# Fourier–Stieltjes category
for twisted groupoid actions

Alcides Buss  Departamento de Matemática, Universidade Federal de Santa
Catarina, 88.040-900 Florianópolis SC, Brazil<EMAIL_ADDRESS>,
Bartosz Kwaśniewski  Faculty of Mathematics, University of Białystok, ul. K.
Ciołkowskiego 1M, 15-245 Białystok, Poland<EMAIL_ADDRESS>,  Andrew
McKee  Faculty of Mathematics, University of Białystok, ul. K. Ciołkowskiego
1M, 15-245 Białystok, Poland<EMAIL_ADDRESS>and  Adam Skalski  Institute
of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656
Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

We extend the theory of Fourier–Stieltjes algebras to the category of twisted
actions by étale groupoids on arbitrary $\mathrm{C}^{*}$-bundles, generalizing
theories constructed previously by Bédos and Conti for twisted group actions
on unital $\mathrm{C}^{*}$-algebras, and by Renault and others for groupoid
$\mathrm{C}^{*}$-algebras, in each case motivated by the classical theory of
Fourier–Stieltjes algebras of discrete groups. To this end we develop a
toolbox including, among other things, a theory of multiplier
$\mathrm{C}^{*}$-correspondences, multiplier $\mathrm{C}^{*}$-correspondence
bundles, Busby–Smith twisted groupoid actions, and the associated crossed
products, equivariant representations and Fell’s absorption theorems. For a
fixed étale groupoid $\mathcal{G}$ a Fourier–Stieltjes multiplier is a family
of maps acting on fibers, arising from an equivariant representation. It
corresponds to a certain fiber-preserving strict completely bounded map
between twisted full (or reduced) crossed products. We establish a KSGNS-type
dilation result which shows that the correspondence above restricts to a
bijection between positive-definite multipliers and a particular class of
completely positive maps. Further we introduce a subclass of Fourier
multipliers, that enjoys a natural absorption property with respect to
Fourier–Stieltjes multipliers and gives rise to ‘reduced to full’ multiplier
maps on crossed products. Finally we provide several applications of the
theory developed, for example to the approximation properties, such as weak
containment or nuclearity, of the crossed products and actions in question,
and discuss outstanding open problems.

###### Key words and phrases:

étale groupoids; twisted groupoid actions; crossed products; Fourier–Stieltjes
algebras; multipliers; $\mathrm{C}^{*}$-correspondences

###### 2010 Mathematics Subject Classification:

Primary 46L05; Secondary 20E26, 22A22, 43A22, 46L55

###### Contents

  1. 1 Multiplier C*-correspondences
  2. 2 Dilation/Decomposition theorem
  3. 3 Bundles of C*-correspondences
  4. 4 Fell bundles and twisted actions
  5. 5 Equivariant correspondences
  6. 6 Induced regular representations and absorption principles
  7. 7 Fourier–Stieltjes multipliers for groupoid twisted actions
  8. 8 Positive-definiteness and multipliers on section C*-algebras
  9. 9 Fourier multipliers
  10. 10 Approximation properties for multipliers
  11. 11 Some further applications: Haagerup trick and decomposable norms
  12. 12 Open questions and further perspectives

## Introduction

The notions of Fourier–Stieltjes and Fourier algebras of general locally
compact groups, motivated by the concepts appearing earlier in classical,
abelian harmonic analysis were first introduced sixty years ago in the thesis
of Eymard [Eym64]. Recall that the _Fourier–Stieltjes algebra_ $B(\Gamma)$ of
a given discrete group $\Gamma$ can be viewed in at least three different
guises:

  1. (1)

as the algebra of functions given by coefficients of unitary representations
of $\Gamma$, with the pointwise multiplication;

  2. (2)

as the dual of the universal group $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\Gamma)$, equipped with the ‘convolution’ type product induced
by the comultiplication of $\mathrm{C}^{*}(\Gamma)$;

  3. (3)

as the span of completely positive (in short cp) Herz–Schur multipliers of
$\mathrm{C}^{*}(\Gamma)$, with the multiplication given as the composition of
maps.

Similarly, the _Fourier algebra_ $A(\Gamma)$, which is an ideal in
$B(\Gamma)$, can be seen as the algebra of coefficients of the left regular
representation, as the predual of the group von Neumann algebra
$\mathrm{vN}(\Gamma)$, or as a certain subclass of Herz–Schur multipliers of
$\mathrm{C}^{*}(\Gamma)$ (or of $\mathrm{vN}(\Gamma)$). Note that developing
the equivalence between the first and last points of view displayed above
requires proving a version of a dilation result for positive-definite
functions, which in this context is nothing but the GNS construction. On the
other hand, the fact that $A(\Gamma)$ is an ideal inside $B(\Gamma)$ is a
consequence of Fell’s absorption principle: the tensor product of a regular
representation and an arbitrary representation is unitarily equivalent to a
multiple of the regular representation.

This flexibility is one of the reasons why the aforementioned Banach algebras
have played a significant role in the study of operator algebras related to
groups and in the noncommutative harmonic analysis in general (we refer to the
recent book [KL18] for an introduction to the topic and several applications).
It is thus natural that together with the development of various
generalizations of group operator algebras the literature has seen several
attempts to investigate analogues of Fourier–Stieltjes algebras in these new
contexts. We would like to mention here for example the study of
Fourier–Stieltjes algebras of locally compact quantum groups à la
Kustermans–Vaes [KV00], as discussed for example in [Daw10]. The study of
Fourier algebras, Fourier–Stieltjes algebras and related multiplier maps for
measure groupoids was initiated in the articles [Ren97, RW97, Oty99], with the
main focus on trying to provide the equivalence between the first and the
third point of view offered above; in particular on proving suitable versions
of dilation and absorption results. The two specific cases which are most
relevant for our paper are however the ones encoding various types of non-
commutative topological dynamics, i.e. that of the étale groupoid
$\mathrm{C}^{*}$-algebras and that of the $\mathrm{C}^{*}$-algebraic twisted
crossed products, which we will discuss next.

Locally compact Hausdorff étale grupoids were introduced to the
$\mathrm{C}^{*}$-algebraic world in the early 1980s in [Ren80], largely
motivated by the developments related to equivalence relations and von Neumann
algebra theory, that later culminated in the celebrated characterisation of
Cartan inclusions [Ren08]. An important special case of the étale groupoid
$\mathrm{C}^{*}$-algebra construction is that of the crossed product
$C(X)\rtimes\Gamma$, where $X$ is a compact space equipped with an action of a
discrete group $\Gamma$. It has however taken relatively long until the theory
of Fourier–Stieltjes algebras and related multiplier maps was developed in the
context of general $\mathrm{C}^{*}$-algebraic crossed products (although
[Ped18, Section 7.6] provides a description of the state space of the crossed
product $A\rtimes\Gamma$, and an attempt to introduce the algebraic structure
on this space was undertaken in [Fuj79]). During the last few years we have
seen a flow of work, notably by Bédos and Conti, see [BC16, BC21] and
references therein, which studied ‘multiplier type’ maps on crossed products
by twisted actions of discrete groups on _unital_ $\mathrm{C}^{*}$-algebras.
In particular, in the articles [BC15, BC16] both the dilation, for ‘kernels of
positive type’, and the absorption results for the so-called ‘induced
representations’ were established. The new context necessitated the use of
Hilbert modules and $\mathrm{C}^{*}$-correspondences, similarly to the quantum
group framework mentioned earlier, as seen for example in [Daw11].

The articles [DR12] and [MSTT18] use multiplier maps similar to those
considered by Bédos and Conti to study approximation properties of the crossed
products in question. We should mention here that the key technique which
makes the last application possible, and goes back to the context of group
$\mathrm{C}^{*}$-algebras, is the so-called _Haagerup trick_ , which provides
a way to ‘average’ arbitrary bounded linear maps on the crossed product
algebra into multipliers. In the context of twisted groupoid actions a version
of the Haagerup trick was applied very recently in [KLS22] by two of the
authors of this work, to study the relative Haagerup property of
$\mathrm{C}^{*}$-algebras in connection with the UCT property. This was
possibly the first time multiplier type maps appeared for the groupoid crossed
products.

$\left(\begin{array}[]{c}\text{{Multiplier}}\\\
\text{$C^{*}$-correspondences}\\\ \text{Section
\ref{sec:multiplier}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Categories}
of }\\\ \text{multipliers}\\\ \text{Sections \ref{Sec:FSmultipliers},
\ref{Sec:Fourierapproximation}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Fell's
absorption}}\\\ \text{Theorems \ref{thm:Fell_absorption},
\ref{thm:Fell_absorption
II}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{KSGNS/Kolmogorov }}\\\
\text{dilation/decomposition}\\\ \text{Theorem
\ref{thm:Murphy_Stinespring}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Functors}
from multiplier }\\\ \text{to $C^{*}$-algebra categories}\\\ \text{Theorems
\ref{thm:FSmultiplier_extends_to_reduced_and_full},
\ref{thm:Fmultipliers_are_reduced_to_full}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Characterisation
of cp}}\\\ \text{fiber preserving maps}\\\ \text{Theorem
\ref{thm:FSmultiplier_positive_definite}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Approximation}}\\\
\text{properties}\\\ \text{Sections \ref{Sec:Applicationapproximation},
\ref{Sec:Applications}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{Decomposable}}\\\
\text{norm of mutipliers}\\\ \text{Theorem
\ref{thm:FSmultipliers_decomposable_discrete}}\end{array}\right)$$\left(\begin{array}[]{c}\text{Haagerup
Trick}\\\ \text{Lemma \ref{lem:Haagerup trick discrete}}\\\ \text{Proposition
\ref{prop:Haagerup_trick}}\end{array}\right)$$\left(\begin{array}[]{c}\text{{nuclearity}
criteria}\\\ \text{and characterisation}\\\ \text{Theorems
\ref{thm:approximation_prop_implies_weak_containment},
\ref{thm:characterisation of nuclearity}}\end{array}\right)$

Figure 1. Structure and relationship between developed steps/results

In this paper, inspired by the developments mentioned above, we undertake a
systematic study of Fourier–Stieltjes and Fourier multipliers for crossed
products by twisted actions of étale groupoids on arbitrary
$\mathrm{C}^{*}$-bundles. To include the work of Bédos and Conti [BC12, BC15,
BC16] as a special case, we consider here twisted groupoid actions á la
Busby–Smith [BS70] with twists taking values in multiplier bundles. Such
actions appear in [BMZ13] but their crossed products, apart from the group
case, have not yet been studied. In particular, they are different from the
Green–Renault twisted groupoid actions considered in [Ren87, Ren91], where the
twist is abelian (usually scalar valued). It has also occurred to us that,
instead of looking at Banach algebras of multipliers for a single action, it
is more natural and useful to study _Banach categories of multipliers_ between
different actions of the same groupoid. The main steps in our development, as
well as the relationships between them, are schematically presented in Figure
1.

Apart from the novelties mentioned above we also for the first time consider
crossed product multipliers for actions on _non-unital_ $C^{*}$-algebras. In
order to do that we develop systematically a theory of _multiplier
$C^{*}$-correspondences_. It seems that so far this notion was considered only
in the special case of Hilbert bimodules [ER95, Sch02], and we believe that
the results of this investigation are of independent interest and will be
applicable in a variety of contexts. With that tool in hand, establishing a
general framework of _bundles of $C^{*}$-correspondences_ and of _equivariant
correspondences_ that generalizes notions appearing in [LG99, EKQR06] and
[Dea18], we construct Banach categories of _Fourier–Stieltjes_ and _Fourier
multipliers_ between twisted actions of a fixed groupoid. This is a far
reaching generalization of the first picture (1) mentioned above for groups.
We prove three different _absorption results_ to achieve our goals. The first
of them implies that the Fourier category is an _ideal_ in the
Fourier–Stieltjes category. The remaining two provide different
generalisations of classical _Fell’s absorption principle_ for groups. The
first implies that Fourier–Stieltjes multipliers act as completely bounded
maps both on the reduced and universal crossed product algebras, which
provides functors into the respective categories. The second one ensures that
the Fourier multipliers even act ‘reduced-to-full’. In the context of group
actions similar versions of Fell’s absorption principle were proved in [BC12,
BC15]. For groupoid actions our second Fell’s absorption principle is closely
related, but not exactly comparable with, the corresponding recent results in
[Kra23] and [BM23].

Having proved the above results we then focus on (3) — the third of the
pictures mentioned in the beginning of the introduction (as finding the right
generalization of the second, ‘functional’ avatar is problematic even in the
context of group crossed products, as can be seen by looking at [Ped18,
Section 7.6]). In order to obtain the desired equivalence we prove a dilation
result, generalizing simultaneously the KSGNS construction of Kasparov
([Lan95]) and Murphy’s Kolmogorov decomposition of positive kernels for
Hilbert modules from [Mur97], and which might be of interest for its own sake.
Using this we prove a core result of the paper. It states that for any two
twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of a given étale
groupoid $\mathcal{G}$ we have natural bijections between the following
objects:

  1. (1)

strict, bounded and positive-definite multipliers from $(\alpha,u_{\alpha})$
to $(\beta,u_{\beta})$;

  2. (2)

strict completely positive fiber preserving maps between the reduced crossed
products;

  3. (3)

strict compleletely positive fiber preserving maps between the full crossed
products;

  4. (4)

Fourier–Stieltjes multipliers from $(\alpha,u_{\alpha})$ to
$(\beta,u_{\beta})$ of the form $T_{\mathcal{E},L,\xi,\xi}$ for some
equivariant correspondence $(L,\mathcal{E})$ and a section
$\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$.

This together with the standard polarization formula applied to (4) gives the
appropriate generalization of the equivalence between the guises (1) and (3).
The strictness of completely positive maps appearing in conditions (1)–(3)
above is a very weak replacement for unitality of the algebras and maps in
question. In fact for the approximation properties that we study strictness
plays no role, as we can always conjugate with approximate units to
‘strictify’ a given map. Our categorical approach allows us to investigate
_approximation properties_ of arbitrary multipliers. When applied to the
identity this leads to a fruitful interaction between properties of the
Fourier–Stieltjes algebra and the approximation properties of the
corresponding crossed product algebra. For example we provide natural
approximation conditions guaranteeing that the relevant universal and reduced
crossed product $\mathrm{C}^{*}$-algebras coincide and are nuclear.

We also return to the Haagerup trick. For general groupoids and general cp
maps it is not completely clear what is the right replacement for this device,
so we decided to consider two cases where we know that it works (this is
reflected in Figure 1 by the use of thinner arrows). Namely, we assume that
either the groupoid is discrete (which is still interesting, as for example it
covers the case of discrete groups) or the maps in question have a certain
bimodule property. Then we can use it to characterise the relevant
approximation properties, and in the discrete case to identify the natural
Fourier–Stieltjes norm with the decomposable norm of [Haa85] of the associated
decomposable map, as well as give full characterisations of nuclearity of the
associated crossed products.

The construction of the Fourier–Stieltjes category in the generality described
above is not yet the end of the story. One could think of further extensions
to multipliers of Fell bundles, with the latter already put to use for example
in [Exe97, BM23]. On the other hand, even the context of twisted crossed
products given by groupoid actions offers many new perspectives. These are on
one hand related to the analysis of more subtle approximation properties, and
on the other to potential reconstruction results, related to the question: to
what extent does the Fourier–Stieltjes or Fourier algebra remember the
underlying action? At the end of the text we indicate several related avenues
of future research and mention also specific problems left open in our work.

The specific plan of the paper is as follows: in Section 1, we introduce
multiplier modules, $\mathrm{C}^{*}$-correspondences and Hilbert bimodules. In
Section 2 we prove a dilation result for a particular form of ‘positive
kernels’. Next, in Section 3, we return to correspondences in the context of
bundles over a locally compact space and describe direct sums, tensor products
and natural morphisms between the respective bundles. Section 4 passes to Fell
bundles, and specifically to these which arise from twisted groupoid actions
and whose cross sectional $C^{*}$-algebras give the relevant crossed products.
Here we also connect the latter to inverse semigroup twisted actions in the
sense of [BE12]. In Section 5 we focus on equivariant correspondences for a
pair of groupoid twisted actions, setting up the appropriate framework and
proving that an _equivariant correspondence induce_ representations from one
crossed product to another. The process of inducing representations from
correspondences without equivariant action is introduced in Section 6. Such
representations are called _induced regular_ because they always descend to
reduced crossed products, while for equivariant induction this holds only if
the equivariant representation is regular or if the representation from which
we induce is regular. Our two counterparts of Fell’s absorption principle
describe the fundamental relationships between the two types of induction.
Section 7 uses the stage set up in the first part of the paper to finally
introduce the Fourier–Stieltjes Banach category and prove that
Fourier–Stieltjes multipliers act both on reduced and full crossed product
$\mathrm{C}^{*}$-algebras. The following Section 8 specialises to
Fourier–Stieltjes multipliers of positive type; here we also establish the
advertised equivalence between different points of view on such multipliers. A
short Section 9 treats Fourier multipliers, using earlier absorption theorems
to deduce that they form ideal in the class of Fourier-Stieltjes multipliers
and extend to reduced-to-full maps. Section 10 introduces the concepts of a
Fourier and Fourier-Stieltjes approximation of a given multiplier and uses it
to provide applications to approximation properties of actions and resulting
crossed products. In Section 11 we discuss appropriate versions of the
Haagerup trick, apply them to find another descriptions of approximation
properties as well as to provide an alternative view of the Fourier–Stieltjes
norm, which in turn again has certain consequences for the approximation
properties. Finally, in Section 12, we set up open problems and present
further research perspectives.

Notation: We will introduce the notation gradually throughout the text, but
the most important conventions are as follows. We denote by $X$ a locally
compact Hausdorff space and by $\mathcal{G}$ a Hausdorff étale groupoid with
the unit space $X$. $C^{*}$-algebras are denoted with $A$, $B$, $\dots$, Fell
bundles with $\mathcal{A}$, $\mathcal{B}$, $\dots$, modules with $E$, $F$ and
bundles of modules with $\mathcal{E}$, $\mathcal{F}$. If $p$ is a logical
statement — mostly an equality — then we use the symbol $[p]$ to mean the
boolean logical value of $p$, that is $[p]=1$ if $p$ is true and $[p]=0$ if
$p$ is false.

##  1\. Multiplier C*-correspondences

In this section we discuss the notion of multiplier Banach bimodules,
multiplier $\mathrm{C}^{*}$-correspondences and multiplier Hilbert bimodules,
extending the theory from [Daw10].

###  1.1. Multiplier Banach bimodules

Throughout this paper $A$ and $B$ will be $\mathrm{C}^{*}$-algebras (although
in this particular subsection it suffices to assume that $A$ and $B$ are
Banach algebras with contractive approximate units). For Banach spaces $E$,
$F$, we denote by $\mathbb{B}(E,F)$ the space of bounded operators $E\to F$.
Let $E$ be a Banach $A$-$B$-bimodule. In this paper Banach modules by
definition will be _nondegenerate_ , i.e. we always assume that $AE=EB=E$.
Note that by the Cohen–Hewitt factorization theorem these products may be
viewed either purely algebraically or as closed linear spans. Then $E$ is also
naturally a Banach $\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule where
$\mathcal{M}(A)$ and $\mathcal{M}(B)$ are the multiplier algebras of $A$ and
$B$, respectively. We recall the definition of the multiplier bimodule from
[Daw10] (note that [Daw10] considers $A$-bimodules, but the constructions and
results there are also valid for $A$-$B$-bimodules). Namely, the direct sum
$\mathbb{B}(A,E)\oplus\mathbb{B}(B,E)$ with the norm
$\|(R,L)\|_{\infty}=\max\\{\|R\|,\|L\|\\}$ becomes a Banach
$\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule with actions given by

$(aR)(a_{0})=R(a_{0}a),\quad(Rb)(a_{0})=R(a_{0})b,\quad(Lb)(b_{0})=L(bb_{0}),\quad(aL)(b_{0})=aL(b_{0}),$

for $(R,L)\in\mathbb{B}(A,E)\oplus\mathbb{B}(B,E)$, $a_{0}\in
A,a\in\mathcal{M}(A)$, $b_{0}\in B,b\in\mathcal{M}(B)$. A _multiplier_ for $E$
is a pair of maps $(R,L)$ where $R:A\to E$, $L:B\to E$ and

$aL(b)=R(a)b,\qquad a\in A,\ b\in B.$

By [Daw10, Proposition 2.5], the set of all multipliers for $E$, denoted
$\mathcal{M}(E)$, is a closed $\mathcal{M}(A)$-$\mathcal{M}(B)$-sub-bimodule
of $\mathbb{B}(A,E)\oplus\mathbb{B}(B,E)$. We have an isometric
$\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule map
$E\ni\xi\mapsto(R_{\xi},L_{\xi})$ where $R_{\xi}(a):=a\xi$ and
$L_{\xi}(b):=\xi b$, which allows us to treat $E$ as a subbimodule of
$\mathcal{M}(E)$. We call the $\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule
$\mathcal{M}(E)$ the _multiplier bimodule_ of the $A$-$B$-bimodule $E$. Below
we will characterise it as the largest essential multiplier extension of $E$.

