# Banach algebras associated to twisted étale groupoids:
inverse semigroup disintegration
and representations on $L^{p}$-spaces

Krzysztof Bardadyn  ,  Bartosz Kwaśniewski  and  Andrew McKee

###### Abstract.

We introduce Banach algebras associated to a twisted étale groupoid
$(\mathcal{G},\mathcal{L})$ and to twisted inverse semigroup actions. This
provides a unifying framework for numerous recent papers on $L^{p}$-operator
algebras and the theory of groupoid $C^{*}$-algebras. We prove a
disintegration theorem that allows us to study Banach algebras associated to
$(\mathcal{G},\mathcal{L})$ as universal Banach algebras generated by
$C_{0}(X)$ and a twisted inverse semigroup $S$ of partial isometries subject
to some relations. Our theory works best when the groupoid is ample or when
the target of a representation is a dual Banach algebra $B$, so for instance
$B=B(E)$ where $E$ is a reflexive Banach space. In particular, it implies that
in the constructions of $L^{p}$-analogues of Cuntz or graph algebras, by
Phillips and Cortiñas–Rodríguez, the use of _spatial partial isometries_ is
not an assumption, in fact it is forced by the relations.

We establish fundamental norm estimates and hierarchy for full and reduced
$L^{p}$-operator algebras for $(\mathcal{G},\mathcal{L})$ and
$p\in[1,\infty]$, whose special cases have been studied recently by
Gardella–Lupini, Choi–Gardella–Thiel and Hetland–Ortega. We also introduce
tight inverse semigroup Banach algebras that cover ample groupoid Banach
algebras, and we discuss Banach algebras associated to twisted partial group
actions, twisted Renault–Deaconu groupoids and directed graphs. Our Banach
Cuntz algebras are related to the construction of Daws–Horváth.

We aim for a high degree of generality throughout: our results cover non-
Hausdorff étale groupoids and both real and complex algebras. Some of our
results seem to be absent from the literature even for $C^{*}$-algebras.

###### 2000 Mathematics Subject Classification:

47L10, 46H15, 22A22

##  1\. Introduction

The theory of Banach algebras generated by groups is a classical part of
harmonic analysis. In the realm of $C^{*}$-algebras it was extended with huge
success to transformation groups, groupoids and even more general structures
and actions. This inspired a number of attempts to generalize this theory to
Banach algebras or at least to operator algebras acting on $L^{p}$-spaces. A
series of initial preprints about Banach algebra crossed products and
$L^{p}$-operator algebras appeared in the 2010s, see [26], [59], [61], [60].
Phillips’s program to create an analogue of the $C^{*}$-theory for
$L^{p}$-operator algebras has noticeably sparked in recent years, see the
survey paper [35] and/or [42], [36], [22], [19], [39], [5], [43]. A number of
significant $C^{*}$-algebraic constructions and results have been carried over
to the $L^{p}$-setting. For $p\neq 2$ the proofs usually require different
techniques, and also some new phenomena occur. An initial motivation behind
the present paper was to solve [36, Problem 8.2] which asks whether a standard
simplicity criterion for groupoid $C^{*}$-algebras works for $L^{p}$-groupoid
algebras. Working on this, we realized we can develop a neat theory of Banach
algebras associated to twisted étale groupoids that not only allows us to
prove strong results, such as simplicity and pure infiniteness, see [6], [7],
but also gives a deep insight into representation theory that allows us to
define and study interesting Banach algebras in terms of generators and
relations. This yields a general theoretical background that covers
$L^{p}$-analogs of Cuntz algebras [61], graph $L^{p}$-algebras [22], Cuntz-
like Banach algebras [25], (twisted) transformation group Banach algebras
[26], [60], AF-algebras, $\ell^{p}$-uniform Roe algebras [76], [20], groupoid
$L^{p}$-operator algebras [36], [19], or reduced twisted groupoid Banach
algebras [5], [43], which have so far often been studied using ad hoc methods.
This theory has a potential to be applied to numerous other constructions. In
fact, in the present paper we introduce Banach algebras associated to inverse
semigroups, directed graphs, twisted partial local homeomorphisms, and twisted
partial group actions. Also, since all classifiable simple $C^{*}$-algebras
are modeled by twisted étale groupoids [54], it is reasonable to expect such
models to play a crucial role in other contexts. On top of that we consider
general Banach algebras over the field $\mathbb{F}=\mathbb{R},\mathbb{C}$ of
real or complex numbers, so in particular our results apply to real
$C^{*}$-algebras, which has recently have been experiencing a renaissance due
to numerous important applications, see [65], [10] and references therein.
Real crossed products and real Cuntz $C^{*}$-algebras were already considered,
see [67], [65].

One of the fundamental ideas, that stands behind the success of the theory of
$C^{*}$-algebras associated to étale groupoid $C^{*}$-algebras, and Cartan
$C^{*}$-subalgebras in particular, is that algebras associated to such
groupoids can be viewed as crossed products by actions of (discrete) inverse
semigroups, cf. [63], [57], [70], [46], [31], [15], [16], [51]. We make this
idea a starting point and the main tool of our analysis. In particular, we
generalize the construction of crossed products by twisted inverse semigroup
actions for $C^{*}$-algebras [69], [15], to the realm of Banach algebras.
Since the category of (real or complex) $C^{*}$-algebras with
$*$-homomorphisms as morphisms is a full subcategory of Banach algebras with
contractive homomorphisms as morphisms, we decided to work in this Banach
algebra category. Thus by a _representation of a Banach algebra_ $A$ in a
Banach algebra $B$ we mean a contractive homomorphism $\pi:A\to B$. For an
inverse semigroup $S$ a representation is a semigroup homomorphisms $v:S\to
B_{1}\subseteq B$ into contractive elements of $B$. Then the operators
$\\{v_{t}\\}_{t\in S}\subseteq B$ are _partial isometries_ in the sense of
Mbekhta [55] (which are the usual partial isometries when $B$ is a
$C^{*}$-algebra). For a _twisted action_ $(\alpha,u)$, à la Buss–Exel [15], of
the inverse semigroup $S$ on a Banach algebra $A$ we define the crossed
product $A\rtimes_{(\alpha,u)}S$ as a Banach algebra universal for _covariant
representations_ $(\pi,v)$ of $(\alpha,u)$, where $\pi$ is a representation of
$A$ in $B$ and $v$ is a twisted representation of $S$ in $B$ satisfying
natural relations similar to those in [69], [15]. To get uniqueness of
_disintegration_ of representations of $A\rtimes_{(\alpha,u)}S$ we need to
impose a _normalization_ condition on covariant representations. In different
situations this could mean different things. When the domains of the action
are not unital this requires weak limits and so normalization works well for
instance when $(B,B_{*})$ is a dual Banach algebra in the sense of Runde, see
[66], [24]. Reducing to covariant representations on Banach spaces $E$ is even
more delicate. We discuss these issues in detail in Section 3.

For a twisted étale groupoid $(\mathcal{G},\mathcal{L})$, with a locally
compact Hausdorff unit space $X$, the Connes $*$-algebra
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ of quasi-continuous compactly
supported sections has plenty of natural Banach algebra completions. We
decided to work only with norms that coincide with the supremum norm
$\|\cdot\|_{\infty}$ for sections with supports in open bisections (apparently
all norms considered so far in the literature have this property). For any
inverse semigroup $S$ of open bisections covering $\mathcal{G}$ we define the
_twisted groupoid Banach algebra_ $F^{S}(\mathcal{G},\mathcal{L})$ of
$(\mathcal{G},\mathcal{L})$. If $S$ is wide, we have an inverse semigroup
action $(\alpha,u)$ of $S$ on $C_{0}(X)$ that can be used to recover
$(\mathcal{G},\mathcal{L})$ (this is implicit in [15], [16], and we make it
explicit in subsection 4.3). One of our main results (see Theorem 4.27) shows
that then we have an isometric isomorphism

$F^{S}(\mathcal{G},\mathcal{L})\cong C_{0}(X)\rtimes_{(\alpha,u)}S$

that establishes a bijective correspondence between representations of
$F^{S}(\mathcal{G},\mathcal{L})$ and normalized covariant representations of
$(\alpha,u)$. This result can be applied to Banach algebras
$F_{\mathcal{R}}^{S}(\mathcal{G},\mathcal{L})$ which are universal for
representations of $F^{S}(\mathcal{G},\mathcal{L})$ for a fixed class
$\mathcal{R}$ of representations. For $p\in[1,\infty]$ we define the
_universal twisted groupoid $L^{p}$-operator algebra_
$F^{p}(\mathcal{G},\mathcal{L})$ as
$F_{\mathcal{R}}(\mathcal{G},\mathcal{L})$, where $\mathcal{R}$ is the class
of all representations on spaces $L^{p}(\mu)$ whose restriction to $C_{0}(X)$
acts by multiplication operators. Here the main structural result (Theorem
5.10) is that this definition does not depend on the choice of $S$, and in the
complex case $F^{p}(\mathcal{G},\mathcal{L})$ is universal for representations
of $F^{S}(\mathcal{G},\mathcal{L})$ in all $L^{p}$-operator algebras.
Moreover, denoting by $\|\cdot\|_{L^{p}}$ the norm in
$F^{p}(\mathcal{G},\mathcal{L})$, for all $p,q\in[1,\infty]$ with $1/p+1/q=1$,
we have the following estimates

$\|f\|_{L^{p}}\leq\|f\|_{L^{1}}^{1/p}\,\|f\|_{L^{\infty}}^{1/q}\leq\|f\|_{I},\qquad
f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$

where

$\|f\|_{L^{1}}:=\max_{x\in
X}\sum_{d(\gamma)=x}|f(\gamma)|,\qquad\|f\|_{L^{\infty}}:=\max_{x\in
X}\sum_{r(\gamma)=x}|f(\gamma)|$

and $\|f\|_{I}:=\max\\{\|f\|_{L^{1}},\|f\|_{L^{\infty}}\\}$ is Hahn’s _$I$
-norm_. These non-trivial estimates solve in particular the problem formulated
below [19, Propositon 6.5]. The algebra $F^{p}(\mathcal{G},\mathcal{L})$ is a
universal model for _the reduced twisted groupoid $L^{p}$-operator algebra_
$F^{p}_{\operatorname{r}}(\mathcal{G},\mathcal{L})$ given by a concrete
representation which was recently studied by Hetland–Ortega [43], and by
Choi–Gardella–Thiel [19]. This also covers untwisted groupoid $L^{p}$-operator
algebra $F^{p}(\mathcal{G})$ introduced by Gardella–Lupini [36] (in the
complex, separable, finite $p$ case).

The hierarchy of the discussed algebras is summarized in Figure 1, where all
algebras are completions of $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, in
particular,
$F_{I}(\mathcal{G},\mathcal{L}):=\overline{C_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{I}}$,
and $C^{*}(\mathcal{G},\mathcal{L})$ and
$C^{*}_{\operatorname{r}}(\mathcal{G},\mathcal{L})$ are standard
$C^{*}$-algebras associated to $(\mathcal{G},\mathcal{L})$. The algebras in
the middle column are Banach $*$-algebras, the horizontal anti-isomorphisms
are given by the involution in $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$,
$1/p+1/q=1$, and the downward arrows are representations extending identities
on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$:

$\textstyle{F^{S}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{I}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{p}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\stackrel{{\scriptstyle\mathrm{anti}}}{{\cong}}}$$\textstyle{F^{2}(\mathcal{G},\mathcal{L})=C^{*}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F^{q}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\infty}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{p}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{2}_{\operatorname{r}}(\mathcal{G},\mathcal{L})=C^{*}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{q}_{\operatorname{r}}(\mathcal{G},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\stackrel{{\scriptstyle\mathrm{anti}}}{{\cong}}}$$\textstyle{F^{\infty}_{\operatorname{r}}(\mathcal{G},\mathcal{L})}$

Figure 1. Hierarchy of twisted groupoid Banach algebras

We also give a geometric description of representations of
$F^{p}(\mathcal{G},\mathcal{L})$ on spaces $L^{p}(\mu)$, where $\mu$ is a
localizable measure. Following Phillips [59] and Gardella [35] we define
_spatial partial isometries_ as certain _weighted composition operators_ on
$L^{p}(\mu)$. They form a natural inverse semigroup
$\operatorname{SPIso}(L^{p}(\mu))$ of partial isometries in $B(L^{p}(\mu))$.
In fact, using the well established notion of an _$L^{p}$ -projection_, see
[3], [8], we define _$L^{p}$ -partial isometries_ on an arbitrary Banach space
$E$. For $p\neq 2$, they form an inverse semigroup, and when $E=L^{p}(\mu)$
they coincide with spatial partial isometries by the (generalized) Lamperti
Theorem (Theorem 2.27). Our third main result (Theorem 5.15) says that for
every $p\in[1,\infty]$, the algebra $F^{p}(\mathcal{G},\mathcal{L})$ is
universal for _spatial covariant representations_ , i.e. covariant
representations $(\pi,v)$ where $v:S\to\operatorname{SPIso}(L^{p}(\mu))$ takes
values in spatial partial isometries and $\pi:C_{0}(X)\to B(L^{p}(\mu))$ acts
by multiplication operators (by functions from $L^{\infty}(\mu)$). This in
particular gives a natural bijective correspondence between spatial
representations of $F^{p}(\mathcal{G},\mathcal{L})$ and
$F^{p^{\prime}}(\mathcal{G},\mathcal{L})$ for any $p,p^{\prime}\in(1,\infty)$,
which is quite surprising as in many cases there is no non-zero continuous
homomorphism from $F^{p}(\mathcal{G},\mathcal{L})$ to
$F^{p^{\prime}}(\mathcal{G},\mathcal{L})$. In the complex case, for
$p\in(1,\infty)\setminus\\{2\\}$, every non-degenerate representation of
$F^{p}(\mathcal{G},\mathcal{L})$ on $L^{p}(\mu)$ is spatial.

As an illustration, and having future applications in mind, we introduce the
universal and the _tight Banach algebra of an inverse semigroup_ $S$. The new
paradigm that appears here is that for general representations $v:S\to B$ into
a Banach algebra $B$ we need to impose a condition that certain associated
projections are _jointly $\mathbb{F}$-contractive_ (see Definitions 6.4 and
6.6), which is related to a number of similar conditions studied by various
authors, cf. [3], [9], [45], [72]. This condition is automatic when $B$ is a
$C^{*}$-algebra or when $v:S\to\operatorname{SPIso}(L^{p}(\mu))$ takes values
in spatial partial isometries, and therefore has not been visible in previous
works. The tight algebras are modeled by the tight groupoid of Exel [31].
Based on the algebraic setting of Steinberg–Szakács [74] and our results, we
characterise when representations of such algebras are isometric on the unit
subalgebra, which leads to convenient improvements even for the already
existing theory of tight $C^{*}$-algebras.

We define the _Banach algebra $F(Q)$ of a directed graph_ $Q$ as the universal
algebra for a Banach $Q$-family of partial isometries and projections. It is
naturally a groupoid Banach algebra and a tight inverse semigroup Banach
algebra. Our results imply that non-degenerate representations of the complex
Banach algebra $F(Q)$ on $L^{p}$-spaces, for $p\in(1,\infty)\setminus\\{2\\}$,
are necessarily given by $Q$-families with values in spatial partial
isometries $\operatorname{SPIso}(L^{p}(\mu))$. Thus the use of spatial partial
isometries in the definition of the graph $L^{p}$-operator algebras $F^{p}(Q)$
in [22] and [59] is not an assumption — it is forced by the groupoid model.
When $Q$ has only one vertex, $F(Q)$ is a Banach Cuntz algebra which seems to
be more tractable than the associated Cuntz-like Banach $*$-algebra of
Daws–Horváth [25] (cf. Remark 7.24).

In addition we describe representations of Banach algebras associated with a
_twisted partial action of a discrete group_ and with a _twisted
Deaconu–Reanult groupoid_. In the presented generality the results are new
even for $C^{*}$-algebras.

The paper is organized as follows. After preliminaries on $L^{p}$-operator
algebras, groupoids and partial isometries in Banach algebras, in Section 2,
we define crossed products for twisted inverse semigroup actions on Banach
algebras in Section 3. Section 4 introduces the title concept — twisted
groupoid Banach algebras, and presents the general inverse semigroup
disintegration theorem. In Section 5 we analyze $L^{p}$-operator algebras
associated to $(\mathcal{G},\mathcal{L})$. Section 6 is devoted to inverse
semigroup algebras and tight groupoids. Finally, in Section 7 we discuss
partial crossed products, Deaconu–Reanult groupoids and graph algebras.

### Acknowledgements

The first two named authors were supported by the National Science Center
(NCN) grant no. 2019/35/B/ST1/02684. The second named author would like to
thank Chris Phillips, Eusebio Gardella and Hannes Thiel for sharing their
expert knowledge so generously, during various meetings and via e-mail. We
also thank Jan Gundelach for the careful reading of the first draft of the
paper and his comments.

##  2\. Preliminaries

Most of the facts presented here are probably known to experts, so we included
only sketches of, or coordinates leading to, the arguments when we were not
able to find a reference in appropriate generality. We discuss representations
of $C_{0}(X)$ on $L^{p}$-spaces with $p\in[1,\infty]$, and the key result here
is Theorem 2.12. For an introduction to $L^{p}$-operator algebras we refer to
[35]. We generalize spatial partial isometries of Phillips and Gardella by
defining $L^{p}$-partial isometries on an arbitrary Banach space $E$. The link
between the two is provided by a version of Banach-Lamperti theorem (Theorem
2.27). Also we recall the fundamental relationship between actions of inverse
semigroups on topological spaces and étale groupoids.

###  2.1. Spatial isometries on $L^{p}$-spaces

Throughout we fix $p\in[1,\infty]$ and the field
$\mathbb{F}=\mathbb{R},\mathbb{C}$ of real or complex numbers. If
$(\Omega,\Sigma,\mu)$ is a measure space we write $L^{p}(\mu)$ for the
corresponding Lebesgue space viewed as a Banach space over $\mathbb{F}$.
Occasionally, to emphasize which field is used, we will write
$L^{p}(\mu,\mathbb{F})$. Recall that the Banach space $L^{p}(\mu)$ can be
recovered from $\mu:\Sigma\to[0,+\infty]$, or in fact from the measure
$[\mu]:[\Sigma]\to[0,+\infty]$ defined on the Boolean algebra $[\Sigma]$
obtained from $\Sigma$ by identifying sets of measure zero (the presentation
in [35], [39] uses this ‘point-free’ approach to $L^{p}$-spaces, but we stay
with the old-fashioned approach). Also if $p<\infty$, then $L^{p}(\mu)\cong
L^{p}(\mu_{f})$ where $\mu_{f}(A)=\sup\\{\mu(B):A\supseteq
B\in\Sigma,\mu(B)<\infty\\}$ is the _semi-finite_ part of $\mu$, and the
isomorphism is identity on integrable simple functions. Recall that a measure
$\mu$ is called _localizable_ if it is semi-finite and $[\Sigma]$ is Dedekind
complete. In general, if $\mu$ is semi-finite, then $[\mu]$ extends uniquely
to the Dedekind completion of $[\Sigma]$ giving a localizable measure on the
completion [34]. Using the Loomis–Sikorski Theorem we may represent this
abstract measure by a concrete localizable measure space
$(\overline{\Omega},\overline{\Sigma},\overline{\mu})$ (see [34], [11]) and
again $L^{p}(\mu)\cong L^{p}(\overline{\mu})$ by the isometric isomorphism
sending characteristic functions to characteristic functions. Thus when
considering $L^{p}$-spaces with $p<\infty$ we may always assume the measure is
localizable, or even decomposable (strictly localizable), see [52, Corollary
on page 136] or [11, Theorem 7.17]). For $p=\infty$ the assumption that $\mu$
is localizable is also natural. For instance, $\mu$ is localizable if and only
if $L^{\infty}(\mu)$ is canonically isomorphic to the dual of $L^{1}(\mu)$ if
and only if $L^{\infty}(\mu)$ is canonically a von Neumann algebra acting on
$L^{2}(\mu)$ (and every abelian $W^{*}$-algebra is of this form), see [11] and
references therein.

######  Definition 2.1.

A _set isomorphism_ from a measure space $(\Omega_{\mu},\Sigma_{\mu},\mu)$
onto a measure space $(\Omega_{\nu},\Sigma_{\nu},\nu)$ is a map
$\Phi:\Sigma_{\nu}\to\Sigma_{\mu}$ that descends to an isomorphism of Boolean
algebras $[\Phi]:[\Sigma_{\nu}]\to[\Sigma_{\mu}]$. Thus, since the Boolean
structure is determined by its preorder, it suffices to assume

  1. (1)

$\nu(C\setminus D)=0$ if and only if $\mu(\Phi(C)\setminus\Phi(D))=0$ for all
$C,D\in\Sigma_{\nu}$;

  2. (2)

for each $C\in\Sigma_{\mu}$ there is $\Phi^{*}(C)\in\Sigma_{\nu}$ such that
$\Phi(\Phi^{*}(C))=C$ up to a $\mu$-null set.

