,\Sigma,S)$ be a locally compact twisted groupoid extension with Haar
systems. Let $H=G/S$ be the quotient groupoid. Then the twisted groupoid
C*-algebra $C^{*}(G,\Sigma)$ is canonically isomorphic to the sectional
C*-algebra $C^{*}(H,{\mathcal{C}})$ of the tautological Fell bundle.

###### Proof.

Recall that the topology of $\mathcal{C}$ has been defined by the fundamental
family of continuous sections $\Gamma$. The bijective map which associates to
$f\in C_{c}(G,{\mathcal{L}})$ the corresponding section $j(f)\in\Gamma$ is an
isomorphism of $*$-algebras: let $f,g\in C_{c}(G,{\mathcal{L}})$. On one hand,

$\displaystyle f*_{\lambda}g(\gamma)$
$\displaystyle=\int_{G^{r(\gamma)}}f(\gamma^{\prime})g(\gamma^{\prime-1}\gamma)d\lambda^{r(\gamma)}(\gamma^{\prime})$
$\displaystyle=\int_{H^{r(\gamma)}}\int_{G(\dot{\gamma}^{\prime})}f(\gamma^{\prime})g(\gamma^{\prime-1}\gamma)d\beta^{\dot{\gamma}^{\prime}}(\gamma^{\prime})d\alpha^{r(\gamma)}(\dot{\gamma}^{\prime})$

On the other hand, for $h\in H$,

$(j(f)*_{\alpha}j(g))_{h}=\int_{H^{r(h)}}j(f)(h^{\prime})*_{\beta}j(g)(h^{\prime-1}h)d\alpha^{r(h)}(h^{\prime})$

Hence, for $\gamma\in G(h)$

$(j(f)*_{\alpha}j(g))_{h}(\gamma)=\int_{H^{r(\gamma)}}\int_{G(\dot{\gamma}^{\prime})}f(\gamma^{\prime})g(\gamma^{\prime-1}\gamma)d\beta^{\dot{\gamma}^{\prime}}(\gamma^{\prime})d\alpha^{r(\dot{\gamma})}(\dot{\gamma}^{\prime})$

The involution in $C_{c}(G,{\mathcal{L}})$ and in $C_{c}(H,{\mathcal{C}})$ are
given by the same formula. The $*$-homomorphism $j:C_{c}(G,\Sigma)\to
C^{*}(H,{\mathcal{C}})$ extends to $C^{*}(G,\Sigma)$ by continuity. It is
surjective since its range contains the dense subalgebra $\Gamma$. To show its
injectivity, one shows by using the disintegration theorem [16, Théorème 4.1.]
that for every representation $L$ of $C^{*}(G,\Sigma)$, there exists a
representation $L^{\prime}$ of $C^{*}(H,{\mathcal{C}})$ such that
$L=L^{\prime}\circ j$. ∎

###  3.4. The tautological groupoid dynamical system

The tautological Fell bundle of the twisted extension $(G,\Sigma,S)$ is in
fact the Fell bundle of the following groupoid dynamical system
$(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$. It can be shown just
as in the untwisted case (see for example [11, Theorem 5.5]) that the twisted
group bundle $(S,\Sigma_{|S})$ with Haar system $\beta$ defines an upper semi-
continuous bundle of C*-algebras over $G^{(0)}$, with fibres
$C^{*}(S_{x},\Sigma_{|S_{x}})$ which we denote by
${\mathcal{C}}^{*}(S,\Sigma_{|S})$. Moreover, when the groups are amenable, it
is a continuous bundle. It is endowed with an action
$\Sigma*{\mathcal{C}}^{*}(S,\Sigma_{|S})\to{\mathcal{C}}^{*}(S,\Sigma_{|S})$,
where

$(\tau.f)(\sigma)=\delta(p(\tau))f(\tau^{-1}\sigma\tau),\quad{\rm
for}\quad\tau\in\Sigma,\,f\in
C_{c}(S_{s(\tau)},\Sigma_{|S_{s(\tau)}}),\,\sigma\in p^{-1}(S_{r(\tau)})$

The introduction of $\delta$ is necessary in order to preserve the convolution
product of the twisted group C*-algebras $C^{*}(S_{x},\Sigma_{|S_{x}})$.

######  Proposition 3.2.

The triple $(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$ is a
groupoid C*-dynamical system, which we call the tautological groupoid
dynamical system of the twisted extension.

###### Proof.

The continuity of the action map
$\Sigma*{\mathcal{C}}^{*}(S,\Sigma_{|S})\to{\mathcal{C}}^{*}(S,\Sigma_{|S})$
is proved just as [6, Proposition 2.7]. The action of $\Sigma_{|S}$ on
${\mathcal{C}}^{*}(S,\Sigma_{|S})$ is implemented by the bundle homomorphism
$\chi:\Sigma_{|S}\to UM({\mathcal{C}}^{*}(S,\Sigma_{|S})$ which associates to
$\sigma\in p^{-1}(S_{x})$ the canonical unitary $\chi(\sigma)$ in the
multiplier algebra of $C^{*}(S_{x},p^{-1}(S_{x}))$. Explicitly, for $f\in
C_{c}(S_{x},\Sigma_{|S_{x}})$ and $\sigma,\tau\in p^{-1}(S_{x})$,

$(\chi(\sigma)f)(\tau)=\delta^{1/2}(p(\sigma))f(\sigma^{-1}\tau)$

One checks just as in [6, Section 2] that the conditions of Definition 2.3 are
satisfied. ∎

######  Theorem 3.3.

Let $(G,\Sigma,S)$ be a locally compact twisted groupoid extension with Haar
systems. Its tautological Fell bundle $\mathcal{C}$ is isomorphic to the Fell
bundle $\mathcal{B}$ of its tautological groupoid dynamical system.

###### Proof.

As before, we denote by $\mathcal{L}=\Sigma\otimes_{\mathbf{T}}{\mathbf{C}}$
the Fell line bundle over $G$ associated with $\Sigma$. Here, the sections of
this bundle are viewed as functions $f:\Sigma\to{\mathbf{C}}$ such that
$f(\theta\tau)=f(\tau){\overline{\theta}}$ for all
$(\theta,\tau)\in{\mathbf{T}}\times\Sigma$. Recall that
${\mathcal{B}}=({\mathcal{C}}^{*}(S,\Sigma_{|S})*\Sigma)/\Sigma_{|S}$. Let
$\tau\in\Sigma$ and $h=\pi_{\Sigma}(\tau)$. Given $f\in
C_{c}(G(h),{\mathcal{L}}(h))$, we define $\rho_{\tau}(f)\in
C_{c}(G(r(h)),{\mathcal{L}}(r(h)))$ by

$\rho_{\tau}(f)(\sigma)=\delta^{1/2}(p(\tau))f(\sigma\tau),\quad\forall\sigma\in
p^{-1}(G(r(h))$

Then $(\rho_{\tau}(f),\tau)\in{\mathcal{C}}^{*}(S,\Sigma_{|S})*\Sigma$ and for
all $\sigma\in\Sigma_{|S}$, we have:

$\displaystyle(\rho_{\sigma\tau}(f),\sigma\tau)$
$\displaystyle=(\rho_{\tau}(f)\chi(\sigma^{-1}),\sigma\tau)$
$\displaystyle=\sigma(\rho_{\tau}(f),\tau)$

Therefore, for all $h\in H$, there is a map
$j_{h}:C_{c}(G(h),{\mathcal{L}}(h))\to B_{h}$ such that

$j_{h}(f)=[\rho_{\tau}(f),\tau]$

where $\pi_{\Sigma}(\tau)=h$. By definition of the norm on
$C_{c}(G(h),{\mathcal{L}}(h))$, $j_{h}$ extends to a Banach space isomorphism
from $C_{h}$ onto $B_{h}$ and that this defines a bundle isomorphism $j$ from
$\mathcal{C}$ to $\mathcal{B}$. Then one deduces from [4, Propositions 13.6
and 13.7] that $j$ is a Banach bundle isomorphism. Let us check that $j$
preserves the product and the involution. Let $(h_{1},h_{2})\in H^{(2)}$.
Choose $\tau_{i}\in\Sigma$ such that $\pi_{\Sigma}(\tau_{i})=h_{i}$ for
$i=1,2$. Let $f_{i}\in C_{c}(G(h_{i}),{\mathcal{L}}(h_{i}))$ for $i=1,2$. The
equality

$j_{h_{1}h_{2}}(f_{1}*_{\beta}f_{2})=j_{h_{1}}(f_{1})j_{h_{2}}(f_{2})$

amounts to the equality

$\rho_{\tau_{1}\tau_{2}}(f_{1}*_{\beta}f_{2})=\rho_{\tau_{1}}(f_{1})[\tau_{1}.\rho_{\tau_{2}}(f_{2})]$

We have for $\sigma\in p^{-1}(G(r(h_{1}))$:

$\displaystyle\rho_{\tau_{1}\tau_{2}}(f_{1}*_{\beta}f_{2})(\sigma)$
$\displaystyle=\delta^{1/2}(p(\tau_{1}\tau_{2}))(f_{1}*_{\beta}f_{2})(\sigma\tau_{1}\tau_{2})$
$\displaystyle=\delta^{1/2}(p(\tau_{1}\tau_{2}))\int
f_{1}(\tau)f_{2}(\tau^{-1}\sigma\tau_{1}\tau_{2})d\beta^{h_{1}}(\dot{\tau})$

where $\dot{\tau}=p(\tau)$. On the other hand

$\displaystyle\rho_{\tau_{1}}(f_{1})[\tau_{1}.\rho_{\tau_{2}}(f_{2})](\sigma)$
$\displaystyle=\int\rho_{\tau_{1}}(f_{1})(\sigma^{\prime})[\tau_{1}.\rho_{\tau_{2}}(f_{2})](\sigma^{\prime-1}\sigma)d\beta^{r(h_{1})}(\dot{\sigma}^{\prime})$
$\displaystyle=\int\delta^{1/2}(\dot{\tau}_{1})f_{1}(\sigma^{\prime}\tau_{1})\delta(\dot{\tau}_{1})\delta^{1/2}(\dot{\tau}_{2})f_{2}(\tau_{1}^{-1}\sigma^{\prime-1}\sigma\tau_{1}\tau_{2})d\beta^{r(h_{1})}(\dot{\sigma}^{\prime})$
$\displaystyle=\delta^{1/2}(\dot{\tau}_{1})\delta^{1/2}(\dot{\tau}_{2})\int
f_{1}(\sigma^{\prime}\tau_{1})f_{2}(\tau_{1}^{-1}\sigma^{\prime-1}\sigma\tau_{1}\tau_{2})\delta(\dot{\tau}_{1})d\beta^{r(h_{1})}(\dot{\sigma}^{\prime})$

This last integral is of the form $\int
g(s\dot{\tau}_{1})\delta(\dot{\tau}_{1})d\beta^{r(h_{1})}(s)$ where
$g(\dot{\tau})=f_{1}(\tau)f_{2}(\tau^{-1}\sigma\tau_{1}\tau_{2})$. The change
of variable $s=\dot{\tau}_{1}t\dot{\tau}_{1}^{-1}$ gives

$\displaystyle\int
g(s\dot{\tau}_{1})\delta(\dot{\tau}_{1})d\beta^{r(h_{1})}(u)$
$\displaystyle=\int g(\dot{\tau}_{1}t)d\beta^{s(h_{1})}(t)$
$\displaystyle=\int g(t)d\beta^{h_{1}}(t)$

which is the desired equality.
Let $f\in C_{c}(G(h),{\mathcal{L}}(h))$. The equality

$(j_{h}(f))^{*}=j_{h^{-1}}(f^{*})$

amounts to the equality

$\tau^{-1}.(\rho_{\tau}(f))^{*}=\rho_{\tau^{-1}}(f^{*})$

where $\pi_{\Sigma}(\tau)=h$. For $\sigma\in p^{-1}(G(s(h))$, we have

$\displaystyle\tau^{-1}.(\rho_{\tau}(f))^{*}(\sigma)$
$\displaystyle=\delta(\dot{\tau}^{-1})(\rho_{\tau}(f))^{*}(\tau\sigma\tau^{-1})$
$\displaystyle=\delta(\dot{\tau}^{-1})(\rho_{\tau}(f)(\tau\sigma^{-1}\tau^{-1}))^{*}$
$\displaystyle=\delta(\dot{\tau}^{-1})\delta^{1/2}(\dot{\tau})f(\tau\sigma^{-1})^{*}$
$\displaystyle=\delta^{1/2}(\dot{\tau}^{-1})f(\tau\sigma^{-1})^{*}$

On the other hand,

$\displaystyle\rho_{\tau^{-1}}(f^{*})(\sigma)$
$\displaystyle=\delta^{1/2}(\dot{\tau}^{-1})f^{*}(\sigma\tau^{-1})$
$\displaystyle=\delta^{1/2}(\dot{\tau}^{-1})f(\tau\sigma^{-1})^{*}$

∎

######  Corollary 3.4.

Let $(G,\Sigma,S)$ be a locally compact twisted groupoid extension with Haar
systems. Then the twisted groupoid C*-algebra $C^{*}(G,\Sigma)$ is isomorphic
to the crossed product C*-algebra
$C^{*}(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$ of the dynamical
system $(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$.

######  Remark 3.1.

This result can be obtained directly (without introducing the tautological
Fell bundle) by using the disintegration theorem of [16] (or rather its
generalization to a non abelian extension). This is the road followed in [6],
where this result (in the untwisted case) appears as Theorem 2.11. The authors
also consider the reduced C*-algebras. The main advantage of the Fell bundle
is that it gives simple and natural formulas.

##  4\. Abelian Fell bundles

When the normal subgroupoid $S$ in the locally compact twisted groupoid
extension with Haar systems $(G,\Sigma,S)$ is abelian and the restriction
$\Sigma_{|S}$ of the twist is also abelian, one can go one step further by
using the Gelfand transform for the bundle of abelian C*-algebras
${\mathcal{C}}^{*}(S,\Sigma_{|S}))$. Again, instead of doing this step
directly, we make a detour via abelian Fell bundles.

