# Approximation properties of Fell bundles over inverse semigroups and non-
Hausdorff groupoids

Alcides Buss 1 Departamento de Matemática, Universidade Federal de Santa
Catarina, 88.040-900 Florianópolis-SC, Brazil<EMAIL_ADDRESS>and  Diego
Martínez 2 Mathematisches Institut, WWU Münster, Einsteinstr. 62, 48149
Münster, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the nuclearity and weak containment property of reduced
cross-sectional $C^{*}$-algebras of Fell bundles over inverse semigroups. In
order to develop the theory, we first prove an analogue of Fell’s absorption
principle in the context of Fell bundles over inverse semigroups. In parallel,
the approximation property of Exel can be reformulated in this context, and
Fell’s absorption principle can be used to prove that the approximation
property, as defined here, implies that the full and reduced cross-sectional
$C^{*}$-algebras are isomorphic via the left regular representation, i.e., the
Fell bundle has the weak containment property.

We then use this machinery to prove that a Fell bundle with the approximation
property and nuclear unit fiber has a nuclear cross-sectional $C^{*}$-algebra.
This result gives nuclearity of a large class of $C^{*}$-algebras as,
remarkably, all the machinery in this paper works for étale non-Hausdorff
groupoids just as well.

###### Key words and phrases:

Inverse semigroup; Fell bundle; Approximation property; Nuclearity

###### 2020 Mathematics Subject Classification:

46L55, 46L06, 20M18

1 Partially supported by CNPq, CAPES - Brazil & Humboldt Fundation and
Germany’s Excellence Strategy - University of Münster - Germany.

2 Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC 2044 – 390685587,
Mathematics Münster – Dynamics – Geometry – Structure; the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
427320536 – SFB 1442, and ERC Advanced Grant 834267 - AMAREC

##  1\. Introduction

There is an abundant number of examples of $C^{*}$-algebras that have been
studied in the almost 80 years of $C^{*}$-literature, along with numerous
different properties. Of these properties, one of the most ubiquitous ones is
_nuclearity_. In general, a $C^{*}$-algebra $A$ is _nuclear_ if there is
exactly one $C^{*}$-norm on $A\odot B$, the algebraic tensor product of $A$
and $B$, for any $C^{*}$-algebra $B$ (see [Brown-Ozawa:Approximations] for
many other equivalent characterizations). Since it was first introduced,
nuclearity has played a crucial role in the field, for it provides a natural
finite-dimensional approximation of $A$ in terms of finite-dimensional
$C^{*}$-algebras (see, e.g., [Brown-Ozawa:Approximations]*Section 2.3). Thus,
as the class of $C^{*}$-algebras is vast, criteria that guarantees when a
$C^{*}$-algebra is nuclear is desirable, as it allows to tackle many different
problems by various means. The present paper has the goal of giving sufficient
criteria for the nuclearity of a large class of $C^{*}$-algebras, namely those
arising as _cross-sectional_ $C^{*}$-algebras from a _Fell bundle_ over an
_inverse semigroup_.

_Inverse semigroups_ were introduced independently by Wagner [Wagner1952,
Wagner1953] and Preston [Preston1954-1, Preston1954-2, Preston1954-3] in the
50’s. If a (discrete) group can be understood as the set of _global_
symmetries of a space, then an _inverse semigroup_ can be understood as an
algebraic structure that captures the _local_ symmetries of a space [lledo-
martinez-2021]. Following this idea, one can also define _actions_ of inverse
semigroups on locally compact and Hausdorff spaces, which generalize group
_partial_ actions [KellendonkLawson:PartialActions, lledo-martinez-2021]. In
this setting, such an action gives rise to _full_ and _reduced_ crossed
product $C^{*}$-algebras. One may even _twist_ the action, in the sense of
[Exel:TwistedActions], and construct _full_ and _reduced twisted_ crossed
products. This class of $C^{*}$-algebras is, already, quite large, as it
contains every _classifiable_ $C^{*}$-algebra [Li:Classifiable], among many
others. In this paper we shall go one step further, and study cross-sectional
$C^{*}$-algebras arising from _Fell bundles_ over inverse semigroups.

A _Fell bundle over an inverse semigroup_ $S$ is a bundle
$\mathcal{A}=(A_{s})_{s\in S}$ of Banach spaces $A_{s}$ endowed with
associative multiplication maps $A_{s}\times A_{t}\to A_{st}$, involution maps
$A_{s}\to A_{s^{*}}$ and inclusion maps $A_{s}\hookrightarrow A_{t}$ for
$s\leq t$ in $S$ satisfying certain natural axioms that are reminiscent of
$C^{*}$-algebraic structures (see Definition 2.4 below). These objects were
first introduced by Sieben in the unpublished paper [SiebenFellbundles], and
first formally introduced in [Exel:noncomm.cartan] by Exel. In fact, Exel
introduced them in order to deal with _noncommutative Cartan subalgebras_ (see
[Exel:noncomm.cartan] for precise definitions). Indeed, the Fell bundle data
generalizes both twisted actions on spaces and actions on noncommutative
$C^{*}$-algebras, and hence it is the right setting where one may study
noncommutative Cartan subalgebras. We shall usually assume that the inverse
semigroup $S$ has a unit $1\in S$, as this amounts to no loss in generality.
In this case $A_{1}$ is a distinguished $C^{*}$-algebra, and the fibres
$A_{s}$ are Hilbert bimodules over $A_{1}$ with respect to the Fell bundle
operations. In this setting, the Fell bundle $\mathcal{A}$ can be viewed as a
(generalized) action of $S$ on $A_{1}$ by Hilbert bimodules; this is the point
of view taken in [Buss-Meyer:Actions_groupoids] and in the sequel.