Condition (2) gives a map $\Phi^{*}:\Sigma_{\mu}\to\Sigma_{\nu}$, which is a
set isomorphism that descends to an isomorphism
$[\Phi^{*}]:[\Sigma_{\mu}]\to[\Sigma_{\nu}]$ inverse to
$[\Phi]:[\Sigma_{\nu}]\to[\Sigma_{\mu}]$.

We denote by $L_{0}(\mu)$ the algebra of $\mathbb{F}$-valued measurable
functions identified up to equality $\mu$-almost everywhere. A set isomorphism
$\Phi:\Sigma_{\nu}\to\Sigma_{\mu}$ uniquely defines a linear operator
$T_{\Phi}:L_{0}(\nu)\to L_{0}(\mu)$ such that

$T_{\Phi}1_{C}=1_{\Phi(C)},\qquad C\in\Sigma_{\nu},$

and $T_{\Phi}$ preserves (monotone) limits, cf. [59, Proposition 5.6]. This is
an algebra isomorphism $L_{0}(\nu)\cong L_{0}(\mu)$ whose inverse is
$T_{\Phi^{*}}$. We call $T_{\Phi}$ a (generalized) _composition operator_. It
preserves the range of functions and in particular it restricts to an
isometric isomorphism $L^{\infty}(\nu)\cong L^{\infty}(\mu)$ [11, Proposition
7.16]. However, unless $\Phi$ preserves measures, it does not preserve the
$L^{p}$-spaces for $p<\infty$. To force this, we will consider _weighted
composition operators_ $aT_{\Phi}:L_{0}(\nu)\to L_{0}(\mu)$ where $a\in
L_{0}(\mu)$ and $[aT_{\Phi}]\xi(x):=a(x)T_{\Phi}(\xi)(x)$. Note that
$\mu\circ\Phi$ is a measure equivalent to $\nu$. Thus if $\nu$ is localizable
then the Radon–Nikodym derivative $\frac{\mu\circ\Phi}{\nu}$ exists, and is
strictly positive $\nu$-almost everywhere, see [39, Theorem 2.7], [11, Theorem
4.4]. Similar comments apply to the measures $\nu\circ\Phi^{*}$ and $\mu$.

######  Lemma 2.2.

Consider localizable measures $\mu$ and $\nu$ and $p\in[1,\infty]$. For any
set isomorphism $\Phi:\Sigma_{\nu}\to\Sigma_{\mu}$ the weighted composition
operator

$U_{\Phi}:=\left(\frac{d\nu\circ\Phi^{*}}{d\mu}\right)^{\frac{1}{p}}T_{\Phi}$

is an invertible isometry $U_{\Phi}:L^{p}(\nu)\to L^{p}(\mu)$. Moreover, the
group $UL^{\infty}(\mu)$ of measurable modulus one functions $\omega\in
L_{0}(\mu)$ embeds into the group of invertible isometries of $L^{p}(\mu)$ as
multiplication operators. Thus for any $\omega\in UL^{\infty}(\mu)$ the
weighted composition operator $\omega U_{\Phi}:L^{p}(\nu)\to L^{p}(\mu)$ is an
invertible isometry.

###### Proof.

This is straightforward and known, see, for instance, [39, Lemmas 3.2, 3.3]. ∎

######  Definition 2.3 ([59]).

We call the invertible isometries $\omega U_{\Phi}$ described in Lemma 2.2
_spatial_.

######  Proposition 2.4.

We have $(\omega U_{\Phi})^{-1}=T_{\Phi^{*}}(\overline{\omega})U_{\Phi^{*}}$
and $\omega U_{\Phi}\circ\upsilon T_{\Psi}=\omega
T_{\Phi}(\upsilon)U_{\Phi\circ\Psi}$ for all localizable measures $\mu$,
$\nu$, $\eta$, all $p\in[1,\infty]$, set isomorphisms
$\Phi:\Sigma_{\nu}\to\Sigma_{\mu}$, $\Psi:\Sigma_{\nu}\to\Sigma_{\eta}$ and
functions $\omega\in UL^{\infty}(\mu)$, $\upsilon\in UL^{\infty}(\nu)$.

In particular, invertible spatial isometries on $L^{p}(\mu)$, for localizable
$\mu$ and $p\in[1,\infty]$, form a group which is isomorphic to the semi-
direct product $UL^{\infty}(\mu)\rtimes\operatorname{Aut}([\Sigma_{\mu}])$
where $[\Phi]\omega=T_{\Phi}(\omega)$ for
$[\Phi]\in\operatorname{Aut}([\Sigma_{\mu}])$ and $\omega\in
UL^{\infty}(\mu)$.

###### Proof.

The first part follows from (the calculations in the proof of) [39, Lemmas
3.3, 3.4]. It readily implies the second part of the assertion, see also [39,
Theorem 3.7] or [35, Proposition 2.10]. ∎

######  Proposition 2.5 (Banach–Lamperti theorem).

Let $p\in[1,\infty]\setminus\\{2\\}$ and $\mu,\nu$ be localizable measures.
Then every invertible isometry from $L^{p}(\mu)$ to $L^{p}(\nu)$ is spatial.

###### Proof.

For $p\in(1,\infty)\setminus\\{2\\}$, $\mathbb{F}=\mathbb{C}$, and $\mu=\nu$
this is proved in [39, Theorem 3.7], see also [35, Theorem 2.12], but the
proof works for two localizable measures $\mu$ and $\nu$, and also for
$\mathbb{F}=\mathbb{R}$ and $p=1$. In fact since $L^{\infty}(\mu)\cong
L^{1}(\mu)^{\prime}$ the case $p=1$ can be deduced from the case $p=\infty$.
The assertion for $p=\infty$ follows from the Banach–Stone Theorem combined
with the Gelfand Theorem. Indeed, we have $L^{\infty}(\mu)\cong C(X)$ and
$L^{\infty}(\nu)\cong C(Y)$ where $X$, $Y$ are compact Hausdorff
(hyperstonean) spaces, and every invertible isometry $C(Y)\to C(X)$ is of the
form $aT$ where $a\in C(X)$, $|a|=1$, and $T:C(Y)\to C(X)$ is an isometric
algebra isomorphism. The Boolean algebras of idempotent elements in $C(Y)$ and
$C(X)$ are canonically isomorphic to $[\Sigma_{\nu}]$ and $[\Sigma_{\mu}]$,
respectively. Thus $T$ induces a set isomorphism
$\Phi:\Sigma_{\nu}\to\Sigma_{\mu}$ such that the action of
$T_{\Phi}:L^{\infty}(\nu)\to L^{\infty}(\mu)$ on idempotents in
$L^{\infty}(\nu)$ agrees with the action of $T$ on the corresponding
idempotents in $C(Y)$. As the idempotents are linearly dense in the considered
spaces we conclude that the isometry $L^{\infty}(\nu)\to L^{\infty}(\mu)$
corresponding to $aT$ is $\omega T_{\Phi}$ where $\omega\in L^{\infty}(\mu)$
is the function corresponding to $a\in C(X)\cong L^{\infty}(\mu)$. ∎

###  2.2. $L^{p}$-operator algebras and representations of $C_{0}(X)$

Throughout this paper $A$ and $B$ will be Banach algebras, and $B(E)$ will
denote the Banach algebra of bounded operators on a Banach space $E$.

######  Definition 2.6.

A _representation of $A$ in $B$_ is a contractive algebra homomorphism
$\pi:A\to B$. If $B=B(E)$ for some Banach space $E$, we call $\pi:A\to B(E)$ a
_representation on the space $E$_, and we say $\pi$ is _non-dedegenerate_ if
$\overline{\pi(A)E}=E$.

######  Definition 2.7 ([60]).

Let $p\in[1,\infty]$. A Banach algebra $A$ is an _$L^{p}$ -operator algebra_
if there is an isometric representation $\pi:A\to B(L^{p}(\mu))$ for some
measure $\mu$.

By an _approximate unit_ in a Banach algebra we will always mean a contractive
two-sided approximate unit. If $p\in(1,\infty)$, every $L^{p}$-operator
algebra $A$ with an approximate unit has an isometric non-degenerate
representation $\pi:A\to B(L^{p}(\mu))$, [38, Theorem 4.3]. Unlike for
$C^{*}$-algebras there is still no axiomatic definiton of an $L^{p}$-operator
algebra.

######  Definition 2.8.

A _$C^{*}$ -algebra_ is a Banach $*$-algebra $A$ such that
$\|a^{*}a\|=\|a\|^{2}$, $a\in A$, and if $\mathbb{F}=\mathbb{R}$ we also
require $\|a^{*}a\|\leq\|a^{*}a+b^{*}b\|$ for all $a,b\in A$. We call a norm
with these properties a _$C^{*}$ -norm_.

######  Remark 2.9.

It is well known that a Banach $*$-algebra $A$ (complex or real) is a
$C^{*}$-algebra if and only if it is isometrically $*$-isomorphic to a
$*$-subalgebra of bounded operators on some $L^{2}$-space (complex or real).
Also a real Banach algebra $A$ is a real $C^{*}$-algebra if and only if its
complexification $A_{\mathbb{C}}:=A+iA$ admits a (necessarily unique)
$C^{*}$-norm that extends that of $A$, see [53]. A homomorphism $A\to B$
between two $C^{*}$-algebras is contractive if and only if it is
$*$-preserving (these well known facts for complex $C^{*}$-algebras [12,
A.5.8] also hold in the real case [53, Proposition 5.4.1 and Theorem 5.6.4]).

Let $X$ be a locally compact Hausdorff space. The algebra $C_{0}(X)$ of
continuous functions on $X$ which vanish at infinity, equipped with the
supremum norm $\|\cdot\|_{\infty}$, is an $L^{p}$-operator algebra for any
$p\in[1,\infty]$. By [38, Theorem 5.3] a complex $C^{*}$-algebra $A$ is an
$L^{p}$-operator algebra for $p\in[1,\infty)\setminus\\{2\\}$ if and only if
it is commutative and hence of the form $C_{0}(X)$. A real commutative Banach
algebra $A$ is isometrically isomorphic to $C_{0}(X)$ for some locally compact
Hausdorff space if and only if $\|a\|^{2}=\|a^{2}\|\leq\|a^{2}+b^{2}\|$ for
all $a,b\in A$, see [4].

######  Proposition 2.10 (Kaplansky–Bonsall).

Let $A$ be a $C^{*}$-algebra (real or complex). The $C^{*}$-norm on $A$ is
minimal among all submultiplicative norms on $A$. Thus every injective
representation $\pi:A\to B$ in a Banach algebra $B$ is automatically
isometric. More generally, every representation $\pi:A\to B$ descends to an
isometric representation $\pi:A/\ker(\pi)\to B$.

###### Proof.

The first part of the assertion in the complex case is [14, Theorems 10], but
the proof also works in the real case by applying [53, Lemma 5.1.8]. The
second part is equivalent to the first. For the last part it suffices to note
that the quotient $A/\ker(\pi)$ is again a $C^{*}$-algebra, which is well
known when $\mathbb{F}=\mathbb{C}$, but also holds when
$\mathbb{F}=\mathbb{R}$, see [53, Proposition 5.4.1]. ∎

We now slightly extend [19, Proposition 2.7], see also [13, Proposition 2.12]
and references therein, which describes hermitian operators acting on complex
$L^{p}$-spaces. This forces the representations of the complex algebra
$C_{0}(X)$ on $L^{p}$ and $C_{0}$-spaces to act by multiplication operators.

######  Proposition 2.11.

Assume $\mathbb{F}=\mathbb{C}$ and let $E=L^{p}(\mu)$ for a localizable $\mu$
and $p\in[1,\infty]\setminus\\{2\\}$, or $E=C_{0}(\Omega)$ for a locally
compact Hausdorff space $\Omega$. An element $a\in B(E)$ is hermitian (i.e.
$\|e^{ita}\|=1$ for all $t\in\mathbb{R}$) if and only if $a$ is a
multiplication operator by a bounded real-valued function (from
$L^{\infty}(\mu,\mathbb{R})$ if $E=L^{p}(\mu)$ and from
$C_{b}(\Omega,\mathbb{R})$ if $E=C_{0}(\Omega)$).

###### Proof.

When $E=L^{p}(\mu)$ and $p<\infty$ this is proved in [19, Proposition 2.7].
The proof relies on the Banach–Lamperti Theorem and it also works for
$p=\infty$, with the Banach–Lamperti theorem extended as in Proposition 2.5.
We show it also works for $E=C_{0}(\Omega)$ with the Banach–Lamperti theorem
replaced by the Banach–Stone theorem. We recall the argument for the ‘only if’
part (the ‘if’ part is straightforward). Let $a\in B(C_{0}(\Omega))$ be
hermitian. We may assume that $\|a\|\leq\frac{\pi}{2}$. The elements
$u_{t}:=e^{ita}$, $t\in\mathbb{R}$, form a continuous group of invertible
isometries. By the Banach–Stone theorem for each $t\in\mathbb{R}$ there is
$\omega_{t}\in C_{b}(\Omega)$ with $|\omega_{t}|=1$ and a homeomorphism
$\varphi_{t}:\Omega\to\Omega$ such that
$[u_{t}\xi](x)=\omega_{t}(x)\xi(\varphi_{t}(x))$. Using this, a simple
calculation, for $s,t\in\mathbb{R}$, gives

$\|u_{t}-u_{s}\|=\max\big{\\{}\|\omega_{t}-\omega_{s}\|_{\infty},2(1-\delta_{\varphi_{t},\varphi_{s}})\big{\\}}.$

Since $t\mapsto u_{t}$ is norm continuous and $u_{0}=1$ is the identity
operator, we get that $\varphi_{t}=\mathrm{id}$ for all $t\in\mathbb{R}$.
Hence each $u_{t}$ is the operator of multiplication by $\omega_{t}$. Since
$\|a\|\leq\frac{\pi}{2}$ the spectrum of $u_{1}=e^{ia}$ is contained in the
half circle $\\{e^{it}:t\in[-\pi/2,\pi/2]\\}$ where the holomorphic inverse
$\log$ of $\exp$ is defined. Applying analytic functional calculus to $u_{1}$
we get $ia=\log(u_{1})$. Since $m:C_{b}(\Omega)\to B(C_{0}(\Omega)));\
m(h)\xi(x):=h(x)\xi(x)$ is a unital homomorphism, we obtain
$a=-i\log(u_{1})=-i\log(m(\omega_{1}))=m(-i\log(\omega_{1}))$. Thus $a$ is a
multiplication operator by $-i\log(\omega_{1})$, and since $a$ is bounded and
hermitian we must have $-i\log(\omega_{1})\in C_{b}(\Omega,\mathbb{R})$. ∎

Recall that $L^{p}$-spaces and $C_{0}$-spaces are naturally Banach lattices
with the lattice structure given by positive (non-negative) functions.
Operators preserving positive functions (and therefore the lattice structure)
are called _positive_.

######  Theorem 2.12.

Let $\pi:C_{0}(X)\to B(E)$ be a non-degenerate representation where
$E=L^{p}(\mu)$ for a localizable $\mu$ and $p\in[1,\infty]\setminus\\{2\\}$,
or $E=C_{0}(\Omega)$ for a locally compact Hausdorff space $\Omega$. If
$\mathbb{F}=\mathbb{C}$ then $\pi$ is positive in the sense that it maps
positive functions to positive operators. If $\pi$ is positive, then it is
given by multiplication operators: there is a representation
$\pi_{0}:C_{0}(X)\to L^{\infty}(\mu)$ if $E=L^{p}(\mu)$, or
$\pi_{0}:C_{0}(X)\to C_{b}(\Omega)$ if $E=C_{0}(\Omega)$, such that

(2.1) $[\pi(a)\xi](x)=\pi_{0}(a)(x)\xi(x),\qquad a\in C_{0}(X),\ \xi\in E.$

###### Proof.

If $X$ is not compact we may extend $\pi:C_{0}(X)\to B(E)$ to a unital
representation $\pi:C(X^{+})\to B(E)$, where $X^{+}$ is the one-point
compactification of $X$, see [38, Theorem 4.1]. Then $\pi$ maps hermitian
elements to hermitian ones by [19, Lemma 2.4]. Hermitian elements in
$C(X^{+})$ are real valued functions $C(X^{+},\mathbb{R})$. Hence
$\pi(C(X^{+},\mathbb{R}))$ consists of multiplication operators by Proposition
2.11. Since $C(X^{+})=C(X^{+},\mathbb{R})+iC(X^{+},\mathbb{R})$, we conclude
that there is $\pi_{0}:C(X^{+})\to L^{\infty}(\mu)$ (or $\pi_{0}:C(X^{+})\to
C_{b}(X)$) satisfying (2.1). Since $\pi$ is a representation it follows that
$\pi_{0}$ is a representation and therefore a $*$-homomorphism between two
$C^{*}$-algebras. Hence both $\pi_{0}$ and $\pi$ are positive.

Suppose now that $\mathbb{F}=\mathbb{R}$. We may pass to complexifications
$C_{0}(X,\mathbb{C})$ and $L^{p}(\mu,\mathbb{C})$ of $C_{0}(X,\mathbb{R})$ and
$L^{p}(\mu,\mathbb{R})$ respectively. Namely, we put

$\big{(}\pi_{\mathbb{C}}(a+ib)\big{)}(\xi+i\eta):=\pi(a)\xi-\pi(b)\eta+i\big{(}\pi(a)\eta+\pi(b)\xi\big{)},$

for $\xi,\eta\in L^{p}(\mu,\mathbb{R}),\ a,b\in C_{0}(X,\mathbb{R})$. This
defines a non-degenerate algebra homomorphism
$\pi_{\mathbb{C}}:C_{0}(X,\mathbb{C})\to B(L^{p}(\mu,\mathbb{C}))$, and for
$a\in C_{0}(X,\mathbb{R})$ we have $\|\pi_{\mathbb{C}}(a)\|=\|\pi(a)\|$, cf.
[1, Corollary 3.23]. Thus
$\|\pi_{\mathbb{C}}(a+ib)\|\leq\|\pi(a)\|+\|\pi(b)\|\leq\|\pi(a)\|_{\infty}+\|\pi(b)\|_{\infty}\leq
2\|a+ib\|_{\infty}$, and so $\|\pi_{\mathbb{C}}\|\leq 2$. However, in general
$\pi_{\mathbb{C}}$ is not contractive, cf. Example 2.13 below. If $\pi$ is
positive, then $\pi$ and $\pi_{\mathbb{C}}$ take values in the lattice of
regular operators, see [1, Section 3.2], and we clearly have
$|\pi_{\mathbb{C}}(a+ib)|\leq\pi_{\mathbb{C}}(|a+ib|)$. Also by the
Riesz–Kantorovich formula [1, Corollary 3.25],
$|\pi_{\mathbb{C}}(a+ib)\xi|\leq\pi_{\mathbb{C}}(|a+ib|)|\xi|$ for all $\xi\in
E_{\mathbb{C}}$. This implies that
$\|\pi_{\mathbb{C}}(a+ib)\|\leq\|\pi_{\mathbb{C}}(|a+ib|)\|=\|\pi(|a+ib|)\|\leq\||a+ib|\|_{\infty}=\|a+ib\|_{\infty}$.
Hence $\pi_{\mathbb{C}}$ is contractive and therefore it acts by
multiplication operators by the first part of the proof. ∎

Positivity of the representation in Theorem 2.12 in the real case is not
automatic.

######  Example 2.13 (Representations from involutive isometries).

Let $U\in B(E)$ be an involutive isometry on a Banach space $E$, i.e.
$\|U\|=1$ and $U^{2}=1$. Then $P:=\frac{1+U}{2}$ and $1-P:=\frac{1-U}{2}$ are
contractive projections, so $P$ is a _bicontractive projection_ (cf. Remark
6.5 below). If $E$ is an $L^{p}$-space or a predual of an $L^{1}$-space, then
every bicontractive projection is of this form, see [9]. The formula

$\pi(a)=a(1)P+a(2)(1-P)=\frac{a(1)+a(2)}{2}1+\frac{a(1)-a(2)}{2}U$

defines a homomorphism $\pi:C(\\{1,2\\})\to B(E)$ satisfying
$\|\pi(a)\|\leq\frac{|a(1)+a(2)|}{2}+\frac{|a(1)-a(2)|}{2}$. If
$\mathbb{F}=\mathbb{R}$, then
$\frac{|a(1)+a(2)|}{2}+\frac{|a(1)-a(2)|}{2}=\max\\{|a(1)|,|a(2)|\\}=\|a\|_{\infty}$
and thus $\pi$ is contractive. For $\mathbb{F}=\mathbb{C}$ this is not true.
In particular, let $\mathbb{F}=\mathbb{R}$,
$E:=\ell^{p}(\\{1,2\\})\cong\mathbb{R}^{2}$, $p\in[1,\infty]$ and
$U\xi:=(\xi(2),\xi(1))$. Then

$\pi(a)\xi=\left(\frac{a(1)+a(2)}{2}\xi(1)+\frac{a(1)-a(2)}{2}\xi(2),\frac{a(1)-a(2)}{2}\xi(1)+\frac{a(1)+a(2)}{2}\xi(2)\right)$

is an isometric unital representation $\pi:C(\\{1,2\\})\to
B(\ell^{p}(\\{1,2\\}))$ of the real algebra $C(\\{1,2\\})$ that does not act
by multiplication operators. For $p\neq 2$, $\pi$ is not even isometrically
conjugate to a representation given by multiplication operators (there are
only four isometries on $\ell^{p}(\\{1,2\\})$ by Proposition 2.5). For $p=2$,
see Lemma 2.16 below.