###  4.1. Abelian Fell bundles

The structure of saturated abelian Fell bundles over a groupoid has been
established by V. Deaconu, A. Kumjian and B. Ramazan [2, Theorem 5.6]. It is a
direct consequence of the well-known following facts about Morita equivalence
of commutative C*-algebras.

######  Lemma 4.1.

[14, Appendix A]

  1. (i)

Let $X,Y$ be locally compact spaces, $\varphi:Y\to X$ a homeomorphism and
${\mathcal{L}}$ a hermitian line bundle over $X$ (with scalar product linear
in the first variable). Then
$(A=C_{0}(X),E=C_{0}(X,{\mathcal{L}}),B=C_{0}(Y))$ is an imprimitivity
bimodule where

    1. (a)

for $(a,\xi,b)\in A\times E\times B$,

$(b\xi)(x)=b(x)\xi(x)\qquad(\xi b)(x)=b(\varphi^{-1}(x))\xi(x)$

    2. (b)

for $(\xi,\eta)\in E\times E$,

$\,{}_{A}\langle\xi,\eta\rangle(x)=(\xi(x)\,|\,\eta(x)_{x}\qquad\langle\xi,\eta\rangle_{B}(y)=(\eta(\varphi(y))\,|\,\xi(\varphi(y))_{\varphi(y)}$

  2. (ii)

Conversely, every imprimitivity bimodule $(A,E,B)$, where the C*-algebras $A$
and $B$ are abelian, is isomorphic to an imprimitivity bimodule

$(C_{0}(X),C_{0}(X,{\mathcal{L}}),C_{0}(Y))$

with $(X,Y,h,{\mathcal{L}})$ as in (i). The homeomorphism $\varphi:Y\to X$ is
uniquely determined by the relation $\xi b=\alpha(b)\xi$ for all $\xi\in E$
and $b\in B$ and $\alpha(b)=b\circ\varphi^{-1}$. The quadruple
$(X,Y,\varphi,{\mathcal{L}})$, which is unique up to isomorphism, is called
the spatial realization of the Morita equivalence.

  3. (iii)

The spatial realization of the composition $(A,E\otimes_{B}F,C)$ of two Morita
equivalences $(A,E,B)$ and $(B,F,C)$ having $(X,Y,\varphi,{\mathcal{L}})$ and
$(Y,Z,\psi,{\mathcal{M}})$ as spatial realizations is the quadruple
$(X,Z,\varphi\circ\psi,{\mathcal{L}}\otimes_{X}\varphi^{*}{\mathcal{M}})$.

  4. (iv)

If the imprimitivity bimodule $(A,E,B)$ admits the spatial realization
$(X,Y,\varphi,{\mathcal{L}})$, then its inverse $(B,E^{*},A)$ admits the
spatial realization $(Y,X,\varphi^{-1},\varphi^{*}\overline{\mathcal{L}})$,
where $\overline{\mathcal{L}}$ is the conjugate line bundle.

###### Proof.

The assertions (i), (iii) and (iv) are straightforward. A direct proof of the
assertion (ii) is given in [2]. It can also be obtained by introducing the
imprimitivity algebra and by using [17], to which we refer the reader for
unexplained notations. There is a C*-algebra $C$ and complementary full
projections $p,q\in M(C)$ such that $(A,E,B)$ is isomorphic to
$(pCp,pCq,qCq)$. It is easily checked that $D=pCp+qCq$ is a Cartan subalgebra
of $C$ having the map $c\mapsto pcp+qcq$ as an expectation onto it. The
spectrum $Z$ of $D$ is the disjoint union of the open subsets $X$ and $Y$,
which are respectively the spectra of $A$ and $B$. Thus, according to [17,
Theorem 5.9], there is an isomorphism $\Phi$ from $C$ onto
$C_{r}^{*}(G(D),\Sigma(D))$, the reduced groupoid C*-algebra of the Weyl twist
$(G(D),\Sigma(D))$, which carries $D$ onto $C_{0}(Z)$. The subsets
$pCp,pCq,qCp$ and $qCq$ are contained in the normalizer $N(D)$ of $D$. Since
$X$ and $Y$ are open, every germ of a partial homeomorphism induced by an
element of the normalizer can be obtained by an element of one of these
subsets. The partial homeomorphisms obtained from elements of $pCp$ and $qCq$
are partial identity maps. Let $n\in pCq$ and $y\in Y$ such that
$n^{*}n(y)>0$. Then the germ of $\alpha_{n}$ at $y$ does not depend on $n$.
Indeed, let $m\in pCq$ such that $m^{*}m(y)>0$. Without changing the germ of
$\alpha_{m}$ at $y$, we may assume that the closed support of $m^{*}m$ is
contained in an open neighborhood of $y$ on which $n^{*}n\geq\epsilon>0$.
Then, there exists $b\in B$ with $b(y)\not=0$ and $n^{*}m=(n^{*}n)b$. Then,
the equality $(nn^{*})m=(nn^{*})nb$ implies that $\alpha_{m}$ and $\alpha_{n}$
have the same germ at $y$. Moreover, since the projection $q$ is full, every
$y\in Y$ is the domain of some normalizer $n\in pCq$. Therefore there exists a
unique homeomorphism $\varphi:Y\to X$ such that $\alpha_{n}(y)=\varphi(y)$ for
all $n\in pCq$ and $y\in Y$ such that $n^{*}n(y)>0$. This shows that $G(D)$ is
the graph of the equivalence relation on $Z=X\sqcup Y$ whose classes are
$\\{y,\varphi(y)\\}$ for $y\in Y$ and $\\{x,\varphi^{-1}(x)\\}$ for $x\in X$.
It is a closed subset of $Z\times Z$. Let ${\mathcal{F}}$ be the Fell line
bundle associated to the $\Sigma(D)$. The restriction ${\mathcal{L}}$ of
${\mathcal{F}}$ to the subset $\\{(x,\varphi^{-1}(x)),x\in X\\}$, which we
identify to $X$, is a hermitian line bundle with scalar product
$(\xi|\eta)=\xi\eta^{*}$. Then $\Phi$ maps isomorphically $(pCp,pCq,qCq)$ onto
$(C_{0}(X),C_{0}(X,{\mathcal{L}}),C_{0}(Y))$. ∎

We give now a construction of a saturated abelian Fell bundle over a groupoid.
It will turn out that every saturated abelian Fell bundle over a groupoid can
be constructed in this fashion. Let $(H,\alpha)$ be a locally compact groupoid
with Haar system and let $Z$ be a right locally compact $H$-space. As usual,
we write $s:Z\to H^{(0)}$ the projection and $Z_{x}=s^{-1}(x)$ for $x\in
H^{(0)}$. Let $\Sigma$ be a twist over the semi-direct product $Z\rtimes H$.
We denote by $\mathcal{L}$ the associated Fell line bundle. Then, for each
$h\in H$, we obtain by restriction to $Z_{r(h)}\times\\{h\\}$ a line bundle
${\mathcal{L}}_{h}$ over $Z_{r(h)}$ and consider the space of continuous
sections vanishing at infinity $B_{h}:=C_{0}(Z_{r(h)},{\mathcal{L}}_{h})$
endowed with the sup-norm. We turn $\mathcal{B}:=\bigsqcup_{h\in H}B_{h}$ into
a Banach bundle with $C_{c}(Z\rtimes H,{\mathcal{L}})$ as fundamental space of
continuous sections, where we identify $f\in C_{c}(Z\rtimes H,{\mathcal{L}})$
with the section $h\to f_{h}$, where $f_{h}\in
C_{c}(Z_{r(h)},{\mathcal{L}}_{h})$ is defined by $f_{h}(z)=f(z,h)$. Given
$(h,h^{\prime})\in H^{(2)}$, we define the product