######  Definition 1.1.

We say that an $\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule $F$ is a _multiplier
extension_ of an $A$-$B$-bimodule $E$ if $E\subseteq F$ is a submodule such
that $AF=FB=E$. We say that this extension is _essential_ if for every $\xi\in
F$, we have $A\xi=0$ only if $\xi=0$ or equivalently, $\xi B=0$ only if
$\xi=0$ (the equivalence follows from the equality $AF=FB=E$).

######  Remark 1.2.

If either of the algebras $A$ or $B$ is unital then $F=1_{A}F=F1_{B}=E$ for
every multiplier extension $F$ of $E$.

######  Lemma 1.3.

For any $A$-$B$-bimodule $E$, the multiplier bimodule $\mathcal{M}(E)$ is the
largest essential multiplier extension of $E$. More precisely,
$\mathcal{M}(E)$ is an essential multiplier extension of $E$ and for any other
multiplier extension $F$ of $E$ there is a unique (necessarily contractive)
$A$-$B$-bimodule map $\mu:F\to\mathcal{M}(E)$ that extends the identity on
$E$. Moreover, $\mu$ is injective if and only if $F$ is essential.

###### Proof.

For any $(R,L)\in\mathcal{M}(E)$ and $a\in A$, $b\in B$ we have
$a\cdot(R,L)=(R_{R(a)},L_{R(a)})$ and $(R,L)\cdot b=(R_{L(b)},L_{L(b)})$ (note
that $L$ is a right module map and $R$ is a left module map, see [Daw10, Lemma
2.1]). Hence $\mathcal{M}(E)$ is a multiplier extension of $E$ and clearly it
is essential. For any multiplier extension $F$ we have the bimodule map
$F\ni\xi\mapsto(R_{\xi},L_{\xi})\in\mathcal{M}(E)$ where $R_{\xi}(a):=a\xi$
and $L_{\xi}(b):=\xi b$, for $\xi\in F$. By nondegeneracy of bimodules this is
the unique bimodule map $F\to\mathcal{M}(E)$ that restricts to the identity on
$E$. By construction it is contractive, and it is injective if and only if $F$
is essential. ∎

######  Corollary 1.4.

$\mathcal{M}(E)$ is the unique (up to an isometric isomorphism extending the
identity on $E$) multiplier extension of $E$ with the property that for any
multiplier extension $F$ of $E$ there is a unique bimodule map
$F\to\mathcal{M}(E)$ that extends the identity $E\to E$.

###### Proof.

Follows from the usual universality considerations. ∎

We also need the following general morphism extension result. It is a common
generalization of [Daw10, Theorem 2.8] and the known fact that any contractive
homomorphism $\pi:A\to\mathcal{M}(B)$ which is _nondegenerate_ in the sense
that $\pi(A)B=B$, extends uniquely to a homomorphism
$\overline{\pi}:\mathcal{M}(A)\to\mathcal{M}(B)$ which is necessarily
contractive.

######  Definition 1.5.

A _morphism_ from a Banach $A$-$B$-bimodule $E$ to a Banach $C$-$D$-bimodule
$F$ consists of a linear map $\pi:E\to\mathcal{M}(F)$ and nondegenerate
homomorphisms $\pi_{A}:A\to\mathcal{M}(C)$, $\pi_{B}:B\to\mathcal{M}(D)$ such
that $\pi(a\xi b)=\pi_{A}(a)\pi(\xi)\pi_{B}(b)$ for $\xi\in E$, $a\in A$,
$b\in B$. We say that this morphism is _contractive_ if all the maps
$(\pi,\pi_{A},\pi_{B})$ are contractive.

######  Lemma 1.6.

Any contractive morphism $(\pi,\pi_{A},\pi_{B}):E\to F$ extends uniquely to a
morphism
$(\overline{\pi},\overline{\pi}_{A},\overline{\pi}_{B}):\mathcal{M}(E)\to\mathcal{M}(F)$
which is necessarily contractive.

###### Proof.

Take any $(R,L)\in\mathcal{M}(E)$. We claim that the formulas

$\widetilde{R}(c\pi_{A}(a)):=c\pi(R(a)),\qquad\widetilde{L}(\pi_{B}(b)d):=\pi(L(b))d,\quad\mbox{
for $a\in A,b\in B,c\in C,d\in D$}$

yield linear maps $\widetilde{R}:C\to F$, $\widetilde{L}:D\to F$. Indeed,
letting $\\{e_{i}^{A}\\}_{i\in I}$ and $\\{e_{i}^{B}\\}_{i\in I}$ be
approximate units respectively in $A$ and $B$ we get
$c\pi(R(a))=\lim_{i}c\pi(R(ae_{i}^{A}))=\lim_{i}c\pi(aR(e_{i}^{A}))=\lim_{i}c\pi_{A}(a)\pi(R(e_{i}^{A}))$
and similarly $\pi_{B}(b)d=\lim_{i}\pi(L(e_{i}^{B}))\pi_{B}(b)d$. This implies
that $\widetilde{R}$ and $\widetilde{L}$ are well defined and contractive. We
have

$\widetilde{R}(c)=\lim_{i}c\pi(R(e_{i}^{A})),\qquad\widetilde{L}(d)=\lim_{i}\pi(L(e_{i}^{B}))d,\qquad
c\in C,d\in D.$

Moreover, for $a\in A,b\in B,c\in C,d\in D$ we get

$c\pi_{A}(a)\widetilde{L}(\pi_{B}(b)d)=c\pi(aL(b))d=c\pi(R(a)b)d=c\pi(R(a))\pi_{B}(b)d=\widetilde{R}(c\pi_{A}(a))\pi_{B}(b)d.$

Hence $(\widetilde{R},\widetilde{L})\in\mathcal{M}(F)$. Thus putting
$\overline{\pi}(R,L):=(\widetilde{R},\widetilde{L})$ we get a contractive
extension $\overline{\pi}:\mathcal{M}(E)\to\mathcal{M}(F)$ of $\pi$. It can be
readily checked that for $a\in A$, $b\in B$ and $\xi=(R,L)\in\mathcal{M}(E)$
we have $\pi_{A}(a)\overline{\pi}(\xi)\pi_{B}(b)=\pi(a\xi b)$. This property
determines $\overline{\pi}$ uniquely, as for any other
$\tilde{\pi}:\mathcal{M}(E)\to\mathcal{M}(F)$ with this property we have
$(\overline{\pi}(\xi)-\tilde{\pi}(\xi))\pi_{B}(B)=0$ which implies
$(\overline{\pi}(\xi)-\tilde{\pi}(\xi))D=0$ and therefore
$\overline{\pi}(\xi)=\tilde{\pi}(\xi)$ because $\mathcal{M}(F)$ is an
essential extension of $F$.

The above (in the special case where $E=A=B$ and $F=C=D$ are Banach algebras
and $\pi=\pi_{A}=\pi_{B}$) also proves the aforementioned fact that
contractive nondegenerate homomorphisms between Banach algebras extend
uniquely to homomorphisms between their multiplier algebras, and they are
necessarily contractive. Thus we have unique (contractive) homomorphic
extensions $\overline{\pi}_{A}:\mathcal{M}(A)\to\mathcal{M}(C)$ and
$\overline{\pi}_{B}:\mathcal{M}(B)\to\mathcal{M}(D)$ of $\pi_{A}$ and
$\pi_{B}$. We immediately get that
$\overline{\pi}_{A}(a)\overline{\pi}(\xi)\overline{\pi}_{B}(b)=\overline{\pi}(a\xi
b)$ for $a\in\mathcal{M}(A)$, $b\in\mathcal{M}(B)$ and $\xi\in\mathcal{M}(E)$.
Thus $(\overline{\pi},\overline{\pi}_{A},\overline{\pi}_{B})$ is a unique
morphism that extends $(\pi,\pi_{A},\pi_{B})$. ∎

###  1.2. Multiplier C*-correspondences

Let $E$ and $F$ be (right) $B$-Hilbert modules. We use the standard notation
$\mathcal{L}(E,F)$ for the Banach space of adjointable operators from $E$ to
$F$, and $\mathcal{K}(E,F)\subseteq\mathcal{L}(E,F)$ for the closed subspace
of (generalized) compact operators, generated by the ‘rank-one’ operators
$\Theta_{x,y}(z):=x\langle y,z\rangle_{B}$, $x\in F,y,z\in E$. In particular,
$\mathcal{K}(E):=\mathcal{K}(E,E)$ is an ideal in the $\mathrm{C}^{*}$-algebra
$\mathcal{L}(E):=\mathcal{L}(E,E)$. A $\mathrm{C}^{*}$_-correspondence_ from
$A$ to $B$ (or an $A$-$B$-$\mathrm{C}^{*}$-correspondence) is a Hilbert module
$E$ over $B$ together with a nondegenerate $*$-homomorphism from $A$ to
$\mathcal{L}(E)$, which introduces a left $A$-module structure on $E$. In
particular, an $A$-$B$-$\mathrm{C}^{*}$-correspondence is a Banach
$A$-$B$-bimodule (where both actions are nondegenerate), so that the
constructions of the previous subsection apply. Any $\mathrm{C}^{*}$-algebra
$A$ can be treated as a trivial $\mathrm{C}^{*}$-correspondence from $A$ to
$A$ where the bimodule structure comes from $A$ and the $A$-valued inner
product is given by the formula $\langle a,b\rangle:=a^{*}b$, $a,b\in A$. Then
$\mathcal{K}(A)\cong A$ and $\mathcal{L}(A)=\mathcal{M}(A)$ is the multiplier
algebra of $A$. For any $A$-$B$-$\mathrm{C}^{*}$-correspondence $E$ the
actions extend uniquely to multiplier algebras so that we may always treat $E$
as a $\mathcal{M}(A)$-$\mathcal{M}(B)$-$\mathrm{C}^{*}$-correspondence. To
relate it to the multiplier $\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodule
$\mathcal{M}(E)$, let us note that whenever $E_{B}$ is a $B$-Hilbert module,
then $\mathcal{L}(E_{B},E)$ is naturally a $\mathrm{C}^{*}$-correspondence
from $\mathcal{M}(A)$ to $\mathcal{L}(E_{B})$ where

$(aT)(x):=aT(x),\qquad(Tb)(x):=Tb(x),\qquad\langle
S,T\rangle_{\mathcal{L}(E_{B})}:=S^{*}T,$

for $S,T\in\mathcal{L}(E_{B},E)$, $a\in M(A)$, $b\in\mathcal{L}(E_{B})$, $x\in
E_{B}$. In particular, $\mathcal{L}(E_{B},E)$ contains $\mathcal{K}(E_{B},E)$
as a sub-$\mathrm{C}^{*}$-correspondence. If $E_{B}=B$ is the trivial
$B$-Hilbert module, then $\mathcal{L}(B,E)$ is a
$\mathrm{C}^{*}$-correspondence from $\mathcal{M}(A)$ to
$\mathcal{M}(B)=\mathcal{L}(B)$ and $\mathcal{K}(B,E)$ is naturally isomorphic
to $E$; the isomorphism is determined by the map $\Theta_{\xi,b}\mapsto\xi
b^{*}$, $\xi\in E,b\in B$. Under the identifications $B=\mathcal{K}(B)$ and
$E=\mathcal{K}(B,E)$ we have $\mathcal{L}(B,E)\cdot B=E$, but in general
$A\cdot\mathcal{L}(B,E)$ is not contained in $E$, unless for instance $E$ is
_proper_ , i.e. $A$ acts on $E$ by compact operators. Hence in general
$\mathcal{L}(B,E)$ is not a multiplier extension of $E$ in the sense of
Definition 1.1. In fact,

$\mathcal{M}_{\mathrm{C}^{*}}(E):=\\{T\in\mathcal{L}(B,E):aT\in\mathcal{K}(B,E)\text{
for all }a\in A\\}$

is the largest sub-bimodule of $\mathcal{L}(B,E)$ which is a multiplier
extension of $E=\mathcal{K}(B,E)$.

######  Lemma 1.7.

For any $A$-$B$-$\mathrm{C}^{*}$-correspondence $E$, the projection
$\mathcal{M}(E)\ni(R,L)\mapsto L\in\mathcal{M}_{\mathrm{C}^{*}}(E)$ is an
isomorphism of Banach $\mathcal{M}(A)$-$\mathcal{M}(B)$-bimodules.

###### Proof.

If $(R,L)\in\mathcal{M}(E)$, then $L$ is adjointable and $L^{*}(a\xi)=\langle
R(a^{*}),\xi\rangle_{B}$, for $a\in A$ and $\xi\in E$, because for any $b\in
B$ we have

$\langle L(b),a\xi\rangle_{B}=\langle a^{*}L(b),\xi\rangle_{B}=\langle
R(a^{*})b,\xi\rangle_{B}=b^{*}\langle R(a^{*}),\xi\rangle_{B}=\langle
b,\langle R(a^{*}),\xi\rangle_{B}\rangle_{B}.$

Thus $R$ is uniquely determined by $L$ and $\|R\|\leq\|L^{*}\|=\|L\|$. Also
for any $a\in A$, writing $R(a)=\xi b^{*}$ for $\xi\in E$ and $b\in B$ we get
that $aL=\Theta_{\xi,b}$. Thus $L\in\mathcal{M}_{\mathrm{C}^{*}}(E)$ and we
see that $\mathcal{M}(E)\ni(R,L)\mapsto L\in\mathcal{M}_{\mathrm{C}^{*}}(E)$
is an isometric embedding of bimodules. To show it is onto let
$L\in\mathcal{M}_{\mathrm{C}^{*}}(E)$. For any $a\in A$,
$aL\in\mathcal{K}(B,E)$ and therefore for any approximate unit
$\\{e_{i}\\}_{i\in I}$ in $B$ the limit $\lim_{i\in I}aL(e_{i})\in E$ exists
(because $aL$ can be approximated in the norm by a finite sum
$\sum_{k=1}^{n}\Theta_{\xi_{k},b_{k}}$ and $\lim_{i\in
I}\sum_{k=1}^{n}\Theta_{\xi_{k},b_{k}}(e_{i})=\sum_{k=1}^{n}\xi_{k}b_{k}^{*}\in
E$). Putting $R(a):=\lim_{i}aL(e_{i})$ one gets a multiplier $(R,L)$. We leave
the remaining straightforward details to the reader. ∎

In the sequel we shall use the above lemma to identify
$\mathcal{M}(E)=\mathcal{M}_{\mathrm{C}^{*}}(E)\subseteq\mathcal{L}(B,E)$. The
advantage is that then we have also the following equality

$\mathcal{M}(E)=\\{T\in\mathcal{L}(B,E):\langle
S,aT\rangle_{\mathcal{M}(B)}\in B\text{ for }a\in
A,\,S\in\mathcal{L}(B,E)\\},$

see for instance [Kat07, Proposition 1.3], which readily implies that
$\mathcal{M}(E)$ is in fact a sub-$\mathrm{C}^{*}$-correspondence of the
$\mathcal{M}(A)$-$\mathcal{M}(B)$-$\mathrm{C}^{*}$-correspondence
$\mathcal{L}(B,E)$.

######  Remark 1.8.

A $\mathrm{C}^{*}$-correspondence multiplier extension of an
$A$-$B$-$\mathrm{C}^{*}$-correspondence $E$ is a
$\mathcal{M}(A)$-$\mathcal{M}(B)$-$\mathrm{C}^{*}$-correspondence $F$ such
that $E=FB$ and

$\langle\xi,a\eta\rangle_{\mathcal{M}(B)}\in B,\qquad\text{ for all }a\in A,\
\xi,\eta\in F,$

becasue the displayed condition is equivalent to $AF=FB$, by [Kat07,
Proposition 1.3]. Any $\mathrm{C}^{*}$-correspondence multiplier extension is
automatically essential.

######  Proposition 1.9.

The multiplier bimodule $\mathcal{M}(E)$ of an
$A$-$B$-$\mathrm{C}^{*}$-correspondence $E$ has a uniquely determined
structure of an
$\mathcal{M}(A)$-$\mathcal{M}(B)$-$\mathrm{C}^{*}$-correspondence extending
the one from $E$. Moreover, $\mathcal{M}(E)$ is the largest among
$\mathrm{C}^{*}$-correspondences which are multiplier extensions of $E$.
Namely, for any $\mathrm{C}^{*}$-correspondence multiplier extension $F$ of
$E$, the unique bimodule map $F\to\mathcal{M}(E)$ that extends the identity on
$E$ is necessarily an isometry (preserves inner products).

###### Proof.

We already observed that $\mathcal{M}(E)$ has the
$\mathrm{C}^{*}$-correspondence structure inherited from $\mathcal{L}(B,E)$.
By Lemma 1.3, the Banach $A$-$B$-bimodule structure of $\mathcal{M}(E)$
uniquely extends the one from $E$, and the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{M}(B)}$ in $\mathcal{M}(E)$ is determined
by the one $\langle\cdot,\cdot\rangle_{B}$ in $E$ because for any approximate
unit $\\{e_{i}\\}_{i\in I}$ in $B$, and any $\xi,\eta\in F$ we have

$\langle\xi,\eta\rangle_{\mathcal{M}(B)}=\text{s-}\\!\lim_{i,j\in
I}e_{j}\langle\xi,\eta\rangle_{\mathcal{M}(B)}e_{i}=\text{s-}\\!\lim_{i,j\in
I}\langle\xi e_{j},\eta e_{i}\rangle_{B},$

where $\text{s-}\\!\lim$ denotes the strict limit. Let $F$ satisfy the
conditions in the second part of the assertion. Then $F$ is an essential
multiplier extension of $E$, and so the identity on $E$ uniquely extends to a
bimodule map $u:F\to\mathcal{M}(E)$ by Lemma 1.3. It necessarily preserves the
inner products as for an approximate unit $\\{e_{i}\\}_{i\in I}$ in $B$, and
$\xi,\eta\in F$, as above we have $\langle
u\xi,u\eta\rangle_{\mathcal{M}(B)}=\text{s-}\\!\lim_{i,j\in I}e_{j}\langle
u(\xi),u(\eta)\rangle_{\mathcal{M}(B)}e_{i}=\text{s-}\\!\lim_{i,j\in I}\langle
u(\xi e_{j}),u(\eta e_{i})\rangle_{B}=\text{s-}\\!\lim_{i,j\in I}\langle\xi
e_{j},\eta e_{i}\rangle_{B}=\langle\xi,\eta\rangle_{\mathcal{M}(B)}$. ∎

######  Remark 1.10.

An $A$-$B$-_Hilbert bimodule_ is an $A$-$B$-$\mathrm{C}^{*}$-correspondence
$E$ which is also a left $A$-Hilbert module and the inner products satisfy
${{}_{A}\langle}\xi,\zeta\rangle\cdot\eta=\xi\cdot\langle\zeta,\eta\rangle_{B}$,
$\xi,\zeta,\eta\in E$. Equivalently, $E$ is an
$A$-$B$-$\mathrm{C}^{*}$-correspondence such that the left action
$\varphi:A\to\mathcal{L}(E)$ restricts to a $*$-isomorphism
$I\cong\mathcal{K}(E)$ where $I$ is an ideal in $A$ (we then necessarily have
$I=(\ker\varphi)^{\bot}$). If $E$ is an $A$-$B$-Hilbert bimodule, then $M(E)$
is $\mathcal{M}(A)$-$\mathcal{M}(B)$-Hilbert bimodule that agrees with the
multiplier Hilbert bimodule constructed in [Sch02], and earlier for
equivalence bimodules in [ER95]. In particular, Proposition 1.9 together with
Corollary 1.4, generalize and improve [Sch02, Proposition 3.6], [ER95,
Proposition 1.2].