######  Corollary 2.14.

The complex Banach algebra $C_{0}(X)$ admits an isometric non-degenerate
representation on $L^{\infty}(\mu)$ for localizable $\mu$ if and only if $X$
is compact.

###### Proof.

If $X$ is not compact and $\pi:C_{0}(X)\to B(L^{\infty}(\mu))$ is isometric
and non-degenerate then, in the notation of Theorem 2.12, $\pi_{0}(C_{0}(X))$
is a non-unital subalgebra of $L^{\infty}(\mu)$. Hence
$\overline{\pi(C_{0}(X))L^{\infty}(\mu)}=\overline{\pi_{0}(C_{0}(X))L^{\infty}(\mu)}$
is a non-trivial (non-unital) ideal in $L^{\infty}(\mu)$, which contradicts
the non-degeneracy of $\pi$. If $X$ is compact, then the embedding
$\pi_{0}:C(X)\to\ell^{\infty}(X)$ gives an isometric unital representation of
$C(X)$ on $\ell^{\infty}(X)$. ∎

######  Remark 2.15.

Corollary 2.14 is perhaps one of the reasons why Phillips [60, Page 3]
suggests that instead of $L^{\infty}$-algebras “it may well be more
appropriate to consider” algebras isometrically represented on _$C_{0}$
-spaces_, i.e. spaces $C_{0}(\Omega)$ for a locally compact Hausdorff space
$\Omega$. In fact, it is also natural to consider pre-duals of $L^{1}$-spaces,
i.e. _Lindenstrauss spaces_ , cf. Corollary 4.18 below.

For $p=2$, Theorem 2.12 holds up to a unitary conjugacy.

######  Lemma 2.16.

For any representation $\pi:C_{0}(X)\to B(H)$ on a Hilbert space $H$, there is
a localizable measure $\mu$ and a unitary $U:H\to L^{2}(\mu)$ such that
$U\pi(\cdot)U^{*}:C_{0}(X)\to B(L^{2}(\mu))$ is given by multiplication
operators, i.e. there is a representation $\pi_{0}:C_{0}(X)\to
L^{\infty}(\mu)$ such that $U\pi(a)U^{*}\xi=\pi_{0}(a)\cdot\xi$.

###### Proof.

The algebra $\pi(C_{0}(X))$ acts on the orthogonal complement of
$\overline{\pi(C_{0}(X))H}$ as zero, and we have
$\overline{\pi(C_{0}(X))H}\cong\ell^{2}(I)$ for some set $I$. The restriction
of $\pi$ to $\overline{\pi(C_{0}(X))H}$ decomposes into a direct sum of cyclic
representations $\\{\pi_{i}\\}_{i\in I}$. By the GNS construction and the
Markov–Riesz Theorem each cyclic part $\pi_{i}:C_{0}(X)\to B(H_{i})$ is
equivalent to a representation of $C_{0}(X)$ given by multiplication on
$L^{2}(\mu_{i})$, where $\mu_{i}$ is a regular (and hence localizable) measure
on $X$. Taking the direct sum of all the arising measure spaces we get a space
$L^{2}(\mu)$ with localizable $\mu$ and a unitary $U$ with desired properties.
∎

###  2.3. Inverse semigroups and étale groupoids

Let $S$ be a semigroup. An element $t\in S$ is called _partially invertible_
if there is an element $t^{*}\in S$ such that $t=tt^{*}t$ and
$t^{*}=t^{*}tt^{*}$. Then $t^{*}t$ and $tt^{*}$ are idempotents, and we call
$t^{*}$ a _generalized inverse_ for $t$. The semigroup $S$ is called an
_inverse semigroup_ if every element in $S$ has a unique generalized inverse.
Assume this. Then the map $t\mapsto t^{*}$ is an anti-multiplicative
involution, and any semigroup homomorphism between two inverse semigroups is
automatically $*$-preserving. The set of idempotents

$E(S):=\\{e\in S:e^{2}=e\\}=\\{tt^{*}:t\in S\\}=\\{t^{*}t:t\in S\\}$

is an abelian semigroup (see for instance [57, Proposition 2.1.1]). A partial
order on $S$ is defined by $s\leq t$ if and only if $s=ts^{*}s$, if and only
if there is $e\in E(S)$ with $s=te$, $s,t\in S$, see [57]. If $S$ has a unit
$1$ then $e\in E(S)$ if and only if $e\leq 1$.

A _partial bijection_ on a set $X$ is a bijection $h:U\to h(U)$ between
subsets $U$, $h(U)$ of $X$. The set $\operatorname{PBij}(X)$ of all partial
bijections on $X$ form an inverse semigroup with composition of $h:U\to h(U)$
and $f:V\to f(V)$ defined as the bijection $h\circ f:f^{-1}(U\cap f(V))\to
h(U\cap f(V))$. Then $h^{*}=h^{-1}:h(U)\to U$, $h\in
E(\operatorname{PBij}(X))$ if and only if $h=\mathrm{id}|_{U}$ for some
$U\subseteq X$, and $h\leq f$ if and only if $f$ is an extension of $h$. Every
inverse semigroup $S$ can be viewed as a subsemigroup of
$\operatorname{PBij}(S)$, where $h_{t}:S_{t^{*}}\to S_{t}$ is given by
$h_{t}(s):=ts$ and $S_{t}:=\\{s\in S:ss^{*}\leq t^{*}t\\}$ for all $t\in S$,
see [57, Proposition 2.1.3]. A _partial homeomorphism_ on a topological space
$X$ is a homeomorphism $h:U\to h(U)$ between two open subsets $U$ and $h(U)$
of $X$. Partial homeomorphisms form an inverse subsemigroup of
$\operatorname{PBij}(X)$ that we denote by $\operatorname{PHomeo}(X)$.

######  Definition 2.17.

An _action of an inverse semigroup $S$ on a topological space $X$_ is a non-
degenerate semigroup homomorphism $h:S\to\operatorname{PHomeo}(X)$: a family
of partial homeomorphisms $h_{t}:X_{t^{*}}\to X_{t}$ such that $h_{t}\circ
h_{s}=h_{ts}$ for all $s,t\in S$, and $\bigcup_{t\in S}X_{t}=X$.

A groupoid is a small category $\mathcal{G}$ in which every arrow is
invertible. If $\mathcal{G}$ (identified with the set of arrows) is equipped
with a topology that makes the range, domain, composition, and inverse of
arrows continuous, we call $\mathcal{G}$ a _topological groupoid_. We denote
by $X\subseteq\mathcal{G}$ the _unit space_ of $\mathcal{G}$. Let
$\mathcal{G}$ be an _étale groupoid_ with unit space $X$, i.e. $\mathcal{G}$
is a topological groupoid where the range and domain maps $r,d:\mathcal{G}\to
X\subseteq\mathcal{G}$ are local homeomorphisms. In particular, $X$ is an open
subset of $\mathcal{G}$. We will assume that $X$ is a locally compact
Hausdorff space, in which case $\mathcal{G}$ is necessarily locally compact
and locally Hausdorff, but $\mathcal{G}$ is Hausdorff if and only if $X$ is
closed in $\mathcal{G}$, see for instance [70, Lemma 2.3.2]. A _bisection_ of
a groupoid $\mathcal{G}$ is an open set $U\subseteq\mathcal{G}$ such that the
restrictions $r|_{U}$ and $d|_{U}$ are homeomorphisms onto open sets of
$\mathcal{G}$. So a topological groupoid is étale if and only if it can be
covered by bisections.

######  Example 2.18 (Inverse semigroup actions from étale groupoids).

The set of bisections $\operatorname{Bis}(\mathcal{G})$ of an étale groupoid
$\mathcal{G}$ forms a unital inverse semigroup with multiplication and
involution given by

$U\cdot V:=\\{\gamma\eta:\text{$\gamma\in U,\ \eta\in V$ are
composable}\\},\qquad U^{*}=U^{-1}:=\\{\gamma^{-1}:\gamma\in U\\}.$

Then $X$ is a unit of $\operatorname{Bis}(\mathcal{G})$. For each
$U\in\operatorname{Bis}(\mathcal{G})$ we have a homeomorphism

$h_{U}:=r\circ d|_{U}^{-1}:d(U)\to r(U),$

and these homeomorphisms define a _canonical action_
$h:\operatorname{Bis}(\mathcal{G})\to\operatorname{PHomeo}(X)$ on the unit
space $X$ [31, Proposition 5.3]. In fact we may view this action as a
restriction of a partial action
$\tilde{h}:\operatorname{Bis}(\mathcal{G})\to\operatorname{PHomeo}(\mathcal{G})$
on the groupoid $\mathcal{G}$, where $\tilde{h}_{U}:r^{-1}(d(U))\to
r^{-1}(r(U))$ is given by
$\tilde{h}_{U}(\gamma)=d|_{U}^{-1}(r(\gamma))\gamma$. In other words,
$\tilde{h}_{U}(\gamma)=\eta\gamma$, where $\eta$ is the unique element in $U$
that can be composed with $\gamma$. Using this interpretation one sees that
$\tilde{h}$ is indeed an inverse semigroup action. It restricts to the
canonical action $h$ in the sense that $h=r\circ\tilde{h}|_{X}$.

######  Example 2.19 (Étale groupoids from inverse semigroup actions).

For any inverse semigroup action $h:S\to\operatorname{PHomeo}(X)$ the
_transformation groupoid_ $S\ltimes_{h}X$ is defined as follows (see [57, p.
140] or [31, Section 4]). The arrows of $S\ltimes_{h}X$ are equivalence
classes of pairs $(t,x)$ for $x\in X_{t^{*}}\subseteq X$; two pairs $(t,x)$
and $(t^{\prime},x^{\prime})$ are equivalent if $x=x^{\prime}$ and there is
$v\in S$ with $v\leq t,t^{\prime}$ and $x\in X_{v^{*}}$. The range and domain
maps $r,d:S\ltimes_{h}X\rightrightarrows X$ and the multiplication are defined
by $r([t,x]):=h_{t}(x)$, $d([t,x]):=x$, and $[s,h_{t}(x)]\cdot[t,x]=[s\cdot
t,x]$ for $x\in X_{(st)^{*}}$. We give $S\ltimes_{h}X$ the unique topology
such that the sets $U_{t}:=\\{[t,x]:x\in X_{t^{*}}\\}$ are open bisections of
$S\ltimes_{h}X$. Then $S\ltimes_{h}X$ is an étale groupoid and the map $S\ni
t\mapsto U_{t}\in\operatorname{Bis}(S\ltimes_{h}X)$ is a semigroup
homomorphism. Composing it with the canonical homomorphism
$\operatorname{Bis}(S\ltimes_{h}X)\to\operatorname{PHomeo}(X)$ we recover the
action $h$. Composing it with the canonical homomorphism
$\operatorname{Bis}(S\ltimes_{h}X)\to\operatorname{PHomeo}(S\ltimes_{h}X)$ we
get an extension $\tilde{h}$ of $h$ where $\tilde{h}_{t}:\\{[s,x]:x\in
h_{s}^{-1}(X_{t^{*}})\\}\to\\{[s,x]:x\in h_{s}^{-1}(X_{t})\\}$ is given by
$\tilde{h}_{t}([s,x]):=[ts,x].$ In this way every inverse semigroup action can
be viewed as coming from an étale groupoid, and then it can be extended to an
action on this groupoid. Conversely, for any étale groupoid $\mathcal{G}$ and
any inverse subsemigroup $S\subseteq\text{Bis}(\mathcal{G})$, acting
canonically on $X$, we have a continous open groupoid homomorphism
$S\ltimes_{h}X\ni[U,x]\mapsto(d|_{U})^{-1}(x)\in\mathcal{G}$. This is an
isomorphism $S\ltimes_{h}X\cong\mathcal{G}$ if and only if $S$ is _wide_ in
the sense that $S$ covers $G$ and $U\cap V$ is a union of bisections in $S$
for all $U,V\in S$, see [31, Proposition 5.4], [51, Proposition 2.2]. In
particular, every étale groupoid is a transformation groupoid.

###  2.4. $L^{p}$-partial isometries on Banach spaces and spatial partial
isometries

Partial isometries acting on arbitrary Banach spaces were defined by Mbekhta
[55, Definition 4.1], see also [19, 8.1], and in Banach algebras the
definition can be phrased as follows.

######  Definition 2.20.

A _partial isometry in a Banach algebra_ $B$ is a partially invertible element
$t$ in the semigroup $B_{1}$ of contractive elements, so $t\in B_{1}$ and
there exists $t^{*}\in B_{1}$ such that $t=tt^{*}t$ and $t^{*}=t^{*}tt^{*}$.
If $B$ is unital and complex we say that $t\in B$ is an _$MP$ -partial
isometry_ if $t$ is contractive and admits a contractive Moore–Penrose
generalized inverse $t^{*}\in B$, i.e. $t^{*}$ is a generalised inverse of $t$
and the idempotents $tt^{*}$ and $t^{*}t$ are hermitian.

If $B$ is a $C^{*}$-algebra (real or complex) then a contractive generalized
inverse to a partial isometry $t\in B$ is necessarily the adjoint element
$t^{*}\in B$, see [55, 3.1, 3.3] and Remark 2.9, so in this case both notions
in Definition 2.20 coincide with the usual notion of partial isometry in a
$C^{*}$-algebra. However, for many Banach spaces $E$, including $L^{1}(\mu)$,
$c_{0}$, $\ell^{\infty}$, a partial isometry $t\in B(E)$ may have many
contractive generalized inverses. On the other hand, if the Moore–Penrose
generalized inverse exists it is necessarily unique. So the $MP$-partial
isometries have uniquely determined generalized inverse $MP$-partial
isometries. Partial isometries usually do not form a semigroup (the product of
two partial isometries $s,t\in B$ in a $C^{*}$-algebra $B$ is a partial
isometry if and only if the projections $s^{*}s$ and $tt^{*}$ commute, see
[41]). For $p\in[1,\infty]\setminus\\{2\\}$ and any Banach space $E$ we
introduce a natural inverse semigroup of partial isometries in $B(E)$. Recall,
see [8], [3], that an _$L^{p}$ -projection on $E$_ is an idempotent $P\in
B(E)$ such that for all $\xi\in E$ we have

$\|\xi\|^{p}=\|P\xi\|^{p}+\|(1-P)\xi\|^{p}$

when $p<\infty$ and $\|\xi\|=\max\\{\|P\xi\|,\|(1-P)\xi\|\\}$ when $p=\infty$.
The $L^{2}$-projections on a Hilbert space are simply the orthogonal
projections. If $p\neq 2$, then all $L^{p}$-projections on $E$ commute and
they form a Boolean algebra $\mathbb{P}_{p}(E)\subseteq B(E)$. The Banach
algebra $C_{p}(E):=\operatorname{\overline{span}}\mathbb{P}_{p}(E)$ generated
by $\mathbb{P}_{p}(E)$ is called the _$p$ -Cunningham algebra_ of $E$, and we
have $C_{p}(E)\cong C_{0}(X)$ where $X$ is a totally disconnected space (the
Stone dual to $\mathbb{P}_{p}(E)$), see [8] for details. We extend the notion
of $L^{p}$-projection to partial isometries as follows:

######  Definition 2.21.

An _$L^{p}$ -partial isometry_ on a Banach space $E$, $p\in[1,\infty]$, is a
contraction $T\in B(E)$ that has a contractive generalized inverse $T^{*}\in
B(E)$ such that $TT^{*}$ and $T^{*}T$ are $L^{p}$-projections. We denote by
$L^{p}\mathchar 45\relax\operatorname{PIso}(E)$ the set of all $L^{p}$-partial
isometries on $E$.

######  Proposition 2.22.

For $p\in[1,\infty]\setminus\\{2\\}$ the set $L^{p}\mathchar
45\relax\operatorname{PIso}(E)$ is an inverse semigroup whose idempotents are
$\mathbb{P}_{p}(E)$.

###### Proof.

Let $S,T\in L^{p}\mathchar 45\relax\operatorname{PIso}(E)$ and choose the
corresponding generalized inverses $S^{*},T^{*}\in L^{p}\mathchar
45\relax\operatorname{PIso}(E)$, so that the corresponding idempotents
$TT^{*}$, $T^{*}T$, $SS^{*}$, $SS^{*}$ are in $\mathbb{P}_{p}(E)$. By
commutativity of these idempotents, $S^{*}T^{*}$ is a generalized inverse of
$TS$. Using the $L^{p}$-projection property of $SS^{*}$, $T^{*}T$, $TT^{*}$
and that $T^{*}$ is an isometry on the range of $T$ (and vice versa), for any
$x\in E$, we get

$\begin{split}\|TSS^{*}T^{*}x\|^{p}&=\|T^{*}TSS^{*}T^{*}x\|^{p}=\|SS^{*}(T^{*}T)T^{*}x\|^{p}=\|SS^{*}T^{*}x\|^{p}\\\
&=\|T^{*}x\|^{p}-\|(1-SS^{*})T^{*}x\|^{p}=\|TT^{*}x\|^{p}-\|T^{*}x-SS^{*}(T^{*}T)T^{*}x\|^{p}\\\
&=\|x\|^{p}-\|(1-TT^{*})x\|^{p}-\|TT^{*}x-TSS^{*}T^{*}x\|^{p}\\\
&=\|x\|^{p}-\|(1-TSS^{*}T^{*})x\|^{p}.\end{split}$

Hence $TSS^{*}T^{*}\in\mathbb{P}_{p}(E)$. By symmetry we also get
$S^{*}T^{*}TS\in\mathbb{P}_{p}(E)$. Hence $TS\in L^{p}\textup{-PIso}(E)$, and
so $L^{p}\mathchar 45\relax\operatorname{PIso}(E)$ is a semigroup. Assume in
addition that $T$ is the generalized inverse of $S$. Using that $S^{*}S$ is an
$L^{p}$-projection, on one hand we have

$\|TSx\|^{p}=\|S^{*}STSx\|^{p}+\|(1-S^{*}S)TSx\|^{p}=\|S^{*}Sx\|^{p}+\|TSx-S^{*}Sx\|^{p}.$

On the other hand, $\|TSx\|=\|STSx\|=\|Sx\|=\|S^{*}Sx\|$. Hence
$\|TSx-S^{*}Sx\|^{p}=0$ and so $TS=S^{*}S$. Similarly one shows that
$ST=SS^{*}$. This implies that $T=S^{*}$, (cf. [49, Propositon 2.4]), and
proves the uniqueness of the generalized inverse in $L^{p}\mathchar
45\relax\operatorname{PIso}(E)$. ∎

If $E=L^{p}(\mu)$ for a localizable measure $\mu$ and
$p\in[1,\infty]\setminus\\{2\\}$, then $P\in B(L^{p}(\mu))$ is an
$L^{p}$-projection if and only if $P$ is an operator of multiplication by a
characteristic function, see [8, Proposition 4.9] (when
$\mathbb{F}=\mathbb{C}$ this can be inferred from Proposition 2.11). In
particular, we have an isomorphism of Boolean algebras
$\mathbb{P}_{p}(L^{p}(\mu))\cong[\Sigma_{\mu}]$. We now discuss the
relationship between $L^{p}\mathchar 45\relax\operatorname{PIso}(L^{p}(\mu))$
and spatial partial isometries, originally introduced (for $\sigma$-finite
measures) by Phillips [59], [60].