$B_{h}\otimes B_{h^{\prime}}\to B_{hh^{\prime}}:b\otimes b^{\prime}\mapsto
b(hb^{\prime})$

where, for $z\in Z_{r(h)}$, $(b(hb^{\prime}))(z)=b(z)b^{\prime}(zh)$. For
$h\in H$, we define the involution

$B_{h}\to B_{h^{-1}}:b\to b^{*}$

where, for $z\in Z_{s(h)}$, $b^{*}(z)=(b(zh^{-1}))^{*}$.

######  Proposition 4.2.

Let $\Sigma$ be a twist over the semi-direct product $Z\rtimes H$, where
$(H,\alpha)$ is a locally compact groupoid with Haar system and $Z$ a right
locally compact $H$-space $Z$. Construct $\mathcal{B}$ as above. Then

  1. (i)

$\mathcal{B}$ is a saturated abelian Fell bundle over $H$;

  2. (ii)

$C^{*}(H,{\mathcal{B}})$ is isomorphic to $C^{*}(Z\rtimes H,\Sigma)$.

###### Proof.

The first assertion is a straightforward verification. For the second
assertion, one checks that the map which sends $f\in C_{c}(Z\rtimes H,L)$ into
the section $h\mapsto f_{h}$ is a $*$-homomorphism from $C_{c}(Z\rtimes
H,\Sigma)$ to $C_{c}(H,{\mathcal{B}})$. It is continuous in the inductive
limit topology and has a dense image. Hence it extends to a $*$-isomorphism
from $C^{*}(Z\rtimes H,\Sigma)$ to $C^{*}(H,{\mathcal{B}})$. ∎

######  Definition 4.1.

We say that $\mathcal{B}$ is the Fell bundle of the twisted semi-direct
product $(Z\rtimes H,\Sigma)$.

Note that this terminology does not agree with Definition 2.5, because here
$\mathcal{B}$ is a Fell bundle over $H$ and not over $Z\rtimes H$. A coherent
terminology would require the notion of fibration as in [1]. Let us show now
that every saturated abelian Fell bundle is the Fell bundle of a twisted semi-
direct product.

######  Theorem 4.3.

[2, Theorem 5.6] Let $\mathcal{B}$ be a saturated abelian Fell bundle over a
locally compact groupoid $H$. Then there exist a right locally compact
$H$-space $Z$ and a twist $\Sigma$ over the semi-direct product $Z\rtimes H$
such that $\mathcal{B}$ is isomorphic to the Fell bundle of $(Z\rtimes
H,\Sigma)$. The pair $(Z,\Sigma)$ is unique up to isomorphism. We call it the
spatial realization of $\mathcal{B}$.

###### Proof.

Let $\mathcal{B}$ be a saturated abelian Fell bundle over a locally compact
groupoid $H$. Its restriction to $H^{(0)}$ is an abelian C*-bundle
$\mathcal{B}^{(0)}$ over $H^{(0)}$. The sectional C*-algebra
$C_{0}(H^{(0)},\mathcal{B}^{(0)})$ is abelian, hence isomorphic to $C_{0}(Z)$,
where $Z$ is its spectrum. The space $Z$ is fibred above $H^{(0)}$. The bundle
map, which is continuous, open and onto, is denoted by $s:Z\to H^{(0)}$ and
the fibre above $x\in H^{(0)}$ is written $Z_{x}$. We write
$B_{x}=C_{0}(Z_{x})$. For each $h\in H$, $(B_{r(h)},B_{h},B_{s(h)})$ is a
Morita equivalence. We let $(Z_{r(h)},Z_{s(h)},\varphi_{h},{\mathcal{L}}_{h})$
be its spatial realization. For $z\in Z_{s(h)}$, we define
$zh=\varphi_{h}(z)$. This defines a map $Z*H\to Z$. Given $(z,h)\in Z*H$, we
write ${\mathcal{L}}_{(z,h)}:=({\mathcal{L}}_{h})_{z}$ and define the
algebraic line bundle ${\mathcal{L}}=\bigsqcup_{(z,h)\in
Z*H}{\mathcal{L}}_{(z,h)}$ over $Z*H$. A section $\xi\in
C_{c}(H,{\mathcal{B}})$ defines a section $\tilde{\xi}$ of ${\mathcal{L}}$
according to $\tilde{\xi}(z,h)=\xi(h)(z)$. This defines a Banach bundle
structure on ${\mathcal{L}}$. Let us show that $(z,h)\mapsto zh$ is an action
map. The relation $(zh)h^{\prime}=z(hh^{\prime})$ results from the lemma. The
continuity of the action can be obtained by applying the relation

$b(zh)\tilde{\xi}(z,h)=\tilde{(\xi b)}(z,h)$

where $b\in C_{c}(Z)\subset C_{c}(H^{(0)},{\mathcal{B}}^{(0)})$ and $\xi\in
C_{c}(H,{\mathcal{B}})$. The isomorphism
$B_{h}\otimes_{B_{s(h)}}B_{h^{\prime}}\to B_{hh^{\prime}}$ defined by the
product in the Fell bundle $\mathcal{B}$ gives an isomorphism
${\mathcal{L}}_{h}\otimes_{Z_{r(h)}}h_{h}^{*}{\mathcal{L}}_{h^{\prime}}\to{\mathcal{L}}_{hh^{\prime}}$
which defines a product on ${\mathcal{L}}$. Similarly, the involution
$B_{h}\to B_{h^{-1}}$ gives an involution
${\mathcal{L}}_{h}\to{\mathcal{L}}_{h^{-1}}$ on ${\mathcal{L}}$. This turns
${\mathcal{L}}$ into a Fell line bundle over $Z\rtimes H$. We let $\Sigma$ be
the unitary bundle of $\mathcal{L}$. Then, by construction, $\mathcal{B}$ is
isomorphic to the Fell bundle of $(Z\rtimes H,\Sigma)$. ∎

######  Corollary 4.4.

Let $\mathcal{B}$ be a saturated abelian Fell bundle over a locally compact
groupoid $H$. Then the spectrum $Z$ of the sectional algebra
$C_{0}(H^{(0)},{\mathcal{B}}^{(0)})$ is a right locally compact $H$-space and
there exists a twist $\Sigma$ over the semi-direct product $Z\rtimes H$ such
that the sectional algebra $C^{*}(H,\mathcal{B})$ is isomorphic to the twisted
groupoid C*-algebra $C^{*}(Z\rtimes H,\Sigma)$.