As is well known [Lawson2012:duality, chung-martinez-szakacs-2022], inverse
semigroups are strongly connected to _étale groupoids_. In particular, every
action of an inverse semigroup gives rise to an étale groupoid that may be
non-Hausdorff (see Section 4.1 below). Likewise, Fell bundles over inverse
semigroups are strongly related to Fell bundles over étale groupoids, as
defined by Kumjian in [Kumjian:Fell_bundles]. The connection between these
structures is, in fact, so tight that Fell bundles over inverse semigroups may
be interpreted as _(twisted) étale groupoids with noncommutative unit spaces_
, as is done in [BussExel:Fell.Bundle.and.Twisted.Groupoids]. This
relationship is further detailed in Section 4.1, particularly in Theorems 4.16
and 4.25.

Ever since it was known that amenable groups have nuclear reduced
$C^{*}$-algebras, _amenability_ of different types of structures has received
considerable attention from the community (see [Buss-Echterhoff-
Willett:amenability, ExelNg:ApproximationProperty, Kranz:AP-groupooids-Fell-
bundles, martinez-2022, Ozawa-Suzuki:Amenable_examples] and references therein
for examples of _amenable_ actions). Amenability of Fell bundles has, in
particular, also received attention, starting from the work of Exel
[Exel:Amenability] and Exel-Ng [ExelNg:ApproximationProperty] where the notion
of _approximation property_ for a Fell bundle $\mathcal{A}=(A_{g})_{g\in G}$
over a group $G$ was first introduced. This should be viewed as amenability of
the (generalized) action of $G$ on $A_{1}$ implemented by $\mathcal{A}$;
indeed, whenever $\mathcal{A}$ comes from an ordinary action $\alpha$ of $G$
on $A_{1}$ the approximation property of $\mathcal{A}$ turns out to be
equivalent to the amenability of $\alpha$ as defined by Anantharaman-Delaroche
in [Anantharaman-Delaroche:Systemes]. Although Exel’s approximation property
and Anantharaman-Delaroche’s amenability have been around for quite some time
(around 35 years), only recently have they been proved to be equivalent for
Fell bundles associated with ordinary group actions; see the recent papers
[Buss-Echterhoff-Willett:amenability, Abadie-Buss-Ferraro:Amenability] for
further discussions on this.

Exel’s approach in [Exel:Amenability], in fact, shows that the approximation
property of $\mathcal{A}$, together with the nuclearity of the unit fiber
$A_{1}$, imply that $C_{\text{max}}^{*}\left(\mathcal{A}\right)\cong
C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is nuclear. Note that $A_{1}$
needs to be nuclear if $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is to be
nuclear as well, since there always is a canonical conditional expectation
$P\colon C_{\text{red}}^{*}\left(\mathcal{A}\right)\to A_{1}$. Moreover, it
was already known for some time, and proved in [Ara-Exel-
Katsura:Dynamical_systems]*Theorem 6.4, that if
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is nuclear then
$C_{\text{max}}^{*}\left(\mathcal{A}\right)\cong
C_{\text{red}}^{*}\left(\mathcal{A}\right)$, that is, the weak containment
property for $\mathcal{A}$ holds. Only recently, see [Buss-Echterhoff-
Willett:amenability, Abadie-Buss-Ferraro:Amenability], it was proved that the
nuclearity of $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ (or equivalently of
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$) is equivalent to nuclearity of
$A_{1}$ plus the approximation property of $\mathcal{A}$. It is, however, not
yet known whether the weak containment property for $\mathcal{A}$ implies its
approximation property.111 This is known to be false in the more general
setting of locally compact groups, as there are examples of locally compact
groups $G$ and actions on noncommutative $C^{*}$-algebras for which the weak
containment holds but the underlying Fell bundle does not have the
approximation property (i.e., the action is not amenable), see [Buss-
Echterhoff-Willett:amenability]. Lastly, Kranz recently [Kranz:AP-groupooids-
Fell-bundles] generalized the notion of approximation property so as to cover
Fell bundles over second countable locally compact Hausdorff étale groupoids,
and derived some of the expected results.

Our main goal in this paper is to extend the notion of _approximation
property_ from Fell bundles over groups to Fell bundles over inverse
semigroups. This is not trivial due to the subtlety implicitly contained in
these objects. In fact, the most direct translation from groups or groupoids
to inverse semigroups does not capture the correct properties. Nevertheless,
our notion of approximation property (see Definition 4.4) does extend that of
groups and groupoids (see Theorem 4.16). It is worth noting that our
techniques actually allow us to generalize some of the main theorems of
[Kranz:AP-groupooids-Fell-bundles], as we do not require the groupoids to be
Hausdorff or second-countable, nor do we need the Fell bundles to be saturated
or separable (see Corollary 6.9 below). In fact, we generalize the main result
of [Exel:Amenability, Kranz:AP-groupooids-Fell-bundles] in the following way.

######  Theorem 1 (cf. Theorem 6.7).

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
with unit $1\in S$. Suppose that $\mathcal{A}$ has the approximation property,
as in Definition 4.4, and that $A_{1}$ is nuclear. Then
$C_{\text{max}}^{*}\left(\mathcal{A}\right)\cong
C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is nuclear.

In order to prove 1 we, in fact, need to prove a version of Fell’s absorption
principle (see [Exel:Partial_dynamical]*Proposition 18.4) for Fell bundles
over inverse semigroups (see 2 below). In the case of groups, Fell’s
absorption principle states that a representation $\pi$ of $\mathcal{A}$, once
tensored with the (left) regular representation $\lambda$ of the underlying
group, yields a new representation $\pi\otimes\lambda$ whose extension to
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$ factors through
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$. Again, for inverse semigroups
this theorem is much more subtle, since the simple-minded tensor product
$\pi\otimes\lambda$ does not necessarily yield a representation of
$\mathcal{A}$, due to the relations one has to impose on those representations
coming from the inclusions $A_{s}\hookrightarrow A_{t}$ for $s\leq t$. We are
going to construct instead a different representation $\pi^{\Lambda}$ that
will play the role of $\pi\otimes\lambda$. It will act on a Hilbert space
$\ell^{2}_{\pi}(S,\mathcal{H})$ that is a Hausdorff completion of a certain
space of finitely supported sections $S\to\mathcal{H}$. Our version of Fell’s
absorption principle then reads as follows.