######  Definition 2.23.

A _subspace_ of a measure space $(\Omega,\Sigma_{\mu},\mu)$ is
$(D,\Sigma_{D},\mu|_{D})$ where $D\in\Sigma$, $\Sigma_{D}:=\\{C\cap
D:C\in\Sigma\\}$ and $\mu|_{D}:=\mu|_{\Sigma_{D}}$. A _partial set
automorphism_ of $(\Omega,\Sigma,\mu)$ is a set isomorphism between two
subspaces of $(\Omega,\Sigma,\mu)$. So a partial set automorphism is a map
$\Phi:\Sigma_{D_{\Phi^{*}}}\to\Sigma_{D_{\Phi}}$
($D_{\Phi},D_{\Phi^{*}}\in\Sigma$) that descends to a Boolean isomorphism
$[\Phi]:[\Sigma_{D_{\Phi^{*}}}]\to[\Sigma_{D_{\Phi}}]$. We define
$\operatorname{PAut}([\Sigma_{\mu}]):=\\{[\Phi]:\text{$\Phi$ is a partial set
automorphism}\\}$ for the set of partial automorphisms of the Boolean algebra
$[\Sigma_{\mu}]$.

By definition $\operatorname{PAut}([\Sigma_{\mu}])$ is the set of isomorphisms
between ideals in the Boolean algebra $[\Sigma_{\mu}]$. It forms an inverse
semigroup that can be identified with an inverse semigroup of operators on
$L_{0}(\mu)$. Indeed, a partial automorphism $\Phi$ defines the composition
operator isomorphism $T_{\Phi}:L_{0}(\mu|_{D_{\Phi^{*}}})\to
L_{0}(\mu|_{D_{\Phi}})$ whose inverse is given by $T_{\Phi^{*}}$. Identifying
$L_{0}(\mu|_{D_{\Phi^{*}}})$ and $L_{0}(\mu|_{D_{\Phi}})$ with subspaces of
$L_{0}(\mu)$ we see that $\Phi$ determines uniquely a linear operator
$T_{\Phi}:L_{0}(\mu)\to L_{0}(\mu)$ such that $T_{\Phi}$ preserves (monotone)
limits and

$T_{\Phi}1_{C}=1_{\Phi(C\cap D_{\Phi^{*}})},\qquad C\in\Sigma.$

We still call $T_{\Phi}:L_{0}(\mu)\to L_{0}(\mu)$ a (generalized) _composition
operator_. For any two partial automorphisms
$\Phi:\Sigma_{D_{\Phi^{*}}}\to\Sigma_{D_{\Phi}}$ and
$\Psi:\Sigma_{D_{\Psi^{*}}}\to\Sigma_{D_{\Psi}}$ we have $T_{\Phi}\circ
T_{\Psi}=T_{\Phi\circ\Psi}$, where
$\Phi\circ\Psi:\Sigma_{\Psi^{*}(D_{\Phi^{*}}\cap
D_{\Psi})}\to\Sigma_{\Phi(D_{\Phi^{*}}\cap D_{\Psi})}$ is a well defined
partial automorphism of $(\Omega,\Sigma,\mu)$. Also $T_{\Phi^{*}}$ is the
unique generalized inverse for $T_{\Phi}$ among the composition operators. In
particular, composition operators on $L_{0}(\mu)$ form an inverse semigroup
naturally isomorphic to $\operatorname{PAut}([\Sigma_{\mu}])$. By the very
definition of the partial automorphism
$\Phi:\Sigma_{D_{\Phi^{*}}}\to\Sigma_{D_{\Phi}}$, the map
$\mu\circ\Phi:\Sigma_{D_{\Phi^{*}}}\to[0,+\infty]$ is a measure on
$D_{\Phi^{*}}$ equivalent to $\mu|_{D_{\Phi^{*}}}$. Thus we may consider the
Radon–Nikodym derivative $\frac{d\mu\circ\Phi}{d\mu|_{D_{\Phi^{*}}}}$ as a
function on $\Omega$ which is zero outside $D_{\Phi^{*}}$. Similar remarks
apply to the inverse partial automorphism
$\Phi^{*}:\Sigma_{D_{\Phi}}\to\Sigma_{D_{\Phi^{*}}}$.

######  Proposition 2.24.

Let $p\in[1,\infty]$ and $\mu$ be localizable. For any
$[\Phi]\in\operatorname{PAut}([\Sigma_{\mu}])$ the weighted composition
operator
$U_{\Phi}:=\left(\frac{d\mu\circ\Phi^{*}}{d\mu|_{D_{\Phi}}}\right)^{\frac{1}{p}}T_{\Phi}$
is a well defined partial isometry on $L^{p}(\mu)$, and the map
$\operatorname{PAut}([\Sigma_{\mu}])\ni[\Phi]\mapsto U_{\Phi}\in
B(L^{p}(\mu))$ is a semigroup embedding. The semigroup generated by
$U_{\Phi}\in B(L^{p}(\mu))$, $[\Phi]\in\operatorname{PAut}([\Sigma_{\mu}])$,
and $UL^{\infty}(\mu)\subseteq B(L^{p}(\mu))$ is an inverse semigroup of
partial isometries of the form

(2.2) $\omega
U_{\Phi}:=\omega\left(\frac{d\mu\circ\Phi^{*}}{d\mu|_{D_{\Phi}}}\right)^{\frac{1}{p}}T_{\Phi}$

where $\Phi:\Sigma_{D_{\Phi^{*}}}\to\Sigma_{D_{\Phi}}$ is a partial set
automorphism and $\omega:D_{\Phi}\to\\{z\in\mathbb{F}:|z|=1\\}$ is measurable.
The operator $\omega U_{\Phi}$ determines $\Phi$ and $\omega$ uniquely up to
sets of measure zero. Moreover,

(2.3) $(\omega
U_{\Phi})^{*}=T_{\Phi^{*}}(\overline{\omega})U_{\Phi^{*}},\qquad(\omega
U_{\Phi})\circ(\upsilon U_{\Psi})=\omega T_{\Phi}(\upsilon)U_{\Phi\circ\Psi}.$

###### Proof.

This follows from [59, Lemma 6.12, 6.17] where $\sigma$-finite measures were
considered, but the arguments work for localizable measures. ∎

######  Definition 2.25.

We denote by $\operatorname{SPIso}(L^{p}(\mu))$ the inverse subsemigroup of
operators (2.2) and call them _spatial partial isometries_ on $L^{p}(\mu)$
[59, Definition 6.4], [35, Definition 6.2].

######  Remark 2.26.

The semilattice of idempotents in $\operatorname{SPIso}(L^{p}(\mu))$ is
isomorphic to $[\Sigma_{\mu}]$, as it consists of multiplication operators of
idempotent elements in $L^{\infty}(\mu)$. The group of invertible elements in
$\operatorname{SPIso}(L^{p}(\mu))$ is isomorphic to
$UL^{\infty}(\mu)\rtimes\operatorname{Aut}([\Sigma_{\mu}])$, see Proposition
2.4.

######  Theorem 2.27 (Banach–Lamperti theorem for partial isometries).

Let $p\in[1,\infty]\setminus\\{2\\}$. A contraction $T\in B(L^{p}(\mu))$ is a
spatial partial isometry if and only if it has a contractive generalized
inverse $T^{*}\in B(L^{p}(\mu))$ such that both $TT^{*}$ and $T^{*}T$ are
multiplication operators on $L^{p}(\mu)$. Thus spatial partial isometries
coincide with $L^{p}$-partial isometries on $L^{p}(\mu)$:

$\operatorname{SPIso}(L^{p}(\mu))=L^{p}\mathchar
45\relax\operatorname{PIso}(L^{p}(\mu)),$

and in the complex case also with $MP$-partial isometries in $B(L^{p}(\mu))$.

###### Proof.

Using (2.3) we see that

$(\omega U_{\Phi})^{*}\omega
U_{\Phi}=T_{\Phi^{*}}(\overline{\omega})U_{\Phi^{*}}\circ\omega
U_{\Phi}=U_{\Phi^{*}\circ\Phi}=U_{\mathrm{id}_{\Sigma_{D_{\Phi^{*}}}}}=1_{D_{\Phi^{*}}}$

is the operator of multiplication by the characteristic function of the domain
$D_{\Phi^{*}}$ of $\Phi$. Conversely, if $T\in B(L^{p}(\mu))$ is a partial
isometry with a contractive generalized inverse $T^{*}\in B(L^{p}(\mu))$ such
that both $T^{*}T$ and $TT^{*}$ are multiplication operators on $L^{p}(\mu)$,
say by functions $1_{D^{*}}$ and $1_{D}$ respectively, then $T$ restricts to
an invertible isometry $T:L^{p}(\mu|_{D^{*}})\to L^{p}(\mu|_{D})$. Hence $T$
is of the form (2.2) by Proposition 2.5. In particular,
$\operatorname{SPIso}(L^{p}(\mu))=L^{p}\mathchar
45\relax\operatorname{PIso}(L^{p}(\mu))$. If $\mathbb{F}=\mathbb{C}$ and $T\in
B(L^{p}(\mu))$ is an $MP$-partial isometry and $T^{*}$ is its Moore–Penrose
generalized inverse, then the idempotents $T^{*}T$ and $T^{*}T$ are operators
of multiplication by characteristic functions of some measurable sets, by
Proposition 2.11. ∎

##  3\. Twisted inverse semigroup actions and their crossed products

We introduce Banach algebra crossed products by twisted inverse semigroup
actions, that generalize $C^{*}$-algebraic constructions introduced by Buss
and Exel in [15].

###  3.1. The crossed product

By an _ideal_ in a Banach algebra we always mean a closed two-sided ideal. A
_partial automorphism_ on a Banach algebra $A$ is an isometric isomorphism
$\alpha:I\to J$ between two ideals $I,J$ of $A$. The partial automorphisms on
$A$ form an inverse subsemigroup of $\operatorname{PBij}(A)$ that we denote by
$\operatorname{PAut}(A)$.

######  Definition 3.1.

An _action of a unital inverse semigroup $S$ on a Banach algebra $A$_ is a
unital semigroup homomorphism $\alpha:S\to\operatorname{PAut}(A)$. Thus an
action of $S$ on $A$ consists of (closed, two-sided) ideals $I_{t}$ of $A$ and
isometric isomorphisms $\alpha_{t}:I_{t^{*}}\to I_{t}$ for all $t\in S$, such
that $\alpha_{1}=\mathrm{id}_{A}$ and $\alpha_{s}\circ\alpha_{t}=\alpha_{st}$
for all $s,t\in S$.

######  Remark 3.2.

If $A$ is a $C^{*}$-algebra then all ideals $I_{t}$ are closed under
involution, and all isometric isomorphisms $\alpha_{t}:I_{t^{*}}\to I_{t}$ are
necessarily $*$-preserving. Hence Definition 3.1 is consistent with the well
established $C^{*}$-algebraic version [68].

######  Example 3.3.

If $A=C_{0}(X)$ for a locally compact Hausdorff space $X$ then
$\operatorname{PAut}(A)\cong\operatorname{PHomeo}(X)$. Indeed, every ideal $I$
in $A$ is of the form $I=C_{0}(U)$ for an open set $U\subseteq X$ (we treat
elements in $C_{0}(U)$ as functions in $C_{0}(X)$ that vanish outside $U$),
and any partial automorphism $\alpha:I\to J$ is given by the composition
$\alpha(a)=a\circ h^{-1}$ with a partial homeomorphism $h:U\to V$, where
$I=C_{0}(U)$ and $J=C_{0}(V)$. Thus inverse semigroup actions on the Banach
algebra $C_{0}(X)$ are equivalent to inverse semigroup actions on the
topological space $X$.

Twisted inverse semigroup actions on $C^{*}$-algebras were first introduced by
Sieben [69], but a more general definition given by Buss and Exel in [15] is
needed for a number of problems. To generalize this to Banach algebras it
seems reasonable to assume that all ideals involved in an action have
_(contractive, two-sided) approximate units_. This in particular guarantees
that the corresponding multiplier algebras have good properties and various
approaches to multipliers coincide. We refer to [23], see also [24] and
references therein, for more details. Following [24, Section 2], for a Banach
algebra $A$ we define the _multiplier algebra_ as

$\mathcal{M}(A):=\big{\\{}(L,R):\text{$L,R:A\to A$ satisfy $aL(b)=R(a)b$ for
all $a,b\in A$}\big{\\}}.$

Then $\mathcal{M}(A)$ is automatically a unital closed subalgebra of the
$\ell^{\infty}$-direct sum $B(A)\oplus B(A)^{\operatorname{op}}$, so in
particular it is a Banach algebra with the norm
$\|(L,R)\|=\max\\{\|L\|,\|R\|\\}$. Also $\mathcal{M}(A)$ is complete in the
strict topology, which is defined by the seminorms
$(L,R)\mapsto\|L(a)\|+\|R(a)\|$, $a\in A$. Every $a\in A$ yields a multiplier
$(L_{a},R_{a})$ where $L_{a}(b):=ab$, $R_{a}(b):=ba$, and if $A$ has a
contractive approximate unit the resulting map is isometric and allows us to
treat $A$ as an ideal of $\mathcal{M}(A)$. Then any isometric automorphism
$\alpha:A\to A$ extends uniquely to an isometric automorphism
$\overline{\alpha}:\mathcal{M}(A)\to\mathcal{M}(A)$. We denote by

$\mathcal{U}M(A):=\\{u\in\mathcal{M}(A):\text{$u$ is invertible and
$\|u\|=\|u^{-1}\|=1$}\\},$

the group of invertible isometries in $\mathcal{M}(A)$.

######  Definition 3.4 (cf. [15, Definition 4.1]).

A _twisted action_ of an inverse semigroup $S$ on a Banach algebra $A$ is a
pair $(\alpha,u)$, where $\alpha=\\{\alpha_{t}\\}_{t\in S}$ is a family of
partial automorphisms $\alpha_{t}:I_{t^{*}}\to I_{t}$ of $A$, such that each
ideal $I_{t}$ contains an approximate unit and their union is linearly dense
in $A$, and $u=\\{u(s,t)\\}_{s,t\in S}$ is a family of unitary multipliers
$u(s,t)\in\mathcal{U}M(I_{st})$, $t,s\in S$, such that the following holds for
all $r,s,t\in S$ and $e,f\in E(S)$:

  1. (A1)

$\alpha_{s}\circ\alpha_{t}=\textup{Ad}_{u(s,t)}\alpha_{st}$;

  2. (A2)

$\alpha_{r}\big{(}au(s,t)\big{)}u(r,st)=\alpha_{r}(a)u(r,s)u(rs,t)$ for $a\in
I_{r^{*}}\cap I_{st}$;

  3. (A3)

$u(e,f)=1_{ef}$ and $u(t,t^{*}t)=u(tt^{*},t)=1_{t}$, where $1_{t}$ is the unit
of $\mathcal{M}(I_{t})$;

  4. (A4)

$u(t^{*},e)u(t^{*}e,t)a=u(t^{*},t)a$ for all $a\in I_{t^{*}et}$.

######  Remark 3.5.

All properties in [15, Lemma 4.6] hold for the maps and ideals involved in a
twisted action on a Banach algebra as defined above. In particular, for all
$s,t\in S$ and $e\in E(S)$ we have

$I_{t}=I_{tt^{*}},\quad\overline{\alpha}_{t}\big{(}u(t^{*},t)\big{)}=u(t,t^{*}),\quad\alpha_{e}=\mathrm{id}|_{I_{e}},\quad\alpha_{s}(I_{s^{*}}\cap
I_{t})=I_{st},$

and $I_{s}\subseteq I_{t}$ whenever $s\leq t$. We will use these relations,
often without warning.

The authors of [68], [69], [15] consider covariant representations on Hilbert
spaces. We propose a more general definition of representations in Banach
algebras. Our definition works best when the target algebra has a predual,
which we explain in detail in Subsection 3.2 below. Therefore in general we
will use the _double dual_ $B^{\prime\prime}$ of a Banach algebra $B$ (which
has a predual $B^{\prime}$ by construction). Recall that $B^{\prime\prime}$ is
again naturally a Banach algebra with either of the Arens products, in which
$B$ sits as a $B^{\prime}$-weakly dense Banach algebra. We will consider
$B^{\prime\prime}$ equipped with the _second Arens product_ , which for
$a,b\in B^{\prime\prime}$ is determined by the formula $a\cdot
b:=B^{\prime}\mathchar 45\relax\lim_{\beta}B^{\prime}\mathchar
45\relax\lim_{\alpha}a_{\alpha}b_{\beta}$ where $\\{a_{\alpha}\\}_{\alpha}$,
$\\{b_{\beta}\\}_{\beta}\subseteq B$ are nets that are convergent to $a$ and
$b$ respectively, in the weak∗ topology on $B^{\prime\prime}$ induced by
$B^{\prime}$. The first Arens product is given by $a\square
b:=B^{\prime}\mathchar 45\relax\lim_{\alpha}B^{\prime}\mathchar
45\relax\lim_{\beta}a_{\alpha}b_{\beta}$, and $B$ is called _Arens regular_ if
the two products coincide, see [23] for more details. Our decision to use the
second Arens product is arbitrary, and in any case we will be primarily
interested in products $av\in B$ where $a\in B$ and $v\in B^{\prime\prime}$,
in which case the first and second Arens products always agree. Also recall
that every $C^{*}$-algebra (complex or real) is Arens regular.

For any representation $\pi:A\to B$ the double adjoint
$\pi^{\prime\prime}:A^{\prime\prime}\to B^{\prime\prime}$ is a
$B^{\prime}$-weakly continuous representation that extends $\pi$, and
$\pi^{\prime\prime}$ is isometric whenever $\pi$ is. In particular, for every
Banach subalgebra $A\subseteq B$ we may identify $A^{\prime\prime}$ with a
subalgebra of $B^{\prime\prime}$. Also any contractive approximate unit
$\\{\mu_{i}\\}$ in $A$ converges $A^{\prime}$-weakly to a left identity
$1_{A}$ in $A^{\prime\prime}$ (and a right identity for the first Arens
product in $A^{\prime\prime}$) and the map $(L,R)\mapsto
L^{\prime\prime}(1_{A})$ is an isometric embedding allowing us to assume that
$\mathcal{M}(A)\subseteq\\{b\in A^{\prime\prime}:\text{$ba,ab\in A$ for all
$a\in A$}\\}\subseteq A^{\prime\prime}$, see [23, Proposition 2.9.16 and
Theorem 2.9.49]. In particular, for each ideal $I$ in $A$, which has an
approximate unit, we will adopt the identifications
$I\subseteq\mathcal{M}(I)\subseteq I^{\prime\prime}\subseteq
A^{\prime\prime}$.

######  Definition 3.6.

Let $(\alpha,u)$ be a twisted action of $S$ on $A$. A _covariant
representation of $(\alpha,u)$ in a Banach algebra $B$_ is a pair $(\pi,v)$,
where $\pi:A\to B$ is a representation and $v:S\to(B^{\prime\prime})_{1}$ is a
map into contractive elements in $B^{\prime\prime}$, such that:

  1. (CR1)

$v_{t}\pi(a)=\pi(\alpha_{t}(a))v_{t}\in B$ for all $a\in I_{t^{*}}$;

  2. (CR2)

$\pi(a)v_{s}v_{t}=\pi(au(s,t))v_{st}$ for all $a\in I_{st}$, $s,t\in S$;

  3. (CR3)

$\pi(a)v_{e}=\pi(a)$ for all $a\in I_{e}$, $e\in E(S)$.

We call $B(\pi,v):=\operatorname{\overline{span}}\\{\pi(a_{t})v_{t}:a_{t}\in
I_{t},\,t\in S\\}$ the _range_ of $(\pi,v)$. We say that $(\pi,v)$ is
injective, isometric, etc. if $\pi$ has that property. We also say that
$(\pi,v)$ is _$B^{\prime}$ -normalized_ if

  1. (CR4)

$v_{t}=B^{\prime}\mathchar 45\relax\lim_{i}(\pi(\mu_{i}^{t})v_{t})$, for all
$t\in S$, where $\\{\mu^{t}_{i}\\}_{i}$ is an approximate unit in
$I_{t}=I_{tt^{*}}$.

######  Remark 3.7.

Condition (CR1) is equivalent to $\pi(a)v_{t}=v_{t}\pi(\alpha_{t}^{-1}(a))\in
B$ for all $a\in I_{t}$, $t\in S$. It implies that each $v_{t}$ can be viewed
as a “partial mutliplier” of $B$, and $B(\pi,v)\subseteq B$. Conditions (CR1)
and (CR3) imply that $v_{e}\pi(a)=\pi(a)$ for all $a\in I_{e}$, $e\in E(S)$.
Employing also (CR2) we get

(3.4) $v_{t}\pi(a)v_{t^{*}}=\pi\big{(}\alpha_{t}(a)u(t,t^{*})\big{)}$

for all $a\in I_{t^{*}}$, $t\in S$. Also, (CR3) and (CR4) imply that
$v_{e}=\pi^{\prime\prime}(1_{e})$ for every $e\in E(S)$.

Notice that the elements $v_{t}$ of Definition 3.6 are not required to be
partial isometries. The definition of the generalised inverse $v_{t}^{*}$ of
$v_{t}$ in the proof below involves both $v_{t^{*}}$ and the twist $u$, so the
covariance conditions (3.4) above and (1) below look different.