###  4.2. Abelian groupoid dynamical system

We say that a groupoid dynamical system $(G,S,{\mathcal{A}})$ is abelian when
$\mathcal{A}$ is a bundle of commutative C*-algebras over $G^{(0)}$. Note that
we do not assume here that the groups $S_{x}$ are abelian. Then the associated
Fell bundle $\mathcal{B}$ over $H=G/S$ is abelian and admits the above spatial
realization. The twisted semi-direct product $(Z\rtimes H,{\Sigma})$ of the
previous section admits a convenient description. The sectional C*-algebra
$A=C_{0}(G^{(0)},{\mathcal{A}})$ is abelian, hence isomorphic to $C_{0}(Z)$,
where $Z$ is the spectrum of $A$. As said earlier, the space $Z$ is fibred
above $G^{(0)}$. The bundle map is denoted by $s:Z\to G^{(0)}$ and the fibre
above $x\in G^{(0)}$ is written $Z_{x}$. We have $A=C_{0}(Z)$ and
$A_{x}=C_{0}(Z_{x})$. The action of $G$ on ${\mathcal{A}}$ induces an action
on $Z$ which we write as a right action so that $(\sigma.f)(z)=f(z\sigma)$
where $f\in C_{0}(Z_{s(\sigma)})=A_{s(\sigma)}$ and $z\in Z_{r(\sigma)}$.
Because the action of $S$ is unitarily implemented and $\mathcal{A}$ is
abelian, $S$ acts trivially on $\mathcal{A}$. Therefore, the action of $G$ is
in fact an action of $H=G/S$. This gives the semi-direct product $Z\rtimes H$.
Let us describe now the twist $\Sigma$ over this semi-direct product. It is
given by a pushout construction. We first observe that the homomorphism
$\chi:S\to UM({\mathcal{A}})$ which implements the restriction of the action
to $S$ gives a map

${\underline{\chi}}:Z*S\to{\mathbf{T}}\qquad(z,t)\mapsto(\chi(t))(z)$

It is a continuous groupoid homomorphism which satisfies
${\underline{\chi}}(z,\gamma t\gamma^{-1})={\underline{\chi}}(z\gamma,t)$ for
all $(\gamma,t)\in G*S$. Here is a general definition.

######  Definition 4.2.

Given a groupoid extension $S\rightarrowtail G\twoheadrightarrow H$ and an
$H$-bundle of abelian groups $T$, we say that a group bundle morphism
$\varphi:S\to T$ is equivariant if $\varphi(\gamma
s\gamma^{-1})=\dot{\gamma}\varphi(s)$ for all $(\gamma,s)\in G*S$.

We give now the general pushout construction (the reader is directed to [7]
for a full exposition). It is summarized by the following diagram.

$\begin{CD}S@>{}>{}>G@>{}>{}>H\\\
@V{\varphi}V{}V@V{\varphi_{*}}V{}V\Big{\|}\\\
T@>{}>{}>\underline{G}@>{}>{}>H\end{CD}$

Here are the details.

######  Proposition 4.5.

Let $S\rightarrowtail G\twoheadrightarrow H$ be a groupoid extension, let $T$
be locally compact abelian group bundle endowed with an $H$-action and let
$\varphi:S\to T$ be an equivariant group bundle morphism. Then there is an
extension $T\rightarrowtail\underline{G}\twoheadrightarrow H$ and a morphism
$\varphi_{*}:G\to\underline{G}$ that is compatible with $\varphi$. They are
unique up to isomorphism.

###### Proof.

We define

$T*G=\\{\,(t,\gamma)\in T\times G,|\,p_{T}(t)=r(\gamma)\,\\}$

It is a groupoid over $G^{(0)}$ with multiplication

$(t,\gamma)(t^{\prime},\gamma^{\prime})=(t(\dot{\gamma}t^{\prime}),\gamma\gamma^{\prime})$

and inverse

$(t,\gamma)^{-1}=(\dot{\gamma}^{-1}(t^{-1}),\gamma^{-1})$

Endowed with the relative topology, it is a locally compact topological
groupoid. Then $S$ embeds into it as a closed normal subgroupoid via $i:S\to
T*G$ given by $i(s)=(f(s^{-1}),s)$. We define $\underline{G}:=T*G/i(S)$.
Equivalently, $\underline{G}$ is the quotient of $T*G$ for the left action of
$S$ given by $s(t,\gamma)=(t\varphi(s^{-1}),s\gamma)$. Its elements are of the
form $[t,\gamma]$ where $(t,\gamma)\in T*G$ and satisfy
$[t,\gamma]=[t\varphi(s^{-1}),s\gamma]$ for $s\in S_{r(\gamma)}$. Let us spell
out its groupoid structure. Its unit space is $G^{(0)}$ with obvious range and
source maps. The multiplication is given by

$[t,\gamma][t^{\prime},\gamma^{\prime}]=[t(\dot{\gamma}t^{\prime}),\gamma\gamma^{\prime}]$

and its inverse map is given

$[t,\gamma]^{-1}=[\dot{\gamma}^{-1}(t^{-1}),\gamma^{-1}]$

The map $\underline{\pi}:\underline{G}\to H$ given by
$\underline{\pi}[t,\gamma]=\pi(\gamma)$ is a surjective homomorphism and
$\underline{\pi}^{(0)}$ is the identity map. Its kernel is identified to $T$
via the map $\underline{j}:T\to\underline{G}$ defined by
$\underline{j}(t)=[t,p_{T}(t)]$. The map $\varphi_{*}:G\to\underline{G}$ is
given by $\varphi_{*}(\gamma)=[r(\gamma),\gamma]$ for $\gamma\in G$.

∎

######  Definition 4.3.

The above extension $T\rightarrowtail\underline{G}\twoheadrightarrow H$ is
called the pushout of the extension $S\rightarrowtail G\twoheadrightarrow H$
by the morphism $\varphi:S\to T$.

To apply this construction to our abelian groupoid C*-dynamical system
$(G,S,{\mathcal{A}})$, we first consider the extension

$S\rightarrowtail G\twoheadrightarrow H$

Taking the semi-direct product, we obtain a new extension

$Z*S\rightarrowtail Z\rtimes G\twoheadrightarrow Z\rtimes H$

We view $Z\times{\mathbf{T}}$ as a group bundle over $Z$ with the trivial
action of $Z\rtimes H$. The map

$\varphi:Z*S\to Z\times{\mathbf{T}}\quad\hbox{given
by}\quad\varphi(z,t)=(z,{\underline{\chi}}(z,t))$

is a group bundle morphism which is equivariant in the above sense. We define
the extension

$Z\times{\mathbf{T}}\rightarrowtail\Sigma\twoheadrightarrow Z\rtimes H$

as the pushout by this morphism. Explicitly,

$\Sigma=\\{[\theta,z,\gamma]:\theta\in{\mathbf{T}},(z,\gamma)\in Z\rtimes
G\\}$

where

$[\theta,z,t\gamma]=[\theta\,{\underline{\chi}}(z,t),z,\gamma],\quad\forall(t,\gamma)\in
S*G$

Note that ${\Sigma}$ is a twist over the semi-direct product $Z\rtimes H$

######  Theorem 4.6.

Let $(G,S,{\mathcal{A}})$ a groupoid C*-dynamical system where ${\mathcal{A}}$
is abelian. Let $Z$ be the spectrum of the abelian C*-algebra
$C_{0}(G^{(0)},{\mathcal{A}})$. Then the twisted crossed product
$C^{*}(G,S,{\mathcal{A}})$ is isomorphic to $C^{*}(Z\rtimes H,\Sigma)$, where
$\Sigma$ is the above twist.