######  Theorem 2 (cf. Theorems 3.20 and 3.23).

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$, and let $\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$.
Then $\pi^{\Lambda}$ is unitarily equivalent to the representation
$\operatorname{Ind}\pi_{1}^{\prime\prime}=\Lambda\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}}\colon\mathcal{A}\to\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H})$
induced from the regular representation
$\Lambda\colon\mathcal{A}\to\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime}))$.
In particular $\pi^{\Lambda}$ extends to a representation of
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$. Moreover, if
$\pi_{1}^{\prime\prime}$ is faithful as a representation of
$A_{1}^{\prime\prime}$, then $\pi^{\Lambda}$ yields a faithful representation

$\pi^{\Lambda}\colon
C_{\text{red}}^{*}\left(\mathcal{A}\right)\hookrightarrow\mathcal{L}(\ell^{2}_{\pi}(S,\mathcal{H})).$

2 above will allow us to describe induced representations from the regular
representation, and hence is the main technical tool in order to prove 1. We
now briefly mention that in Corollary 6.11 we also prove that the
approximation property of $\mathcal{A}$ implies that the _essential
$C^{*}$-algebra_ $C_{\text{ess}}^{*}\left(\mathcal{A}\right)$ is nuclear (see
[Kwaniewski2019EssentialCP]*Theorem 4.11 or Theorem 6.10 below). In fact, in
an upcoming paper, the authors will use similar methods to those in this text
in order to introduce an _essential approximation property_ , proving the same
results we now show but in the context of
$C_{\text{ess}}^{*}\left(\mathcal{A}\right)$ instead.

We end the introduction with the structure of the paper. In Section 2 we
introduce and recall the necessary background needed throughout the rest of
the paper. Later, in Section 3 we recall the construction of the _reduced_ and
_full_ cross-sectional $C^{*}$-algebras and present and prove our version of
Fell’s absorption principle (see 2 above). In Section 4 we introduce the
_approximation property_ for Fell bundles over inverse semigroups. Moreover,
we also relate it to its companion property for Fell bundles over étale
groupoids. Section 5 then deals with maximal and minimal tensor products of a
Fell bundle $\mathcal{A}$ with a fixed $C^{*}$-algebra $B$, and we prove the
expected compatibility properties. These are technical tools used in the final
Section 6, where we prove 1.

Conventions: throughout the paper $S$ will denote a (discrete) inverse
semigroup with unit $1\in S$ (i.e., an inverse _monoid_), and its set of
idempotents will be denoted by $E$. Likewise, $\mathcal{A}=(A_{s})_{s\in S}$
will be a Fell bundle over $S$. Moreover, $\mathcal{L}(\mathcal{H}{})$ denotes
the set of linear bounded operators on the Hilbert space $\mathcal{H}{}$.
Lastly, given a $C^{*}$-algebra $A$, its double dual, or von Neumann
enveloping algebra, will be denoted by $A^{\prime\prime}$.

Acknowledgements: we would like to thank Julian Kranz for his useful comments
on a previous version of this manuscript. We would also like to thank the
anonymous referee for their comments.

##  2\. Preliminaries

We recall here all the basic theory necessary for this paper, including
inverse semigroups and groupoids (see Section 2.1); Fell bundles (see Section
2.2); and cross-sectional $C^{*}$-algebras and their representations (see
Sections 2.3 and 2.4).

###  2.1. Inverse semigroups and groupoids

Recall that a _semigroup_ is a set $S$ equipped with a binary associative
operation $S\times S\to S$, where $(s,t)\mapsto st$. As expected, the main
difference between _semigroups_ and _groups_ is the lack of the inverse
$s^{-1}$ of some elements, which means that the maps given by
$\lambda_{s}\colon S\to S$, where $t\mapsto st$, may be non-injective. In
particular, the map $\lambda_{s}$ may not define a _bounded_ operator on
$\ell^{2}(S)$. The following notion, introduced independently by Wagner
[Wagner1952, Wagner1953] and Preston [Preston1954-1, Preston1954-2,
Preston1954-3], will fix this caveat, albeit only _locally_ (see Remark 2.2).

######  Definition 2.1.

A semigroup $S$ is called an _inverse semigroup_ if for all $s\in S$ there is
a unique $s^{*}\in S$ such that $ss^{*}s=s$ and $s^{*}ss^{*}=s^{*}$.

Henceforth $S$ will denote a discrete inverse semigroup. We now briefly recall
the inner structure of a general inverse semigroup, and set some notation. We
refer the reader to [Lawson:InverseSemigroups] for a more comprenhensive
introduction to inverse semigroups. The set of _idempotents of $S$_, denoted
by $E$, is the set $E\coloneqq\\{e\in S\mid e^{2}=e\\}$. It is well known (see
[Lawson:InverseSemigroups]*Theorem 3) that $E\subseteq S$ forms a
_commutative_ sub-semigroup, and hence we may define a partial order on $E$ by
stating that $e\leq f$ if, and only if, $ef=e$. This endows $E$ with a meet
semi-lattice structure. In fact, the partial order $\leq$ can be extended to
the whole semigroup $S$ by stating that $s\leq t$ if, and only if, $te=s$ for
some idempotent $e\in E$. It is also well known (and follows from the
uniqueness of $s^{*}$ in Definition 2.1, see [Lawson:InverseSemigroups]) that
$E$ is precisely the set $\\{s^{*}s\mid s\in S\\}$, and hence every idempotent
$e\in E$ is self-adjoint. We say that an element $z\in S$ is _a zero_ if
$zt=tz=z$ for every $t\in S$. If there is a zero element then it is clear it
must be unique, and in such case we will denote it by $0\in E\subseteq S$.
Likewise, a _unit_ , or an _identity of $S$_, is an element $u\in S$ such that
$ut=tu=t$ for every $t\in S$. Again, if there is a unit in $S$ then it must be
unique, and we shall henceforth denote it by $1\in E\subseteq S$. In fact, a
unit may be always adjoined to $S$ and for convenience we shall assume $S$ has
a unit throughout.