######  Proposition 3.8.

A pair $(\pi,v)$ is a $B^{\prime}$-normalized covariant representation of
$(\alpha,u)$ if and only if $\pi:A\to B$ is a representation and
$v:S\to(B^{\prime\prime})_{1}$ takes values in partial isometries such that
every $v_{t}$, $t\in S$, admits a contractive generalised inverse $v_{t}^{*}$,
satisfying $\pi(a)v_{t}\in B$, for $a\in I_{t}$, $t\in S$, and

  1. (1)

$v_{t}\pi(a)v_{t}^{*}=\pi(\alpha_{t}(a))$ for all $a\in I_{t^{*}}$, $t\in S$;

  2. (2)

$v_{s}v_{t}=\pi^{\prime\prime}(u(s,t))v_{st}$ for all $s,t\in S$;

  3. (3)

$v_{t}v_{t}^{*}=\pi^{\prime\prime}(1_{t})$ and
$v_{t}^{*}v_{t}=\pi^{\prime\prime}(1_{t^{*}})$ for all $t\in S$.

Moreover, for any covariant representation $(\pi,v)$ of $(\alpha,u)$ in $B$
there is a unique $B^{\prime}$-normalized covariant representation
$(\pi,\tilde{v})$ such that $\pi(a)v_{t}=\pi(a)\tilde{v}_{t}$ for all $a\in
I_{t}$ and $t\in S$, namely $\tilde{v}_{t}=B^{\prime}\mathchar
45\relax\lim_{i}(\pi(\mu_{i}^{t})v_{t})=v_{t}\pi^{\prime\prime}(1_{t^{*}})$,
$t\in S$.

###### Proof.

Assume $(\pi,v)$ is a pair with the properties described in the assertion. For
every $t\in S$ and $a\in I_{t^{*}}$, using (3) and (1) we have

$v_{t}\pi(a)=v_{t}\pi(a)\pi^{\prime\prime}(1_{t^{*}})=v_{t}\pi(a)v_{t}^{*}v_{t}=\pi\big{(}\alpha_{t}(a)\big{)}v_{t},$

which is (CR1). Condition (2) immediately gives (CR2). By (3) we have
$v_{t}=v_{t}v_{t}^{*}v_{t}=\pi^{\prime\prime}(1_{t})v_{t}=v_{t}\pi^{\prime\prime}(1_{t^{*}})$.
This implies (CR4) and that for $e\in E(S)$ we get
$v_{e}v_{e}=\pi^{\prime\prime}(u(e,e))v_{e}=\pi^{\prime\prime}(1_{e})v_{e}=v_{e}$.
Therefore
$\pi^{\prime\prime}(1_{e})=v_{e}^{*}v_{e}=v_{e}^{*}v_{e}v_{e}=\pi^{\prime\prime}(1_{e})v_{e}=v_{e}$,
and using this one gets (CR3). Hence $(\pi,v)$ is a $B^{\prime}$-normalized
covariant representation of $(\alpha,u)$.

For the converse let $(\pi,v)$ be any $B^{\prime}$-normalized covariant
representation. For each $t\in S$ choose an approximate unit
$\\{\mu^{t}_{i}\\}_{i}$ in $I_{t}=I_{tt^{*}}$. Using (CR1) and
$B^{\prime}$-continuity of multiplication in $B^{\prime\prime}$ in the second
variable we get

$B^{\prime}\mathchar
45\relax\lim_{i}\big{(}\pi(\mu^{t}_{i})v_{t}\big{)}=B^{\prime}\mathchar
45\relax\lim_{i}\big{(}v_{t}\pi(\alpha_{t}^{-1}(\mu^{t}_{i}))\big{)}=v_{t}B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}\alpha_{t}^{-1}(\mu^{t}_{i})\big{)}=v_{t}\pi^{\prime\prime}(1_{t^{*}}),$

because $\\{\alpha_{t}^{-1}(\mu^{t}_{i})\\}_{i}$ is an approximate unit in
$I_{t^{*}}$ which is $B^{\prime}$-convergent to $1_{t^{*}}$. Thus
$v_{t}=B^{\prime}\mathchar
45\relax\lim_{i}(\pi(\mu^{t}_{i})v_{t})=v_{t}\pi^{\prime\prime}(1_{t^{*}})$.
Also (CR3) and (CR4) imply that $v_{e}=\pi^{\prime\prime}(1_{e})$ for all
$e\in E(S)$. We put $v_{t}^{*}:=v_{t^{*}}\pi^{\prime\prime}(u(t,t^{*})^{-1})$.
Using (CR1), (A1) and (CR2), and being careful to use only
$B^{\prime}$-continuity of multiplication in $B^{\prime\prime}$ in the second
variable, we get

$\begin{split}v_{t}v_{t}^{*}&=B^{\prime}\mathchar
45\relax\lim_{i}v_{t}v_{t^{*}}\pi\big{(}\mu^{t}_{i}u(t,t^{*})^{-1}\big{)}\\\
&=B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}\alpha_{t}\circ\alpha_{t^{*}}\big{(}\mu^{t}_{i}u(t,t^{*})^{-1}\big{)}\big{)}v_{t}v_{t^{*}}\\\
&=B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}u(t,t^{*})\mu^{t}_{i}u(t,t^{*})^{-1}u(t,t^{*})^{-1}\big{)}v_{t}v_{t^{*}}\\\
&=B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}u(t,t^{*})\mu^{t}_{i}u(t,t^{*})^{-1}u(t,t^{*})^{-1}u(t,t^{*})\big{)}v_{tt^{*}}\\\
&=B^{\prime}\mathchar
45\relax\lim_{i}\pi^{\prime\prime}\big{(}u(t,t^{*})\mu^{t}_{i}u(t,t^{*})^{-1}\big{)}v_{tt^{*}}=B^{\prime}\mathchar
45\relax\lim_{i}v_{tt^{*}}\pi^{\prime\prime}\big{(}u(t,t^{*})\mu^{t}_{i}u(t,t^{*})^{-1}\big{)}\\\
&=v_{tt^{*}}\pi^{\prime\prime}(1_{t})=\pi^{\prime\prime}(1_{t}).\end{split}$

By Remark 3.5 we have
$\overline{\alpha}_{t^{*}}(u(t,t^{*})^{-1})=u(t^{*},t)^{-1}$, so a similar
calculation gives

$\begin{split}v_{t}^{*}v_{t}&=B^{\prime}\mathchar
45\relax\lim_{i}v_{t^{*}}\pi\big{(}u(t,t^{*})^{-1}\mu^{t}_{i}\big{)}v_{t}=B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}u(t^{*},t)^{-1}\alpha_{t^{*}}(\mu^{t}_{i})\big{)}v_{t^{*}}v_{t}\\\
&=B^{\prime}\mathchar
45\relax\lim_{i}\pi\big{(}u(t^{*},t)^{-1}\alpha_{t^{*}}(\mu^{t}_{i})u(t^{*},t)\big{)}v_{t^{*}t}=B^{\prime}\mathchar
45\relax\lim_{i}v_{t^{*}t}\pi\big{(}u(t^{*},t)^{-1}\alpha_{t^{*}}(\mu^{t}_{i})u(t^{*},t)\big{)}\\\
&=v_{t^{*}t}\pi^{\prime\prime}(1_{t^{*}})=\pi^{\prime\prime}(1_{t^{*}}).\end{split}$

This finishes the proof of (3) and shows that $v_{t}$ and $v_{t}^{*}$ are
mutual generalized inverses. For $a\in I_{t^{*}}$ we get

$\begin{split}v_{t}\pi(a)v_{t}^{*}&=v_{t}\pi(a)v_{t^{*}}\pi^{\prime\prime}\big{(}u(t,t^{*})^{-1}\big{)}=\pi\big{(}\alpha_{t}(a)\big{)}v_{t}v_{t^{*}}\pi^{\prime\prime}\big{(}u(t,t^{*})^{-1}\big{)}\\\
&=\pi\big{(}\alpha_{t}(a)\big{)}\pi^{\prime\prime}\big{(}u(t,t^{*})\big{)}v_{tt^{*}}\pi^{\prime\prime}(u(t,t^{*})^{-1})\\\
&=\pi\big{(}\alpha_{t}(a)\big{)}\pi^{\prime\prime}\big{(}u(t,t^{*})u(t,t^{*})^{-1}\big{)}=\pi\big{(}\alpha_{t}(a)\big{)},\end{split}$

which is (1). Furthermore, $\alpha_{s}^{\prime\prime}(1_{s^{*}}1_{t})=1_{st}$
because $\alpha_{s}(I_{s^{*}}\cap I_{t})=I_{st}$ and therefore

$v_{s}v_{t}=v_{s}\pi^{\prime\prime}(1_{s^{*}})\pi^{\prime\prime}(1_{t})v_{t}=\pi^{\prime\prime}(1_{st})v_{s}v_{t}=\pi^{\prime\prime}\big{(}u(s,t)\big{)}v_{st},$

which proves (2). Hence $(\pi,v)$ satisfies all properties in the assertion.

Now let $(\pi,v)$ be any covariant representation of $(\alpha,u)$ in $B$. As
above we see that the limit $\tilde{v}_{t}:=B^{\prime}\mathchar
45\relax\lim_{i}(\pi(\mu_{i}^{t})v_{t})=v_{t}\pi^{\prime\prime}(1_{t^{*}})$
exists and does not depend on the choice of an approximate unit
$\\{\mu^{t}_{i}\\}_{i}$ in $I_{t}$ ($t\in S$). Clearly if $(\pi,\tilde{v})$ is
a $B^{\prime}$-normalized covariant representation satisfying
$\pi(a)v_{t}=\pi(a)\tilde{v}_{t}$ for all $a\in I_{t}$ and $t\in S$, then each
$\tilde{v}_{t}$ must be given by such a limit, giving (CR4). For $a\in
I_{t^{*}}$ we get

$\tilde{v}_{t}\pi(a)=v_{t}\pi^{\prime\prime}(1_{t^{*}})\pi(a)=v_{t}\pi(a)=\pi\big{(}\alpha_{t}(a)\big{)}v_{t}=\lim_{i}\pi\big{(}\alpha_{t}(a)\big{)}\pi(\mu_{i}^{t})v_{t}=\pi\big{(}\alpha_{t}(a)\big{)}\tilde{v}_{t}.$

This shows that $(\pi,\tilde{v})$ satisfies (CR1) and
$\pi(a)v_{t}=\pi(a)\tilde{v}_{t}$ for all $a\in I_{t}$ and $t\in S$. Using
this we get $\pi(a)\tilde{v}_{e}=\pi(a)v_{e}=\pi(a)$ for all $a\in I_{e}$,
$e\in E(S)$, so (CR3) holds. Finally, for all $s,t\in S$ and $a\in I_{st}$ the
calculation

$\pi(a)\tilde{v}_{s}\tilde{v}_{t}=v_{s}\pi\big{(}\alpha_{s^{*}}(a)\big{)}\tilde{v}_{t}=\pi(a)v_{s}v_{t}=\pi\big{(}au(s,t)\big{)}v_{st}=\pi\big{(}au(s,t)\big{)}\tilde{v}_{st}$

gives (CR2). Thus $(\pi,\tilde{v})$ is a $B^{\prime}$-normalized covariant
representation of $(\alpha,u)$. ∎

######  Remark 3.9.

If each $I_{t}$, $t\in S$, is unital then a covariant representation $(\pi,v)$
is $B^{\prime}$-normalized if and only if $v_{t}=\pi(1_{t})v_{t}\in B$ for
each $t\in S$, and Proposition 3.8 could be formulated without the use of the
bidual algebra $B^{\prime\prime}$ and the extended representation
$\pi^{\prime\prime}$.

######  Lemma 3.10.

Let $(\pi,v)$ be a covariant representation of $\alpha$ in a Banach algebra
$B$. The range $B(\pi,v)$ is a Banach subalgebra of $B$. The spaces
$A_{t}=\\{\pi(a_{t})v_{t}:a_{t}\in I_{t}\\}$, $t\in S$, form a grading of
$B(\pi,v)$ over the inverse semigroup $S$ in the sense that

$B(\pi,v)=\operatorname{\overline{span}}\\{A_{t}:t\in S\\},\qquad
A_{s}A_{t}\subseteq A_{st},\qquad\text{$s\leq t$ implies $A_{s}\subseteq
A_{t}$},$

for all $s,t\in S$. In fact, we have

  1. (1)

$\pi(a_{s})v_{s}\cdot\pi(a_{t})v_{t}=\pi\big{(}\alpha_{s}(\alpha_{s}^{-1}(a_{s})a_{t})u(s,t)\big{)}v_{st}$
and $\alpha_{s}\big{(}\alpha_{s}^{-1}(a_{s})a_{t}\big{)}\in I_{st}$ for all
$a_{t}\in I_{t},a_{s}\in I_{s}$, $s,t\in S$;

  2. (2)

$s\leq t$ implies $I_{s}\subseteq I_{t}$ and
$\pi(a)v_{s}=\pi\big{(}au(ss^{*},t)^{-1}\big{)}v_{t}$ for any $a\in I_{s}$;

If in addition $A$ and $B$ are $C^{*}$-algebras then
$(\pi(a_{t})v_{t})^{*}=\pi\big{(}\alpha_{t}^{-1}(a_{t}^{*})u(t^{*},t)^{-1}\big{)}v_{t^{*}}$
for all $a_{t}\in I_{t}$, $t\in S$, so $A_{t}^{*}=A_{t^{*}}$, and
$\\{A_{t}\\}_{t\in S}$ is a saturated grading of $B(\pi,v)$ over the inverse
semigroup of $S$ in the sense of [50, Definition 6.15], see also [31,
Definition 7.1].

###### Proof.

(1) For any $a_{t}\in I_{t},a_{s}\in I_{s}$, $s,t\in S$, we have
$\alpha_{s}^{-1}(a_{s})a_{t}\in I_{s^{*}}I_{t}\subseteq I_{s^{*}}\cap I_{t}$
and so $\alpha_{s}(\alpha_{s}^{-1}(a_{s})a_{t})\in I_{st}$ because
$\alpha_{s}(I_{s^{*}}\cap I_{t})=I_{st}$, see Remark 3.5. Using (CR1) we get

$\pi(a_{s})v_{s}\cdot\pi(a_{t})v_{t}=v_{s}\pi\big{(}\alpha_{s}^{-1}(a_{s})\big{)}\cdot\pi(a_{t})v_{t}=v_{s}\pi\big{(}\alpha_{s}^{-1}(a_{s})a_{t}\big{)}v_{t}=\pi\Big{(}\alpha_{s}\big{(}\alpha_{s}^{-1}(a_{s})a_{t}\big{)}u(s,t)\Big{)}v_{st}.$

(2) If $s\leq t$ then $I_{s}\subseteq I_{t}$ by Remark 3.5. Using (CR3) and
(CR2), for any $a\in I_{s}$ we get
$\pi(a)v_{t}=\pi(a)v_{ss^{*}}v_{t}=\pi(au(ss^{*},t))v_{ss^{*}t}=\pi(au(ss^{*},t))v_{s}$.

If $B$ is a $C^{*}$-algebra then $B^{\prime\prime}$ is again a $C^{*}$-algebra
– the enveloping $W^{*}$-algebra of $B$ (this also holds in the real case, see
[53, Theorem 5.5.3]). In particular,
$\pi^{\prime\prime}(1_{t})v_{t}=v_{t}\pi^{\prime\prime}(1_{t^{*}})$ is a
partial isometry whose adjoint is
$\pi^{\prime\prime}(u(t^{*},t)^{-1})v_{t^{*}}$, see (the proof of) Proposition
3.8. Thus for $a_{t}\in I_{t}$, $t\in S$, we get
$\big{(}\pi(a_{t})v_{t}\big{)}^{*}=\Big{(}v_{t}\pi^{\prime\prime}(1_{t^{*}})\pi\big{(}\alpha_{t}^{-1}(a_{t})\big{)}\Big{)}^{*}=\pi\big{(}\alpha_{t}^{-1}(a_{t})\big{)}^{*}\big{(}\pi^{\prime\prime}(1_{t^{*}})v_{t}\big{)}^{*}=\pi\big{(}\alpha_{t}^{-1}(a_{t})\big{)}^{*}\pi^{\prime\prime}\big{(}u(t^{*},t)^{-1}\big{)}v_{t^{*}}.$
If in addition $A$ is a $C^{*}$-algebra then both $\pi$ and $\alpha_{t}$ are
necessarily $*$-preserving, and thus
$(\pi(a_{t})v_{t})^{*}=\pi(\alpha_{t}^{-1}(a_{t}^{*})u(t^{*},t)^{-1})v_{t^{*}}$.
∎

For any twisted action $(\alpha,u)$ the subspace
$\ell^{1}(\alpha,u):=\\{f\in\ell^{1}(S,A):\text{$f(t)\in I_{t}$ for all $t\in
S$}\\}$ of $\ell^{1}(S,A)$ is a Banach algebra with multiplication

$(f*g)(r):=\sum_{st=r}\alpha_{s}\Big{(}\alpha_{s^{*}}\big{(}f(s)\big{)}g(t)\Big{)}u(s,t),$

see (the proof of) [69, Proposition 3.1]. If $A$ is a $C^{*}$-algebra then in
fact $\ell^{1}(\alpha,u)$ is a Banach $*$-algebra with involution given by
$f^{*}(t):=\alpha_{t}^{-1}(f(t^{*})^{*})u(t^{*},t)^{-1}$, $t\in S$. Lemma 3.10
(see also [69, Proposition 3.7]) implies that every covariant representation
$(\pi,v)$ of $\alpha$ in a Banach algebra $B$ ‘integrates’ to a representation
$\pi\times v:\ell^{1}(\alpha,u)\to B$ given by

$\pi\times v(f):=\sum_{t\in S}\pi\big{(}f(t)\big{)}v_{t},\qquad
f\in\ell^{1}(\alpha,u),$

and $\overline{\pi\times v(\ell^{1}(\alpha,u))}=B(\pi,v)$ is a Banach algebra.

######  Definition 3.11.

For a fixed class $\mathcal{R}$ of covariant representations of $(\alpha,u)$
we define the _$\mathcal{R}$ -crossed product_ for $(\alpha,u)$, denoted
$A\rtimes_{(\alpha,u),\mathcal{R}}S$, as the Hausdorff completion of
$\ell^{1}(\alpha,u)$ in the seminorm

$\|f\|_{\mathcal{R}}:=\sup\\{\|\pi\times v(f)\|:(\pi,v)\in\mathcal{R}\\}.$

The _universal Banach algebra crossed product_ $A\rtimes_{(\alpha,u)}S$ is the
crossed product associated to the class of all covariant representations of
$(\alpha,u)$.

######  Remark 3.12.

If $S=G$ is a group then we necessarily have $I_{t}=A$ for all $t\in G$, so
$\alpha:G\to\operatorname{Aut}(A)$, and
$A\rtimes_{(\alpha,u)}G=\ell^{1}(\alpha,u)$ is $\ell^{1}(G,A)$ as a Banach
space. When $A=C_{0}(X)$ we study such crossed products in [6], and in
Subsection 7.1 below we discuss their generalizations to partial actions. For
general $A$, but when the twist $u\equiv 1$ is trivial, $(A,G,\alpha)$ is a
Banach algebra dynamical system in the sense of [26], and the algebras
$A\rtimes_{\alpha,\mathcal{R}}G$ are examples of crossed products considered
in [26].

###  3.2. Covariant representations in dual Banach algebras and on Banach
spaces

We fix a twisted action $(\alpha,u)$ of an inverse semigroup $S$ on $A$. We
will now discuss situations, beyond the one described in Remark 3.9, when in
the study of algebras generated by covariant representations we do not need to
pass through the bidual algebra $B^{\prime\prime}$.

######  Definition 3.13.

Let $(\pi,v)$ be a covariant representation of $(\alpha,u)$ in a Banach
algebra $B$. If $B$ has a predual Banach space $B_{*}$, we say that $(\pi,v)$
is _$B_{*}$ -normalized_ if, for every $t\in S$, we have $v_{t}=B_{*}\mathchar
45\relax\lim_{i}(\pi(\mu_{i}^{t})v_{t})$, for an approximate unit
$\\{\mu^{t}_{i}\\}_{i}$ in $I_{t}=I_{tt^{*}}$. Here $B_{*}\mathchar
45\relax\lim$ denotes the limit in the weak∗ topology on $B$ induced by
$B_{*}$.

######  Lemma 3.14.

If $B_{*}^{\prime}\cong B$ and $(\pi,v)$ is a $B_{*}$-normalized covariant
representation in $B$, then $v:S\to B$ takes values in $B$ (rather than in
$B^{\prime\prime}$).