######  Example 1.11 (Matrix $\mathrm{C}^{*}$-correspondences).

Consider a $\mathrm{C}^{*}$-$A$-$B$-correspondence $E$ and $n\in\mathbb{N}$.
Then, as described in [Ble97, Section 3], the space

$\mathbb{M}_{n}(E):=\\{[\xi_{ij}]_{i,j=1}^{n}:\xi_{ij}\in E,\
i,j=1,\ldots,n\\}.$

becomes a Hilbert $\mathbb{M}_{n}(B)$-$\mathrm{C}^{*}$-module with the natural
right matrix multiplication and the inner product of
$\xi=[\xi_{ij}]_{i,j=1}^{n}$,
$\zeta=[\zeta_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(E)$ is given by

$\langle\xi,\zeta\rangle_{B}=\left[\sum_{k=1}^{n}\langle\xi_{ki},\zeta_{kj}\rangle_{B}\right]_{i,j=1,\ldots,n}.$

If the left action of $A$ on $E$ is given by $\phi:A\to\mathcal{L}(E)$, then
the left action of $\mathbb{M}_{n}(A)$ on $\mathbb{M}_{n}(E)$ is given by
$\phi^{(n)}:\mathbb{M}_{n}(A)\to\mathbb{M}_{n}(\mathcal{L}(E))\subseteq\mathcal{L}(\mathbb{M}_{n}(E))$,
where the last inclusion is given by
$T=[T_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{L}(E))$ and
$\xi\in\mathbb{M}_{n}(E)$ is as above, then

$T\xi=\left[\sum_{k=1}^{n}T_{ik}(\xi_{kj})\right]_{i,j=1,\ldots,n}.$

Furthermore, we have the expected identification
$\mathcal{M}(\mathbb{M}_{n}(E))\cong\mathbb{M}_{n}(\mathcal{M}(E))$,
compatible with the standard isomorphisms
$\mathcal{M}(\mathbb{M}_{n}(A))\cong\mathbb{M}_{n}(\mathcal{M}(A))$,
$\mathcal{M}(\mathbb{M}_{n}(B))\cong\mathbb{M}_{n}(\mathcal{M}(B))$. This is
easiest to verify via Corollary 1.4. Indeed, if $F$ is a multiplier extension
of the $\mathbb{M}_{n}(A)$-$\mathbb{M}_{n}(B)$-bimodule $\mathbb{M}_{n}(E)$,
the desired map $F\to\mathbb{M}_{n}(\mathcal{M}(E))$ can be constructed from
entrywise embeddings $\psi_{ij}:E\to\mathbb{M}_{n}(E)\subseteq F$, which are
compatible with the ‘diagonal’ actions of $A$ and $B$ — so that via the
universal property of $\mathcal{M}(E)$ one obtains unique maps
$\gamma_{ij}:F\to E$. Naturally combining these yields a bimodule map
$\gamma:F\to\mathbb{M}_{n}(E)$ which is a unique extension of the identity.

Let $E$ and $F$ be $\mathrm{C}^{*}$-correspondences from $A$ to $B$ and from
$B$ to $C$, respectively. The (internal) _tensor product_ $E\otimes_{B}F$ is a
$\mathrm{C}^{*}$-correspondence from $A$ to $C$ constructed as the completion
of the algebraic tensor product of $E$ and $F$ balanced over $B$, with inner
product determined by

$\langle\xi_{1}\otimes\zeta_{1},\xi_{2}\otimes\zeta_{2}\rangle_{C}=\langle\zeta_{1},\langle\xi_{1},\xi_{2}\rangle_{B}\cdot\zeta_{2}\rangle_{C},$

$\xi_{i}\in E$, $\zeta_{i}\in F$, $i=1,2$. This readily implies that we have a
natural $\mathrm{C}^{*}$-correspondence isomorphism
$\mathcal{M}(E)\otimes_{\mathcal{M}(B)}F\cong E\otimes_{B}F$ that sends
$\xi\otimes b\zeta$ to $\xi b\otimes\zeta$ for $\xi\in\mathcal{M}(E)$,
$\zeta\in F$ and $b\in B$. Similarly, we also have the isomorphism
$E\otimes_{\mathcal{M}(B)}\mathcal{M}(F)\cong E\otimes_{B}F$. Therefore we may
and we will assume the identifications

$E\otimes_{B}F=\mathcal{M}(E)\otimes_{\mathcal{M}(B)}F=E\otimes_{\mathcal{M}(B)}\mathcal{M}(F)\subseteq\mathcal{M}(E)\otimes_{\mathcal{M}(B)}\mathcal{M}(F).$

Then $\mathcal{M}(E)\otimes_{\mathcal{M}(B)}\mathcal{M}(F)$ is clearly a
multiplier extension of $E\otimes_{B}F$.

######  Lemma 1.12.

For $\mathrm{C}^{*}$-correspondences $E$ from $A$ to $B$ and $F$ from $B$ to
$C$ we have an embedding of $\mathrm{C}^{*}$-correspondences

$\mathcal{M}(E)\otimes_{\mathcal{M}(B)}\mathcal{M}(F)\rightarrowtail\mathcal{M}(E\otimes_{B}F)$

which is given by the unique bimodule map that extends the identity on
$E\otimes_{B}F$. More specifically, the formula
$(\xi\otimes\zeta)(c):=\xi\otimes\zeta(c)\in\mathcal{M}(E)\otimes_{\mathcal{M}(B)}F=E\otimes_{B}F$
for $c\in C,\xi\in\mathcal{M}(E)$ and $\zeta\in\mathcal{M}(F)$ defines an
operator $\xi\otimes\zeta\in\mathcal{L}(C,E\otimes_{B}F)$ and this gives a map

$\mathcal{M}(E)\times\mathcal{M}(F)\ni(\xi,\zeta)\longmapsto\xi\otimes\zeta\in\mathcal{M}(E\otimes_{B}F)\subseteq\mathcal{L}(C,E\otimes_{B}F)$

that induces the aforementioned embedding.

###### Proof.

The first part follows from Proposition 1.9. For the second part, note that
the operator $\xi\otimes\zeta$, for $\xi\in\mathcal{M}(E)$,
$\zeta\in\mathcal{M}(F)$ is adjointable, where
$(\xi\otimes\zeta)^{*}(\xi_{0}\otimes\zeta_{0})=\zeta^{*}(\langle\xi,\xi_{0}\rangle_{\mathcal{M}(B)}\zeta_{0})$
for $\xi_{0}\in\mathcal{M}(E)$, $\zeta_{0}\in\mathcal{M}(F)$. Hence
$\xi\otimes\zeta\in\mathcal{L}(C,E\otimes_{B}F)$. Clearly, for $b\in B$ we
have $\xi b\otimes\zeta=\xi\otimes b\zeta\in E\otimes_{B}F$. In particular,
for any $a\in A$ we have $a(\xi\otimes\zeta)=(a\xi\otimes\zeta)\in
E\otimes_{\mathcal{M}(B)}\mathcal{M}(F)=E\otimes_{B}F$. Thus the map
$\mathcal{M}(E)\times\mathcal{M}(F)\ni(\xi,\zeta)\longmapsto\xi\otimes\zeta\in\mathcal{M}(E\otimes_{B}F)$
is well defined. Since it is bilinear, bimodule and $B$-balanced, it descends
to a bimodule map on the algebraic $B$-balanced tensor product of
$\mathcal{M}(E)$ and $\mathcal{M}(F)$, which in turn extends to the embedding
$\mathcal{M}(E)\otimes_{\mathcal{M}(B)}\mathcal{M}(F)\rightarrowtail\mathcal{M}(E\otimes_{B}F)$
by the first part of the assertion. ∎

######  Remark 1.13.

The above embedding is, in general, not surjective. For instance, let $A$ be a
non-unital $\mathrm{C}^{*}$-algebra, and let $E=A$ and $F=A$ be viewed as
$\mathrm{C}^{*}$-correspondences from $A$ to $\mathcal{M}(A)$ and from
$\mathcal{M}(A)$ to $A$, respectively, so that $M(E)=M(F)=A$. Then
$\mathcal{M}(E)\otimes_{\mathcal{M}(A)}\mathcal{M}(F)=E\otimes_{\mathcal{M}(A)}F=A\subsetneq\mathcal{M}(A)=\mathcal{M}(E\otimes_{\mathcal{M}(A)}F)$
as $\mathrm{C}^{*}$-correspondences from $\mathcal{M}(A)$ to $\mathcal{M}(A)$.

##  2\. Dilation/Decomposition theorem

Let $A,B$ be $\mathrm{C}^{*}$-algebras. In the sequel we will study maps
$T:A\to B$ that are of the form
$T(a)=\langle\xi,a\zeta\rangle_{\mathcal{M}(B)}=\xi^{*}a\zeta$ for
$\xi,\zeta\in\mathcal{M}(E)\subseteq\mathcal{L}(B,E)$, with $E$ a
$\mathrm{C}^{*}$-correspondence from $A$ to $B$. To this end, we develop in
this section the following common generalization of Kasparov’s Stinespring
dilation theorem (called the KSGNS construction in [Lan95, Theorem 5.6]) and
Murphy’s Kolmogorov decomposition of positive kernels for Hilbert modules
[Mur97, Theorem 2.3].

######  Theorem 2.1.

Let $k:S\times S\to\mathbb{B}(A,\mathcal{L}(E_{B}))$ be a map where $A$ and
$B$ are $\mathrm{C}^{*}$-algebras, $E_{B}$ is a right $B$-Hilbert module, and
$S$ is a set. The following are equivalent:

  1. (1)

there exists an $A$-$B$-$\mathrm{C}^{*}$-correspondence $E$ and a map
$\zeta:S\to\mathcal{L}(E_{B},E)$ such that

$k(s,t)(a)=\zeta(s)^{*}\varphi_{E}(a)\zeta(t),\qquad s,t\in S,\ a\in A,$

where $\varphi_{E}:A\to\mathcal{L}(E)$ is the homomorphism defining the left
action of $A$ on $E$.

  2. (2)

$k$ is a _positive-definite kernel_ in the sense that for all
$n\in\mathbb{N}$, $s_{1},\ldots,s_{n}\in S$, $a_{1},\ldots,a_{n}\in A$ and
$\xi_{1},\ldots,\xi_{n}\in E_{B}$, we have

$\sum_{i,j=1}^{n}\langle\xi_{i},k(s_{i},s_{j})(a_{i}^{*}a_{j})\xi_{j}\rangle_{B}\geq
0,$

and in addition $k$ is _strict_ in the sense that for each $s\in S$ the map
$k(s,s):A\to\mathcal{L}(E_{B})$ is strict, i.e. for an approximate unit
$\\{e_{i}\\}_{i\in I}$ in $A$, the net $\\{k(s,s)(e_{i})\\}_{i\in I}$
converges strictly in $\mathcal{L}(E_{B})$.

If the above conditions hold, the $\mathrm{C}^{*}$-correspondence $E$ in (1)
may be chosen so that $\\{a\zeta(s)\xi:a\in A,s\in S,\xi\in E_{B}\\}$ is
linearly dense in $E$; we call such pairs $(\zeta,E)$ _minimal_. Given another
minimal pair $(\zeta^{\prime},E^{\prime})$ as above we have a uniquely
determined unitary isomorphism of $\mathrm{C}^{*}$-correspondences $U:E\to
E^{\prime}$ such that $U\zeta(s)=\zeta^{\prime}(s)$, $s\in S$.

###### Proof.

Assume (1). Using the notation in (2) we have

$\displaystyle\sum_{i,j=1}^{n}\langle\xi_{i},k(s_{i},s_{j})(a_{i}^{*}a_{j})\xi_{j}\rangle_{B}$
$\displaystyle=\langle\sum_{i=1}^{n}a_{i}\zeta(s_{i})\xi_{i},\sum_{i=1}^{n}a_{i}\zeta(s_{i})\xi_{i}\rangle_{B}\geq
0,$

so $k$ is positive-definite. Also for any $\xi\in E_{B}$, $s\in S$ and an
approximate unit $(e_{i})_{i\in I}$ of $A$, the net
$k(s,s)(e_{i})\xi=\zeta(s)^{*}e_{i}\zeta(s)\xi$ converges to
$\zeta(s)^{*}\zeta(s)\xi$. Hence $k$ is strict.

Now assume (2). Begin by defining a new positive-definite kernel, now in the
usual sense of [Mur97], $\tilde{k}:(S\times A)\times(S\times
A)\to\mathcal{L}(E_{B})$, as follows:

$\tilde{k}((s,a),(t,b)):=k(s,t)(a^{*}b),\qquad s,t\in S,\ a,b\in A.$

By [Mur97, Theorem 2.3] there exists a Hilbert $B$-module $E$ and a map
$\tilde{\zeta}:S\times A\to\mathcal{L}(E_{B},E)$ (with
$\|\tilde{\zeta}(s,a)\|\leq\|k(s,s)\|^{1/2}\|a\|$ for all $s\in S,a\in A$)
such that for each $s,t\in S,a,b\in A$ we have

(2.1)
$k(s,t)(a^{*}b)=\tilde{k}((s,a),(t,b))=\tilde{\zeta}(s,a)^{*}\tilde{\zeta}(t,b).$

Moreover, we may and we do assume that $\tilde{\zeta}$ is minimal, i.e. the
set $\bigcup_{s\in S,a\in A}\tilde{\zeta}(s,a)E_{B}$ is linearly dense in $E$.
We claim that there is a natural $*$-homomorphism
$\varphi_{E}:\mathcal{M}(A)\to\mathcal{L}(E)$ determined by the formula

$\varphi_{E}(u)\tilde{\zeta}(s,a)\xi:=\tilde{\zeta}(s,ua)\xi,\qquad
u\in\mathcal{M}(A),\ s\in S,\ a\in A,\ \xi\in E_{B}.$

To see that $\varphi_{E}(u)$ exists, by linearity and the Russo–Dye theorem
([RD66]) it suffices to consider the case when $u\in\mathcal{M}(A)$ is
unitary. Let then $u\in\mathcal{UM}(A)$. It is then easy to see that the map
$\tilde{\zeta}_{u}:S\times A\to\mathcal{L}(E_{B},E)$ given by

$\tilde{\zeta}_{u}(s,a)=\tilde{\zeta}(s,ua),\qquad s\in S,\ a\in A,$

yields another minimal Kolmogorov dilation of the kernel $\tilde{k}$. Hence,
by [Mur97, Theorem 2.3], there exists a unitary
$\varphi_{E}(u)\in\mathcal{L}(E)$ such that
$\varphi_{E}(u)(\tilde{\zeta}(s,a)=\tilde{\zeta}(s,ua)$ for all $s\in S$ and
$a\in A$. It is then routine to check that the prescription
$u\mapsto\varphi_{E}(u)$ extends to a unital ∗-homomorphism from
$\mathcal{M}(A)$ to $\mathcal{L}(E)$. Restriction of $\varphi_{E}$ to $A$
gives $E$ the structure of a $\mathrm{C}^{*}$-correspondence from $A$ to $B$.
Indeed, as $k$ is bounded-operator valued, we have for any approximate unit
$\\{e_{i}\\}_{i\in I}$ in $A$, $s,t\in S$ and $a,b\in A$ that
$(k(s,t)(be_{i}a))_{i\in I}$ converges (pointwise) to $k(s,t)(ba)$. This it
suffices to verify that $\varphi_{E}|_{A}:A\to\mathcal{L}(E)$ is
nondegenerate.

Fix an approximate unit $\\{e_{i}\\}_{i\in I}$ in $A$ and $s\in S$. For any
$\xi\in E_{B}$ and $i,j\in I,i\geq j$, using (2.1), we get (noting that
$k(s,s)$ is positive)

$\|\tilde{\zeta}(s,e_{i})\xi-\tilde{\zeta}(s,e_{j})\xi\|^{2}=\|\langle\xi,k(s,s)\big{(}(e_{i}-e_{j})^{2}\big{)}\xi\rangle_{B}\|\leq\|\langle\xi,k(s,s)(e_{i}-e_{j})\xi\rangle_{B}\|.$

Since $\\{k(s,s)(e_{i})\\}_{i\in I}$ is strictly Cauchy it follows that
$\\{\tilde{\zeta}(s,e_{i})\\}_{i\in I}$ is strictly Cauchy and hence it is
convergent to an operator $\zeta(s)\in\mathcal{L}(E_{B},E)$. In this way we
get a map $\zeta:S\to\mathcal{L}(E_{B},E)$, and for any $a\in A$

$k(s,t)(a)=\lim_{i,j\in I}k(s,t)(e_{i}ae_{j})=\lim_{i,j\in
I}\tilde{\zeta}(s,e_{i})^{*}\varphi_{E}(a)\tilde{\zeta}(t,e_{j})=\zeta(s)^{*}\varphi_{E}(a)\zeta(t).$

This proves (1). Once again using the convergence of
$\\{k(s,t)(ae_{i}b)\\}_{i\in I}$ to $k(s,t)(ab)$ we deduce that
$a\zeta(s)\xi=\tilde{\zeta}(s,a)\xi$ and hence the pair $(\zeta,E)$ is minimal
in the sense described in the last part of the assertion.

Suppose now that we are given another minimal pair
$(\zeta^{\prime},E^{\prime})$ that dilates $k$ as in (1). Putting
$\tilde{\zeta^{\prime}}(s,a):=a\cdot\zeta^{\prime}(s)$ for all $s\in S$, $a\in
A$, we get a map $\tilde{\zeta^{\prime}}:S\times A\to E^{\prime}$ which is a
minimal Kolmogorov decomposition for the kernel $\tilde{k}$. So once again we
have a unique unitary $U\in\mathcal{L}(E;E^{\prime})$ such that
$U(\tilde{\zeta}(s,a))=\tilde{\zeta^{\prime}}(s,a)$ for all $s\in S$, $a\in
A$. It is routine to check that such a $U$ necessarily intertwines the
respective left actions of $A$ on $E$ and $E^{\prime}$. ∎

######  Remark 2.2.

When $S$ is a singleton, Theorem 2.1 reduces to the KSGNS dilation of strict
completely positive maps $\varrho:A\to\mathcal{L}(E_{B})$, [Lan95, Theorem
5.6]. When $A=\mathbb{C}$ is trivial, then under the identification
$\mathbb{B}(\mathbb{C},\mathcal{L}(E_{B}))\cong\mathcal{L}(E_{B})$, via the
map $\mathbb{B}(\mathbb{C},\mathcal{L}(E_{B}))\ni T\mapsto
T(1)\in\mathcal{L}(E_{B})$, Theorem 2.1 reduces to Murphy’s Kolmogorov
decomposition, [Mur97, Theorem 2.3] (which was used in the proof above).

######  Remark 2.3.

If $k(s,s)\in\mathbb{B}(A,\mathcal{K}(E_{B}))$ for all $s\in S$, then for any
(minimal) decomposition $\zeta:S\to\mathcal{L}(E_{B},E)$ as in Theorem 2.1 and
any $a\in A$ we have $\varphi_{E}(a)\zeta:S\to\mathcal{K}(E_{B},E)$. This
follows for instance from [Kat07, Proposition 1.3] applied to each
$\varphi_{E}(a)\zeta(s)$. In particular, if the kernel in Murphy’s theorem
takes values in compact operators $\mathcal{K}(E_{B})$, then its Kolmogorov
decomposition is also compact valued.

We will need yet another version of dilation that follows from Theorem 2.1.

######  Corollary 2.4.

Let $k:S\times S\to\mathbb{B}(A,B)$ be a map where $A$ and $B$ are
$\mathrm{C}^{*}$-algebras and $S$ is a set. The following are equivalent:

  1. (1)

there exists a $\mathrm{C}^{*}$-correspondence $E$ from $A$ to $B$ and a map
$\zeta:S\to\mathcal{M}(E)$ such that

$k(s,t)(a)=\langle\zeta(s),a\cdot\zeta(t)\rangle_{\mathcal{M}(B)},\qquad
s,t\in S,\ a\in A;$

  2. (2)

$k$ is a _positive-definite kernel_ in the sense that for all
$n\in\mathbb{N}$, $s_{1},\ldots,s_{n}\in S$ and $a_{1},\ldots,a_{n}\in A$ the
matrix

$[k(s_{i},s_{j})(a_{i}^{*}a_{j})]_{i,j=1}^{n}\in\mathbb{M}_{n}(B)$

is positive, and $k$ is _strict_ in the sense that for each $s\in S$ the map
$k(s,s):A\to B$ is strict, i.e. for an approximate unit $\\{e_{i}\\}_{i\in I}$
in $A$, the net $\\{k(s,s)(e_{i})\\}_{i\in I}$ converges strictly in
$\mathcal{M}(B)$.

Moreover if the conditions above hold, then the
$\mathrm{C}^{*}$-correspondence $E$ in (1) may be chosen so that the set
$\\{a\zeta(s)b:s\in S,a\in A,b\in B\\}$ is dense in $E$; we call such pairs
$(\zeta,E)$ _minimal_. Given another minimal pair
$(\zeta^{\prime},E^{\prime})$ as above, we have a uniquely determined unitary
isomorphism of $\mathrm{C}^{*}$-correspondences $U:E\to E^{\prime}$ such that
$U\zeta(s)=\zeta^{\prime}(s)$, $s\in S$.