###### Proof.

This is a particular case of Corollary 4.4 but we give here an independent
proof. We shall identify both $C_{c}(G,S,{\mathcal{A}})$ and $C_{c}(Z\rtimes
H,\Sigma)$ as $*$-algebras of complex-valued functions on $Z\rtimes G$ and
observe that these $*$-algebras essentially coincide.
By definition, an element $f\in C_{c}(G,S,{\mathcal{A}})$ is a map
$f:G\to{\mathcal{A}}$, continuous with compact support modulo $S$ and
satisfying $f(\gamma)\in A_{r(\gamma)}$ and
$f(s\gamma)=f(\gamma)\chi(s^{-1})$. Writing $A_{r(\gamma)}$ as
$C_{0}(Z_{r(\gamma)})$, we define

$f(z,\gamma)=f(\gamma)(z),\qquad(z,\gamma)\in Z\rtimes G$

One can check that this complex-valued function defined on $Z\rtimes G$ is
continuous, satisfies

$f(z,s\gamma)=f(s,\gamma){\underline{\chi}}(z,s^{-1}),\qquad(s,\gamma)\in S*G$

there is a compact subset $K$ of $H$ such that $f(z,\gamma)=0$ for all
$(z,\gamma)\in Z\rtimes G$ such that $\dot{\gamma}\notin K$ and for all
$\epsilon>0$, there exists a compact subset $L$ of $Z$ such that
$|f(z,\gamma)|\leq\epsilon$ for all $(z,\gamma)\in Z\rtimes G$ such that
$z\notin L$. Conversely, every complex-valued function defined on $Z\rtimes G$
satisfying these conditions defines an element of $C_{c}(G,S,{\mathcal{A}})$.
With this identification of $C_{c}(G,S,{\mathcal{A}})$, the $*$-algebra
structure is given by

$f*g(z,\gamma)=\int
f(z,\tau)g(z\tau,\tau^{-1}\gamma)d\alpha^{r(\gamma)}(\dot{\tau}),\qquad
f^{*}(z,\gamma)=\overline{f(z\gamma,\gamma^{-1})}$

On the other hand, an element of $C_{c}(Z\rtimes H,\Sigma)$ is a continuous
function $f:\Sigma\to{\mathbf{C}}$ which is compactly supported modulo
${\mathbf{T}}$ (since ${\mathbf{T}}$ is compact, this equivalent to be
compactly supported) and which satisfies
$f[\theta^{\prime}\theta,z,\gamma]=f[\theta,z,\gamma]\theta^{\prime-1}$ for
$\theta,\theta^{\prime}\in{\mathbf{T}}$ and $(z,\gamma)\in Z\rtimes G$. It is
completely determined by its restriction to $\theta=1$. Thus, with a slight
abuse of notation, we write $f(z,\gamma)=f[1,z,\gamma]$. We obtain a complex-
valued function $f$ defined on $Z\rtimes G$ which is continuous with compact
support and satisfies

$f(z,s\gamma)=f(z,\gamma){\underline{\chi}}(z,s)^{-1}\qquad\forall(s,z,\gamma)\in
S*Z\rtimes G$

Conversely, given such a function $f$, we retrieve the original element of
$C_{c}(Z\rtimes H,\Sigma)$ by defining $f(\theta,s,z)=f(s,z)\theta^{-1}$. When
we express the product and the involution of $C_{c}(Z\rtimes H,\Sigma)$ in
terms of these functions, we obtain the same expressions as above. Thus the
elements of $C_{c}(G,S,{\mathcal{A}})$ and $C_{c}(Z\rtimes H,\Sigma)$ are both
continuous functions on $Z\rtimes G$ satisfying

$f(z,s\gamma)=f(z,\gamma)\gamma(z,s)^{-1}\qquad\forall(s,z,\gamma)\in
S*Z\rtimes G$

The only difference between the elements of $C_{c}(G,S,{\mathcal{A}})$ and
those of $C_{c}(Z\rtimes H,\Sigma)$ is their supports. Note that, for these
functions $f$, the absolute value $|f|$ is defined on $Z\rtimes H$. For $f\in
C_{c}(Z\rtimes H,\Sigma)$, $|f|$ has compact support. Therefore, $f$ belongs
to $C_{c}(G,S,{\mathcal{A}})$. Thus, we have realized $C_{c}(Z\rtimes
H,\Sigma)$ as a $*$-subalgebra of $C_{c}(H,\Sigma,{\mathcal{A}})$. Let us
return to the original description of $C_{c}(G,S,{\mathcal{A}})$ as the space
of compactly supported continuous sections of a Banach bundle $\mathcal{B}$
over $G$. $C_{c}(Z\rtimes G,\Sigma)$ is a linear subspace of
$C_{c}(G,S,{\mathcal{A}})$. It satisfies conditions (I) and (II) of
Proposition 14.6 of [Fell-Doran, vol I, page 139]. Indeed, for $h$ continuous
function on $G$ and $f\in C_{c}(Z\rtimes G,\Sigma)$, the function $hf$ defined
by

$(hf)(z,\gamma)=h(\dot{\gamma})f(z,\gamma)\qquad\forall(z,\gamma)\in Z\rtimes
G$

belongs to $C_{c}(Z\rtimes G,\Sigma)$. The fibre $B_{h}$ of the bundle
$\mathcal{B}$ can be identified to the Banach space $C_{0}(Z_{r(h)})$. In this
identification, the evaluation at $h$ of the elements of $C_{c}(Z\rtimes
G,\Sigma)$ gives the whole subspace $C_{c}(Z_{r(h)})$, which is dense in
$C_{0}(Z_{r(h)})$. Therefore, $C_{c}(Z\rtimes G,\Sigma)$ is dense in
$C_{c}(G,S,{\mathcal{A}})$ in the inductive limit topology. Since
$C_{c}(G,S,{\mathcal{A}})$ is complete in the inductive limit topology,
representations of $C_{c}(Z\rtimes G,\Sigma)$ which are continuous in the
inductive limit topology extend by continuity. Therefore, the inclusion of
$C_{c}(Z\rtimes G,\Sigma)$ into $C_{c}(G,S,{\mathcal{A}})$ gives an
isomorphism $C^{*}(Z\rtimes G,\Sigma)\simeq C^{*}(G,S,{\mathcal{A}})$. ∎

###  4.3. Abelian twisted extensions

After this digression about abelian Fell bundles and abelian groupoid
dynamical systems, we return to our initial problem, which is the analysis of
a twisted groupoid C*-algebra $C^{*}(G,\Sigma)$ in presence of a closed normal
subgroupoid $S$ having a Haar system. As said earlier, we make a further
assumption, whose present form I owe to Alex Kumjian.