###### Proof.

The limit $v_{t}=B_{*}\mathchar 45\relax\lim_{i}\pi(\mu_{i}^{t})v_{t}$ is in
$B$ as $\\{\pi(\mu_{i}^{t})v_{t}\\}_{i}\subseteq B_{1}$ and $B_{1}$ is
$B_{*}$-compact. ∎

A _dual Banach algebra_ is a pair $(B,B_{*})$ where $B$ is a Banach algebra,
$B_{*}$ is a predual of $B$, and in addition multiplication in $B$ is
separately $B_{*}$-continuous, see [66], [24]. Examples of dual Banach
algebras include all $W^{*}$-algebras (complex or real) with their unique
preduals, and all algebras $B(E)$ of all bounded operators on a reflexive
Banach space $E$, equipped with the canonical predual Banach space
$E^{\prime}\widehat{\otimes}E$. By [44, Theorem 5.6], every representation
$\pi:A\to B$ into a dual Banach algebra $(B,B_{*})$ has a unique extension to
a representation $\overline{\pi}:\mathcal{M}(A)\to B$ which is
strictly-$B_{*}$-continuous; it is given by $\overline{\pi}(m):=B_{*}\mathchar
45\relax\lim_{i}\pi(m\mu_{i})$ for an approximate unit $\\{\mu_{i}\\}_{i}$ in
$A$.

######  Proposition 3.15.

Assume $(B,B_{*})$ is a dual Banach algebra. For any covariant representation
$(\pi,v)$ of $(\alpha,u)$ in $B$ there is a unique $B_{*}$-normalized
covariant representation $(\pi,\tilde{v})$ such that
$\pi(a)v_{t}=\pi(a)\tilde{v}_{t}$ for all $a\in I_{t}$ and $t\in S$. A pair
$(\pi,v)$ is a $B_{*}$-normalized covariant representation of $(\alpha,u)$ in
$B$ if and only if $\pi:A\to B$ is a representation and $v:S\to B_{1}$ is a
map such that every $v_{t}$, $t\in S$, admits a contractive generalised
inverse $v_{t}^{*}$, satisfying

$v_{t}\pi(a)v_{t}^{*}=\pi\big{(}\alpha_{t}(a)\big{)},\quad
v_{s}v_{t}=\pi_{st}\big{(}u(s,t)\big{)}v_{st},\quad\text{$v_{t}v_{t}^{*}=\pi_{t}(1_{t})$
and $v_{t}^{*}v_{t}=\pi_{t^{*}}(1_{t^{*}})$},$

where $a\in I_{t^{*}}$, $s,t\in S$ and $\pi_{t}:\mathcal{M}(I_{t})\to B$ is
the unique strictly-$B_{*}$-continuous extension of $\pi|_{I_{t}}$.

###### Proof.

Follow the proof of Proposition 3.8 (some arguments can be simplified as the
multiplication in $B$ is $B_{*}$-continuous in both variables). ∎

######  Remark 3.16.

For any Banach algebra $B$ the pair $(B^{\prime\prime},B^{\prime})$ is a dual
Banach algebra if and only if $B$ is Arens regular. Thus Proposition 3.8
cannot be deduced from Proposition 3.15. We were not able to find a nice
common generalization of these two statements.

######  Remark 3.17.

Proposition 3.15 implies that a $B_{*}$-normalized representation of an
untwisted action $\alpha$ (an action with a trivial twist) in a dual Banach
algebra $B$ is a pair $(\pi,v)$, where $\pi:A\to B$ is a representation and
$v:S\to B_{1}$ is a semigroup homomorphism, such that
$v_{t}\pi(a)v_{t^{*}}=\pi(\alpha_{t}(a))$ and $v_{e}=\pi_{e}(1_{e})$ for $a\in
I_{t^{*}}$, $s,t\in S$, $e\in E(S)$.

We now relate the above discussion to representations on Banach spaces. Note
that if $\pi:A\to B(E)$ is a representation then, for each $t\in S$, $\pi$
yields a non-degenerate representation $I_{t}\to B(\overline{\pi(I_{t})E})$
which thus uniquely extends to a representation

$\overline{\pi}_{t}:\mathcal{M}(I_{t})\to B(\overline{\pi(I_{t})E});\
\overline{\pi}_{t}(m)\big{(}\pi(a)x\big{)}:=\pi(ma)x,\qquad
m\in\mathcal{M}(I_{t}),\ a\in I_{t},\ x\in E,$

see for instance [38, Theorem 4.1].

######  Definition 3.18.

A _covariant representation of $(\alpha,u)$ on a Banach space $E$_ is a pair
$(\pi,v)$ where $\pi:A\to B(E)$ is a representation and $v:S\to B(E)_{1}$ has
the property that every $v_{t}$, $t\in S$, admits a contractive generalised
inverse $v_{t}^{*}$, satisfying:

  1. (SCR1)

$v_{t}\pi(a)v_{t}^{*}=\pi(\alpha_{t}(a))$ for all $a\in I_{t^{*}}$, $t\in S$;

  2. (SCR2)

the ranges of $v_{t}$ and $v_{t}^{*}$ are $\overline{\pi(I_{t})E}$ and
$\overline{\pi(I_{t^{*}})E}$ respectively, for all $t\in S$;

  3. (SCR3)

$v_{s}v_{t}=\overline{\pi}_{st}(u(s,t))v_{st}$ for all $s,t\in S$.

Equality in (SCR3) holds pointwise and it makes sense due to (SCR2). We say
that $(\pi,v)$ is non-degenerate, injective, etc. if $\pi$ has that property.

######  Remark 3.19.

If $\mathbb{F}=\mathbb{C}$, $E=H$ is a Hilbert space and $A$ is a
$C^{*}$-algebra then Definition 3.18 is equivalent to [69, Definition 3.2] and
[15, Definition 6.2].

######  Lemma 3.20.

Let $(\pi,v)$ be a covariant representation of $(\alpha,u)$ on a Banach space
$E$. Then $(\pi,v)$ is a covariant representation in the Banach algebra $B(E)$
such that each $v_{e}\in B(E)$, $e\in E(S)$, is a strong limit of
$\\{\pi(\mu_{i}^{e})\\}_{i}$, where $\\{\mu_{i}^{e}\\}_{i}$ is a contractive
approximate unit in $I_{e}$.

###### Proof.

The arguments in the beginning of the proof of Proposition 3.8 show that
$(\pi,v)$ is a covariant representation in the Banach algebra $B(E)$. Then
each $v_{e}\in B(E)$, $e\in E(S)$, is a projection onto
$\overline{\pi(I_{e})E}$ which commutes with elements of $\pi(I_{e})$, and
thus $v_{e}$ is a strong limit of $\\{\pi(\mu_{i}^{e})\\}_{i}$. ∎

######  Corollary 3.21.

If $A$ is a complex $C^{*}$-algebra and $(\pi,v)$ is a non-degenerate
covariant representation of $(\alpha,u)$ on a complex Banach space $E$ then
each $v_{t}$, $t\in S$, is an $MP$-partial isometry on $E$.

###### Proof.

Since $\pi$ is non-degenerate we may extend it to the unitization of $A$ and
so we may assume that $A$ and $\pi$ are in fact unital. Since $A$ is a
$C^{*}$-algebra, for each $e\in E(S)$ we may choose an approximate unit
$\\{\mu_{i}^{e}\\}_{i}$ in $I_{e}$ consisting of hermitian operators. Then
$\\{\pi(\mu_{i}^{e})\\}_{i}$ are hermitian operators by [19, Lemma 2.4]. As
$v_{e}$ is a strong limit of $\\{\pi(\mu_{i}^{e})\\}_{i}$, this implies that
$v_{e}$ is also hermitian. Indeed, recall that $a\in B(E)$ is hermitian if and
only if its numerical range $V(a)=\\{f(ax):x\in E,\ f\in E^{\prime},\
\|f\|=\|x\|=1=f(x)\\}$ is real. Hence for any $x\in E,\ f\in E^{\prime}$ with
$\|f\|=\|x\|=1=f(x)$ we get
$f(v_{e}x)=f(\lim_{i}\pi(\mu_{i}^{e})x)\in\mathbb{R}$, and so $v_{e}$ is
hermitian. As $v_{t}v_{t}^{*}=v_{tt^{*}}$ and $v_{t}^{*}v_{t}=v_{t^{*}t}$,
where $tt^{*},t^{*}t\in E(S)$, for all $t\in S$, we conclude that
$\\{v_{t}\\}_{t\in S}\subseteq B(E)_{1}$ are $MP$-partial isometries. ∎

######  Lemma 3.22.

Assume that $E$ is a reflexive Banach space, and recall that the projective
tensor product Banach space $B_{*}:=E^{\prime}\mathbin{\widehat{\otimes}}E$ is
naturally a predual of $B:=B(E)$ making it a dual Banach algebra. A covariant
representation $(\pi,v)$ of $(\alpha,u)$ in $B(E)$ is $B_{*}$-normalized if
and only if $(\pi,v)$ is a covariant representation on the Banach space $E$.

###### Proof.

We identify $f\otimes\xi\in E^{\prime}\mathbin{\widehat{\otimes}}E=B_{*}$ with
the functional map $B\ni b\mapsto f\otimes\xi(b):=f(b\xi)\in\mathbb{F}$.
Assume $(\pi,v)$ is $B_{*}$-normalized, and so $\\{v_{t}\\}_{t\in S}\subseteq
B(E)$ are partial isometries with the associated generalized inverses
$\\{v_{t}^{*}\\}_{t\in S}\subseteq B(E)$, satisfying the relations described
in Proposition 3.15, which include (SCR1). For $e\in E(S)$ we have
$v_{e}=\pi_{e}(1_{e})=B_{*}\mathchar 45\relax\lim\pi(\mu_{i}^{e})$, where
$\\{\mu_{i}^{e}\\}_{i}$ is an approximate unit in $I_{e}$. Thus
$f(v_{e}\xi)=\lim_{i}f(\pi(\mu_{i}^{e})\xi)$ for all $(f,\xi)\in
E^{\prime}\times E$, which implies that for any $\xi\in E$ we have
$v_{e}\xi\in\overline{\pi(I_{e})E}^{E^{\prime}}$. As the weak and norm
closures of convex sets coincide this means that the range of $v_{e}$ is
$\overline{\pi(I_{e})E}$. Thus, for each $t\in S$, we have
$v_{tt^{*}}=\pi_{t}(1_{t})$ is a projection onto the space
$\overline{\pi(I_{t})E}=\overline{\pi(I_{tt^{*}})E}$. This implies (SCR2)
because $v_{t}=\pi_{t}(1_{t})v_{t}$ and
$v_{t}^{*}=\pi_{t}(1_{t^{*}})v_{t}^{*}$. We also have
$\pi_{t}(m)=\overline{\pi}_{t}(m)\pi_{t}(1_{t})$ for $m\in\mathcal{M}(I_{t})$,
which immediately gives (SCR3) from $v_{s}v_{t}=\pi_{st}(u(s,t))v_{st}$.

Conversely, assume $(\pi,v)$ is a covariant representation in the sense of
Definition 3.18. By Lemma 3.20 each $v_{tt^{*}}$ is a strong limit of
$\\{\pi(\mu_{i}^{t})\\}_{i}$, where $\\{\mu_{i}^{t}\\}_{i}$ is an approximate
unit in $I_{t}=I_{tt^{*}}$. Using this, one sees that
$v_{tt^{*}}=\pi_{t}(1_{t})$ and
$\pi_{t}(m)=\overline{\pi}_{t}(m)\pi_{t}(1_{t})$ for $m\in\mathcal{M}(I_{t})$.
Now one readily gets the relations in Proposition 3.15. ∎

######  Corollary 3.23.

For any covariant representation $(\pi,v)$ of $(\alpha,u)$ in $B(E)$, where
$E$ is a reflexive Banach space, there is a unique covariant representation
$(\pi,\tilde{v})$ on $E$ such that $\pi(a)v_{t}=\pi(a)\tilde{v}_{t}$ for all
$a\in I_{t}$ and $t\in S$ (in fact $(\pi,\tilde{v})$ is
$E^{\prime}\mathbin{\widehat{\otimes}}E$-normalized).

###### Proof.

Combine Lemma 3.22 and Proposition 3.15. ∎

######  Corollary 3.24.

Let $(\alpha,u)$ be a twisted inverse semigroup action on a $C^{*}$-algebra
$A$. Let $\mathcal{R}_{C^{*}}$ be the class of all covariant representations
of $(\alpha,u)$ in some $C^{*}$-algebra, and let $\mathcal{R}_{\textup{Hil}}$
be the class of all covariant representations of $(\alpha,u)$ on some Hilbert
space. Then

$A\rtimes_{(\alpha,u),\mathcal{R}_{C^{*}}}S=A\rtimes_{(\alpha,u),\mathcal{R}_{\textup{Hil}}}S,$

and when $\mathbb{F}=\mathbb{C}$ this coincides with the crossed product
introduced in [15], [69].

###### Proof.

Since $\mathcal{R}_{\textup{Hil}}\subseteq\mathcal{R}_{C^{*}}$ we get
$\|\cdot\|_{\mathcal{R}_{\textup{Hil}}}\leq\|\cdot\|_{\mathcal{R}_{C^{*}}}$.
This is equality, because for any covariant representation
$(\pi,v)\in\mathcal{R}_{C^{*}}$ in a $C^{*}$-algebra $B$, we may assume that
the enveloping $W^{*}$-algebra $B^{\prime\prime}$ is faithfully represented on
a Hilbert space $H$. Then, by Corollary 3.23, the renormalized covariant
representation $(\pi,\tilde{v})$ is a covariant representation of $\alpha$ on
$H$, so $(\pi,\tilde{v})\in\mathcal{R}_{\textup{Hil}}$. The claim follows
because Corollary 3.23 implies $\pi\times v(f)=\pi\times\tilde{v}(f)$ for all
$f\in\ell^{1}(\alpha,u)$. ∎

##  4\. Groupoid Banach algebras and inverse semigroup disintegration

Let $\mathcal{G}$ be an étale groupoid with locally compact Hausdorff unit
space $X$. By a twist over $\mathcal{G}$ we mean a Fell line bundle [47]. This
is equivalent to twists introduced by Kumjian in [46], see [47, 2.5.iv]. More
specifically, let $\mathcal{L}$ be a line bundle over $\mathcal{G}$.
Equivalently, $\mathcal{L}=\bigsqcup_{\gamma\in\mathcal{G}}L_{\gamma}$ is a
topological space such that the canonical projection
$\mathcal{L}\twoheadrightarrow\mathcal{G}$ is continuous and open, each fiber
$L_{\gamma}\cong\mathbb{F}$ is a one dimensional Banach space with a structure
consistent with the topology of $\mathcal{L}$, in the sense that the maps

$\bigsqcup_{\gamma\in\mathcal{G}}L_{\gamma}\times L_{\gamma}\ni(z,w)\mapsto
z+w\in\mathcal{L},\quad\mathbb{F}\times\mathcal{L}\ni(\lambda,z)\mapsto\lambda
z\in\mathcal{L},\quad\mathcal{L}\ni z\mapsto|z|\in\mathbb{R}$

are continuous. Then $\mathcal{L}$ is necessarily locally trivial. For any
open $U\subseteq\mathcal{G}$ we write $C(U,\mathcal{L})$ for the linear space
of continuous sections in the restricted bundle
$\mathcal{L}|_{U}:=\bigsqcup_{\gamma\in U}L_{\gamma}$ (i.e. continuous maps
$f:U\to\mathcal{L}$ with $f(\gamma)\in L_{\gamma}$ for $\gamma\in U$).
Similarly we denote by $C_{c}(U,\mathcal{L})$, $C_{0}(U,\mathcal{L})$,
$C_{b}(U,\mathcal{L})$, $C_{u}(U,\mathcal{L})$ the spaces of continuous
sections with compact support, vanishing at infinity, bounded and unitary,
respectively. Then $C_{0}(U,\mathcal{L})$ is a Banach space with the norm
$\|f\|_{\infty}:=\max_{\gamma\in U}|f(\gamma)|$. Since $\mathcal{L}$ is
locally trivial, every point in $\mathcal{G}$ has an open neighbourhood $U$
and unitary section $f\in C(U,\mathcal{L})$, i.e. $|f(\gamma)|=1$, for all
$\gamma\in U$. This section induces an isometric isomorphism
$C_{0}(U,\mathcal{L})\cong C_{0}(U)$, as for every $g\in C_{0}(U,\mathcal{L})$
there is a unique element $g/f\in C_{0}(U)$ such that
$g(\gamma)=(g/f)(\gamma)f(\gamma)$, $\gamma\in U$.

######  Definition 4.1.

A _twist over the groupoid_ $\mathcal{G}$ is a line bundle
$\mathcal{L}=\bigsqcup_{\gamma\in\mathcal{G}}L_{\gamma}$ over $\mathcal{G}$
together with a continuous _multiplication_
${\cdot}:\bigsqcup_{d(\gamma)=r(\eta)\in\mathcal{G}}L_{\gamma}\times
L_{\eta}\to\mathcal{L}$, which is associative (whenever the product makes
sense), makes the modulus multiplicative, and induces bilinear maps
$L_{\gamma}\times L_{\eta}\longrightarrow L_{\gamma\eta}$; and a continuous
_involution_ ${}^{*}:\mathcal{L}\to\mathcal{L}$ that restricts to conjugate
linear maps $L_{\gamma}\to L_{\gamma^{-1}}$, preserves the modulus (norm), is
anti-multiplicative and $z\cdot z^{*}=|z|^{2}\in L_{r(\gamma)}\cong\mathbb{F}$
for every $z\in L_{\gamma}$, $\gamma\in\mathcal{G}$. See [47] for more
details.

######  Lemma 4.2.

Let $(\mathcal{G},\mathcal{L})$ be a twisted groupoid. The set
$S(\mathcal{L})$ of bisections $U$ for which the restricted line bundle
$\mathcal{L}|_{U}$ is trivial forms a unital wide inverse subsemigroup
$S(\mathcal{L})\subseteq\operatorname{Bis}(\mathcal{G})$.

###### Proof.

The restricted bundle $\mathcal{L}|_{U}$ is trivial if and only if there is a
continuous unitary section $c_{U}\in C_{u}(U,\mathcal{L})$. If
$\mathcal{L}|_{U}$ and $\mathcal{L}|_{V}$ are trivial, and $c_{U}\in
C_{u}(U,\mathcal{L}),\ c_{V}\in C_{u}(V,\mathcal{L})$, then
$\mathcal{L}|_{UV}$ and $\mathcal{L}|_{U^{-1}}$ are trivial because the
convolution product $c_{U}*c_{V}\in C_{u}(UV,\mathcal{L})$ and $c_{U}^{*}\in
C_{u}(U^{-1},\mathcal{L})$ (in a Fell line bundle the norm $|\cdot|$ is
multiplicative, rather than just submultiplicative). Thus
$S(\mathcal{L})\subseteq\operatorname{Bis}(\mathcal{G})$ is an inverse
semigroup. Since $\mathcal{L}$ is locally trivial, $S(\mathcal{L})$ is wide.
Moreover, $\mathcal{L}$ is trivial over $X$ as each fiber
$L_{x}\cong\mathbb{F}$ is a one-dimensional algebra with unit $1_{x}$, and the
unit section defined by $1(x):=1_{x}$ is unitary and continuous. ∎

According to Lemma 4.2 we may and we shall always assume that

$\mathcal{L}|_{X}=X\times\mathbb{F}$ is trivial.

######  Example 4.3 (Twist by a cocycle).

The trivial bundle $\mathcal{L}=\mathcal{G}\times\mathbb{F}$ with pointwise
operations is a _trivial twist_. More generally, if $\sigma$ is a _continuous
normalised $2$-cocycle_ on $\mathcal{G}$, that is a continuous map
$\sigma:\mathcal{G}^{(2)}\to\\{z\in\mathbb{F}:|z|=1\\}$ satisfying

$\sigma\big{(}r(\gamma),\gamma\big{)}=1=\sigma\big{(}\gamma,d(\gamma)\big{)},\qquad\sigma(\alpha,\beta)\sigma(\alpha\beta,\gamma)=\sigma(\beta,\gamma)\sigma(\alpha,\beta\gamma)$

for every composable triple $\alpha,\beta,\gamma\in\mathcal{G}$, then we may
consider a twist $\mathcal{L}_{\sigma}$ given by the trivial bundle
$\mathcal{G}\times\mathbb{F}$ with operations given by

$(\alpha,w)\cdot(\beta,z):=\big{(}\alpha\beta,\sigma(\alpha,\beta)wz\big{)},\qquad(\alpha,w)^{*}:=\big{(}\alpha^{-1},\overline{\sigma(\alpha^{-1},\alpha)w}\big{)}.$

Every twist based on a trivial line bundle is of this form.