###### Proof.

Let $E_{B}:=B$ be the trivial Hilbert module over $B$, so that
$B\cong\mathcal{K}(E_{B})\subseteq\mathcal{L}(E_{B})$. Then clearly (2) in
Theorem 2.1 and (2) in Corollary 2.4 are equivalent. Also by Remark 2.3, one
readily sees that (1) in Theorem 2.1 is equivalent to (1) in Corollary 2.4. ∎

##  3\. Bundles of C*-correspondences

Throughout this paper $X$ will stand for a locally compact Hausdorff space. In
this section we introduce bundles of $\mathrm{C}^{*}$-correspondences between
two $\mathrm{C}^{*}$-bundles over $X$ and present several relevant
constructions.

A _Banach bundle_ $\mathcal{E}$ over $X$ is a topological space with an open
continuous surjection $p:\mathcal{E}\to X$ such that each fiber
$E_{x}=p^{-1}(x)$ is a Banach space whose topology agrees with that of
$\mathcal{E}$, the linear operations are continuous, and the norm
$\mathcal{E}\ni a\mapsto\|a\|\in[0,\infty)$ is _upper semicontinuous_ and
nondegenerate, see, for instance, [BE12, Definition 2.1] for a detailed
definition. If the norm is continuous then we say $\mathcal{E}$ is a
_continuous Banach bundle_. We will often write $\mathcal{E}=\\{E_{x}\\}_{x\in
X}$ for a Banach bundle. We denote by $\mathrm{C_{c}}(\mathcal{E})$,
$\mathrm{C_{0}}(\mathcal{E})$, $\mathrm{C_{b}}(\mathcal{E})$ the spaces of
continuous sections that are compactly supported, vanishing at infinity, and
bounded, respectively. By _Fell’s reconstruction theorem_ , any of these
spaces is enough to reconstruct the topology of $\mathcal{E}$. More
specifically (see [FD88, Theorem II.13.18] or [BE12, Proposition 2.4]), for
any family of Banach spaces $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ and any linear
space $\Gamma$ of sections of $\mathcal{E}$ such that
$\overline{\\{\xi(x):\xi\in\Gamma\\}}=E_{x}$ for each $x\in X$ and $X\ni
x\mapsto\|\xi(x)\|\in[0,\infty)$ is upper semicontinuous for each
$\xi\in\Gamma$, there exists a unique topology on $\mathcal{E}$ turning it
into a Banach bundle with $\Gamma\subseteq C(\mathcal{E})$. A net
$\\{f_{i}\\}_{i\in I}$ of elements of $\mathcal{E}$ converges in this topology
to $f\in\mathcal{E}$ if and only if $\lim_{i\in I}p(f_{i})=p(f)$ and for every
$\varepsilon>0$ there is $\xi\in\Gamma$ such that
$\|f-\xi(p(f))\|<\varepsilon$ and $\|f_{i}-\xi(p(f_{i}))\|<\varepsilon$ for
large enough $i\in I$.

It is well known that (upper semicontinuous) bundles of
$\mathrm{C}^{*}$-algebras $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ are equivalent to
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-algebras, see, for instance, [Wil07,
Appendix C]. More specifically, a $\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-algebra
is a $\mathrm{C}^{*}$-algebra $A$ equipped with a nondegenerate homomorphism
$\mathrm{C_{0}}(X)\to Z(\mathcal{M}(A))$ turning $A$ into a
$\mathrm{C_{0}}(X)$-module. Given a bundle of $\mathrm{C}^{*}$-algebras
$\mathcal{A}$ we write $A=\mathrm{C_{0}}(\mathcal{A})$ for its
$\mathrm{C}^{*}$-algebra of $\mathrm{C_{0}}$-sections, equipped with the
$\mathrm{C_{0}}(X)$-module structure given by

$(fa)(x)=f(x)a(x),\qquad f\in\mathrm{C_{0}}(X),\qquad
a\in\mathrm{C_{0}}(\mathcal{A}).$

We now briefly explain how this equivalence extends to
$\mathrm{C}^{*}$-correspondences. We begin by introducing a bundle version of
Le Gall’s notion of a $\mathrm{C_{0}}(X)$-representation [LG99, Definition
4.1], which was renamed a _$\mathrm{C_{0}}(X)$
-$\mathrm{C}^{*}$-correspondence_ by Deaconu in [Dea18, Definition 5.8].

######  Definition 3.1.

A _$\mathrm{C}^{*}$ -correspondence bundle_ from a $\mathrm{C}^{*}$-bundle
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to a $\mathrm{C}^{*}$-bundle
$\mathcal{B}=\\{B_{x}\\}_{x\in X}$ is a Banach bundle
$\mathcal{E}=\\{E_{x}\\}_{x\in X}$, where each $E_{x}$ is a
$\mathrm{C}^{*}$-correspondence from $A_{x}$ to $B_{x}$, and the sections

$X\ni x\mapsto\langle\zeta(x),\xi(x)\rangle_{E_{x}}\in\mathcal{B},\qquad X\ni
x\mapsto a(x)\xi(x)\in\mathcal{E},\qquad X\ni
x\mapsto\xi(x)b(x)\in\mathcal{E}$

are continuous for every $\zeta,\xi\in\mathrm{C_{c}}(\mathcal{E})$,
$a\in\mathrm{C_{c}}(\mathcal{A})$, $b\in\mathrm{C_{c}}(\mathcal{B})$.

######  Proposition 3.2.

For any $\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}=\\{E_{x}\\}_{x\in
X}$ from $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to $\mathcal{B}=\\{B_{x}\\}_{x\in
X}$ the space $E:=\mathrm{C_{0}}(\mathcal{E})$ is a
$\mathrm{C}^{*}$-correspondence from $A:=\mathrm{C_{0}}(\mathcal{A})$ to
$B:=\mathrm{C_{0}}(\mathcal{B})$ with the additional property that

(3.1) $(fa)\cdot\xi\cdot b=a\cdot\xi\cdot(fb)\quad\text{for all
$f\in\mathrm{C_{0}}(X),\ a\in A,\ b\in B,\ \xi\in E$,}$

i.e. $E$ is a $\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence from $A$ to
$B$ in the sense of [Dea18]. Moreover, every
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence from $A$ to $B$ is of the
above form for some $\mathrm{C}^{*}$-correspondence bundle
$\mathcal{E}=\\{E_{x}\\}_{x\in X}$.

###### Proof.

The first part of the assertion is straightforward. Let $E$ be any
$\mathrm{C}^{*}$-correspondence from $A$ to $B$ satisfying (3.1). Recall that
the fibers of the corresponding bundles $\mathcal{A}$ and $\mathcal{B}$ are
given by $A_{x}:=A/I_{x}A$ and $B_{x}:=B/I_{x}B$ where
$I_{x}:=\\{f\in\mathrm{C_{0}}(X):f(x)=0\\}$, $x\in X$. By (3.1) for each $x\in
X$, we have $(I_{x}A)E\subseteq E(I_{x}B)$. Thus the quotient
$E_{x}:=E/(EI_{x}B)$ is naturally a $\mathrm{C}^{*}$-correspondence from
$A_{x}=A/I_{x}$ to $B_{x}=B/I_{x}$, cf. [FMR03, Lemma 2.3] (note that $E_{x}$
is canonically isomorphic to the
$A_{x}$-$B_{x}$-$\mathrm{C}^{*}$-correspondence $E\otimes_{B}B_{x}$ considered
in [LG99, Dea18]). Denoting by $\xi(x)$ the image of $\xi$ in the quotient
$E_{x}=E/(EI_{x}B)$, we get
$\|\xi(x)\|^{2}=\|\langle\xi(x),\xi(x)\rangle_{B_{x}}\|=\|\langle\xi,\xi\rangle(x)\|$,
where $\langle\xi,\xi\rangle$ is in the $\mathrm{C_{0}}(X)$-algebra $B$. Hence
the map $X\ni x\mapsto\|\xi(x)\|$ is upper semicontinuous, vanishes at
infinity and its maximum is $\|\xi\|$. Thus by Fell’s reconstruction theorem,
there is a unique topology on $\mathcal{E}:=\\{E_{x}\\}_{x\in X}$ such that
sections $X\ni x\mapsto\xi(x)\in E_{x}$ for $\xi\in E$ are continuous. The
proof of [Wil07, Proposition C.24] can be readily adapted to show that the
image of $E$ in $\mathrm{C_{0}}(\mathcal{E})$ is dense. Since the embedding
$E\hookrightarrow\mathrm{C_{0}}(\mathcal{E})$ is isometric, and $E$ is
complete, we conclude it must be an isomorphism. ∎

######  Remark 3.3.

For any $\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}=\\{E_{x}\\}_{x\in
X}$ from $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to $\mathcal{B}=\\{B_{x}\\}_{x\in
X}$ the space $\mathrm{C_{b}}(\mathcal{E})$ is naturally a
$\mathrm{C}^{*}$-correspondence from $\mathrm{C_{b}}(\mathcal{A})$ to
$\mathrm{C_{b}}(\mathcal{B})$, and so $\mathrm{C_{b}}(\mathcal{E})$ is also a
$\mathrm{C_{b}}(X)$-$\mathrm{C}^{*}$-correspondence. Moreover,
$\mathrm{C_{0}}(\mathcal{E})$ can also be treated as a
$\mathrm{C}^{*}$-correspondence from $\mathrm{C_{b}}(\mathcal{A})$ to
$\mathrm{C_{b}}(\mathcal{B})$, a $\mathrm{C}^{*}$-subcorrespondence of
$\mathrm{C_{b}}(\mathcal{E})$.

######  Example 3.4.

Every $\mathrm{C}^{*}$-bundle $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ can be viewed
as a (trivial) $\mathrm{C}^{*}$-correspondence bundle over $\mathcal{A}$, by
viewing each fiber $A_{x}$ as the trivial $\mathrm{C}^{*}$-correspondence.
Proposition 3.2 applied to such $\mathrm{C}^{*}$-correspondence bundles
recovers the correspondence between $\mathrm{C_{0}}(X)$-algebras and
$\mathrm{C}^{*}$-bundles, extending the arguments used to establish the latter
fact.

######  Example 3.5.

If $\mathcal{A}=\\{\mathbb{C}\\}_{x\in X}$ is the trivial one-dimensional
$\mathrm{C}^{*}$-bundle, so that
$\mathrm{C_{0}}(\mathcal{A})=\mathrm{C_{0}}(X)$, then a
$\mathrm{C}^{*}$-correspondence bundle from $\mathcal{A}$ to a
$\mathrm{C}^{*}$-bundle $\mathcal{B}=\\{B_{x}\\}_{x\in X}$ (with the
associated $\mathrm{C}^{*}$-algebra $B$), is simply a _Hilbert module bundle_
over $\mathcal{B}=\\{B_{x}\\}_{x\in X}$, see [Kum98, 1.7]. In view of
Proposition 3.2 such Hilbert module bundles correspond to _$\mathrm{C_{0}}(X)$
-Hilbert modules_ $E$ over $B$, which by definition are right Hilbert modules
$E$ over $B$ which are also nondegenerate left $\mathrm{C_{0}}(X)$-modules
such that $f\cdot\xi\cdot b=\xi\cdot(fb)$, $f\in\mathrm{C_{0}}(X)$, $b\in B$,
$\xi\in E$. If, in addition, $\mathcal{B}=\\{\mathbb{C}\\}_{x\in X}$ is a
trivial bundle, then all Hilbert modules over $B=C(X)$ are
$\mathrm{C_{0}}(X)$-Hilbert modules and they correspond to _continuous bundles
of Hilbert spaces_ $\mathcal{H}=\\{H_{x}\\}_{x\in X}$, see [FD88, Definition
II.13.5] or [Ren97].

######  Example 3.6.

Let $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ be a $\mathrm{C}^{*}$-bundle and let
$n\in\mathbb{N}$. The $\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-algebra picture
yields immediately that
$\mathbb{M}_{n}(\mathcal{A})=\\{\mathbb{M}_{n}(A_{x})\\}_{x\in X}$ is again a
$\mathrm{C}^{*}$-bundle; and if $A$ is the
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-algebra associated to $\mathcal{A}$, then
$\mathbb{M}_{n}(A)$ is the one associated to $\mathbb{M}_{n}(\mathcal{A})$.
Next if $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ is a
$\mathrm{C}^{*}$-correspondence bundle from $\mathcal{A}$ to $\mathcal{B}$ as
in Definition 3.1, then
$\mathbb{M}_{n}(\mathcal{E})=\\{\mathbb{M}_{n}(E_{x})\\}_{x\in X}$, with
$\mathbb{M}_{n}(E_{x})$ constructed as in Example 1.11 and the natural
topology on $\mathbb{M}_{n}(\mathcal{E})$, becomes a
$\mathrm{C}^{*}$-correspondence bundle from $\mathbb{M}_{n}(\mathcal{A})$ to
$\mathbb{M}_{n}(\mathcal{B})$.

######  Definition 3.7.

Let $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ be a $\mathrm{C}^{*}$-correspondence
bundle from $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to
$\mathcal{B}=\\{B_{x}\\}_{x\in X}$. We say that a section $\xi$ of the bundle
$\mathcal{M}(\mathcal{E}):=\\{\mathcal{M}(E_{x})\\}_{x\in X}$ of multiplier
$\mathrm{C}^{*}$-correspondences is _strictly continuous_ if
$a\cdot\xi,\xi\cdot b\in\mathrm{C_{0}}(\mathcal{E})$ for any
$a\in\mathrm{C_{0}}(\mathcal{A})$, $b\in\mathrm{C_{0}}(\mathcal{B})$. We
denote by $\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ the set of all bounded
strictly continuous sections of $\mathcal{M}(\mathcal{E})$.

For a $\mathrm{C}^{*}$-bundle $\mathcal{A}=\\{A_{x}\\}_{x\in X}$, treated as a
trivial correspondence bundle from $\mathcal{A}$ to $\mathcal{A}$, the above
definition of the multiplier bundle
$\mathcal{M}(\mathcal{A})=\\{\mathcal{M}(A_{x})\\}_{x\in X}$ is consistent
with the one given in [APT73, Section 3]. In particular, the following
proposition generalizes [APT73, Theorem 3.3] from $\mathrm{C}^{*}$-bundles to
$\mathrm{C}^{*}$-correspondence bundles.

######  Proposition 3.8.

Let $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ be a $\mathrm{C}^{*}$-correspondence
bundle from $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to
$\mathcal{B}=\\{B_{x}\\}_{x\in X}$. Equipped with pointwise operations, the
section algebras $\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$,
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))$ are $\mathrm{C}^{*}$-algebras and
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ is a
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$-$\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))$-$\mathrm{C}^{*}$-correspondence
isomorphic to the
$\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$-$\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}))$-$\mathrm{C}^{*}$-correspondence
$\mathcal{M}(\mathrm{C_{0}}(\mathcal{E}))$ where the isomorphism

$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{E}))$

is the unique morphism that extends identities on
$\mathrm{C_{0}}(\mathcal{E})$, $\mathrm{C_{0}}(\mathcal{A})$ and
$\mathrm{C_{0}}(\mathcal{B})$. In particular,
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$
and
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}))$
as $\mathrm{C}^{*}$-algebras.

###### Proof.

It is immediate that bounded strictly continuous sections of
$\mathcal{M}(\mathcal{E})$ form a Banach
$\mathrm{C_{0}}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{B})$-bimodule with
pointwise operations and norm $\|\xi\|:=\sup\|\xi(x)\|$. When this
construction is applied to the $\mathrm{C}^{*}$-bundle $\mathcal{A}$, we also
see that $\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$ is closed under pointwise
multiplication, involution, and contains the unit section. So
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$ is naturally a
$\mathrm{C}^{*}$-algebra that contains $\mathrm{C_{0}}(\mathcal{A})$ as a
$\mathrm{C}^{*}$-subalgebra, in fact, an ideal. We have a unital
$*$-homomorphism
$\pi_{A}:\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))\to\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$
where $[\pi_{A}(b)a](x)=b(x)\cdot a(x)$,
$b\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$,
$a\in\mathrm{C_{0}}(\mathcal{A})$, $x\in X$. It is isometric and extends the
identity on $\mathrm{C_{0}}(\mathcal{A})$. To show it is surjective, let $x\in
X$ and note that the evaluation map $\pi_{A,x}:\mathrm{C_{0}}(\mathcal{A})\to
A_{x}$, $\pi_{A,x}(a)=a(x)$, extends (uniquely) to a $*$-homomorphism
$\overline{\pi}_{A,x}:\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))\to\mathcal{M}(A_{x})$.
For $a\in\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$ the formula
$\widehat{a}(x)=\overline{\pi}_{A,x}(a)$ defines
$\widehat{a}\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$ with
$\pi_{A}(\widehat{a})=a$. Hence we have $*$-isomorphisms

$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{A})),\qquad\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}))$

which are in fact given by unique homomorphisms extending the identities on
$\mathrm{C_{0}}(\mathcal{A})$ and $\mathrm{C_{0}}(\mathcal{B})$, respectively.
A similar argument works for the
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$-$\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))$-bimodule
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$. Namely, the map
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))\ni\xi\mapsto(R_{\xi},L_{\xi})\in\mathcal{M}(\mathrm{C_{0}}(\mathcal{E}))$
where $R_{\xi}(a):=a\xi$ and $L_{\xi}(b):=\xi b$, is a linear isometry
$\pi:\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))\to\mathcal{M}(\mathrm{C_{0}}(\mathcal{E}))$
that extends the identity on $\mathrm{C_{0}}(\mathcal{E})$. To see that $\pi$
is surjective, note that, by Lemma 1.6, for any $x\in X$, the map
$\pi_{x}:\mathrm{C_{0}}(\mathcal{E})\to E_{x}$, $\pi_{x}(\xi)=\xi(x)$, extends
(uniquely) to a linear map
$\overline{\pi}_{x}:\mathcal{M}(\mathrm{C_{0}}(\mathcal{E}))\to\mathcal{M}(E_{x})$
such that $\overline{\pi}_{x}(a\xi
b)=\overline{\pi}_{A,x}(a)(x)\overline{\pi}_{x}(\xi)\overline{\pi}_{B,x}$ for
$a\in\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$,
$b\in\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}))$. For $\xi\in
M(\mathrm{C_{0}}(\mathcal{E}))$ the formula
$\widehat{\xi}(x)=\overline{\pi}_{x}(\xi)$ defines
$\widehat{\xi}\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$ with
$\pi(\widehat{\xi}):=\xi$. Hence $(\pi,\pi_{A},\pi_{B})$ is an isometric
isomorphism from the
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$-$\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))$-bimodule
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ to the
$\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$-$\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}))$-bimodule
$\mathcal{M}(\mathcal{E})$. It is the unique morphism that extends the
identity morphism on $\mathrm{C_{0}}(\mathcal{E})$. Its inverse maps the inner
product in $\mathcal{M}(\mathcal{E})$ to the inner product in
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ defined pointwise. ∎

######  Remark 3.9.

For a $\mathrm{C}^{*}$-bundle $\mathcal{A}=\\{A_{x}\\}_{x\in X}$, the
multiplier bundle $\mathcal{M}(\mathcal{A})=\\{\mathcal{M}(A_{x})\\}_{x\in X}$
is usually not a Banach bundle with the ‘strict topology’. It may also happen
that $\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))\neq\mathrm{C_{b}}(\mathcal{A})$
even when $\mathcal{M}(\mathcal{A})=\mathcal{A}$ as sets. For instance, let
$\mathcal{A}$ be the bundle corresponding to the $C([0,1])$-algebra

$A=\left\\{a\in
C([0,1],\mathbb{M}_{2}(\mathbb{C})):a(0)=\left(\begin{array}[]{cc}\lambda&0\\\
0&0\end{array}\right),\ \lambda\in\mathbb{C}\right\\}.$

Then $\mathcal{A}=\mathcal{M}(\mathcal{A})$, because the fibers $A_{x}$,
$x\in[0,1]$ are unital. But $\mathrm{C_{b}}(\mathcal{A})=A$ is smaller than

$\mathcal{M}(A)=\left\\{a\in
C([0,1],\mathbb{M}_{2}(\mathbb{C})):a(0)=\left(\begin{array}[]{cc}\lambda&0\\\
0&\mu\end{array}\right),\
\lambda,\mu\in\mathbb{C}\right\\}=\mathrm{C_{b}}(\mathcal{M}(A)).$

In general, the strictly continuous sections of $\mathcal{M}(\mathcal{A})$
coincide with continuous sections of $\mathcal{A}$ if and only if the bundle
$\mathcal{A}$ has a _continuous unital section_ , i.e. each fiber $A_{x}$,
$x\in X$ is unital and the unit section $X\ni x\mapsto 1_{x}\in
A_{x}\subseteq\mathcal{A}$ is continuous. This latter condition was made a
standing assumption in [KLS22].

The tensor product of $\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondences is
a $\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence, cf. [LG99, Proposition
4.1]. In terms of bundles this corresponds to the following construction.