######  Remark 2.2.

We briefly highlight that, even though the map $\lambda_{s}\colon S\to S$
defined above is still not injective even if $S$ is an inverse semigroup, it
will be injective when restricted to $\lambda_{s}\colon s^{*}s\cdot S\to
ss^{*}\cdot S$. This observation leads to a theory of _actions_ of inverse
semigroups (see, e.g., [AraLledoMartinez-2020] and references therein), and
will permeate throughout the present paper, although not explicitly. In fact,
the canonical action given by $\lambda_{s}\colon s^{*}s\cdot S\to ss^{*}\cdot
S$ is known as the _Wagner-Preston representation_. Likewise, inverse
semigroups are then realized by partial isometries of Hilbert spaces with
commuting domains and ranges.

We may now shift our point of view, and discuss _groupoids_. However, observe
that the main objects of study of the text (see Definitions 2.4 and 4.4)
revolve around inverse semigroups, and so we will keep the discussions about
groupoids brief. We refer the reader to [SimsNotes2020] and references therein
for a more in-depth discussion, as well as natural examples of groupoids
arising in the literature.

######  Definition 2.3.

A _groupoid_ $G$ is a small category with inverses, that is, it is a set $G$
together with a distinguished set $G^{(2)}\subseteq G\times G$ and maps
$\cdot\colon G^{(2)}\to G$ and ${}^{-1}\colon G\to G$ such that

  1. (1)

$(g^{-1})^{-1}=g$;

  2. (2)

if $(g,h),(h,k)\in G^{(2)}$ then $(g,hk),(gh,k)\in G^{(2)}$ and $(gh)k=g(hk)$;

  3. (3)

$(g,g^{-1})\in G^{(2)}$, and whenever $(g,h)\in G^{(2)}$ (resp. $(h,g)\in
G^{(2)}$) we have $g^{-1}gh=h$ (resp. $h=hgg^{-1}$),

for all $g,h,k\in G$.

We usually think of elements in a groupoid as _arrows between units_. Observe
that every groupoid has a distinguished subset $G^{0}\subseteq G$, whose
elements are called _units_ , which are precisely those elements that, as
arrows, form loops. In fact, the rest of the elements of the groupoid can be
thought of as arrows $g\colon g^{-1}g\to gg^{-1}$. This way we may define maps
$r,s\colon G\to G^{0}$ by $s(g)\coloneqq g^{-1}g$ and $r(g)\coloneqq gg^{-1}$,
which we call the _range_ and _source_ map respectively. Henceforth, every
groupoid here considered will be endowed with a topology for which the
multiplication and inverse maps are continuous. Hence, we say $G$ is a
_locally compact groupoid_ if it is locally compact as a topological space.
Moreover, the notion of _étale_ groupoid will be discussed in Section 4.1, and
thus we will not discuss it here any further.

It is by now well known that _étale groupoids_ and _inverse semigroups_ are
dual to each other (see [Lawson2012:duality] for precise statements). On one
hand, given an inverse semigroup $S$ one may construct the _universal_
groupoid associated to $S$ (see [Paterson1999]*Chapter 4). As is expected from
the name, this groupoid encapsulates the representation theory of $S$, and is
_universal_ in some suitable sense [Paterson1999]*Theorems 4.4.1 and 4.4.2.
However, we will not use this construction in the present paper, and hence we
will refrain from discussing it further.

On the other hand, given a locally compact étale groupoid $G$, one may look at
the set of open _bisections_ $\operatorname{\mathrm{Bis}}(G)$, which carries a
natural inverse semigroup structure (see Section 4.1 below, particularly
Definition 4.11). We may then recover the groupoid $G$ as the _groupoid of
germs $G^{0}\rtimes\operatorname{\mathrm{Bis}}(G)$_, where
$\operatorname{\mathrm{Bis}}(G)$ acts on $G^{0}$ canonically (see Proposition
4.12). All in all, elements in an inverse semigroup may be seen as
appropriately chosen open sets of a groupoid, and arrows in a groupoid may be
seen as germs of actions of inverse semigroups on spaces.

###  2.2. Fell bundles

This section introduces the main object of study of the paper, and fixes
notation that will be of use throughout the paper.

######  Definition 2.4.

Let $S$ be an inverse semigroup with unit $1\in S$, and let $A$ be a
$C^{*}$-algebra. A _Fell bundle over $S$ with unit fibre $A$_ consists of

  * •

Hilbert $A$-$A$-bimodules $A_{s}$ for every $s\in S$;

  * •

bimodule embeddings $\mu_{s,t}\colon A_{s}\otimes_{A}A_{t}\hookrightarrow
A_{st}$ for all $s,t\in S$;

such that

  1. (i)

$A_{1}=A$ (as a $A$-$A$-bimodule);

  2. (ii)

$\mu_{s,1}\colon A_{s}\otimes_{A}A\rightarrow A$ and $\mu_{1,s}\colon
A\otimes_{A}A_{s}\rightarrow A_{s}$ are the canonical isomorphisms;

  3. (iii)

for all $s,t,r\in S$ the diagram

${A_{s}\otimes_{A}A_{t}\otimes_{A}A_{r}}$${A_{st}\otimes_{A}A_{r}}$${A_{s}\otimes_{A}A_{tr}}$${A_{str}}$$\scriptstyle{\operatorname{\mathrm{id}}_{A_{s}}\otimes\mu_{t,r}}$$\scriptstyle{\mu_{s,t}\otimes\operatorname{\mathrm{id}}_{A_{r}}}$$\scriptstyle{\mu_{st,r}}$$\scriptstyle{\mu_{s,tr}}$

commutes, and the common composition is denoted by

$\mu_{s,t,r}\colon A_{s}\otimes_{A}A_{t}\otimes_{A}A_{r}\to A_{srt};$

  4. (iv)

and $\mu_{s,s^{*},s}\colon A_{s}\otimes_{A}A_{s^{*}}\otimes_{A}A_{s}\to A_{s}$
is an isomorphism for every $s\in S$.