###  4.1. The algebras

Let us fix a twisted groupoid $(\mathcal{G},\mathcal{L})$. Recall our
convention that $\mathcal{G}$ is locally compact and only locally Hausdorff.
Thus the associated $*$-algebra will be defined on the set of quasi-continuous
compactly supported sections:

$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}):=\operatorname{span}\\{f\in
C_{c}(U,\mathcal{L}):U\in\operatorname{Bis}(\mathcal{G})\\},$

where we treat sections of $\mathcal{L}|_{U}$ as sections of $\mathcal{L}$
that vanish outside $U$. Note that
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})=\operatorname{span}\\{f\in
C_{c}(U,\mathcal{L}):U\in\mathcal{C}\\}$ for any cover
$\mathcal{C}\subseteq\operatorname{Bis}(\mathcal{G})$ of $\mathcal{G}$, cf.
[31, Proposition 3.10]. In particular, we may always take
$\mathcal{C}=S(\mathcal{L})$. If the groupoid $\mathcal{G}$ is Hausdorff, then
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})=C_{c}(\mathcal{G},\mathcal{L})$ is
the usual space of continuous compactly supported sections. We define a
$*$-algebra structure on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ by

(4.5) $(f*g)(\gamma):=\sum_{r(\eta)=r(\gamma)}f(\eta)\cdot
g(\eta^{-1}\gamma),\qquad(f^{*})(\gamma):=f(\gamma^{-1})^{*},$

where $f,g\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, cf. [31, Proposition
3.11]. Since we assume that $\mathcal{L}|_{X}$ is trivial, we get that
$C_{c}(X)=C_{c}(X,\mathcal{L})$ is a $*$-subalgebra of
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$.

The domain and range maps $d,r:\mathcal{G}\to X$ induce the following three
submultiplicative norms on $\mathfrak{C}_{c}(\mathcal{G})$:

$\|f\|_{d_{*}}:=\max_{x\in
X}\sum_{d(\gamma)=x}|f(\gamma)|,\qquad\|f\|_{r_{*}}:=\max_{x\in
X}\sum_{r(\gamma)=x}|f(\gamma)|,\qquad\|f\|_{I}:=\max\\{\|f\|_{d_{*}},\|f\|_{r_{*}}\\},$

see [57, Section 2.2] or [63, II, 1.4]. The norm $\|\cdot\|_{I}$ is called the
_$I$ -norm_ and was introduced by Hahn [40]. It has the advantage that it is
preserved by the involution $*$, while for the remaining norms we only have
$\|f^{*}\|_{d_{*}}=\|f\|_{r_{*}}$. Also it is well known (cf. [70, Lemma
3.2.3]) and we prove it in Theorem 5.5 below, that the largest $C^{*}$-norm
$\|\cdot\|_{C^{*}\mathrm{max}}$ on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$
exists and it does not exceed $\|\cdot\|_{I}$.

######  Definition 4.4.

We write
$F_{I}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{I}}$,
$F_{d_{*}}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{d_{*}}}$,
$F_{r_{*}}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{r_{*}}}$,
and
$C^{*}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{C^{*}\mathrm{max}}}$
for the completions in the corresponding norms.

The important feature of the above norms is that when restricted to any
subspace
$C_{c}(U,\mathcal{L})\subseteq\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$,
where $U\in\operatorname{Bis}(\mathcal{G})$, they coincide with the supremum
norm $\|\cdot\|_{\infty}$. Hence each space $C_{0}(U,\mathcal{L})$ embeds
naturally into the completions of $\mathfrak{C}_{c}(\mathcal{G})$ in each of
these norms. We introduce yet another norm which is maximal with respect to
this property. In fact we will associate norms to subsemigroups of
$\operatorname{Bis}(\mathcal{G})$.

######  Lemma 4.5.

Let $S\subseteq\operatorname{Bis}(\mathcal{G})$ be a semigroup covering
$\mathcal{G}$. There is a submultiplicative norm on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ given by

(4.6)
$\|f\|_{\max}^{S}:=\inf\left\\{\sum_{k=1}^{n}\|f_{k}\|_{\infty}:f=\sum_{k=1}^{n}f_{k},\
f_{k}\in C_{c}(U_{k},\mathcal{L}),\ U_{k}\in S\right\\}.$

This is the largest norm on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ which
agrees with $\|\cdot\|_{\infty}$ on each
$C_{c}(U,\mathcal{L})\subseteq\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, for
$U\in S$. If $S$ is an inverse semigroup, then involution is isometric in
$\|\cdot\|_{\max}^{S}$.

###### Proof.

Let $f,g\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$. For any $\varepsilon>0$
there are $f_{k}\in C_{c}(U_{k},\mathcal{L})$ and $g_{l}\in
C_{c}(V_{k},\mathcal{L})$, where $U_{k},V_{l}\in S$, such that
$f=\sum_{k=1}^{n}f_{k}$, $g=\sum_{l=1}^{m}g_{l}$ and
$\|f\|_{\max}^{S}+\varepsilon>\sum_{k=1}^{n}\|f_{k}\|_{\infty}$ and
$\|g\|_{\max}^{S}+\varepsilon>\sum_{l=1}^{m}\|g_{l}\|_{\infty}$. Then
$f*g=\sum_{k,l}f_{k}*g_{l}$, and $f_{k}*g_{l}\in
C_{c}(U_{k}V_{l},\mathcal{L})$, $U_{k}V_{l}\in S$. Therefore

$\|f*g\|_{\max}^{S}\leq\sum_{k,l}\|f_{k}*g_{l}\|_{\infty}\leq\sum_{k,l}\|f_{k}\|_{\infty}\|g_{l}\|_{\infty}<\big{(}\|f\|_{\max}^{S}+\varepsilon\big{)}\big{(}\|g\|_{\max}^{S}+\varepsilon\big{)}.$

This shows that $\|\cdot\|_{\max}^{S}$ is submultiplicative. The proof of the
triangle inequality is even simpler. If in addition $f\in
C_{c}(U,\mathcal{L})$ for some $U\in S$, then there is $\gamma_{0}\in U$ where
$|f|$ attains its maximum, and then (retaining the above choice of $f_{k}$’s)
$\|f\|_{\infty}=|f(\gamma_{0})|\leq\sum_{k=1}^{n}|f_{k}(\gamma_{0})|\leq\sum_{k=1}^{n}\|f_{k}\|_{\infty}<\|f\|_{\max}^{S}+\varepsilon,$
which shows that $\|f\|_{\infty}\leq\|f\|^{S}_{\max}$. The converse inequality
$\|f\|_{\max}^{S}\leq\|f\|_{\infty}$ holds by (4.6). Hence
$\|\cdot\|^{S}_{\max}$ coincides with $\|\cdot\|_{\infty}$ on
$C_{c}(U,\mathcal{L})$ for $U\in S$. If $\|\cdot\|$ is any norm on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ that agrees with
$\|\cdot\|_{\infty}$ on
$C_{c}(U,\mathcal{L})\subseteq\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$, for
$U\in S$, then for any $f=\sum_{k=1}^{n}f_{k}$, $f_{k}\in
C_{c}(U_{k},\mathcal{L})$, $U_{k}\in S$, we have
$\|f\|\leq\sum_{k=1}^{n}\|f_{k}\|=\sum_{k=1}^{n}\|f_{k}\|_{\infty}$, which
implies $\|f\|\leq\|f\|_{\max}^{S}$. If $S$ is an inverse semigroup then
$f^{*}=\sum_{k=1}^{n}f_{k}^{*}$, $f_{k}^{*}\in C_{c}(U_{k}^{*},\mathcal{L})$,
where $U_{k}^{*}\in S$. Thus $\|f^{*}\|_{\max}^{S}=\|f\|_{\max}^{S}$ by (4.6).
∎

######  Definition 4.6.

For any inverse semigroup $S\subseteq\operatorname{Bis}(\mathcal{G})$ covering
$\mathcal{G}$, we denote by
$F^{S}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{\max}^{S}}$
the associated Banach $*$-algebra. We also denote by $\|\cdot\|_{\max}$ the
norm $\|\cdot\|_{\max}^{\operatorname{Bis}(\mathcal{G})}$ and put
$F(\mathcal{G},\mathcal{L}):=F^{\operatorname{Bis}(\mathcal{G})}(\mathcal{G},\mathcal{L})$.

######  Remark 4.7.

For any inverse semigroup $S\subseteq\operatorname{Bis}(\mathcal{G})$ covering
$\mathcal{G}$, we have

$\|\cdot\|_{\infty}\leq\|\cdot\|_{d_{*}},\|\cdot\|_{C^{*}\mathrm{max}},\|\cdot\|_{r_{*}}\leq\|\cdot\|_{I}\leq\|\cdot\|_{\max}\leq\|\cdot\|_{\max}^{S}$

on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$. In particular,
$\|\cdot\|_{\max}$ is the minimal norm amongst all norms
$\|\cdot\|_{\max}^{S}$ on $\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$
($S\subseteq
T\subseteq\operatorname{Bis}(\mathcal{G})\,\,\Longrightarrow\,\,\|\cdot\|_{\max}^{T}\leq\|\cdot\|_{\max}^{S}$).
In general there is no maximal amongst the norms $\|\cdot\|_{\max}^{S}$
(unless $\mathcal{G}$ is discrete and then this maximal norm is the
$\ell^{1}$-norm), see Example 4.10 below. In this analysis we may restrict
only to wide inverse semigroups as the ‘saturation’
$\widetilde{S}:=\\{U:U\subseteq V\in S\\}$ of $S$ is a wide inverse
subsemigroup of $\operatorname{Bis}(\mathcal{G})$ and
$\|\cdot\|_{\max}^{S}=\|\cdot\|_{\max}^{\widetilde{S}}$. Moreover
$\|\cdot\|_{\max}^{S}$ is majorized by a norm coming from an inverse
subsemigroup of $S(\mathcal{L})$ from Lemma 4.2. Namely, $\widetilde{S}\cap
S(\mathcal{L})$ is a wide inverse semigroup and

(4.7) $\|\cdot\|_{\max}^{S}\leq\|\cdot\|_{\max}^{\widetilde{S}\cap
S(\mathcal{L})}.$

######  Example 4.8 (Discrete group).

When $\mathcal{G}=G$ is a group, then
$\operatorname{Bis}(\mathcal{G})=\\{g\\}_{g\in G}\cong G$ and
$\mathcal{L}=G\times\mathbb{F}$, and so all the norms
$\|\cdot\|_{d_{*}},\|\cdot\|_{r_{*}},\|\cdot\|_{I},\|\cdot\|_{\max}^{G}$
coincide with the $\ell^{1}$-norm in $\ell^{1}(G)$.

######  Example 4.9 (Pair groupoid).

Let $\mathcal{G}=X\times X$ be the pair groupoid over the finite set
$X=\mathbb{Z}_{n}=\\{0,\ldots,n-1\\}$. Then we may identify
$C_{c}(\mathcal{G})$ with the $*$-algebra $M_{n}(\mathbb{F})$ of $n\times n$
matrices. For any matrix $f=[f_{ij}]_{i,j\in X}$ we have
$\|f\|_{\infty}=\max_{i,j\in\mathbb{Z}_{n}}|f_{ij}|$ and

$\|f\|_{d_{*}}=\max_{i\in\mathbb{Z}_{n}}\sum_{j\in\mathbb{Z}_{n}}|f_{ij}|,\quad\|f\|_{r_{*}}=\max_{j\in\mathbb{Z}_{n}}\sum_{i\in\mathbb{Z}_{n}}|f_{ij}|,\quad\|f\|_{I}=\max\max_{i}\\{\sum_{j}|f_{i,j}|,\sum_{j}|f_{j,i}|\\}.$

Consider inverse semigroups $S_{1}$, $S_{2}$ and $S_{3}$ of bisections where
$S_{1}$ consists of $\\{(i,j):i-j=k\,(\text{mod }n)\\}$, $k\in\mathbb{Z}_{n}$;
$S_{2}$ consists of $\\{(i,j):i-j=k\,\\}$, $k=0,\pm 1,\ldots,\pm(n-1)$; and
$S_{3}$ consists of singletons $\\{(i,j)\\}$, $(i,j)\in X\times X$. Then

$\|f\|_{\max}^{S_{1}}=\max_{k\in\mathbb{Z}_{n}}\sum_{i-j=k\,(\text{mod
}n)}|f_{ij}|,\qquad\|f\|_{\max}^{S_{2}}=\max_{k=0,\pm
1,\ldots,\pm(n-1)}\sum_{i-j=k}|f_{ij}|$

and $\|f\|_{\max}^{S_{3}}=\sum_{i,j=1}^{n}|f_{ij}|$. The gradings $S_{1}$ and
$S_{2}$ are natural when viewing $\mathcal{G}$ as the transformation groupoid
associated to the permutation homeomorphism $h(i)=i-1\,(\text{mod }n)$ on
$\mathbb{Z}_{n}$, or the partial homeomorphism $h|_{\\{1,\ldots,n-1\\}}$ of
$\mathbb{Z}_{n}$, respectively. The grading $S_{3}$ gives a maximum amongst
the norms we consider.

######  Example 4.10.

(Topological space) Let $\mathcal{G}=X:=\\{0\\}\cup\\{1/n:n\in\mathbb{N}\\}$
with the usual topology. For any $n\in\mathbb{N}$ the set
$S_{n}:=[0,1/n]\cup\\{\\{1/k\\}:k=1,\ldots,n-1\\}$ is a wide inverse
subsemigroup of $\operatorname{Bis}(\mathcal{G})$ and we have
$\|1\|_{\max}^{S_{n}}=n$, which tends to infinity as $n\to\infty$.

We fix an inverse semigroup $S\subseteq\operatorname{Bis}(\mathcal{G})$
covering $\mathcal{G}$. By (4.7), in the following definition we may always
assume that $S$ is a wide inverse subsemigroup of $S(\mathcal{L})$.

######  Definition 4.11.

For a class $\mathcal{R}$ of representations of
$F^{S}(\mathcal{G},\mathcal{L})$ we denote by
$F_{\mathcal{R}}^{S}(\mathcal{G},\mathcal{L}):=\overline{\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})}^{\|\cdot\|_{\mathcal{R}}}$
the Hausdorff completion in the seminorm
$\|f\|_{\mathcal{R}}:=\sup\\{\|\psi(f)\|:\psi\in\mathcal{R}\\}.$ If
$\|\cdot\|_{\mathcal{R}}\leq\|\cdot\|_{\max}$, then we will write
$F_{\mathcal{R}}(\mathcal{G},\mathcal{L})$ for
$F_{\mathcal{R}}^{S}(\mathcal{G},\mathcal{L})$.

######  Remark 4.12.

If $\|\cdot\|_{\mathcal{R}}\leq\|\cdot\|_{\max}$, then the algebra
$F_{\mathcal{R}}^{S}(\mathcal{G},\mathcal{L})$ does not depend on the choice
of $S$, as it can be viewed as a Hausdorff completion of
$F(\mathcal{G},\mathcal{L})$. In this paper we will be mainly interested in
the case when $\|\cdot\|_{\mathcal{R}}\leq\|\cdot\|_{I}$ (and hence all the
more $\|\cdot\|_{\mathcal{R}}\leq\|\cdot\|_{\max}$).

All the aforementioned Banach algebras have approximate units.

######  Lemma 4.13.

Let $S\subseteq\operatorname{Bis}(\mathcal{G})$ be an inverse semigroup
covering $\mathcal{G}$. An approximate unit $\\{\mu_{i}\\}_{i}$ in $C_{0}(X)$
is also an approximate unit in $F^{S}(\mathcal{G},\mathcal{L})$, and for any
representation $\psi:F^{S}(\mathcal{G},\mathcal{L})\to B$ the net
$\\{\psi(\mu_{i})\\}_{i}$ is an approximate unit in
$\overline{\psi(F^{S}(\mathcal{G},\mathcal{L}))}$. In particular, if $X$ is
compact then $F^{S}(\mathcal{G},\mathcal{L})$ and
$\overline{\psi(F^{S}(\mathcal{G},\mathcal{L}))}$ are unital.

###### Proof.

It suffices to prove that $\\{\mu_{i}*f\\}_{i}$ converges to $f$ in
$\|\cdot\|_{\max}^{S}$ for $f\in C_{c}(U,\mathcal{L})$, $U\in S$. But then
$\mu_{i}*f-f\in C_{c}(U,\mathcal{L})$ and therefore
$\|\mu_{i}*f-f\|_{\max}^{S}=\|\mu_{i}*f-f\|_{\infty}=\max_{\gamma\in
U}|\mu_{i}(r(\gamma))f(\gamma)-f(\gamma)|$, which obviously tends to zero. ∎

We write $A\cong B$ to indicate that two Banach algebras $A$ and $B$ are
isometrically isomorphic. We write
$A\stackrel{{\scriptstyle\mathrm{anti}}}{{\cong}}B$ if there is an isometry
from $A$ onto $B$ which is anti-linear and anti-multiplicative. We write
$A\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}B$ if there is
a representation of $A$ in $B$ with dense range. By Remark 4.7 the identity on
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ extends to canonical
representations

$F^{S}(\mathcal{G},\mathcal{L})\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F_{I}(\mathcal{G},\mathcal{L})\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F_{d_{*}}(\mathcal{G},\mathcal{L}),\
C^{*}(\mathcal{G},\mathcal{L}),\ F_{r_{*}}(\mathcal{G},\mathcal{L}).$

###  4.2. Opposite algebras and twists

We denote by $A^{\operatorname{op}}$ the _opposite algebra_ to an algebra $A$,
meaning that the linear structure of $A^{\operatorname{op}}$ is the same as
that of $A$ but the multiplication is performed in the reverse order. When $A$
is a Banach algebra then $A^{\operatorname{op}}$ is a Banach algebra with the
same norm. Fix a twisted groupoid $(\mathcal{G},\mathcal{L})$ and an inverse
semigroup $S\subseteq\operatorname{Bis}(\mathcal{G})$ covering $\mathcal{G}$.
An _opposite twist_ $\mathcal{L}^{\operatorname{op}}$ on $\mathcal{G}$ is
defined as follows (see [17]). The fibers are
$L_{\gamma}^{\operatorname{op}}:=L_{\gamma^{-1}}$, $\gamma\in\mathcal{G}$, and
the topology is induced by the sections

(4.8) $(\widecheck{f})(\gamma):=f(\gamma^{-1}),$

where $f\in C_{c}(U,\mathcal{L})$, $U\in\operatorname{Bis}(\mathcal{G})$. The
involution on $\mathcal{L}^{\operatorname{op}}$ is given by the involution
maps $L_{\gamma^{-1}}\to L_{\gamma}$ from $\mathcal{L}$, and the
multiplication maps $L_{\gamma}^{\operatorname{op}}\times
L_{\eta}^{\operatorname{op}}\longrightarrow
L_{\gamma\eta}^{\operatorname{op}}$ are given by $(z,w)\mapsto w\cdot z$ where
$z\in L_{\gamma}^{\operatorname{op}}=\mathcal{L}_{\gamma^{-1}}$, $w\in
L_{\eta}^{\operatorname{op}}=\mathcal{L}_{\eta^{-1}}$ and $d(\gamma)=r(\eta)$.
The map $f\mapsto\widecheck{f}$ yields a $*$-isomorphism

$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}}).$

In particular, the untwisted algebra
$\mathfrak{C}_{c}(\mathcal{G})^{\operatorname{op}}\cong\mathfrak{C}_{c}(\mathcal{G})$
is self-opposite. It follows from the formulas for the norms
$\|\cdot\|_{d_{*}},\|\cdot\|_{r_{*}},\|\cdot\|_{I},\|\cdot\|_{\max}^{S}$ that
the above isomorphism extends to isometric isomorphisms

$F^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}}),\qquad
F_{I}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F_{I}(\mathcal{G},\mathcal{L}^{\operatorname{op}}),\qquad
F_{d_{*}}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F_{r_{*}}(\mathcal{G},\mathcal{L}^{\operatorname{op}}).$

Thus the algebras $F^{S}(\mathcal{G})$ and $F_{I}(\mathcal{G})$ are self-
opposite, while for the remaining two algebras we have only
$F_{d_{*}}(\mathcal{G})^{\operatorname{op}}\cong F_{r_{*}}(\mathcal{G})$. Let
us consider a slightly more general situation.

######  Definition 4.14.

For a class $\mathcal{E}$ of Banach spaces we write
$F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}):=F_{\mathcal{R}}^{S}(\mathcal{G},\mathcal{L})$
and $\|\cdot\|_{\mathcal{E}}:=\|\cdot\|_{\mathcal{R}}$ for $\mathcal{R}$
consisting of all representations of $F^{S}(\mathcal{G},\mathcal{L})$ on
Banach spaces in $\mathcal{E}$.

If $\mathcal{E}$ is a class of Banach spaces we denote by
$\mathcal{E}^{\prime}$ the class of dual spaces to spaces in $\mathcal{E}$.
For any other class $\mathcal{F}$ of Banach spaces we write
$\mathcal{E}\subseteq_{\operatorname{iso}}\mathcal{F}$ if for every
$E\in\mathcal{E}$ there is $F\in\mathcal{F}$ such that $E\cong F$.