######  Definition 4.4.

We say that a twisted extension $(G,\Sigma,S)$ is abelian if

  1. (i)

the group bundle $S$ is abelian and

  2. (ii)

the group bundle $\Sigma_{|S}$ is abelian.

We stated condition (i) for convenience only since it is implied by condition
(ii). When the twisted extension is abelian, the C*-algebra
$C^{*}(S,\Sigma_{|S}))$ is abelian and Corollary 3.4 can be completed. This
gives our main result.

######  Theorem 4.7.

Let $(G,\Sigma,S)$ be a locally compact abelian twisted groupoid extension.
Then the twisted groupoid C*-algebra $C^{*}(G,\Sigma)$ is isomorphic to the
twisted groupoid C*-algebra $C^{*}(Z\rtimes H,\underline{\Sigma})$ where $Z$
is the spectrum of $C^{*}(S,\Sigma_{|S}))$, $H=G/S$ and the twist
$\underline{\Sigma}$ is obtained by a pushout construction.

###### Proof.

We have seen that $C^{*}(G,\Sigma)$ is isomorphic to the crossed product
C*-algebra $C^{*}(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$ of the
tautological dynamical system
$(\Sigma,\Sigma_{|S},{\mathcal{C}}^{*}(S,\Sigma_{|S}))$. Since the bundle of
C*-algebras ${\mathcal{C}}^{*}(S,\Sigma_{|S})$ is abelian, we can apply
Theorem 4.6. ∎

For applications, it is necessary to be more explicit about the space $Z$, the
action of $H$ on it and the twist $\underline{\Sigma}$. Recall that we assume
that the abelian group bundle $S$ has a Haar system. Therefore, we endow its
dual group bundle $\hat{S}$ with the topology of the spectrum of $C^{*}(S)$,
as in [13, Section 3]. Since the abelian group bundle $\Sigma_{|S}$, as an
extension of $S$ by $G^{(0)}\times{\mathbf{T}}$ has also a Haar system, its
dual group bundle $\hat{\Sigma}_{|S}$ has also a natural locally compact
topology. An element of $\hat{S}$ [resp. $\hat{\Sigma}_{|S}$] will be denoted
by $(x,\chi)$, where $x$ is a base point and $\chi\in\hat{S}_{x}$ [resp.
$(\hat{\Sigma}_{|S})_{x}$ . We first consider the general case of an abelian
twist $(S,\Sigma)$.

######  Definition 4.5.

Let $X\times{\mathbf{T}}\rightarrowtail\Sigma\twoheadrightarrow S$ be an
abelian twist over an abelian group bundle $S$. Its twisted spectrum is
defined as

$\hat{S}^{\Sigma}=\\{(x,\chi)\in\hat{\Sigma}\quad\hbox{such
that}\quad\chi(\theta)=\theta\quad\forall\theta\in{\mathbf{T}}\\}$

The twisted spectrum $\hat{S}^{\Sigma}$ is an affine space over the dual group
bundle $\hat{S}$: the action of $\hat{S}$ on $\hat{S}^{\Sigma}$ is the usual
multiplication: given $\chi\in\hat{S}_{x}^{\Sigma}$ and $\rho\in\hat{S}_{x}$,
$\chi\rho\in\hat{S}_{x}^{\Sigma}$ is defined by
$(\chi\rho)(\sigma)=\chi(\sigma)\rho(\dot{\sigma})$ for $\sigma\in\Sigma_{x}$
and where $\dot{\sigma}$ is the image of $\sigma$ in $S_{x}$.

######  Remark 4.1.

In [3], the authors give a similar description of the twisted spectrum when
the twist $\Sigma$ is given by a symmetric 2-cocycle. Then, the twist is
obviously abelian. It may be useful to recall here that, according to [8,
Lemma 7.2], a Borel 2-cocycle on a locally compact abelian group is trivial if
and only if it is symmetric. Moreover, if the topology of the group is second
countable, every twist is given by a Borel 2-cocycle. Thus, a twist over a
bundle of second countable locally compact abelian groups is abelian if and
only if it is pointwise trivial.

######  Lemma 4.8.

Let $(S,\Sigma)$ be an abelian twist over an abelian group bundle $S$. Assume
that $S$ has a Haar system $\beta$. Then the twisted spectrum
$\hat{S}^{\Sigma}$ is the spectrum of the abelian C*-algebra
$C^{*}(S,\Sigma)$.

###### Proof.

When $S$ is an abelian group and the twist $\Sigma$ is trivial, this is, by
the choice of a trivialization, the well-known result that $\hat{S}$ is the
spectrum of $C^{*}(S)$. The explicit correspondence between $\hat{S}^{\Sigma}$
and the spectrum of $C^{*}(S,\Sigma)$ is given by

$\chi(f)=\int f(\sigma)\chi(\sigma)d\beta(\dot{\sigma})$

for $\chi\in S^{\Sigma}$ and $f\in C_{c}(S,\Sigma)$.

When $S$ is an abelian group bundle, the C*-algebra $C^{*}(S,\Sigma)$ is the
C*-algebra defined by the continuous field of C*-algebras $x\mapsto
C^{*}(S_{x},\Sigma_{|S_{x}})$. As a set, its spectrum is the disjoint union
over $X$ of the above spectra, which is $\hat{S}^{\Sigma}$. ∎

This turns $\hat{S}^{\Sigma}$ into a locally compact space (in fact, a locally
compact affine bundle).
We now return to our situation. We denote by $\hat{S}^{\Sigma}$ rather than
$\hat{S}^{\Sigma_{|S}}$ the twisted spectrum of $(S,\Sigma_{|S})$. Let us
describe the action of $H=G/S$ on $\hat{S}^{\Sigma}$. The groupoid $H$ acts on
the group bundle $\Sigma_{|S}$ by conjugation: $h.\sigma=\tau\sigma\tau^{-1}$,
where $\pi_{\Sigma}(\tau)=h$. The transposed action on the dual group bundle
$\widehat{\Sigma_{|S}}$, defined by $(\chi
h)(\sigma)=\chi(\tau\sigma\tau^{-1})$, preserves the twisted spectrum
$\hat{S}^{\Sigma}$. It is easily checked that this is the action arising from
the action of $H$ on the bundle of C*-algebras
${\mathcal{C}}^{*}(S,\Sigma_{|S})$.

The above pushout diagram defining the twist $\underline{\Sigma}$ becomes:

$\begin{CD}\hat{S}^{\Sigma}*\Sigma_{|S}@>{}>{}>\hat{S}^{\Sigma}\rtimes\Sigma
@>{}>{}>\hat{S}^{\Sigma}\rtimes H\\\ @V{\varphi}V{}V@V{}V{}V\Big{\|}\\\
\hat{S}^{\Sigma}\times{\mathbf{T}}@>{}>{}>\underline{\Sigma}@>{}>{}>\hat{S}^{\Sigma}\rtimes
H\end{CD}$

where $\varphi(\chi,\sigma)=(\chi,\chi(\sigma))$ for
$(\chi,\sigma)\in\hat{S}^{\Sigma}*\Sigma_{|S}$.
Explicitly, $\underline{\Sigma}$ is the quotient of the groupoid
$(\hat{S}^{\Sigma}\rtimes\Sigma)\times{\mathbf{T}}$ by the equivalence
relation

$(\chi,\sigma\tau,\theta)\sim(\chi,\tau,\overline{\chi(\sigma)}\theta),\quad\forall\sigma\in\Sigma_{|S}.$

######  Remark 4.2.

In [7, Proposition 3.5], the above Theorem 4.7 (the twisted case) is deduced
from the similar result for the untwisted case, established in the previous
work [6, Theorem 3.3].