The following remarks are in order.

######  Remark 2.5.

Historically, the notion of Fell bundles has first been defined by Sieben in
an unpublished manuscript [SiebenFellbundles]. This definition was then
published and used in the paper [Exel:noncomm.cartan] by Exel. We use the
above notion from [Buss-Meyer:Actions_groupoids] by the first-named author and
Meyer, where it is shown that both definitions are equivalent. In [Buss-
Meyer:Actions_groupoids], in fact, a Fell bundle over $S$ with unit fibre $A$
is viewed as a (generalized) _action of $S$ on $A$ by Hilbert bimodules_.

######  Remark 2.6.

The notion in Definition 2.4 is actually what is known as a _non-saturated_
Fell bundle in the literature, since the maps $\mu_{s,t}\colon
A_{s}\otimes_{A}A_{t}\rightarrow A_{st}$ are assumed to be only embeddings of
Hilbert $A$-$A$-bimodules. _Saturated_ Fell bundles, on the other hand, assume
that $\mu_{s,t}$ is, actually, an isomorphism. It is well known (see
[BussExel:InverseSemigroupExpansions]) that given a non-saturated Fell bundle
$\mathcal{A}=(A_{s})_{s\in S}$ one can enlarge $S$ to an inverse semigroup
$T\coloneqq\mathrm{Pr}(S)$ (with the same unit, the so-called _prefix
expansion of $S$_) and endow $\mathcal{A}$ with a natural saturated Fell
bundle structure over $T$ in such a way that the full and reduced
$C^{*}$-algebras of $\mathcal{A}$ over $S$ and $T$ are canonically isomorphic
(see [BussExel:InverseSemigroupExpansions]*Theorem 7.2).

######  Remark 2.7.

We restrict ourselves to the case where $S$ has a unit for convenience of both
the readers and the authors, but this is not necessary. In general, if $S$ is
_not_ a monoid then a Fell bundle over $S$ is defined as in Definition 2.4,
but conditions (i) and (ii) are dropped. As $S$ is always assumed to have a
unit $1\in S$ throughout the paper and the $C^{*}$-algebra $A_{1}=A$ is then
already part of the structure of $\mathcal{A}$, we shall usually just say that
$\mathcal{A}$ is a _Fell bundle over $S$_, as opposed to a Fell bundle over
$S$ with unit fibre $A$.

The following theorem was proved in [Buss-Meyer:Actions_groupoids]*Theorem
4.8, and describes more structure inherently present in any Fell bundle (given
as axioms in [Exel:noncomm.cartan]). In fact, the following was only proved
for _saturated_ Fell bundles (recall Remark 2.6), but the same ideas carry
over to the non-saturated setting.

######  Theorem 2.8.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over $S$. Then there are

  * •

unique Hilbert bimodule embeddings $\iota_{t,s}\colon A_{t}\rightarrow A_{s}$,
where $t\leq s$, such that

    1. (i)

for all $s_{1},t_{1},s_{2},t_{2}\in S$ such that $t_{1}\leq s_{1}$ and
$t_{2}\leq s_{2}$ the diagram

${A_{t_{1}}\otimes_{A}A_{t_{2}}}$${A_{t_{1}t_{2}}}$${A_{s_{1}}\otimes_{A}A_{s_{2}}}$${A_{s_{1}s_{2}}}$$\scriptstyle{\iota_{t_{1},s_{1}}\otimes\iota_{t_{2},s_{2}}}$$\scriptstyle{\mu_{t_{1},t_{2}}}$$\scriptstyle{\iota_{t_{1}t_{2},s_{1}s_{2}}}$$\scriptstyle{\mu_{s_{1},s_{2}}}$

commutes;

    2. (ii)

$\iota_{s,s}=\textup{id}_{A_{s}}$ for every $s\in S$; and

    3. (iii)

$\iota_{s,r}\circ\iota_{t,s}=\iota_{t,r}$ for every $t\leq s\leq r$;

  * •

unique Hilbert bimodule isomorphisms $J_{s}\colon A_{s}^{*}\rightarrow
A_{s^{*}}$ such that

$\mu_{s,s^{*},s}\left(a\otimes J_{s}\left(a\right)\otimes
a\right)=a\cdot\langle a\\!\mid\\!a\rangle_{A}={}_{A}\langle
a\\!\mid\\!a\rangle\cdot a$

for all $s\in S$ and $a\in A_{s}$; here $A_{s}^{*}$ denotes the dual Hilbert
bimodule of $A_{s}$.

######  Remark 2.9.

Henceforth, the embeddings $\mu_{s,t}\colon A_{s}\otimes_{A}A_{t}\rightarrow
A_{st}$ are assumed to be part of the data of a Fell bundle. Likewise, so are
the bimodule maps $\iota_{t,s}$ and $J_{s}$ from Theorem 2.8. Thus, for
convenience of the reader, a Fell bundle over $S$ will just be denoted by
$\mathcal{A}=(A_{s})_{s\in S}$. Furthermore, we shall use the following
abbreviations:

  * •

$a_{s}a_{t}\coloneqq\mu_{s,t}(a_{s}\otimes a_{t})$ whenever $a_{s}\in A_{s}$
and $a_{t}\in A_{t}$;

  * •

the map $\iota_{t,s}$, where $t\leq s$, will just be understood as an
inclusion $A_{t}\subseteq A_{s}$;

  * •

$a_{s}^{*}\coloneqq J_{s}(a_{s})\in A_{s^{*}}$ for every $s\in S$.