######  Example 3.10 (Tensor product of $\mathrm{C}^{*}$-correspondence
bundles).

If $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ and $\mathcal{F}=\\{F_{x}\\}_{x\in X}$
are $\mathrm{C}^{*}$-correspondence bundles from
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to $\mathcal{B}=\\{B_{x}\\}_{x\in X}$ and
from $\mathcal{B}=\\{B_{x}\\}_{x\in X}$ to $\mathcal{C}=\\{C_{x}\\}_{x\in X}$
respectively, then the collection of $\mathrm{C}^{*}$-correspondence tensor
products
$\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F}:=\\{E_{x}\otimes_{B_{x}}F_{x}\\}_{x\in
X}$ is naturally a $\mathrm{C}^{*}$-correspondence bundle from $\mathcal{A}$
to $\mathcal{C}$. Indeed, for every $\xi_{i}\in\mathrm{C_{c}}(\mathcal{E})$,
$\zeta_{i}\in\mathrm{C_{c}}(\mathcal{F})$, $i=1,\ldots,n$, consider the
section
$x\mapsto(\sum_{i=1}^{n}\xi_{i}\otimes\zeta_{i})(x):=\sum_{i=1}^{n}\xi_{i}(x)\otimes\zeta_{i}(x)\in\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F}$.
It follows from Definition 3.1 that the map
$x\mapsto\|\sum_{i=1}^{n}\xi_{i}\otimes\zeta_{i}(x)\|^{2}=\|\sum_{i,j=1}^{n}\langle\zeta_{i}(x),\langle\xi_{i}(x),\xi_{j}(x)\rangle_{B_{x}}\zeta_{j}(x)\rangle_{C_{x}}\|$
is upper semicontinuous. Hence $\mathcal{E}\otimes\mathcal{F}$ can be equipped
with a unique topology such that the sections $\xi\otimes\zeta$,
$\xi\in\mathrm{C_{c}}(\mathcal{E})$, $\zeta\in\mathrm{C_{c}}(\mathcal{F})$,
are continuous. Then $\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F}$ is a
$\mathrm{C}^{*}$-correspondence bundle from $\mathcal{A}$ to $\mathcal{C}$ and
we have a canonical isomorphism

$\mathrm{C_{0}}(\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F})\cong\mathrm{C_{0}}(\mathcal{E})\otimes_{\mathrm{C_{0}}(\mathcal{B})}\mathrm{C_{0}}(\mathcal{F}).$

of $\mathrm{C}^{*}$-correspondences from $\mathrm{C_{0}}(\mathcal{A})$ to
$\mathrm{C_{0}}(\mathcal{C})$. Combining Lemma 1.12 and Proposition 3.8 we
also have the embedding
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))\otimes_{\mathrm{C_{b}}(\mathcal{M}(\mathcal{B}))}\mathrm{C_{b}}(\mathcal{M}(\mathcal{F}))\rightarrowtail\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F}))$
of $\mathrm{C}^{*}$-correspondences from
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{A}))$
to
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{C}))\cong\mathcal{M}(\mathrm{C_{0}}(\mathcal{C}))$.

######  Example 3.11 (Direct sums of $\mathrm{C}^{*}$-correspondence bundles).

Let $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ and $\mathcal{F}=\\{F_{x}\\}_{x\in X}$
be $\mathrm{C}^{*}$-bundles from $\mathcal{A}=\\{A_{x}\\}_{x\in X}$ to
$\mathcal{B}=\\{B_{x}\\}_{x\in X}$. We equip the family
$\mathcal{E}\oplus\mathcal{F}:=\\{E_{x}\oplus F_{x}\\}_{x\in X}$ of direct
sums of $\mathrm{C}^{*}$-correspondences with the unique topology such that
for all $\xi\in\mathrm{C_{c}}(\mathcal{E})$,
$\zeta\in\mathrm{C_{c}}(\mathcal{F})$ the section
$x\mapsto\xi\oplus\zeta(x):=\xi(x)\oplus\zeta(x)$ is in
$\mathrm{C_{c}}(\mathcal{E}\oplus\mathcal{F})$. This is a
$\mathrm{C}^{*}$-correspondence bundle over the $\mathrm{C}^{*}$-bundle
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$ and we have a canonical isomorphism of
$\mathrm{C}^{*}$-correspondences

$\mathrm{C_{0}}(\mathcal{E}\oplus\mathcal{F})\cong\mathrm{C_{0}}(\mathcal{E})\oplus\mathrm{C_{0}}(\mathcal{F}),\qquad\mathrm{C_{b}}(\mathcal{E}\oplus\mathcal{F})\cong\mathrm{C_{b}}(\mathcal{E})\oplus\mathrm{C_{b}}(\mathcal{F}).$

More generally, if $\\{\mathcal{E}_{i}\\}_{i\in I}$ is a family of
$\mathcal{A}$-$B$-$\mathrm{C}^{*}$-correspondence, then we define the direct
sum $\mathrm{C}^{*}$-correspondence bundle by putting $\oplus_{i\in
I}\mathcal{E}_{i}:=\\{\oplus_{i\in I}E_{x}^{i}\\}_{x\in X}$ and equipping it
with the unique topology such that for sections
$\xi_{i}\in\mathrm{C_{c}}(\mathcal{E}_{i})$, $i\in I$, such that $\sum_{i\in
I}\|\xi_{i}(x)\|^{2}<\infty$ for every $x\in X$, the section $X\ni
x\to\sum_{i\in I}\xi_{i}(x)\in\oplus_{i\in I}\mathcal{E}_{i}$ is continuous.

Finally we comment on morphisms between $\mathrm{C}^{*}$-correspondence
bundles.

######  Lemma 3.12.

Let $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ and $\mathcal{E}=\\{F_{x}\\}_{x\in X}$
be Banach bundles. The relation

$L(\xi)(x)=L_{x}(\xi(x)),\qquad x\in X,\ \xi\in\mathrm{C_{0}}(\mathcal{E}),$

establishes a bijective correspondence between bounded
$\mathrm{C_{0}}(X)$-linear maps
$L:\mathrm{C_{0}}(\mathcal{E})\to\mathrm{C_{0}}(\mathcal{F})$ and bundles
$\\{L_{x}\\}_{x\in X}$ of operators $L_{x}:E_{x}\to F_{x}$, such that
$\sup_{x\in X}\|L_{x}\|<\infty$ and the map $\mathcal{E}\supseteq
E_{x}\ni\xi\longmapsto L_{x}(\xi)\in F_{x}\in\mathcal{F}$ is continuous.

###### Proof.

This is standard, see for instance (the proofs of) [Kwa16, Propositions 3.2,
3.5]. ∎

##  4\. Fell bundles and twisted actions

In this section we introduce twisted actions of groupoids on
$\mathrm{C}^{*}$-bundles and describe the associated crossed products. From
now on, throughout this paper,

> $\mathcal{G}$ is a _locally compact Hausdorff étale groupoid_ with the _unit
> space_ $X:=\mathcal{G}^{(0)}$.

We denote by $r,s:\mathcal{G}\to X\subseteq\mathcal{G}$ the _range_ and
_source_ maps and by
$\mathcal{G}^{(2)}=\\{(g,h)\in\mathcal{G}\times\mathcal{G}:s(g)=r(h)\\}$ the
set of _composable pairs_. Similarly we define _composable triples_
$\mathcal{G}^{(3)}$, and so on. We refer to [Ren80, Exe08, Sim20] for more
information on étale groupoids. In this section we will define their (twisted)
actions on $\mathrm{C}^{*}$-bundles. We will however begin the discussion in
the context of general Fell bundles.

Let $\mathcal{A}$ be a _Fell bundle over the groupoid $\mathcal{G}$_, see
[BE12, Section 2] or [Kum98, Tak14]. For the convenience of the reader we
recall some basic facts here. By definition, $\mathcal{A}$ is an upper-
semicontinuous Banach bundle $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$
equipped with a continuous multiplication
${\cdot}:\\{(a,b)\in\mathcal{A}\times\mathcal{A}:a\in A_{g},\ b\in A_{h},\
(g,h)\in\mathcal{G}^{(2)}\\}\to\mathcal{A}$ and a continuous involution
${}^{*}:\mathcal{A}\to\mathcal{A}$ such that for all $g,h\in\mathcal{G}$

$A_{g}\times A_{h}\ni(a,b)\mapsto ab\in A_{gh},\qquad\mathcal{A}\ni A_{g}\ni
a\mapsto a^{*}\in A_{g^{-1}},$

satisfying the standard set of axioms. The linear space
$\mathrm{C_{c}}(\mathcal{A})$ is naturally a $*$-algebra with algebraic
operations coming from the Fell bundle: for
$a,b\in\mathrm{C_{c}}(\mathcal{A})$ we put

$(a*b)(g):=\sum_{h\in\mathcal{G}_{s(g)}}a(gh^{-1})b(h)=\sum_{h\in\mathcal{G}^{r(g)}}a(h)b(h^{-1}g)\quad\text{and}\quad
a^{*}(g):=a(g^{-1})^{*}.$

Here, for $x\in X$, we put $\mathcal{G}_{x}:=\\{g\in\mathcal{G}:s(g)=x\\}$ and
$\mathcal{G}^{x}:=\\{g\in\mathcal{G}:r(g)=x\\}$.

######  Definition 4.1.

The _full section $\mathrm{C}^{*}$-algebra_ $\mathrm{C}^{*}(\mathcal{A})$ of
the Fell bundle $\mathcal{A}$ is the completion of
$\mathrm{C_{c}}(\mathcal{A})$ in the maximal $\mathrm{C}^{*}$-norm.

One shows that the largest $\mathrm{C}^{*}$-(semi)norm indeed exists on
$\mathrm{C_{c}}(\mathcal{A})$, see [BE11]. To prove that it is a
$\mathrm{C}^{*}$-norm, one constructs the reduced norm as follows. The
restriction $\mathcal{A}|_{X}=\\{A_{x}\\}_{x\in X}$ of $\mathcal{A}$ to $X$ is
a $\mathrm{C}^{*}$-bundle and
$\mathrm{C_{c}}(\mathcal{A}|_{X})\subseteq\mathrm{C_{c}}(\mathcal{A})$ is a
$*$-subalgebra. The inclusion
$\mathrm{C_{c}}(\mathcal{A}|_{X})\subseteq\mathrm{C_{c}}(\mathcal{A})$ extends
to a $\mathrm{C}^{*}$-inclusion
$\mathrm{C_{0}}(\mathcal{A}|_{X})\subseteq\mathrm{C}^{*}(\mathcal{A})$. The
restriction map $a\mapsto a|_{X}$, $a\in\mathrm{C_{c}}(\mathcal{A})$, extends
to a conditional expectation
$\mathbb{E}:\mathrm{C}^{*}(\mathcal{A})\to\mathrm{C_{0}}(\mathcal{A}|_{X})$.
Consider the Hilbert $A_{x}$-module direct sum

$\ell^{2}(\mathcal{A})_{x}:=\Big{\\{}\xi:\mathcal{G}_{x}\to\mathcal{A}:\xi(g)\in
A_{g},\,g\in\mathcal{G}_{x},\;\sum_{g\in\mathcal{G}_{x}}\xi(g)^{*}\xi(g)\text{
converges in $A_{x}$}\Big{\\}}=\bigoplus_{g\in\mathcal{G}_{x}}A_{g}.$

Then we have a representation
$\Lambda_{x}:\mathrm{C}^{*}(\mathcal{A})\to\mathcal{L}(\ell^{2}(\mathcal{A})_{x})$
determined by the formula

$[\Lambda_{x}(a)\xi](g)=\sum_{h\in\mathcal{G}_{x}}a(gh^{-1})\xi(h),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}),\ \xi\in\ell^{2}(A_{x}),\
g\in\mathcal{G}_{x}.$

There is a unique topology on
$\ell^{2}(\mathcal{A})=\\{\ell^{2}(\mathcal{A})_{x}\\}_{x\in X}$ making it a
Hilbert module bundle over $\mathcal{A}|_{X}$, and such that
$\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))$ contains
$x\mapsto\bigoplus_{g\in\mathcal{G}_{x}}a(g)$, for all
$a\in\mathrm{C_{c}}(\mathcal{A})$, cf. [Kum98, 3.3]. The field of
representations $\\{\Lambda_{x}\\}_{x\in X}$ induces a representation
$\Lambda:\mathrm{C}^{*}(\mathcal{A})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(\mathcal{A})))$

$\big{(}\Lambda(a)\xi\big{)}(x)=\Lambda_{x}(a)\xi(x),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}),\
\xi\in\mathrm{C_{0}}(\ell^{2}(\mathcal{A})),\ x\in X,$

that we call the _regular representation_ of $\mathrm{C}^{*}(\mathcal{A})$.
This representation is faithful on $\mathrm{C_{c}}(\mathcal{A})$, but not on
$\mathrm{C}^{*}(\mathcal{A})$, in general. When $\Lambda$ is faithful we say
that $\mathcal{A}$ has the _weak containment property_. This is implied by
_amenability_ of $\mathcal{G}$ [SW13] or some sort of amenability of
$\mathcal{A}$ [Kra23, BM23], but there are examples of non-amenable groupoids
with the weak containment property [Wil15].

Notice that $\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))$ is a
$\mathrm{C}^{*}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{A}|_{X})$-$\mathrm{C}^{*}$-correspondence
that we call the _regular $\mathrm{C}^{*}$-correspondence_ for $\mathcal{A}$.
It can be defined equivalently as the completion of the pre-Hilbert module
$\mathrm{C_{c}}(\mathcal{A})$ with respect to the inner product $\langle
a,b\rangle_{\mathrm{C_{0}}(\mathcal{A}|_{X})}:=\mathbb{E}(a^{*}b)$. Then
$\Lambda:\mathrm{C}^{*}(\mathcal{A})\to\mathcal{L}\left(\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))\right)$
is determined by the multiplication in $\mathrm{C_{c}}(\mathcal{A})$:
$\Lambda(a)b=a*b$ for
$a\in\mathrm{C_{c}}(\mathcal{A})\subseteq\mathrm{C}^{*}(\mathcal{A})$ and
$b\in\mathrm{C_{c}}(\mathcal{A})\subseteq\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))$.

######  Definition 4.2.

The _reduced section $\mathrm{C}^{*}$-algebra_
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})$ is the completion of
$\mathrm{C_{c}}(\mathcal{A})$ in the norm
$\|a\|_{\operatorname{red}}:=\sup_{x\in X}\|\Lambda_{x}(a)\|$. Thus
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})\cong\Lambda(\mathrm{C}^{*}(\mathcal{A}))$
and the restriction map
$\mathbb{E}:\mathrm{C_{c}}(\mathcal{A})\to\mathrm{C_{c}}(\mathcal{A}|_{X})$
extends to a faithful conditional expectation
$\mathbb{E}:\mathrm{C}^{*}_{r}(\mathcal{A})\to\mathrm{C_{0}}(\mathcal{A}|_{X})$.

We briefly comment on the relationship between Fell bundles over the groupoid
$\mathcal{G}$ and Fell bundles over inverse semigroups. This will be useful
when studying representations of the section $\mathrm{C}^{*}$-algebras and
also checking that twisted groupoid actions yield Fell bundles. To this end
recall that the family of (open) bisections of the groupoid $\mathcal{G}$,
i.e.

$\operatorname{Bis}(\mathcal{G})=\\{U\subseteq\mathcal{G}:\text{$U$ is open
and $r|_{U},s|_{U}$ are injective}\\}.$

with operations induced by composition and inverse from $\mathcal{G}$ forms
naturally a unital inverse semigroup. The unit space $X$ is the unit for the
inverse semigroup $\operatorname{Bis}(\mathcal{G})$.

Let now $S$ be any unital inverse semigroup with unit $1$. A _Fell bundle over
the inverse semigroup_ $S$ is a family $\\{A_{t}\\}_{t\in S}$ of Hilbert
$A$-$A$-bimodules, where $A:=A_{1}$ is a $\mathrm{C}^{*}$-algebra, and hence
also a trivial Hilbert $A$-bimodule, equipped with a multiplication given by
bimodule embeddings $A_{t}\otimes_{A}A_{u}\to A_{tu}$, $t,u\in S$, which are
associative in an appropriate sense and induce isomorphisms $A_{t}\otimes
A_{t^{*}}\otimes_{A}A_{t}\cong A_{t}$, $t\in S$. See [BM23] for details and
relationship with previous equivalent definitions. In particular, this data
induces inclusion maps $j_{t,u}:A_{t}\to A_{u}$ and involutions $*:A_{t}\to
A_{t^{*}}$ compatible with the above structures.

######  Example 4.3 (Inverse semigroup Fell bundle from a groupoid Fell
bundle).

Every Fell bundle $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ over
$\mathcal{G}$ _induces_ a Fell bundle
$\mathcal{A}_{\operatorname{sem}}=\\{A_{U}\\}_{U\in\operatorname{Bis}(\mathcal{G})}$
over $\operatorname{Bis}(\mathcal{G})$ where fibers are the spaces
$A_{U}:=\mathrm{C_{0}}(\mathcal{A}|_{U})$,
$U\in\operatorname{Bis}(\mathcal{G})$, and multiplication maps $A_{U}\times
A_{V}\to A_{UV}$ and involution maps $A_{U}\to A_{U^{*}}$ are given by
multiplication and involution on $\mathcal{A}$. We have natural inclusions
$A_{U}\subseteq A_{V}$ for $U\subseteq V$. Moreover, $\mathcal{A}$ is
saturated, that is $\overline{A_{g}\cdot A_{h}}=A_{gh}$ for all
$(g,h)\in\mathcal{G}^{(2)}$, if and only if $\mathcal{A}_{\operatorname{sem}}$
is saturated, that is $\overline{A_{U}\cdot A_{V}}=A_{UV}$ for all
$U,V\in\operatorname{Bis}(\mathcal{G})$.

In the saturated case we have the following elegant characterisation of
inverse semigroup Fell bundles coming from groupoid Fell bundles as described
in Example 4.3:

######  Theorem 4.4 ([BM17, Theorem 6.1]).

Let $S:=\operatorname{Bis}(\mathcal{G})$ be the inverse semigroup of
bisections of $\mathcal{G}$. A saturated Fell bundle
$\mathcal{A}_{\operatorname{sem}}=\\{A_{U}\\}_{U\in S}$ over $S$ is induced
from a Fell bundle $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ over
$\mathcal{G}$, as described in Example 4.3, if and only the map $X\supseteq
U\mapsto A_{U}\triangleleft A_{X}$ from the lattice of open sets in $X$ to the
lattice ideals in $A_{X}$ preserves arbitrary suprema.

If this is the case, the Fell bundle $\mathcal{A}$ over $\mathcal{G}$ is
unique up to an isomorphism.

######  Remark 4.5.

For further reference we describe how the bundle $\mathcal{A}$ over
$\mathcal{G}$ in Theorem 4.4 is constructed from
$\mathcal{A}_{\operatorname{sem}}$ in Theorem 4.4. If the map $X\supseteq
U\mapsto A_{U}\triangleleft A_{X}$ preserves suprema, then (since it
automaticaly preserves infima) by general Stone duality $A_{X}$ is a
$\mathrm{C_{0}}(X)$-algebra where $A_{U}=A_{X}\mathrm{C_{0}}(U)$ for every
open $U\subseteq X$. In particular,
$A_{X}\cong\mathrm{C_{0}}(\\{A_{x}\\}_{x\in X})$ where $A_{x}:=A_{X}/I_{x}$
and $I_{x}:=A_{X}\mathrm{C_{0}}(X\setminus\\{x\\})$ for $x\in X$. To extend
this construction to $A_{U}$ for $U\in S$, we use that $A_{U}$ is naturally a
Morita equivalence $A_{r(U)}$-$A_{s(U)}$-bimodule. Namely, for any $g\in U$ we
have $A_{U}I_{s(g)}=I_{r(g)}A_{U}=I_{r(g)}A_{U}I_{s(g)}$ and this this
subspace of $A_{U}$ denoted by $I_{U,g}$ is naturally an
$I_{r(g)}$-$I_{s(g)}$-Hilbert bimodule. Hence the quotient
$A_{U,g}:=A_{U}/I_{U,g}$ is naturally an $A_{r(g)}$-$A_{s(g)}$-Hilbert
bimodule. Writing $a_{U}(g)$ for the image of $a\in A_{U}$ in $A_{U,g}$ we
have a unique topology on the bundle $\\{A_{U,g}\\}_{g\in U}$ such that we
have an isomorphism

$A_{U}\cong\mathrm{C_{0}}(\\{A_{U,g}\\}_{g\in U})\quad\text{ where }\quad
A_{U}\ni a\longmapsto a_{U}\in\mathrm{C_{0}}(\\{A_{U,g}\\}_{g\in U}).$

Using the inclusion maps from $\mathcal{A}_{\operatorname{sem}}$ one may piece
together the above structures to obtain a Banach bundle
$\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ where $A_{g}\cong A_{U,g}$
canonically for any $U\in\operatorname{Bis}(\mathcal{G})$ contatining
$g\in\mathcal{G}$. Then the formulas

$a_{U}(g)^{*}:=a^{*}_{U^{*}}(g^{-1}),\qquad a_{U}(g)\cdot b_{V}(h):=(a\cdot
b)_{UV}(gh),\qquad\text{ for }a\in A_{U},b\in A_{V},$

induce the desired Fell bundle structure on $\mathcal{A}$.