##  5\. An application: deformation quantization

Rieffel has introduced a notion of C*-algebraic deformation quantization and
illustrated it by a number of examples in [18]. On the other hand, Ramazan,
generalizing Connes’ tangent groupoid, has produced deformation quantization
of Lie-Poisson manifolds by using groupoid techniques (see [15, 11]). Our
Theorem 4.7 shows that the two approaches agree on some important examples. We
consider here the basic example of a symplectic finite-dimensional real vector
space $(V,\omega)$. Then, for every $\hbar\in{\mathbf{R}}$,
$\sigma_{\hbar}=e^{i\hbar\omega/2}$ is a ${\mathbf{T}}$-valued 2-cocycle on
$(V,+)$. It is shown in [19] that $\hbar\mapsto C^{*}(V,\sigma_{\hbar})$ can
be made into a continuous field of C*-algebras and in [20] that it gives a
C*-algebraic deformation quantization of the Lie-Poisson manifold
$(V,\omega)$. Note that its sectional algebra can be viewed as a twisted
groupoid C*-algebra $C^{*}(G,\Sigma)$, where $G$ is the trivial group bundle
${\mathbf{R}}\times V$ over ${\mathbf{R}}$ and $\Sigma$ is the twist defined
by the 2-cocycle $\sigma(\hbar,.)=\sigma_{\hbar}$. Let $V=L\oplus L^{\prime}$
be a direct sum decomposition, where $L$ and $L^{\prime}$ are complementary
Lagrangian subspaces. This gives the extension

${\mathbf{R}}\times L\rightarrowtail G\twoheadrightarrow{\mathbf{R}}\times
L^{\prime}$

The abelian group bundle $S={\mathbf{R}}\times L$ satisfies the conditions of
Definition 3.2 with respect to the twisted groupoid $(G,\Sigma)$. Therefore,
by Theorem 4.7, $C^{*}(G,\Sigma)$ is isomorphic to
$C^{*}(Z\rtimes({\mathbf{R}}\times L^{\prime}),\underline{\Sigma})$, where $Z$
is the twisted spectrum and the twist $\underline{\Sigma}$ is obtained by the
pushout construction. Let us determine them explicitly. The action of
$H={\mathbf{R}}\times L^{\prime}$ on $\Sigma_{|S}={\mathbf{R}}\times
L\times{\mathbf{T}}$ is given by

$(\hbar,y).(\hbar,x,\theta)=(\hbar,x,e^{-i\hbar\omega(x,y)}\theta),\quad{\rm
where}\quad\hbar\in{\mathbf{R}},\,\quad y\in L^{\prime},\,x\in
L,\,\theta\in{\mathbf{T}}$

Since $\Sigma_{|S}={\mathbf{R}}\times L\times{\mathbf{T}}$, the twisted
spectrum $Z$ is ${\mathbf{R}}\times\hat{L}$, where $\hat{L}$ denotes the dual
group of the abelian locally compact group $L$. The action of $H$ on $Z$ is
given by

$(\hbar,\chi)(\hbar,y)=(\hbar,\chi\varphi_{\hbar}(y)),\quad{\rm
where}\quad\hbar\in{\mathbf{R}},\,\chi\in\hat{L},\,y\in L^{\prime}$

and for $\hbar\in{\mathbf{R}}$, $\varphi_{\hbar}$ is the group homomorphism
from $L^{\prime}$ to $\hat{L}$ such that

$<\varphi_{\hbar}(y),x>=e^{-i\hbar\omega(x,y)}\quad{\rm where}\quad x\in
L,\,y\in L^{\prime}$

The semi-direct product $Z\rtimes H$ is a bundle of semi-direct products
$\hat{L}\rtimes_{\hbar}L^{\prime}$. For $\hbar\not=0$, $\varphi_{\hbar}$ is an
isomorphism and $\hat{L}\rtimes_{\hbar}L^{\prime}$ is isomorphic to the
trivial groupoid $\hat{L}\times\hat{L}$. For $\hbar=0$, we get $\hat{L}\times
L^{\prime}$, where the first term is a space and the second is a group. We use
again $\omega$ to identify $L^{\prime}$ and the dual $L^{*}$, which is the
tangent space of $\hat{L}$. Thus $\hat{L}\times L^{\prime}$ is isomorphic to
the tangent bundle $T\hat{L}$ and the groupoid $Z\rtimes H$ is isomorphic to
the tangent groupoid of the manifold $\hat{L}$. One can check that we have an
isomorphism of topological groupoids.

It remains to determine the twist $\underline{\Sigma}$.

######  Proposition 5.1.

The above twist is trivial.

###### Proof.

By construction, $\underline{\Sigma}$ is the quotient of
$(Z\rtimes\Sigma)\times{\mathbf{T}}$ by the equivalence relation

$(\hbar,\chi,x+v,\sigma_{\hbar}(x,v)\varphi\psi,\chi(x)\varphi\theta)\sim(\hbar,\chi,v,\psi,\theta)$

where $\hbar\in{\mathbf{R}}$, $\chi\in\hat{L}$, $x\in L,v\in V$ and
$\varphi,\psi,\theta\in{\mathbf{T}}$. The map

$(Z\rtimes\Sigma)\times{\mathbf{T}}\to(Z\rtimes H)\times{\mathbf{T}}$

sending $(\hbar,\chi,x+y,\psi,\theta)$ to
$(\hbar,\chi,y,\overline{\psi}\overline{\chi(x)}\sigma_{\hbar}(x,y)\theta)$
where $\hbar\in{\mathbf{R}}$, $\chi\in\hat{L}$, $x\in L$, $y\in L^{\prime}$,
and $\varphi,\theta\in{\mathbf{T}}$ identifies topologically this quotient.
This is also a groupoid homomorphism. Therefore, $\underline{\Sigma}$ is
isomorphic to $(Z\rtimes H)\times{\mathbf{T}}$.

∎

The above example can also be presented via the Heisenberg group
${\mathcal{H}}=V\times{\mathbf{R}}$ with multiplication
$(v,s)(w,t)=(v+w,\omega(v,w)+s+t)$. Mackey’s normal subgroup analysis (i.e.
Theorem 4.7) applied to the center $\\{0\\}\times{\mathbf{R}}$ gives the first
deformation. The second deformation can be obtained by applying this analysis
to the subgroup $L\times{\mathbf{R}}$. In conclusion, we have three isomorphic
C*-algebras: $C^{*}({\mathcal{H}})$, $C^{*}(G,\Sigma)$ and $C^{*}(Z\rtimes
H)$.

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