######  Proposition 4.15.

If $\mathcal{E}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{F}$ then the
map $f\mapsto\widecheck{f}$ induces a contractive homomorphism
$F_{\mathcal{F}}^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$.
If $\mathcal{E}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{F}$ and
$\mathcal{F}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{E}$ then this
homomorphism is an isometric isomorphism
$F_{\mathcal{F}}^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$.

###### Proof.

For any representation
$\pi:F^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\to B(E)$ the formula
$\pi^{\prime}(f):=\pi(\widecheck{f})^{\prime}$,
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$, defines a
representation
$\pi^{\prime}:F^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})\to
B(E^{\prime})$. If
$\mathcal{E}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{F}$ and
$E\in\mathcal{E}$, then we may replace $E^{\prime}$ by $F\in\mathcal{F}$ such
that $E^{\prime}\cong F$, so that we get a representation
$\pi^{\prime}:F^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})\to B(F)$.
This implies that
$\|\widecheck{f}\|_{\mathcal{E}}^{S}\leq\|f\|_{\mathcal{F}}^{S}$ for
$f\in\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$. Thus the
isomorphism
$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$
extends to a representation
$F_{\mathcal{F}}^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\stackrel{{\scriptstyle\mathrm{dense}}}{{\twoheadrightarrow}}F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$.
Similarly, $\mathcal{F}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{E}$
implies $\|\cdot\|_{\mathcal{F}}^{S}\leq\|\cdot\|_{\mathcal{E}}^{S}$. Hence
$F_{\mathcal{F}}^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$ if
$\mathcal{E}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{F}$ and
$\mathcal{F}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{E}$. ∎

######  Corollary 4.16.

If $\mathcal{E}$ consists of reflexive Banach spaces, then
$F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
F_{\mathcal{E}^{\prime}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$.

###### Proof.

Take $\mathcal{F}=\mathcal{E}^{\prime}$ in Proposition 4.15. ∎

######  Corollary 4.17.

We have $C^{*}(\mathcal{G},\mathcal{L})^{\operatorname{op}}\cong
C^{*}(\mathcal{G},\mathcal{L}^{\operatorname{op}})$ and
$C^{*}(\mathcal{G})^{\operatorname{op}}\cong C^{*}(\mathcal{G})$.

###### Proof.

Apply Corollary 4.16 to the case where $\mathcal{E}$ consists of Hilbert
spaces. ∎

The untwisted version of Corollary 4.16 is known (see [37], [36]) and
Corollary 4.17 is a special case of the main result in [17]. We can apply
Proposition 4.15 beyond reflexive spaces.

######  Corollary 4.18.

Let $\mathcal{E}$ be the class of $L^{1}$-spaces, $\mathcal{F}_{1}$ the class
of $L^{\infty}$-spaces, $\mathcal{F}_{2}$ the class of $C_{0}$-spaces and
$\mathcal{F}_{3}$ the class of Lindenstrauss spaces, cf. Remark 2.15. Then

$F_{\mathcal{F}_{1}}^{S}(\mathcal{G},\mathcal{L})\cong
F_{\mathcal{F}_{2}}^{S}(\mathcal{G},\mathcal{L})\cong
F_{\mathcal{F}_{3}}^{S}(\mathcal{G},\mathcal{L})\cong
F_{\mathcal{E}}^{S}(\mathcal{G},\mathcal{L}^{\operatorname{op}})^{\operatorname{op}}.$

###### Proof.

We clearly have
$\mathcal{E}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{F}_{1}\subseteq_{\operatorname{iso}}\mathcal{F}_{2}$
and $\mathcal{F}_{3}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{E}$. Also
$\mathcal{F}_{2}\subseteq\mathcal{F}_{3}$, which follows from the
Riesz–Markov–Kakutani theorem and the abstract characterization of
$L^{1}$-spaces, see for instance [58, page 261]. Thus
$\mathcal{F}_{2}^{\prime}\subseteq\mathcal{F}_{3}^{\prime}\subseteq_{\operatorname{iso}}\mathcal{E}$
and the assertion follows from the second part of Proposition 4.15. ∎

###  4.3. From twisted groupoids to twisted actions and back

We describe the correspondence between twisted groupoids and twisted actions
of inverse semigroups on $C_{0}(X)$, which implicitly can be found in [15],
[16]. Let $(\mathcal{G},\mathcal{L})$ be a twisted groupoid. Take any wide
inverse subsemigroup $S\subseteq
S(\mathcal{L})\subseteq\operatorname{Bis}(\mathcal{G})$ and let
$h:S\to\operatorname{PHomeo}(X)$ be the canonical action. Then we have the
inverse semigroup action $\alpha:S\to\operatorname{PAut}(C_{0}(X))$ given by
$\alpha_{U}(a):=a\circ h_{U^{*}}$ for $a\in I_{d(U)}:=C_{0}(d(U))$, $U\in S$.
For each $U\in S(\mathcal{L})$ take any continuous unitary section $c_{U}\in
C_{u}(U,\mathcal{L})$, but if $U\subseteq X$ let $c_{U}(x):=(x,1)\subseteq
X\times\mathbb{F}$, $x\in U$. For any $U,V\in S(\mathcal{L})$ there is a
unique unitary function $u(U,V)\in C_{u}(r(UV))$ satisfying
$u(U,V)\big{(}r(\gamma)\big{)}c_{UV}(\gamma\eta)=c_{U}(\gamma)\cdot
c_{V}(\eta)$, $\gamma\in U,\ \eta\in V,\ d(\gamma)=r(\eta)$. In fact we have

(4.9) $\alpha_{U}(a)=c_{U}*a*c_{U}^{*},\qquad
u(U,V)=c_{U}*c_{V}*c_{UV}^{*},\qquad a\in C_{0}(d(U)),\ U,V\in S,$

with operations given by (4.5).

######  Lemma 4.19.

For any twisted groupoid $(\mathcal{G},\mathcal{L})$ the pair $(\alpha,u)$
defined above is a twisted action of $S$ on $C_{0}(X)$.

###### Proof.

Since $C_{0}(X)$ is commutative (A1) is equivalent to $\alpha$ being an
action. Axioms (A2), (A3), (A4) can be checked exactly as in the proof of [15,
Proposition 3.6] using (4.9). ∎

######  Example 4.20.

If the twist $\mathcal{L}=\mathcal{L}_{\sigma}$ comes from a $2$-cocycle
$\sigma$ then $S(\mathcal{L}_{\sigma})=\operatorname{Bis}(\mathcal{G})$ and
for any $\eta\in U$,$\gamma\in V$, we have
$u(U,V)(\eta\gamma)=\sigma(\eta,\gamma)\overline{\sigma((\eta\gamma)^{-1},\eta\gamma)}$.

Now let $(\alpha,u)$ be any twisted action of $S$ on $C_{0}(X)$. Since
$C_{0}(X)$ is commutative $\alpha:S\to\operatorname{PAut}(C_{0}(X))$ is a
semigroup homomorphism, and hence there is a semigroup homomorphism
$h:S\to\operatorname{PHomeo}(X)$ with $\alpha_{t}(a)=a\circ h_{t^{*}}$ for
$a\in I_{t^{*}}=C_{0}(X_{t^{*}})$, $t\in S$. Let $\mathcal{G}=S\ltimes_{h}X$
be the associated transformation groupoid. We define a bundle $\mathcal{L}$
over $\mathcal{G}$ whose elements are equivalence classes of triples $(a,t,x)$
for $a\in C_{0}(X_{t})$, $x\in X_{t^{*}}\subseteq X$, $t\in S$, and two
triples $(a,t,x)$ and $(a^{\prime},t^{\prime},x^{\prime})$ are equivalent if
$x=x^{\prime}$ and there is $v\in S$ with $v\leq t,t^{\prime}$, where $x\in
X_{v^{*}}$, and $(a\cdot u(vv^{*},t))(h_{v}(x))=(a^{\prime}\cdot
u(vv^{*},t^{\prime}))(h_{v}(x))$. In particular, we have a canonical
surjection

$\mathcal{L}\to\mathcal{G};\ [a,t,x]\mapsto[t,x],$

and we denote by $L_{[t,x]}$ the corresponding fibers. This makes
$\mathcal{L}$ a line bundle with operations defined by

$[a,t,x]+[b,t,x]:=[a+b,t,x],\qquad\lambda\cdot[a,t,x]:=[\lambda
a,t,x],\quad\big{|}[a,t,x]\big{|}:=\big{|}a\big{(}h_{t}(x)\big{)}\big{|},$

and the unique topology on $\mathcal{L}$ such that the local sections
$[t,x]\mapsto[a,t,x]$, for $x\in X_{t^{*}}$, $a\in C_{0}(X_{t})$, $t\in S$,
are continuous. Define the partial multiplication and involution on
$\mathcal{L}$ by

$\displaystyle[a,s,h_{t}(x)]\cdot[b,t,x]$
$\displaystyle:=[\alpha_{s}(\alpha_{s}^{-1}(a)b)u(s,t),st,x]=[a(b\circ
h_{s^{*}})u(s,t),st,x],$ $\displaystyle[b,t,x]^{*}$
$\displaystyle:=[\overline{b\circ
h_{t}}\cdot\overline{u(t,t^{*})},t^{*},h_{t}(x)]$

for all $a\in C_{0}(X_{s})$, $b\in C_{0}(X_{t})$, $x\in X_{(st)^{*}}$, $s,t\in
S$.

######  Proposition 4.21.

The pair $(\mathcal{G},\mathcal{L})$ described above is a well-defined twisted
groupoid.

###### Proof.

This follows from [16, Theorem 3.22], as the above construction is a special
case of the construction in [16, Section 3] applied to a Fell bundle over $S$
associated to $(\alpha,u)$, as described in [15, p. 250]. ∎

######  Definition 4.22.

We say that a twisted inverse semigroup action $(\alpha,u)$ _models a twisted
étale groupoid_ $(\mathcal{G},\mathcal{L})$ if $(\mathcal{G},\mathcal{L})$ is
isomorphic to the twisted groupoid associated to $(\alpha,u)$ as in
Proposition 4.21.

######  Lemma 4.23.

Let $(\mathcal{G},\mathcal{L})$ be a twisted étale groupoid. Let $S\subseteq
S(\mathcal{L})\subseteq\operatorname{Bis}(\mathcal{G})$ be any wide inverse
semigroup and $(\alpha,u)$ a twisted action of $S$ as in Lemma 4.19, obtained
from a choice of unitary sections $c_{U}\in C_{u}(U,\mathcal{L})$, $U\in S$.
Let $(S\ltimes X,\mathcal{L}_{u})$ be the twisted groupoid associated to
$(\alpha,u)$, as in Proposition 4.21. Then

$[a,U,x]\mapsto
a\big{(}h_{U}(x)\big{)}c_{U}\big{(}d|_{U}^{-1}(x)\big{)},\qquad a\in
C_{0}(r(U)),\ U\in S,\ x\in d(U),$

is an isomorphism of twisted groupoids $(S\ltimes
X,\mathcal{L}_{u})\cong(\mathcal{G},\mathcal{L})$. Thus $(\alpha,u)$ models
$(\mathcal{G},\mathcal{L})$.

###### Proof.

The map $S\ltimes X\ni[U,x]\mapsto d|_{U}^{-1}(x)\in\mathcal{G}$ is an
isomorphism of topological groupoids. The sections $[U,x]\mapsto[a,U,x]$ and
$\gamma\mapsto a(r(\gamma))c_{U}(\gamma)$, $\gamma\in U$, $x\in d(U)$, $a\in
C_{0}(r(U))$, $U\in S$, determine the topology of $\mathcal{L}_{u}$ and
$\mathcal{L}$ respectively. Thus the map described in the assertion is a
homeomorphism. It is straightforward to check that it preserves the algebraic
operations. ∎

###  4.4. Integration and disintegration of representations

We fix a twisted action $(\alpha,u)$ of an inverse semigroup $S$ on $C_{0}(X)$
and let $(\mathcal{G},\mathcal{L})$ be the associated twisted groupoid as
described in the previous subsection. By Lemma 4.23 every twisted étale
groupoid arises in this way. It follows from the construction that the sets
$U_{t}=\\{[t,x]:x\in X_{t}\\}$, $t\in S$, form a wide inverse semigroup of
bisections of $\mathcal{G}$, which by abuse of notation we will identify with
$S$ in the notation of $F^{S}(\mathcal{G},\mathcal{L})$. For each $t\in S$ we
have a linear isometric isomorphism

(4.10) $C_{c}(X_{t})\ni a_{t}\mapsto a_{t}\delta_{t}\in
C_{c}(U_{t},\mathcal{L}),\qquad a_{t}\delta_{t}[t,x]:=[a_{t},t,x],\qquad x\in
X_{t},$

and we view $(C_{c}(U_{t},\mathcal{L}),\|\cdot\|_{\infty})$ as a subspace of
$(\mathfrak{C}_{c}(\mathcal{G},\mathcal{L}),\|\cdot\|_{\max})$.

######  Lemma 4.24.

$\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})$ is spanned by elements
$a_{t}\delta_{t}$, $a_{t}\in C_{c}(X_{t})$, $t\in S$, and

  1. (1)

$a_{s}\delta_{s}\cdot a_{t}\delta_{t}=a_{s}(a_{t}\circ
h_{s^{*}})u(s,t)\delta_{st}$ and $(a_{t}\delta_{t})^{*}=\overline{a_{t}\circ
h_{t}}\cdot\overline{u(t,t^{*})}\delta_{t^{*}}$;

  2. (2)

$s\leq t$ implies $X_{s}\subseteq X_{t}$ and
$a\delta_{s}=a\overline{u(ss^{*},t)}\delta_{t}$ for any $a\in C_{c}(X_{s})$.

###### Proof.

We only explain (2) as the rest is straightforward. Let $s\leq t$. Then
$X_{s}\subseteq X_{t}$ by the composition law (Remark 3.5) and so also
$U_{s}\subseteq U_{t}$. For $a\in C_{c}(X_{s})$ we have
$a\overline{u(ss^{*},t)}\in C_{c}(X_{s})\subseteq C_{c}(X_{t})$ and so
$a\delta_{s}\in C_{c}(U_{s},\mathcal{L})$ and
$a\overline{u(ss^{*},t)}\delta_{t}\in C_{c}(U_{t},\mathcal{L})$. Take $x\in
X_{t^{*}}$. If $x\in X_{s^{*}}$ then $[t,x]=[s,x]$ and

$a\overline{u(ss^{*},t)}\delta_{t}[t,x]=[a\overline{u(ss^{*},t)},t,x]=[a,s,x]=a\delta_{s}[t,x]$

by the equivalence relations defining $\mathcal{G}$ and $\mathcal{L}$. If
$x\notin X_{s^{*}}$ then $a\delta_{s}[t,x]=0$ by convention, and
$a\overline{u(ss^{*},t)}\delta_{t}[t,x]=[a\overline{u(ss^{*},t)},t,x]=[0,t,x]=0$
because $a(h_{t}(x))=0$. Hence
$a\delta_{s}=a\overline{u(ss^{*},t)}\delta_{t}$. ∎

######  Proposition 4.25.

Every covariant representation $(\pi,v)$ of $(\alpha,u)$ in a Banach algebra
$B$ extends uniquely to a representation $\pi\rtimes
v:F^{S}(\mathcal{G},\mathcal{L})\to B$ such that

$\pi\rtimes v(a_{t}\delta_{t})=\pi(a_{t})v_{t},\qquad a_{t}\in C_{c}(X_{t}),\
t\in S.$

If $B$ is a $C^{*}$-algebra then $\pi\rtimes v$ is $*$-preserving. If
$(\pi,\tilde{v})$ is a $B^{\prime}$-normalization (or $B_{*}$-normalization)
of $(\pi,v)$ as in Proposition 3.8 (or Proposition 3.15) then $\pi\rtimes
v=\pi\rtimes\tilde{v}$.

###### Proof.

We claim that the formula $\pi\rtimes v(\sum_{t\in
F}a_{t}\delta_{t})=\sum_{t\in F}\pi(a_{t})v_{t}$, where $F\subseteq S$ is
finite, yields a well-defined map $\pi\rtimes
v:\mathfrak{C}_{c}(\mathcal{G},\mathcal{L})\to B$, or equivalently that
$\sum_{t\in F}a_{t}\delta_{t}=0$ implies $\sum_{t\in F}\pi(a_{t})v_{t}=0$.
This is proved in [31, Lemma 8.4] in the untwisted case, under the assumption
that $\mathcal{G}$ is second countable, using measure theoretical methods. We
prove it in general using topological tools and induction on the cardinality
of $F$. Suppose that $\sum_{t\in F}a_{t}\delta_{t}=0$. If $|F|=1$ then
$\sum_{t\in F}\pi(a_{t})v_{t}=0$ (because $a_{t}\delta_{t}=0$ if and only if
$a_{t}=0$). Now assume the claim is true for all sets with cardinality smaller
than that of $F$. Pick any $t_{0}\in F$ and put $F_{0}=F\setminus\\{t_{0}\\}$.
Then $\sum_{t\in F_{0}}a_{t}\delta_{t}=-a_{t_{0}}\delta_{t_{0}}$, so the
closed support of $a_{t_{0}}\delta_{t_{0}}$, which we denote by $K$, is a
compact set covered by $\\{U_{t}\cap U_{t_{0}}\\}_{t\in F_{0}}$. By
construction $U_{t}\cap U_{t_{0}}=\bigcup_{s\leq t,t_{0}}U_{s}$. So for each
$t\in F_{0}$ we may find $s_{1,t},...,s_{n_{t},t}\leq t,t_{0}$ such that
$K\subseteq\bigcup_{t\in F}\bigcup_{i=1}^{n_{t}}U_{s_{i,t}}$. Let
$\\{\varrho_{s_{i,t}}\\}_{i=1...,n_{t},t\in F_{0}}$ be a partition of unity on
$K$ subordinate to this open cover. Then by Lemma 4.24(2) and Lemma 3.10(2)
(applied to each $t$, $2n_{t}$-times) we get

$a_{t_{0}}\delta_{t_{0}}=\sum_{t\in
F_{0}}\left(\sum_{i=1}^{n_{t}}\varrho_{s_{i,t}}a_{t_{0}}\frac{u(s_{i,t}s_{i,t}^{*},t_{0})}{u(s_{i,t}s_{i,t}^{*},t)}\right)\delta_{t},\qquad\pi(a_{t_{0}})v_{t_{0}}=\sum_{t\in
F_{0}}\pi\left(\sum_{i=1}^{n_{t}}\varrho_{s_{i,t}}a_{t_{0}}\frac{u(s_{i,t}s_{i,t}^{*},t_{0})}{u(s_{i,t}s_{i,t}^{*},t)}\right)v_{t}.$

Thus $\sum_{t\in F}a_{t}\delta_{t}=\sum_{t\in F_{0}}b_{t}\delta_{t}$ and
$\sum_{t\in F}\pi(a_{t})v_{t}=\sum_{t\in F_{0}}\pi(b_{t})v_{t}$ for some
$b_{t}\in C_{c}(X_{t})$, $t\in F_{0}$. Hence the claim follows by the
inductive hypothesis. Now Lemma 3.10 and Lemma 4.24 readily imply that
$\pi\rtimes v:\mathfrak{C}_{c}(\mathcal{G})\to B$ is an algebra homomorphism,
which is $*$-preserving if $B$ is a $C^{*}$-algebra. It is
$\|\cdot\|_{\max}^{S}$ contractive because $\|\pi\rtimes
v(a_{t}\delta_{t})\|=\|\pi(a_{t})v_{t}\|\leq\|a_{t}\|_{\infty}$. Hence it
extends to a representation $\pi\rtimes v:F^{S}(\mathcal{G},\mathcal{L})\to
B$, which is $*$-preserving if $B$ is a $C^{*}$-algebra. The last statement is
clear. ∎

The following lemma will allow us to show the converse to Proposition 4.25.

######  Lemma 4.26.

Let $\psi:A\to E$ be a contractive linear map from a $C^{*}$-algebra into a
Banach space $E$ which has a predual Banach space $E_{*}$. There is a
contractive element $\psi(1_{A})\in E$ such that for every approximate unit
$\\{\mu_{i}\\}_{i}\subseteq A$ we have $\psi(1_{A})=E_{*}\mathchar
45\relax\lim_{i}\psi(\mu_{i})\in E$.

###### Proof.

For simplicity we assume $\mathbb{F}=\mathbb{C}$ (the real case can be reduced
to the complex case by complexification). We identify $E_{*}$ with a subspace
of $E^{\prime}$. Let $f\in E_{*}$. Then
$\psi^{\prime}(f)=f\circ\psi:A\to\mathbb{C}$ is a bounded functional. Hence it