### Twisted actions

Most of the literature on groupoids only deals with the so-called
Green–Renault twisted actions, with the abelian twist, see [Ren87, Ren91]. We
need to consider here twisted actions of $\mathcal{G}$ á la Busby–Smith
([BS70]), where the twist is not necessarily abelian. This type of twisted
actions have appeared in [BMZ13], where they are interpreted as certain
functors between $2$-categories.

If $A$ is a $\mathrm{C}^{*}$-algebra we denote by $U\mathcal{M}(A)$ the
unitary group in $\mathcal{M}(A)$, and if $\mathcal{A}$ is a
$\mathrm{C}^{*}$-bundle we write
$U\mathcal{M}(\mathcal{A}):=\\{U\mathcal{M}(A_{x})\\}_{x\in
X}\subseteq\mathcal{M}(\mathcal{A})$ for the corresponding group bundle. A
twisted groupoid action on a $\mathrm{C}^{*}$-bundle $\mathcal{A}$ should be
viewed as a ‘continuous twisted functor’ $\alpha$ from $\mathcal{G}$ to the
category of $*$-isomorphisms between the fibers of $\mathcal{A}$ and the twist
$u$ is a ‘strictly continuous 2-cocycle in $U\mathcal{M}(\mathcal{A})$’. A
more precise interpretation is given in [BMZ13] where one views twisted
actions of $\mathcal{G}$ as ‘continuous morphisms’ (or weak functors) from
$\mathcal{G}$ to the $2$-category of $\mathrm{\mathrm{C}^{*}}$-algebras whose
morphisms are nondegenerate homomorphisms $A\to\mathcal{M}(B)$, and
$2$-morphisms are certain unitary multiplier intertwiners between these
homomorphisms. We now make precise exactly what we need.

######  Definition 4.6.

A _twisted action of $\mathcal{G}$ on a $\mathrm{C}^{*}$-bundle_
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$ is a pair $(\alpha,u)$ where

  1. (1)

$\alpha=\\{\alpha_{g}\\}_{g\in\mathcal{G}}$ is a family of $*$-isomorphisms
$\alpha_{g}:A_{s(g)}\to A_{r(g)}$, $g\in\mathcal{G}$, such that the map
$s^{*}\mathcal{A}\ni(a,g)\mapsto(\alpha_{g}(a),g)\in r^{*}\mathcal{A}$ is
continuous, where
$r^{*}\mathcal{A}:=\\{(a,g)\in\mathcal{A}\times\mathcal{G}:a\in A_{r(g)}\\}$
and $s^{*}\mathcal{A}:=\\{(a,g)\in\mathcal{A}\times\mathcal{G}:a\in
A_{s(g)}\\}$ are the pullback bundles;

  2. (2)

$u=\\{u(g,h)\\}_{(g,h)\in\mathcal{G}^{(2)}}$ is a section of the pullback
group bundle
$r^{(2)*}U\mathcal{M}(\mathcal{A})=\\{(w,(g,h))\in\mathcal{M}(\mathcal{A})\times\mathcal{G}^{(2)}:w\in
U\mathcal{M}(A_{r(g)})\\},$ such that for every continuous section $a$ of the
pullback bundle $r^{*}\mathcal{A}^{(2)}:=\\{((a,g),(b,h))\in
r^{*}\mathcal{A}\times r^{*}\mathcal{A}:s(g)=r(h)\\}$ the formula
$au(g,h):=a(g,h)u(g,h)$ for $g,h\in\mathcal{G}$ defines a continuous section
of $r^{*}\mathcal{A}^{(2)}$.

In addition, we require this pair to satisfy the following algebraic
conditions

  1. (1)

$\alpha_{x}=\operatorname{id}_{A_{x}}$, $u(x,x)=1_{\mathcal{M}(A_{x})}$ for
all $x\in X$;

  2. (2)

$\operatorname{Ad}_{u(g,h)}\circ\alpha_{gh}=\alpha_{g}\circ\alpha_{h}$ for all
$(g,h)\in\mathcal{G}^{(2)}$;

  3. (3)

$\overline{\alpha}_{f}(u(g,h))u(f,gh)=u(f,g)u(fg,h)$, for all
$(f,g,h)\in\mathcal{G}^{(3)}$;

where $\overline{\alpha}_{f}:\mathcal{M}(A_{s(f)})\to\mathcal{M}(A_{r(f)})$ is
the unique homomorphic extension of $\alpha_{f}$ for $f\in G$.

######  Remark 4.7.

Relations (1)-(3) have some standard consequences that we will sometimes use
without warning. For instance,
$\alpha_{g}^{-1}=\operatorname{Ad}_{u(g^{-1},g)^{*}}\circ\alpha_{g^{-1}}$,
$u(g,s(g))=u(r(g),g)=1_{\mathcal{M}(A_{r(g)})}$ and
$\alpha_{g}(u(g^{-1},g))=u(g,g^{-1})$, $g\in\mathcal{G}$.

We first note that a twisted action of $\mathcal{G}$ induces a twisted inverse
semigroup action of $\operatorname{Bis}(\mathcal{G})$ in the sense of Sieben
[Sie98]. This is a special case of the inverse semigroup twisted actions of
Buss–Exel [BE11, Section 5], which were also considered in [KLS22, Section 2].
The latter actions are more general and cover twists given by groupoid circle
extensions, see [BE11, Theorem 7.2] or [KLS22, Theorem 2.16].

######  Lemma 4.8.

Let $(\alpha,u)$ be a twisted action of $\mathcal{G}$ on
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$. For any open $U\subseteq X$ put
$A_{U}:=\mathrm{C_{0}}(\mathcal{A}|_{U})$. Then for any
$U,V\in\operatorname{Bis}(\mathcal{G})$, the maps $\alpha_{U}:A_{s(U)}\to
A_{r(U)}$ and sections $\omega(U,V)\in
U\mathcal{M}(A_{r(UV)})=U\mathrm{C_{b}}(\mathcal{A}|_{r(UV)})$ given by the
formulas

$\alpha_{U}(a)(r(g)):=\alpha_{g}(a(s(g)),\qquad\omega(U,V)(r(g)):=u(g,h),\qquad
g\in U,h\in V,r(h)=s(g),$

form a twisted inverse semigroup action of $\operatorname{Bis}(\mathcal{G})$
on $A=\mathrm{C_{0}}(\mathcal{A})$, in the sense of [Sie98] and hence also of
[BE11]. In particular, it gives rise to a saturated inverse semigroup Fell
bundle $\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}=\\{(a,U):a\in A_{r(U)},\
U\in\operatorname{Bis}(\mathcal{G})\\}$, where writing $a\delta_{U}$ for
$(a,U)\in\mathcal{A}_{\operatorname{sem}}$ we put

$(a\delta_{U})\cdot(b\delta_{V}):=\alpha_{U}(\alpha_{U}^{-1}(a)b)\omega(U,V)\delta_{UV},\quad(a\delta_{U})^{*}=\alpha_{U}^{-1}(a^{*})\omega(U^{*},U)^{*}\delta_{U^{*}},$

for all $a\in A_{r(U)}$, $b\in A_{r(V)}$,
$U,V\in\operatorname{Bis}(\mathcal{G})$.

###### Proof.

By (1) and (2) in Definition 4.6, the maps $\alpha_{U}$ are well-defined
isometries, and $\omega(U,V)$ are well defined elements of
$U\mathcal{M}(A_{UV})$. Thus we need to check that for any for $U,V,W\in S$:

  1. (1)

$A_{s(UV)}=A_{s(V)}\cap\alpha_{V}^{-1}(A_{s(U)})$ and
$\alpha_{U}\circ\alpha_{V}=Ad_{\omega(U,V)}\alpha_{UV}$ on this ideal;

  2. (2)

$\alpha_{U}\big{(}a\omega(V,W)\big{)}\omega(U,VW)=\alpha_{U}(a)\omega(U,V)\omega(UV,W)$
for $a\in A_{s(V)}\cap A_{r(VW)}$;

  3. (3)

$\omega(U,V)=1_{\mathcal{M}(A_{r(UV)})}$ if either $U\subseteq X$ or
$V\subseteq Y$.

Let $a\in A_{s(UV)}$, and $g\in U$, $h\in V$ with $s(g)=r(h)$. By Definition
4.6(2) we get

$\begin{split}Ad_{\omega(U,V)}\alpha_{UV}(a)(r(g))&=u(g,h)\alpha_{gh}(a(s(h)))u(g,h)^{*}=\alpha_{g}(\alpha_{h}(a(s(h))))\\\
&=\alpha_{g}(\alpha_{V}(a)(s(g)))=\alpha_{U}(\alpha_{V}(a))(r(g))\end{split}$

This implies (1), as the above calculation implies that
$A_{r(UV)}=\alpha_{U}(A_{r(V)}\cap A_{s(U)})=\alpha_{U}(A_{r(V)})\cap
A_{r(U)}$, which by taking inverses is equivalent to
$A_{s(UV)}=A_{s(V)}\cap\alpha_{V}^{-1}(A_{s(U)})$.

To show (2) let $a\in A_{s(V)}\cap A_{r(VW)}$ and $f\in U$, $g\in V$, $h\in W$
with $(f,g,h)\in\mathcal{G}^{(3)}$. Using Definition 4.6(3) we get

$\begin{split}[\alpha_{U}\big{(}a\omega(V,W)\big{)}\omega(U,VW)](r(f))&=\alpha_{f}\Big{(}a(s(f))u(g,h)\Big{)}u(f,gh)\\\
&=\alpha_{f}\Big{(}a(s(f))\Big{)}u(f,g)u(fg,h)\\\
&=\big{(}\alpha_{U}(a)\omega(U,V)\omega(UV,W)\big{)}(r(f)).\end{split}$

Property (3) follows from Definition 4.6(1), see also Remark 4.7.

Every twisted inverse semigroup action naturally yields a saturated Fell
bundle over the inverse semigroup, see [BE11, page 250], which in the present
context is as described in the assertion. ∎

######  Corollary 4.9.

Let $(\alpha,u)$ be a twisted action of $\mathcal{G}$ on
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$. Write the pullback bundle
$\mathcal{A}^{(\alpha,u)}:=r^{*}\mathcal{A}$, as
$\mathcal{A}^{(\alpha,u)}=\bigsqcup_{g\in\mathcal{G}}A_{g}$ where
$A_{g}:=A_{r(g)}$ for $g\in\mathcal{G}$. Then $\mathcal{A}^{(\alpha,u)}$ is a
Fell bundle over the groupoid $\mathcal{G}$ with multiplication and involution
given by

$\displaystyle A_{g}\times A_{h}\ni(a,b)\longmapsto
a\cdot\alpha_{g}(b)u(g,h)\in A_{gh},$ $\displaystyle A_{g}\ni a\longmapsto
u(g^{-1},g)^{*}\alpha_{g^{-1}}(a^{*})=\alpha_{g}^{-1}(a^{*})u(g^{-1},g)^{*}\in
A_{g^{-1}}.$

###### Proof.

One could check the conditions in [BE12, Definition 2.6], which is a tedious
verification. To avoid that we appeal to Lemma 4.8 and Theorem 4.4. Namely, by
Theorem 4.4 there is a unique Fell bundle $\mathcal{A}^{(\alpha,u)}$ over
$\mathcal{G}$, such that $\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}$ is
induced by $\mathcal{A}^{(\alpha,u)}$. By the description in Remark 4.5, the
bundle $\mathcal{A}^{(\alpha,u)}$ as a Banach bundle can be naturally
identified with the pullback bundle
$\mathcal{A}^{(\alpha,u)}:=r^{*}\mathcal{A}$, and then the multiplication and
involution have to be as in the statement. ∎

######  Definition 4.10.

If $(\alpha,u)$ is a twisted action of $\mathcal{G}$ on
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$, we define the _full_ and the _reduced
crossed product_ for $\alpha$ as the cross-sectional
$\mathrm{\mathrm{C}^{*}}$-algebras $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$
and $C_{\operatorname{red}}^{*}(\mathcal{A}^{(\alpha,u)})$ for the Fell bundle
$\mathcal{A}^{(\alpha,u)}$ described in Corollary 4.9.

More specifically, the full crossed product
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ is the maximal
$\mathrm{C}^{*}$-completion of the $*$-algebra
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ where for
$a,b\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$, $g\in G$, we have

$(a*b)(g)=\sum_{h\in\mathcal{G}_{s(g)}}a(gh^{-1})\alpha_{gh^{-1}}(b(h))u(gh^{-1},h),\quad\quad
a^{*}(g)=u(g,g^{-1})^{*}\alpha_{g}(a(g^{-1})^{*}).$

The _regular representation_ $\Lambda$ of
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ on
$\ell^{2}(\mathcal{A}^{(\alpha,u)})=\ell^{2}(r^{*}\mathcal{A})=\\{\bigoplus_{g\in\mathcal{G}_{x}}A_{r(g)}\\}_{x\in
X}$ is determined by representations $\Lambda_{x}$ on Hilbert $A_{x}$-modules
$\ell^{2}(r^{*}\mathcal{A})_{x}=\bigoplus_{g\in\mathcal{G}_{x}}A_{r(g)}$,
$x\in X$, where $[\xi\cdot a_{x}](g)=\xi(g)\alpha_{g}(a_{x})$,
$\left\langle\xi,\zeta\right\rangle_{A_{x}}=\sum_{g\in\mathcal{G}_{x}}\alpha_{g}^{-1}(\xi(g)^{*}\zeta(g))$,
for $a_{x}\in A_{x}$, $\xi,\zeta\in\bigoplus_{g\in\mathcal{G}_{x}}A_{r(g)}$,
$g\in\mathcal{G}$ and

(4.1)
$\big{(}\Lambda_{x}(a)\xi\big{)}(g)=\sum_{h\in\mathcal{G}_{x}}a(gh^{-1})\alpha_{gh^{-1}}(\xi(h))u(gh^{-1},h),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)}),\ g\in\mathcal{G}_{x}.$

######  Example 4.11.

When $\mathcal{G}=G$ is a discrete group then a twisted action $(\alpha,u)$ of
$\mathcal{G}$ is the same as a twisted group action on a
$\mathrm{C}^{*}$-algebra $\mathcal{A}=A$ in the sense of Busby–Smith [BS70];
this has been considered in many other papers, for example [PR89, BC12, BC15,
BC16], and the algebras $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ and
$C_{\operatorname{red}}^{*}(\mathcal{A}^{(\alpha,u)})$ are the usual full and
reduced crossed products, respectively, usually also denoted by
$A\rtimes_{(\alpha,u)}G$ and $A\rtimes_{(\alpha,u),r}G$ in the literature.

######  Example 4.12.

Twisted actions of $\mathcal{G}$ on a trivial bundle
$\mathcal{A}=\\{\mathbb{C}\\}_{x\in X}$ correspond to continuous groupoid
cocycles, i.e. for any such action $(\alpha,u)$ we necessarily have
$\alpha=\\{\mathrm{id}_{\mathbb{C}}\\}_{x\in X}$, and $u$ is a _continuous
normalised $2$-cocycle_ on $\mathcal{G}$, that is,
$u:\mathcal{G}^{(2)}\to\mathbb{T}$ is a continuous map satisfying
$u\big{(}r(g),g\big{)}=1=u\big{(}g,s(g)\big{)}$ and
$u(f,g)u(fg,h)=u(g,h)u(f,gh)$ for every composable triple
$(f,g,h)\in\mathcal{G}^{(3)}$. The algebras
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ and
$C_{\operatorname{red}}^{*}(\mathcal{A}^{(\alpha,u)})$ are then the usual full
and reduced twisted groupoid $\mathrm{C}^{*}$-algebras, usually also written
as $\mathrm{C}^{*}(\mathcal{G},u)$ and
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{G},u)$, see [Ren80]. In
particular, the (left) regular representation (4.1) for $x\in X$ specializes
in this case to a homomorphism
$\lambda^{u}_{x}\colon\mathrm{C_{c}}(\mathcal{G},u)\to\mathcal{L}(\ell^{2}(\mathcal{G}_{x}))$,
which on the canonical basis
$\\{\delta_{g}\\}_{g\in\mathcal{G}_{x}}\subseteq\ell^{2}(\mathcal{G}_{x})$ is
given by

(4.2)
$\lambda^{u}_{x}(f)\delta_{h}=\sum_{g\in\mathcal{G}_{r(h)}}f(g)u(g,h)\delta_{gh}.$

We use the superscript $u$ above in order to differentiate $\lambda^{u}$ from
the ordinary (left) regular representation $\lambda$, which coincides with
$\lambda^{u}$ if $u=1$ is the trivial $2$-cocycle.

######  Example 4.13.

Consider a twisted action $(\alpha,u)$ of a groupoid $\mathcal{G}$ on a
$\mathrm{C}^{*}$-bundle $\mathcal{A}$ and let $n\in\mathbb{N}$. The lifted
action $(\alpha^{(n)},u^{(n)})$ on $\mathbb{M}_{n}(\mathcal{A})$ is given by
the isomorphisms $\alpha_{g}^{(n)}$ from $\mathbb{M}_{n}(A_{s(g)})$ to
$\mathbb{M}_{n}(A_{r(g)})$ and unitaries $u(g,h)^{(n)}=u(g,h)\otimes
I_{n}\in\mathbb{M}_{n}(\mathcal{M}(A_{r(g)}))\cong\mathcal{M}(\mathbb{M}_{n}(A_{r(g)}))$.
The algebraic and topological conditions appearing in Definition 4.6 are
clearly satisfied by the lifted action. Further checks show that we have the
isomorphisms
$\mathbb{M}_{n}(\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)}))\cong\mathrm{C}^{*}(\mathbb{M}_{n}(\mathcal{A})^{(\alpha^{(n)},u^{(n)})})$
and $\mathbb{M}_{n}(C_{\operatorname{red}}^{*}(\mathcal{A}^{(\alpha,u)}))\cong
C_{\operatorname{red}}^{*}(\mathbb{M}_{n}(\mathcal{A})^{(\alpha^{(n)},u^{(n)})})$.
Defining a matrix lift of an abstract Fell bundle is more complicated. The
question relates to the norm on the matrices over fibers, and it can addressed
via the $\mathrm{C}^{*}$-condition, but we will not need it in this paper.

###  4.1. Representations of crossed products

Let us fix a twisted action $(\alpha,u)$ of $\mathcal{G}$ on
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$, and recall the inverse semigroup Fell
bundle $\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}$ from Lemma 4.8. For
every $U\in\operatorname{Bis}(\mathcal{G})$, and
$a\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$ we may identify
$a\delta_{U}\in\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}$ with an element
of $\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ given by

$(a\delta_{U})(g):=[g\in U]a(r(g)),\qquad g\in\mathcal{G}.$

The product and involutions in $\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}$
and $\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ are compatible, and every
element in $\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ is of the form
$\sum_{U\in F}a_{U}\delta_{U}$ where
$F\subseteq\operatorname{Bis}(\mathcal{G})$ is finite and
$a_{U}\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$, $U\in F$. This extends to an
isomorphism between the algebras $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$
and $C_{\operatorname{red}}^{*}(\mathcal{A}^{(\alpha,u)})$ and the full and
reduced algebras associated to the inverse semigroup Fell bundle
$\mathcal{A}_{\operatorname{sem}}^{(\alpha,u)}$, respectively, see [BM17,
Theorem 5.5] and [BE12, Theorem 2.15, Theorem 4.11]. In particular, the
relation

$\psi(\sum_{U\in F}a_{U}\delta_{U})=\sum_{U\in F}\psi_{U}(a_{U}\delta_{U})$

gives a bijective correspondence between representations
$\psi:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to B$ in a
$\mathrm{C}^{*}$-algebra $B$ and families
$\\{\psi_{U}\\}_{U\in\operatorname{Bis}(\mathcal{G})}$ of linear maps maps
$\psi_{U}:A_{r(U)}\to B$ such that for all $a\in A_{r(U)}$, $b\in A_{r(V)}$,
$U,V\in\operatorname{Bis}(\mathcal{G})$, we have

  1. (SR1)

$\psi_{U}(a\delta_{U})\psi_{V}(b\delta_{V})=\psi_{UV}(\alpha_{U}(\alpha_{U}^{-1}(a_{U})a_{V})\omega(U,V)\delta_{UV})$;

  2. (SR2)

$\psi_{U}(a\delta_{U})^{*}=\psi_{U^{*}}(\alpha_{U}^{-1}(a^{*})\omega(U^{*},U)\delta_{U^{*}})$.