The previous notation should be enlightening. Indeed, $\mu_{s,t}$ will be the
product of elements; $\iota_{t,s}$ is just a natural inclusion; and $J_{s}$
stands for taking the adjoint of an element.

Given a $C^{*}$-algebra $A$ we write $A^{\prime\prime}$ for the von Neumann
enveloping algebra of $A$, that is, the von Neumann algebra generated by the
image of its universal representation. The weak and strong topologies on
$A^{\prime\prime}$ are the ones inherited from this representation. The
ultraweak topology corresponds to the ∗-weak topology of
$A^{\prime\prime}\cong(A^{\prime})^{\prime}$.

The following notion was already introduced in [Buss-Exel-Meyer:Reduced] and
will be important to us also in this paper.

######  Definition 2.10.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ over an inverse semigroup
$S$, the _von Neumann enveloping Fell bundle of $\mathcal{A}$_, denoted by
$\mathcal{A}^{\prime\prime}$, is the Fell bundle whose fibers are the biduals
$A^{\prime\prime}_{s}$ of $A_{s}$ and with multiplication maps
$A_{s}^{\prime\prime}\overline{\otimes}_{A_{1}^{\prime\prime}}A_{t}^{\prime\prime}\to
A_{st}^{\prime\prime}$ being the weakly continuous extensions of the
multiplications on $\mathcal{A}$.

######  Remark 2.11.

The Fell bundle $\mathcal{A}^{\prime\prime}$ is a _von Neumann Fell bundle_ or
a _$W^{*}$ -Fell bundle_ as defined in [Abadie-Buss-
Ferraro:Amenability]*Definition 5.1 for Fell bundles over groups (and the same
definition obviously extends to inverse semigroups), which essentially means
the whole structure of the Fell bundle is compatible with the weak topology.
One can look at the _universal representation_ of the Fell bundle
$\mathcal{A}$ in order to represent every fibre $A_{s}$ with $s\in S$ into a
single Hilbert space. Alternatively, one can view $A_{s}$ as a subspace of
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$ (the full cross-sectional
$C^{*}$-algebra of $\mathcal{A}$, see Section 2.3) and embed everything into
the enveloping von Neumann algebra
$C_{\text{max}}^{*}\left(\mathcal{A}\right)^{\prime\prime}$. This allows us to
endow the fibres $A_{s}$ with a weak topology, as we implicitly did above. And
in this sense $A_{s}^{\prime\prime}$ is just the weak closure of $A_{s}$.

We end this section with a discussion of some notation that will be used
throughout the paper virtually without mention. Note that this will lead to
the useful Proposition 2.17, which allows us to perform some commutation
relations in certain computations.

######  Definition 2.12.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$ with unit $1\in S$. Given any $s,t\in S$, let:

  1. (i)

$I_{s,t}\subseteq A$ be the (closed) ideal generated by $A_{v^{*}v}\subseteq
A_{1}$ with $v\leq s,t$;

  2. (ii)

$1_{s,t}\in I_{s,t}^{\prime\prime}$ be the unit of
$I_{s,t}^{\prime\prime}\subseteq A^{\prime\prime}$;

  3. (iii)

$1_{s}\coloneqq 1_{s,1}\in I_{s,1}^{\prime\prime}$.

######  Remark 2.13.

Note that if $e\in E$ is an idempotent then $I_{e,1}=A_{e}$, which is a two-
sided closed ideal in $A_{1}$. Thus $1_{e}\in A_{e}^{\prime\prime}$ is the
unit of the von Neumann algebra $A_{e}^{\prime\prime}$. We can also view
$1_{e}$ as the unit of the multiplier $C^{*}$-algebra
$\mathcal{M}(A_{e})\subseteq A_{e}^{\prime\prime}$. More generally we may view
$1_{s}$ as the unit of the multiplier $C^{*}$-algebra
$\mathcal{M}(I_{s,1})\subseteq I_{s,1}^{\prime\prime}$. Notice that
$I_{s,t}=I_{s^{*}t,1}=I_{t^{*}s,1}=I_{t,s}$ so that
$1_{s,t}=1_{s^{*}t}=1_{t^{*}s}=1_{t,s}$ for all $s,t\in S$. This follows from
the fact that if $e\leq s^{*}t,1$, then $v\coloneqq se=te\leq s,t$ (see the
proof of Lemma 2.14 below) and, conversely, if $v\leq s,t$, then $e\coloneqq
v^{*}v\leq s^{*}t,1$.

The ideal $I_{s,1}$ is intrinsic to the Fell bundle $\mathcal{A}$ and plays an
important role in its structure. In a sense that will become clear in what
follows, $I_{s,1}$ implements the _trivial_ part of the fibre $A_{s}$. The
following results give certain important computation relations that will be
used later.

######  Lemma 2.14.

Let $S$ be an inverse semigroup with idempotent semilattice $E$. For $s\in S$,
write $E_{s}:=\\{e\in E:e\leq s\\}$ and let $\omega_{s}\colon E\to E$ be the
map $\omega_{s}(e)\coloneqq ses^{*}$. Then:

  1. (i)

$E_{r}=E_{r^{*}}$ for all $r\in S$, in particular $E_{s^{*}t}=E_{t^{*}s}$ for
all $s,t\in S$;

  2. (ii)

$\omega_{s}=\omega_{t}$ as maps $E_{s^{*}t}\to E_{st^{*}}$ and give a
bijection with inverse $\omega_{s^{*}}=\omega_{t^{*}}$.