If $H$ is a Hilbert space then the relation

$\psi(\sum_{U\in F}a_{U}\delta_{U})=\sum_{U\in F}\pi_{U}(a_{U})v_{U}$

gives a bijective correspondence between representations
$\psi:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to\mathbb{B}(H)$ and _covariant
representation of $(\alpha,u)$ on $H$_, i.e. pairs $(\pi,v)$ where
$\pi:A\to\mathbb{B}(H)$ is a representation (recall that $A$ denotes the
$\mathrm{C}^{*}$-algebra associated with the bundle $\mathcal{A}$) and
$\\{v_{U}\\}_{U\in S}\subseteq\mathbb{B}(H)$ is a family of partial isometries
satisfying:

  1. (CR1)

$v_{U}\pi(a)v_{U}^{*}=\pi(\alpha_{U}(a))$ for all $a\in A_{s(U)}$,
$U\in\operatorname{Bis}(\mathcal{G})$;

  2. (CR2)

the ranges of $v_{U}$ and $v_{U}^{*}$ are $\overline{\pi(A_{r(U)})H}$ and
$\overline{\pi(A_{s(U)})H}$, for all $U\in\operatorname{Bis}(\mathcal{G})$;

  3. (CR3)

$\pi(a)v_{U}v_{V}=\pi(a\omega(U,V))v_{UV}$ for all $a\in A_{r(UV)}$,
$U,V\in\operatorname{Bis}(\mathcal{G})$.

##  5\. Equivariant correspondences

Throughout this section $\mathcal{A}$ and $\mathcal{B}$ will be
$\mathrm{C}^{*}$-bundles over $X:=\mathcal{G}^{(0)}$, carrying twisted actions
$(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of $\mathcal{G}$ (where
$\mathcal{G}$ is an étale locally compact Hausdorff groupoid). In the
untwisted case, the following notion was called an _equivariant
representation_ of $\mathcal{G}$ by Le Gall in [LG99, Definition 4.6]. Bédos
and Conti [BC15, BC16] follow this naming, while Echterhoff, Kaliszewski,
Quigg and Raeburn [EKQR06, Definition 2.5] call such an object a _compatible
right-Hilbert module_ (in the context of group actions). Deaconu [Dea18,
Definition 5.10] calls it a _groupoid action on a
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence_. Here we extend this
notion to twisted actions, and call it equivariant correspondence.

######  Definition 5.1.

Let $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ be twisted actions of
$\mathcal{G}$ on $\mathcal{A}$ and $\mathcal{B}$, respectively. An
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ _equivariant correspondence_ is a
$\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}$ from $\mathcal{A}$ to
$\mathcal{B}$ endowed with an _equivariant representation_ of $\mathcal{G}$,
meaning a family $L=\\{L_{g}\\}_{g\in\mathcal{G}}$ satisfying

  1. (1)

for each $g\in\mathcal{G}$ the map $L_{g}:E_{s(g)}\to E_{r(g)}$ is an
invertible isometry, $L_{x}=\mathrm{id}_{E_{x}}$ for $x\in X$, and

$L_{g}(L_{h}(\xi))=u_{\alpha}(g,h)L_{gh}(\xi)u_{\beta}(g,h)^{*},\qquad\xi\in
E_{s(h)},\,(g,h)\in\mathcal{G}^{(2)};$

  2. (2)

for $g\in\mathcal{G}$, $\xi,\zeta\in E_{s(g)}$, $a\in A_{s(g)}$, $b\in
B_{s(g)}$ we have

$\qquad L_{g}(a\xi)=\alpha_{g}(a)(L_{g}\xi),\qquad L_{g}(\xi
b)=L_{g}(\xi)\beta_{g}(b),\qquad\beta_{g}(\langle\xi,\zeta\rangle_{B_{s(g)}})=\langle
L_{g}\xi,L_{g}\zeta\rangle_{B_{r(g)}};$

  3. (3)

for each $\xi,\zeta\in\mathrm{C_{c}}(\mathcal{E})$ the section $\mathcal{G}\ni
g\longmapsto\langle\xi(r(g)),L_{g}\zeta(s(g))\rangle_{B_{r(g)}}\in
r^{*}\mathcal{B}$ is continuous.

######  Remark 5.2.

The relations in Definition 5.1 imply that
$L_{g}^{-1}(\cdot)=u_{\alpha}(g^{-1},g)^{*}L_{g^{-1}}(\cdot)u_{\beta}(g^{-1},g)$,
$L_{g}^{-1}(\alpha_{g}(a)\xi)=aL_{g}^{-1}(\xi)$,
$L_{g}^{-1}(\xi\beta_{g}(b))=L_{g}^{-1}(\xi)b$ and
$\beta_{g}^{-1}(\langle\xi,\zeta\rangle_{B_{s(g)}})=\langle
L_{g}^{-1}\xi,L_{g}^{-1}\zeta\rangle_{B_{r(g)}}$ for all $\zeta\in E_{r(g)}$,
$a\in A_{s(g)}$, $b\in B_{s(g)}$ and $g\in\mathcal{G}$.

######  Lemma 5.3.

Condition (3) in Definition 5.1, assuming (2), is equivalent to continuity of

(5.1) $s^{*}\mathcal{E}\supseteq E_{s(g)}\ni\xi\longmapsto L_{g}(\xi)\in
E_{r(g)}\in r^{*}\mathcal{E}.$

###### Proof.

Clearly, continuity of (5.1) implies (3) in Definition 5.1. Conversely, assume
(3) and the last relation in (2) in Definition 5.1. The latter implies that
all maps $L_{g}:E_{s(g)}\to E_{r(g)}$, $g\in\mathcal{G}$, are isometries and
for all $\xi,\zeta\in\mathrm{C_{c}}(\mathcal{E})$, the section

$\mathcal{G}\ni
g\longmapsto\beta_{g}(\langle\xi,\zeta\rangle_{B_{s(g)}})=\langle
L_{g}\xi,L_{g}\zeta\rangle_{B_{r(g)}}\in B_{r(g)}\subseteq r^{*}\mathcal{B}$

is continuous (because $\beta$ and the inner product are ‘continuous’). Let
$(\xi_{i})_{i\in I}$ be a net of elements of $s^{*}\mathcal{E}$ convergent to
$\xi_{0}\in s^{*}\mathcal{E}$. This means that $g_{i}:=p(\xi_{i})\to
g:=p(\xi_{0})$ in $\mathcal{G}$ and for every
$\xi\in\mathrm{C_{0}}(s^{*}\mathcal{E})$ we have
$\|\xi_{i}-\xi(g_{i})\|\to\|\xi_{0}-\xi(g)\|$. Choose
$\xi\in\mathrm{C_{0}}(s^{*}\mathcal{E})$ and
$\zeta\in\mathrm{C_{0}}(r^{*}\mathcal{E})$ such that $\xi(g)=\xi_{0}$ and
$\zeta(g)=L_{g}(\xi_{0})$. Then

$\|L_{g_{i}}(\xi_{i})-\zeta(g_{i})\|\leq\|\xi_{i}-\xi(g_{i})\|+\|L_{g_{i}}(\xi(g_{i}))-\zeta(g_{i})\|$

where the first summand tends to zero because $\xi_{0}=\xi(g)$. To see the
same for $\|L_{g_{i}}(\xi(g_{i}))-\zeta(g_{i})\|$ note that its square is the
norm of the following element of $B_{g_{i}}\subseteq\mathcal{B}$:

$\langle L_{g_{i}}(\xi(g_{i})),L_{g_{i}}(\xi(g_{i}))\rangle_{B_{r(g)}}-\langle
L_{g_{i}}(\xi(g_{i})),\zeta(g_{i}),\rangle_{B_{r(g)}}-\langle\zeta(g_{i}),L_{g_{i}}(\xi(g_{i}))\rangle_{B_{r(g)}}+\langle\zeta(g_{i}),\zeta(g_{i})\rangle_{B_{r(g)}}$

which tends to

$\langle\zeta(g),\zeta(g)\rangle_{B_{r(g)}}-\langle\zeta(g),\zeta(g)\rangle_{B_{r(g)}}-\langle\zeta(g),\zeta(g)\rangle_{B_{r(g)}}+\langle\zeta(g),\zeta(g)\rangle_{B_{r(g)}}=0$

by continuity of all the maps involved. This proves that
$L_{g_{i}}(\xi_{i})\to L_{g}(\xi_{0})$ in $r^{*}\mathcal{E}$. ∎

For a twisted groupoid action $(\alpha,u_{\alpha})$ on $\mathcal{A}$ we may
think of a pair $(\overline{\alpha},u_{\alpha})$, where
$\overline{\alpha}:=\\{\overline{\alpha}_{g}\\}_{g\in\mathcal{G}}$ is a family
of standard strict extensions of the $*$-isomorphisms
$\alpha=\\{\alpha_{g}\\}_{g\in\mathcal{G}}$, as a ‘strictly continuous’
twisted action on $\mathcal{M}(\mathcal{A})$. Then any equivariant
representation of $\mathcal{G}$ extends to a ‘strictly continuous’ equivariant
representation between the extended actions as follows.

######  Proposition 5.4.

Any $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant representation
$L=\\{L_{g}\\}_{g\in\mathcal{G}}$ of $\mathcal{G}$ on $\mathcal{E}$ extends to
a ‘strictly continuous’
$(\overline{\alpha},u_{\alpha})$-$(\overline{\beta},u_{\beta})$ equivariant
representation $\overline{L}=\\{\overline{L}_{g}\\}_{g\in\mathcal{G}}$ on
$\mathcal{M}(\mathcal{E})$, in the sense that each isometry $L_{g}$ extends
uniquely to an isometry
$\overline{L}_{g}:\mathcal{M}(E_{s(g)})\to\mathcal{M}(E_{r(g)})$,
$g\in\mathcal{G}$, such that
$\overline{L}=\\{\overline{L}_{g}\\}_{g\in\mathcal{G}}$ satisfies the obvious
analogues of conditions (1) and (2) in Definition 5.1, and the following
analog of (3):

  1. (3’)

the section $\mathcal{G}\ni
g\longmapsto\langle\xi(r(g)),a(r(g))\overline{L}_{g}\zeta(s(g))\rangle_{\mathcal{M}(B_{r(g)})}\in
B_{r(g)}\subseteq r^{*}\mathcal{B}$ is continuous for all strictly continuous
sections $\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ and any
$a\in\mathrm{C_{c}}(\mathcal{A})$.

###### Proof.

For each $g\in\mathcal{G}$, $(L_{g},\alpha_{g},\beta_{g})$ is an isomorphism
from the $A_{s(g)}$-$B_{s(g)}$-bimodule $E_{s(g)}$ onto the
$A_{r(g)}$-$B_{r(g)}$-bimodule $E_{r(g)}$. Hence by Lemma 1.6, $L_{g}$ extends
uniquely to a linear map
$\overline{L}_{g}:\mathcal{M}(E_{s(g)})\to\mathcal{M}(E_{r(g)})$ such that

$\overline{L}_{g}(a\xi)=\overline{\alpha}_{g}(a)(\overline{L}_{g}\xi),\qquad\overline{L}_{g}(\xi
b)=\overline{L}_{g}(\xi)\overline{\beta}_{g}(b),$

for all $a\in\mathcal{M}(A_{s(g)})$, $b\in\mathcal{M}(B_{s(g)})$,
$\xi\in\mathcal{M}(E_{s(g)})$. Moreover, $\overline{L}_{g}$ is necessarily an
invertible isometry, and it is strictly continuous. Strict continuity readily
implies that the remaining conditions in Definition 5.1(1), (2) remain valid
when $L$ is replaced by $\overline{L}$. To show (3’) note that any
$a\in\mathrm{C_{c}}(\mathcal{A})$ may be written as the product $a=bc$ for
$b,c\in\mathrm{C_{c}}(\mathcal{A})$. Then for
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ we have

$\langle\xi(r(g)),a\overline{L}_{g}\zeta(s(g))\rangle_{\mathcal{M}(B_{r(g)})}=\langle
b^{*}(r(g))\xi(r(g)),L_{g}(\alpha_{g^{-1}}(c(r(g)))\zeta(s(g)))\rangle_{\mathcal{M}(B_{r(g)})}\in
B_{r(g)}$

and the section $\mathcal{G}\ni
g\longmapsto\langle\xi(r(g)),a(r(g))\overline{L}_{g}\zeta(s(g))\rangle_{\mathcal{M}(B_{r(g)})}\in
r^{*}\mathcal{B}$ is continuous because
$b^{*}\xi\in\mathrm{C_{c}}(\mathcal{E})$ and for each open bisection
$U\subseteq\mathcal{G}$ the map $s(U)\ni
s(g)\mapsto\alpha_{g^{-1}}(a(r(g)))\zeta(s(g))$ is a continuous section of
$\mathcal{E}|_{s(U)}$. ∎

######  Example 5.5.

For a twisted group action $(A,G,\alpha,u)$, that is, when $X=\\{e\\}$ and
$\mathcal{G}=G$ is a discrete group, the $\mathrm{C}^{*}$-correspondence
bundle is a single $\mathrm{C}^{*}$-correspondence $E$ over $A$ and
$(\alpha,u)$-$(\alpha,u)$ equivariant representations of $G$ on $E$, as in
Definition 5.1, generalize equivariant representations considered in [BC15] to
the non-unital case.

######  Example 5.6.

When the bundle $\mathcal{A}=\\{\mathbb{C}\\}_{x\in X}$ is trivial, so that a
twisted action $(\alpha,u)$ on $\mathcal{A}$ reduces to a continuous groupoid
$2$-cocycle $u$ (i.e. $\alpha$ is trivial, see Example 4.12), then
$(\alpha,u)$-$(\alpha,u)$ equivariant correspondences are _continuous Hilbert
$\mathcal{G}$-bundles_, i.e., continuous bundles
$\mathcal{H}=\\{H_{x}\\}_{x\in X}$ of Hilbert spaces together with a
continuous groupoid homomorphism $L$ from $\mathcal{G}$ to the unitary
groupoid of $\mathcal{H}$, meaning that $L_{g}\colon
H_{s(g)}\xrightarrow{\sim}H_{r(g)}$ are unitary isomorphisms with
$L_{g}L_{h}=L_{gh}$ for all $(g,h)\in\mathcal{G}^{2}$, and for continuous
sections $\xi,\eta\in\mathrm{C_{c}}(\mathcal{H})$, the map
$g\mapsto\langle\xi(r(g))\,|\,\mathopen{}L_{g}(\eta(g))\rangle$ defines a
continuous function on $\mathcal{G}$. In particular, we have the _left_ and
_right_ _regular continuous Hilbert $\mathcal{G}$-bundles_
$(\\{\ell^{2}(\mathcal{G}^{x})\\}_{x\in X},\lambda)$ and
$(\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in X},\rho)$ where the topology is
induced by the functions from $\mathrm{C_{c}}(\mathcal{G})$ and for each
$g\in\mathcal{G}$ the unitaries
$\lambda_{g}\colon\ell^{2}(\mathcal{G}^{s(g)})\xrightarrow{\sim}\ell^{2}(\mathcal{G}^{r(g)})$
and
$\rho_{g}\colon\ell^{2}(\mathcal{G}_{s(g)})\xrightarrow{\sim}\ell^{2}(\mathcal{G}_{r(g)})$
are given by

$\lambda_{g}(\delta_{h}):=\delta_{gh},\qquad\text{and}\qquad\rho_{g}(\delta_{h}):=\delta_{hg^{-1}},$

respectively. The inverse operation in $\mathcal{G}$ induces a unitary
equivalence between these bundles. Note that in the groupoid case (as opposed
to the situation for usual groups) the relationship between the equivariant
regular representations of $\mathcal{G}$ introduced here and the usual regular
representations of the associated $\mathrm{C}^{*}$-algebra, as discussed in
Example 4.12 is not straightforward.

######  Remark 5.7.

Equivariant correspondences in Example 5.6 do not depend on the cocycle $u$.
This is compatible with [BC16, Remark 3.2]. We also notice that our notion of
equivariant representation only deals with the ‘continuous part’ of the
representation theory of $\mathcal{G}$. This idea is also considered by
Paterson in [Pat04], while in [RW97] general measurable Hilbert bundles are
also taken into account by Ramsay and Walter in order to define
Fourier–Stieltjes algebras of general locally compact groupoids.

######  Example 5.8.

Consider a $2$-cocycle $u\in Z^{2}(G,\mathbb{T})$ on a group $G$ and the
trivial action $\operatorname{id}$ of $G$ on $A=\mathbb{C}$, so that
$(\operatorname{id},u)$ is a twisted action of $G$ on $\mathbb{C}$. To the
cocycle $u$ we can assign its left $u$-representation $\lambda^{u}\colon
G\to\mathcal{L}(\ell^{2}(G))$ by
$\lambda^{u}_{g}(\delta_{h}):=u(g,h)\delta_{gh}$ (which is a special case of
(4.2)). This is a unitary (projective) $u$-representation in the sense that
$\lambda^{u}_{g}\lambda^{u}_{h}=u(g,h)\lambda^{u}_{gh}$ for all $g,h\in G$.
Viewing $\ell^{2}(G)$ as a correspondence from $\mathcal{K}(\ell^{2}(G))$ to
$\mathbb{C}$, we may view $(\ell^{2}(G),\lambda^{u})$ as an equivariant
correspondence from the (untwisted) $G$-action
$(\mathcal{K}(\ell^{2}(G)),\operatorname{Ad}\lambda^{u})$ to the twisted
action $(\mathbb{C},\operatorname{id},u)$. This is an _equivariant Morita
equivalence_. Indeed, at least in the separable case every twisted group
action is Morita equivalent to an ordinary action by the Packer–Raeburn
stabilization trick [PR89]. A version of this also holds for twisted groupoid
actions, see [BMZ13, Theorem 5.3].

The following canonical constructions apply to equivariant correspondences:

######  Example 5.9.

Any twisted action $(\alpha,u)$ of $\mathcal{G}$ on $\mathcal{A}$ can be
viewed as the $(\alpha,u)$-$(\alpha,u)$-equivariant representation of
$\mathcal{G}$ on the trivial bundle of $\mathrm{C}^{*}$-correspondences
$\mathcal{E}=\mathcal{A}$ where $L=\alpha$. We call it _the trivial
equivariant representation_ of $(\alpha,u)$.

######  Example 5.10 (Tensoring with Hilbert bundles).

We may tensor any $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant
representation $L=\\{L_{g}\\}_{g\in\mathcal{G}}$ of $\mathcal{G}$ on
$\mathcal{E}$ with any continuous Hilbert $\mathcal{G}$-bundle
$(\mathcal{H},\ell)=(\\{H_{x}\\}_{x\in X},\\{\ell_{g}\\}_{g\in\mathcal{G}})$
to get another $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant
representation $L\otimes\ell$ on $\mathcal{E}$. Namely, for each $x\in X$ the
external tensor product $E_{x}\otimes H_{x}$ is naturally a
$\mathrm{C}^{*}$-correspondence from $A_{x}\otimes\mathbb{C}\cong A_{x}$ to
$B_{x}\otimes\mathbb{C}\cong B_{x}$, see [Lan95]. We equip
$\mathcal{E}\otimes\mathcal{H}:=\\{E_{x}\otimes H_{x}\\}_{x\in X}$ with the
unique topology such that the sections $X\ni x\to\xi(x)\otimes h(x)\in
E_{x}\otimes H_{x}$ are continuous for all $\xi\in\mathrm{C_{0}}(\mathcal{E})$
and $h\in\mathrm{C_{0}}(\mathcal{H})$. Then the maps

$L_{g}\otimes\ell_{g}(\xi\otimes h):=L_{g}(\xi)\otimes\ell_{g}(h),\qquad\xi\in
E_{s(g)},h\in H_{s(g)},\ g\in\mathcal{G},$

satisfy the conditions in Definition 5.1. Tensoring $(\mathcal{E},L)$ with
left or right regular Hilbert bundles, viewed as a $\mathcal{G}$-bundle, see
Example 5.6, we get equivariant correspondences equivalent to a _regularized_
version of $L$, see Definition 6.8 and Example 6.12 below.