###### Proof.

Recall that, in general, it is well known (and straightforward to prove) that
$a\leq b$ if and only if $a^{*}\leq b^{*}$ for any $a,b\in S$. Thus
$E_{r}=E_{r^{*}}$, since idempotents are self-adjoint. Next, observe that if
$e\in E$ and $e\leq t^{*}s$ then $t^{*}se=e=s^{*}te$, so that

$te=tt^{*}se=tt^{*}ss^{*}se=ss^{*}tt^{*}se=se.$

From this it also follows that $et^{*}=es^{*}$ and therefore
$\omega_{s}=\omega_{t}$ on $E_{s^{*}t}$. Moreover, if $e\leq s^{*}t$, then
$ses^{*}\leq ts^{*}$ because $ts^{*}ses^{*}=tes^{*}=ses^{*}$ and we have
$\omega_{t^{*}}(ses^{*})=t^{*}ses^{*}t=e$, therefore
$\omega_{t^{*}}=\omega_{s^{*}}\colon E_{st^{*}}\to E_{s^{*}t}$ realizes the
inverse of $\omega_{s}=\omega_{t}$. ∎

Recall the notion of _saturated_ Fell bundles from Remark 2.6, and observe
that the following lemma states that, even in the non-saturated setting we are
working in, some multiplication maps are isomorphisms.

######  Lemma 2.15.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle. The following assertions
hold:

  1. (i)

$\overline{\operatorname{span}}A_{s}\cdot
A_{s^{*}s}=\operatorname{\overline{span}}A_{s}\cdot A_{s^{*}}\cdot A_{s}$; and

  2. (ii)

$A_{se}=\operatorname{\overline{span}}A_{s}\cdot A_{e}$

for every idempotent $e\in E$ and $s\in S$.

###### Proof.

In all assertions the inclusions $\supset$ are clear and follow from the
canonical multiplication maps. Assertion (i) follows from Definition 2.4 (iv).
Assertion (ii) follows from Definition 2.4 (iv) as well, since

$A_{se}=\operatorname{\overline{span}}A_{se}A_{\left(se\right)^{*}}A_{se}\subseteq\operatorname{\overline{span}}A_{se}\cdot
A_{es^{*}s}\subseteq\operatorname{\overline{span}}A_{s}\cdot A_{e},$

as desired. ∎

######  Lemma 2.16.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle. Then
$a_{s}1_{e}=1_{ses^{*}}a_{s}$ for every $a_{s}\in A_{s},s\in S$ and idempotent
$e\in E$ whenever $s^{*}s\geq e$.

###### Proof.

The claim follows from the fact that $A_{se}^{\prime\prime}$ is a Hilbert
$A_{ses^{*}}^{\prime\prime}$-$A_{e}^{\prime\prime}$-bimodule, and hence
$a_{s}1_{e}=1_{ses^{*}}a_{s}1_{e}=1_{ses^{*}}a_{s}$, as desired. ∎

######  Proposition 2.17.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle. Then the following hold:

  1. (i)

$1_{s^{*}t}=1_{t^{*}s}$ and $1_{st^{*}}=1_{ts^{*}}$ for every $s,t\in S$.

  2. (ii)

$1_{s}\in I_{s,1}^{\prime\prime}$ is central in $A_{1}^{\prime\prime}$.

  3. (iii)

$a_{s}1_{s^{*}t}=1_{st^{*}}a_{s}$ for every $s,t\in S$ and $a_{s}\in A_{s}$.

###### Proof.

(i) follows from Lemma 2.14 (i) and the fact that, by construction,
$1_{s^{*}t}\in I_{s^{*}t,1}^{\prime\prime}=\langle A_{e}\mid e\leq
s^{*}t,1\rangle^{\prime\prime}\subseteq A_{1}^{\prime\prime}$. In fact, note
that $I_{s^{*}t,1}^{\prime\prime}=I_{t^{*}s,1}^{\prime\prime}$, and hence
$1_{s^{*}t}=1_{t^{*}s}$. For assertion (ii) just observe that every projection
$1_{e}\in A_{e}^{\prime\prime}$ is central whenever $e\in E$ is an idempotent.

In order to show (iii) denote by $E_{s^{*}t}\coloneqq\\{e\in E\mid e\leq
s^{*}t\\}$. In addition, given any finite subset $K\Subset E$ let
$1_{K}\coloneqq\prod_{e\in K}1_{e}$. With this notation, observe that for any
finite set $F\Subset E_{s^{*}t}$ the element

$p_{F}\coloneqq\sum_{i=1}^{\left|F\right|}\sum_{\begin{subarray}{c}K\subseteq
F\\\ \left|K\right|=i\end{subarray}}\left(-1\right)^{i+1}1_{K}\in\sum_{e\in
F}A_{e}^{\prime\prime}\subseteq I_{s^{*}t,1}^{\prime\prime}$

is a self-adjoint projection (it realizes the join projection $\vee_{e\in
F}1_{e}$). Moreover, $p_{F}\rightarrow 1_{s^{*}t}$ when $F$ grows, where
convergence happens in the weak topology of $I_{s^{*}t,1}^{\prime\prime}$.
Now, for a fixed $F\Subset E_{s^{*}t}$, we have

$a_{s}p_{F}=a_{s}\sum_{i=1}^{\left|F\right|}\sum_{\begin{subarray}{c}K\subseteq
F\\\
\left|K\right|=i\end{subarray}}\left(-1\right)^{i+1}1_{K}=\sum_{i=1}^{\left|F\right|}\sum_{\begin{subarray}{c}K\subseteq
F\\\
\left|K\right|=i\end{subarray}}\left(-1\right)^{i+1}1_{sKs^{*}}a_{s}=p_{sFs^{*}}a_{s},$

where the second equality follows from Lemma 2.16 and the third from Lemma
2.14 (ii). Therefore, since $p_{F}\rightarrow 1_{s^{*}t}$, the proof will be
done once we show that $p_{sFs^{*}}\rightarrow 1_{st^{*}}$ in the weak
operator topology of $I_{st^{*},1}^{\prime\prime}$. But, again, this is clear
by Lemma 2.14 (ii), since conjugation by $s$ defines a bijective map between
finite subsets of $E_{s^{*}t}$ and finite subsets of $E_{st^{*}}$. ∎