######  Example 5.11 (Amplifications of equivariant correspondences).

Let $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ be twisted actions of
$\mathcal{G}$ on $\mathrm{C}^{*}$-bundles $\mathcal{A}$ and $\mathcal{B}$,
respectively. Let $L=\\{L_{g}\\}_{g\in\mathcal{G}}$ be an
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ _equivariant representation_ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}$ from
$\mathcal{A}$ to $\mathcal{B}$. Fix $n\in\mathbb{N}$. Then
$L^{(n)}=\\{L_{g}^{(n)}\\}_{g\in\mathcal{G}}$, with
$L_{g}^{(n)}:\mathbb{M}_{n}(E_{s(g)})\to\mathbb{M}_{n}(E_{r(g)})$ being the
entry-wise matrix lifting of $L_{g}$ becomes an
$(\alpha^{(n)},u_{\alpha}^{(n)})$-$(\beta^{(n)},u_{\beta}^{(n)})$ _equivariant
representation_ of $\mathcal{G}$ on a $\mathrm{C}^{*}$-correspondence bundle
$\mathbb{M}_{n}(\mathcal{E})$ from $\mathbb{M}_{n}(\mathcal{A})$ to
$\mathbb{M}_{n}(\mathcal{B})$ (see Examples 3.6 and 4.13). Once again checking
the algebraic and topological conditions is straightforward; moreover the
strict extensions $\overline{L_{g}^{{}_{(n)}}}$ constructed in Proposition 5.4
are the expected matrix liftings of $\overline{L}_{g}$.

######  Example 5.12 (Tensor product of equivariant correspondences).

Let $L$ be an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant
representation of $\mathcal{G}$ on $\mathcal{E}$ and $K$ a
$(\beta,u_{\beta})$-$(\gamma,u_{\gamma})$ equivariant representation of
$\mathcal{G}$ on $\mathcal{F}$ where $(\alpha,u_{\alpha})$,
$(\beta,u_{\beta})$ and $(\gamma,u_{\gamma})$ are twisted actions on
$\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ respectively. In such a
situation we will say that $L$ and $K$ are ‘composable’. We have an
$(\alpha,u_{\alpha})$-$(\gamma,u_{\gamma})$ equivariant representation
$L\otimes K$ on the tensor product $\mathrm{C}^{*}$-correspondence bundle
$\mathcal{E}\otimes_{\mathcal{B}}\mathcal{F}$, discussed in Example 3.10,
determined by

$(L_{g}\otimes
K_{g})(\xi\otimes\zeta):=(L_{g}\xi)\otimes(K_{g}\zeta),\qquad\xi\in E_{s(g)},\
\zeta\in F_{s(g)},\ g\in\mathcal{G}.$

Indeed, let us first note that $L_{g}\otimes K_{g}$ extends to an isometry
from $E_{s(g)}\otimes_{B_{s(g)}}F_{s(g)}$ to
$E_{r(g)}\otimes_{B_{r(g)}}F_{r(g)}$ because it satisfies the last relation
from (2) in Definition 5.1 on elementary tensors:

$\begin{split}\gamma_{g}(\langle\xi_{1}\otimes\zeta_{1},\xi_{2}\otimes\zeta_{2}\rangle_{C_{s(g)}})&=\gamma_{g}(\langle\zeta_{1},\langle\xi_{1},\xi_{2}\rangle_{B_{s(g)}}\zeta_{2}\rangle_{C_{s(g)}})\\\
&=\langle
K_{g}\zeta_{1},K_{g}(\langle\xi_{1},\xi_{2}\rangle_{B_{s(g)}}\zeta_{2})\rangle_{C_{r(g)}}\\\
&=\langle
K_{g}\zeta_{1},\beta_{g}(\langle\xi_{1},\xi_{2}\rangle_{B_{s(g)}})K_{g}\zeta_{2}\rangle_{C_{r(g)}}\\\
&=\langle K_{g}\zeta_{1},\langle
L_{g}\xi_{1},L_{g}\xi_{2}\rangle_{B_{r(g)}}K_{g}\zeta_{2}\rangle_{C_{r(g)}}\\\
&=\langle(L_{g}\xi_{1})\otimes(K_{g}\zeta_{1}),(L_{g}\xi_{2})\otimes(K_{g}\zeta_{2})\rangle_{C_{r(g)}}\\\
&=\langle(L_{g}\otimes K_{g})\xi_{1}\otimes\zeta_{1},(L_{g}\otimes
K_{g})\xi_{2}\otimes\zeta_{2}\rangle_{C_{r(g)}}.\end{split}$

The remaining relations in (2) in Definition 5.1 are immediate, and the
relation in (1) holds because we consider balanced tensor products. Condition
(3) holds because the section of $r^{*}\mathcal{C}$ sending $g\in\mathcal{G}$
to $\langle(L_{g}\otimes
K_{g})\xi_{1}\otimes\zeta_{1},\xi_{2}\otimes\zeta_{2}\rangle_{A_{r(g)}}=\langle
L_{g}\zeta_{1},\langle
K_{g}\xi_{1},\xi_{2}\rangle_{B_{r(g)}}\zeta_{2}\rangle_{C_{r(g)}}$ is
continuous for every $\xi_{1},\xi_{2}\in\mathrm{C_{c}}(\mathcal{E})$,
$\zeta_{1},\zeta_{2}\in\mathrm{C_{c}}(\mathcal{F})$.

Moreover, it is straightforward to see that the extension
$\overline{L_{g}\otimes
K_{g}}:\mathcal{M}(E_{s(g)}\otimes_{B_{s(g)}}F_{s(g)})\to\mathcal{M}(E_{r(g)}\otimes_{B_{r(g)}}F_{r(g)})$
of $L_{g}\otimes K_{g}$, as described in Proposition 5.4, is given on
elementary tensors defined in Lemma 1.12 by

$\overline{L_{g}\otimes
K_{g}}(\xi\otimes\zeta)=\overline{L}_{g}(\xi)\otimes\overline{K}_{g}(\zeta),\qquad\xi\in\mathcal{M}(E_{r(g)}),\,\zeta\in\mathcal{M}(F_{r(g)}).$

######  Example 5.13 (Direct sums of of equivariant correspondences).

For any family of $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant
representations $\\{L_{i}\\}_{i\in I}$ of the groupoid $\mathcal{G}$ on
$\mathrm{C}^{*}$-bundles $\\{\mathcal{E}_{i}\\}_{i\in I}$ from $\mathcal{A}$
to $\mathcal{B}$ the direct sum $\oplus_{i\in I}L_{i}:=\\{\oplus_{i\in
I}(L_{i})_{g}\\}_{g\in\mathcal{G}}$ is a well-defined
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant representation of
$\mathcal{G}$ on the direct sum bundle $\oplus_{i\in I}\mathcal{E}_{i}$. Also
we have a natural embedding $\oplus_{i\in
I}\mathcal{M}(\mathcal{E}_{i})\subseteq\mathcal{M}(\oplus_{i\in
I}\mathcal{E}_{i})$, which is equality when $I$ is finite, and the extended
action $\overline{\oplus_{i\in I}L_{i}}$ restricted to $\oplus_{i\in
I}\mathcal{M}(\mathcal{E}_{i})$ coincides with $\\{\oplus_{i\in
I}(\overline{L_{i}})_{g}\\}_{g\in\mathcal{G}}$.

###  5.1. Representations induced by equivariant correspondences

Equivariant representations allow us to transfer representations of the
crossed product for $(\beta,u_{\beta})$ to the crossed product for
$(\alpha,u_{\alpha})$. Firstly, let us notice that any such representation
$(\mathcal{E},L)$ can be interpreted in terms of inverse semigroups as
follows. For every open $V\subseteq X$ we put
$E_{V}:=\mathrm{C_{0}}(\mathcal{E}|_{V})$. For every
$U\in\operatorname{Bis}(\mathcal{G})$ we have an isometry $L_{U}:E_{s(U)}\to
E_{r(U)}$ where

$L_{U}(\xi)(r(g))=L_{g}(\xi(s(g))),\qquad g\in U,\ \xi\in E_{s(U)},$

see Lemma 5.3. Conditions (1) and (2) in Definition 5.1 imply that

$L_{U}(L_{V}(\xi))=\omega_{\alpha}(U,V)L_{UV}(\xi)\omega_{\beta}(U,V)^{*}$

for $\xi\in E_{s(UV)}$, and for $\xi,\zeta\in E_{s(U)}$, $a\in A_{s(U)}$,
$b\in B_{s(U)}$, and $\xi,\zeta\in E_{s(U)}$ we have

$L_{U}(a\xi)=\alpha_{U}(a)L_{U}(\xi),\qquad L_{U}(\xi
b)=L_{U}(\xi)\beta_{U}(b),\qquad\beta_{U}(\langle\xi,\zeta\rangle_{B_{s(U)}})=\langle
L_{U}(\xi),L_{U}(\zeta)\rangle_{B_{r(U)}}.$

In particular, for any $\xi\in\mathrm{C_{0}}(\mathcal{E})$ and $a\in A_{r(U)}$
we have a naturally defined element $aL_{U}(\xi)\in E_{r(U)}$, which agrees
with $L_{U}(\alpha_{U}^{-1}(a)\xi)$.

######  Proposition 5.14.

Let $L=\\{L_{g}\\}_{g\in\mathcal{G}}$ be an
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant representation of
$\mathcal{G}$ on $\mathcal{E}$ and let
$\psi:\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})\to\mathcal{L}(\mathcal{H})$
be a representation of $\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$ on a
right Hilbert $C$-module $\mathcal{H}$. Let
$\mathrm{C_{0}}(\mathcal{E})\otimes_{\psi}\mathcal{H}$ be the tensor product
of $\mathrm{C}^{*}$-correspondences where the left action of
$\mathrm{C_{0}}(\mathcal{B})$ on $\mathcal{H}$ is given by $\psi$. Then we get
a representation
$L\operatorname{-Ind}(\psi):\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathcal{L}(\mathrm{C_{0}}(\mathcal{E})\otimes_{\psi}\mathcal{H})$
determined by the formula

$L\operatorname{-Ind}(\psi)(a\delta_{U})\,\xi\otimes\zeta:=aL_{U}(\xi)\otimes\psi(\delta_{U})\zeta,\qquad\xi\in\mathrm{C_{0}}(\mathcal{E}),\zeta\in\mathcal{H},a\in
A_{r(U)},U\in\operatorname{Bis}(\mathcal{G}),$

where
$aL_{U}(\xi)\otimes\psi(\delta_{U})\zeta:=\lim_{i}L_{U}(\alpha_{U}^{-1}(a)\xi)\otimes\psi(e_{i}^{r(U)}\delta_{U})\zeta$
for an approximate unit $\\{e_{i}^{r(U)}\\}_{i}$ in $B_{r(U)}$. This limit
exists and does not depend on the choice of $\\{e_{i}\\}_{i}$.

###### Proof.

If $H$ were a Hilbert space (equivalently $C=\mathbb{C}$), then we could
disintegrate $\psi$ into a covariant representation $(\pi,v)$, as described in
Subsection 4.1, and then put
$L\operatorname{-Ind}(\psi)(a\delta_{U})\,\xi\otimes\zeta=aL_{U}(\xi)\otimes
v_{U}\zeta$ where the corresponding partial isometry $v_{U}\in\mathbb{B}(H)$
is given by the strong limit of $\psi(e_{i}^{r(U)}\delta_{U})$. Here we work
with a general Hilbert module $\mathcal{H}$ (with a view towards one of our
versions of the Fell absorption principle — see Theorem 6.15 below) where such
a strong limit does not necessarily exist, so we need to struggle with
approximate units.

Let us first note that the limit
$\lim_{i}L_{U}(\alpha_{U}^{-1}(a)\xi)\otimes\psi(e_{i}^{r(U)}\delta_{U})\zeta$
exists and does not depend on the choice of the approximate unit. Indeed, we
may write $aL_{U}(\xi)=\kappa b$ where $\kappa\in E_{r(U)}$ and $b\in
B_{r(U)}$, and then
$\lim_{i}aL_{U}(\xi)\otimes\psi(e_{i}^{r(U)}\delta_{U})\zeta=\lim_{i}\kappa\otimes\psi(be_{i}^{r(U)}\delta_{U})\zeta=\kappa\otimes\psi(b\delta_{U})\zeta$.
Let $\xi_{j}\in\mathrm{C_{0}}(\mathcal{E})$ and $\zeta_{j}\in\mathcal{H}$ for
$j=1,\ldots,n$. A simple calculation shows that for $j,k\in\\{1,\ldots,n\\}$
we have

$\langle
L\operatorname{-Ind}(\psi)(a\delta_{U})\,\xi_{j}\otimes\zeta_{j},L\operatorname{-Ind}(\psi)(a\delta_{U})\,\xi_{k}\otimes\zeta_{k}\rangle_{C}=\langle\zeta_{j},\psi(\langle\xi_{j},\alpha_{U}^{-1}(a^{*}a)\xi_{k}\rangle)\zeta_{k}\rangle_{C}.$

Since
$\sum_{j,k=1}^{n}\langle\zeta_{j},\psi(\langle\xi_{j},\alpha_{U}^{-1}(a^{*}a)\xi_{k}\rangle)\zeta_{k}\rangle_{C}\leq\sum_{j,k=1}^{n}\langle\zeta_{j},\psi(\langle\xi_{j},\|a^{*}a\|\xi_{k}\rangle_{C})\zeta_{k}\rangle$
it follows that

$\Bigg{\|}L\operatorname{-Ind}(\psi)(a\delta_{U})\left(\sum_{i=1}^{n}\xi_{i}\otimes\zeta_{i}\right)\Bigg{\|}^{2}\leq\|a\|^{2}\Bigg{\|}\sum_{i=1}^{n}\xi_{i}\otimes\zeta_{i}\Bigg{\|}^{2}.$

Hence $L\operatorname{-Ind}(\psi)(a\delta_{U})$ is a well defined contractive
map on $\mathrm{C_{0}}(\mathcal{E})\otimes_{\psi}\mathcal{H}$. As in Remark
5.2 we get
$L_{U}^{-1}(\xi)=\omega_{\alpha}(U^{*},U)^{*}L_{U^{*}}(\xi)\omega_{\beta}(U^{*},U)$
for $\xi\in E_{r(U)}$. Thus for $\xi,\xi^{\prime}\in
E=\mathrm{C_{0}}(\mathcal{E})$ we have

$\begin{split}\langle aL_{U}(\xi),\xi^{\prime}\rangle_{B}&=\lim_{i}\langle
aL_{U}(\xi),e_{i}^{r(U)}\xi^{\prime}\rangle_{B}=\lim_{i}\langle
L_{U}(\alpha_{U}^{-1}(a)\xi),L_{U}(L_{U}^{-1}(e_{i}^{r(U)}\xi^{\prime}))\rangle_{B_{r(U)}}\\\
&=\lim_{i}\beta_{U}\Big{(}\langle\xi,\alpha_{U}^{-1}(a^{*})L_{U}^{-1}(e_{i}^{r(U)}\xi^{\prime})\rangle_{B_{s(U)}}\Big{)}\\\
&=\beta_{U}\Big{(}\langle\xi,\alpha_{U}^{-1}(a^{*})\omega_{\alpha}(U^{*},U)^{*}L_{U^{*}}(\xi^{\prime})\rangle_{B_{s(U)}}\omega_{\beta}(U^{*},U)\Big{)}.\end{split}$

Hence

$\begin{split}\lim_{i}(e_{i}^{r(U)}\delta_{U})^{*}\langle
aL_{U}(\xi),\xi^{\prime}\rangle_{B}&=\lim_{i}\beta_{U}^{-1}(e_{i}^{r(U)})\omega_{\beta}(U^{*},U)^{*}\delta_{U^{*}}\langle
aL_{U}(\xi),\xi^{\prime}\rangle_{B}\\\ &=\beta_{U}^{-1}(\langle
aL_{U}(\xi),\xi^{\prime}\rangle_{B})\omega_{\beta}(U^{*},U)^{*}\delta_{U^{*}}\\\
&=\langle\xi,\alpha_{U}^{-1}(a^{*})\omega_{\alpha}(U^{*},U)^{*}L_{U^{*}}(\xi^{\prime})\rangle_{B_{s(U)}}\delta_{U^{*}}.\end{split}$

Using this for $\zeta,\zeta^{\prime}\in\mathcal{H}$ we get

$\begin{split}\langle
aL_{U}(\xi)\otimes\psi(\delta_{U})\zeta,\xi^{\prime}\otimes\zeta^{\prime}\rangle_{C}&=\lim_{i}\langle\psi(e_{i}^{r(U)}\delta_{U})\zeta,\psi(\langle
aL_{U}(\xi),\xi^{\prime}\rangle_{B})\zeta^{\prime}\rangle_{C}\\\
&=\lim_{i}\langle\zeta,\psi((e_{i}^{r(U)}\delta_{U})^{*}\langle
aL_{U}(\xi),\xi^{\prime}\rangle_{B})\zeta^{\prime}\rangle_{C}\\\
&=\langle\zeta,\psi(\langle\xi,\alpha_{U}^{-1}(a^{*})\omega_{\alpha}(U^{*},U)^{*}L_{U^{*}}(\xi^{\prime})\rangle_{B_{s(U)}}\delta_{U^{*}})\zeta^{\prime}\rangle_{C}\\\
&=\langle\xi\otimes\zeta,\alpha_{U}^{-1}(a^{*})\omega_{\alpha}(U^{*},U)^{*}L_{U^{*}}(\xi^{\prime})\otimes\psi(\delta_{U^{*}})\zeta^{\prime}\rangle_{C}.\end{split}$

This shows that
$L\operatorname{-Ind}(\psi)(a\delta_{U})\in\mathcal{L}(\mathrm{C_{0}}(\mathcal{E})\otimes_{\psi}\mathcal{H})$
and

$L\operatorname{-Ind}(\psi)(a\delta_{U})^{*}=L\operatorname{-Ind}(\psi)(\alpha_{U}^{-1}(a^{*})\omega_{\alpha}(U^{*},U)^{*}\delta_{U^{*}})=L\operatorname{-Ind}(\psi)((a\delta_{U})^{*}).$

Now if $b\in A_{r(V)}$ for $V\in\operatorname{Bis}(\mathcal{G})$, then

$\begin{split}L\operatorname{-Ind}(\psi)&(a\delta_{U})[L\operatorname{-Ind}(\psi)(b\delta_{V})\xi\otimes\zeta]=\lim_{i}\lim_{j}aL_{U}(bL_{V}(\xi))\otimes\psi(e_{i}^{r(U)}\delta_{U})\psi(e_{j}^{r(V)}\delta_{V})\zeta\\\
&=\lim_{i}\lim_{j}\alpha_{U}(\alpha_{U}^{-1}(a)b)L_{U}(L_{V}(\xi))\otimes\psi\Big{(}\beta_{U}\big{(}\beta_{U}^{-1}(e_{i}^{r(U)})e_{j}^{r(V)}\big{)}\omega_{\beta}(U,V)\delta_{UV}\Big{)}\zeta\\\
&=\lim_{i}\alpha_{U}(\alpha_{U}^{-1}(a)b)\omega_{\alpha}(U,V)L_{UV}(\xi)\otimes\psi\Big{(}\omega_{\beta}(U,V)^{*}e_{i}^{r(UV)}\omega_{\beta}(U,V)\delta_{UV}\Big{)}\zeta\\\
&=L\operatorname{-Ind}(\psi)[(a\delta_{U})\cdot(b\delta_{V})](\xi\otimes\zeta).\end{split}$

Thus $L\operatorname{-Ind}(\psi)(a\delta_{U})\cdot
L\operatorname{-Ind}(\psi)(b\delta_{V})=L\operatorname{-Ind}(\psi)((a\delta_{U})\cdot(b\delta_{V}))$,
and the assertion now follows, cf. Subsection 4.1. ∎

######  Example 5.15.

For group actions on unital $C^{*}$-algebra $\mathrm{\mathrm{C}^{*}}$-algebras
as in [BC12, BC15, BC16, BC21], the above proposition takes the following
form. Let $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ be two twisted actions
of a group $G$ on unital $\mathrm{\mathrm{C}^{*}}$-algebras $A$ and $B$, let
$L$ be an equivariant representation on an
$A$-$B$-$\mathrm{\mathrm{C}^{*}}$-correspondence $E$. Given a representation
$\psi\colon B\rtimes_{(\beta,u_{\beta})}G\to\mathcal{L}(\mathcal{H})$ on a
Hilbert $C$-module $\mathcal{H}$, there is a representation
$L\operatorname{-Ind}(\psi)\colon
A\rtimes_{(\alpha,u_{\alpha})}G\to\mathcal{L}(E\otimes_{\psi}\mathcal{H})$
which on the generators $a\delta_{g}\in A\rtimes_{(\alpha,u_{\alpha})}G$,
$a\in A$, $g\in G$, is given by the formula

$L\operatorname{-Ind}(\psi)(a\delta_{g})(\xi\otimes\zeta)=aL_{g}(\xi)\otimes\psi(\delta_{g})\zeta,\quad\xi\in
E,\eta\in H.$

##  6\. Induced regular representations and absorption principles

In this section we will discuss induced/regular representations of bundle