###  2.3. Cross-sectional $C^{*}$-algebras of Fell bundles

In this section we recall the construction of the _full cross-sectional
$C^{*}$-algebra_ associated to a Fell bundle (see Sections 3 and 3.3 below for
the _reduced_ counterpart). To such end, let $\mathcal{A}=(A_{s})_{s\in S}$ be
a Fell bundle over an inverse semigroup $S$ with unit $1\in S$. Notice that
$A_{e}$ is a $C^{*}$-algebra as it is, in fact, a closed two sided-ideal of
$A=A_{1}$ for every idempotent $e\in E$. We denote by
$C_{c}(\mathcal{A})\coloneqq C_{c}(S,\mathcal{A})$ the space of finitely
supported sections $S\to\mathcal{A}$, that is, the algebraic direct sum
$\oplus_{s\in S}^{\mathrm{alg}}A_{s}$. We shall write $a_{s}\delta_{s}\in
C_{c}(\mathcal{A})$ for the section supported in $\\{s\\}$ taking value
$a_{s}\in A_{s}$ at $s\in S$. Recall that $C_{c}(\mathcal{A})$ is a ∗-algebra
with respect to the operations

$\left(a_{s}\delta_{s}\right)\cdot\left(b_{t}\delta_{t}\right)\coloneqq
a_{s}b_{t}\delta_{st}\quad\text{and}\quad\left(a_{s}\delta_{s}\right)^{*}\coloneqq
a_{s}^{*}\delta_{s^{*}},$

where we implicitly use the Fell bundle and inverse semigroup operations
(recall Theorem 2.8 and the subsequent Remark 2.9). The following definition
is central to the paper.

######  Definition 2.18.

The maximal cross-sectional $C^{*}$-algebra
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$ of a Fell bundle
$\mathcal{A}=(A_{s})_{s\in S}$ is the universal $C^{*}$-algebra of the
quotient $\mathcal{Q}_{c}(\mathcal{A})$ of $C_{c}(\mathcal{A})$ given by

$\mathcal{Q}_{c}\left(\mathcal{A}\right)\coloneqq
C_{c}\left(\mathcal{A}\right)/\mathcal{I}_{\mathcal{A}},$

where $\mathcal{I}_{\mathcal{A}}\subseteq C_{c}(\mathcal{A})$ is the $*$-ideal
generated by the differences $a_{s}\delta_{s}-a_{s}\delta_{t}$ for all $s\leq
t$ in $S$ and $a_{s}\in A_{s}\subseteq A_{t}$.

In other words, $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ is the universal
$C^{*}$-algebra generated by $a_{s}\delta_{s}$ with $a_{s}\in A_{s}$ and $s\in
S$ satisfying the relations that defined the ∗-algebra structure on
$C_{c}(\mathcal{A})$ above and the relation imposing that
$a_{s}\delta_{s}=a_{s}\delta_{t}$ whenever $a_{s}\in A_{s}$ and $s\leq t$ in
$S$.

######  Remark 2.19.

It can be shown that the subspace of $C_{c}(\mathcal{A})$ generated by
differences $a_{s}\delta_{s}-a_{s}\delta_{t}$ with $s\leq t\in S$ and
$a_{s}\in A_{s}\subseteq A_{t}$ is already a ∗-ideal of $C_{c}(\mathcal{A})$,
and therefore this equals $\mathcal{I}_{\mathcal{A}}$. This is already
observed in [Exel:noncomm.cartan], where the above definition was first given.

######  Remark 2.20.

By the universal property of $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ (see
Proposition 2.36) we get a ∗-homomorphism
$\mathcal{Q}_{c}\left(\mathcal{A}\right)\to
C_{\text{max}}^{*}\left(\mathcal{A}\right)$ with dense image. But, as it turns
out, this ∗-homomorphism does not have to be injective in general, that is,
its kernel is not just $\mathcal{I}_{\mathcal{A}}$, but a possibly larger
ideal that we now describe.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ over $S$ and elements
$t,u\in S$, it is proved in [Buss-Exel-Meyer:Reduced]*Lemma 2.5 that there
exists a unique isomorphism of Hilbert bimodules:

(2.21) $\theta_{t,u}\colon A_{u}\cdot I_{t,u}\xrightarrow{\sim}A_{t}\cdot
I_{t,u}$

extending the identity map $A_{u}\cdot A_{v^{*}v}=A_{v}\to A_{v}=A_{t}\cdot
A_{v^{*}v}$ for every $v\in S$ with $v\leq t,u$, so that $uv^{*}v=tv^{*}v=v$.
Recall that $I_{t,u}$ is the ideal of $A$ generated by
$A_{v}^{*}A_{v}\subseteq A_{v^{*}v}$ with $v\leq t,u$ (see Lemma 2.15).

######  Definition 2.22.

Given a Fell bundle $\mathcal{A}$, let $\mathcal{J}_{\mathcal{A}}$ be the
subspace of $C_{c}(\mathcal{A})$ generated by elements of the form
$\theta_{t,u}(a)\delta_{t}-a\delta_{u}$ for $t,u\in S$ and $a\in A_{u}\cdot
I_{t,u}$.

Notice that $\mathcal{J}_{\mathcal{A}}$ obviously contains
$\mathcal{I}_{\mathcal{A}}$, but the inclusion might be strict as
$\mathcal{J}_{\mathcal{A}}$ may also contain some limits of elements of
$\mathcal{I}_{\mathcal{A}}$. As will become clear in what follows,
$\mathcal{J}_{\mathcal{A}}$ is also a ∗-ideal of $C_{c}(\mathcal{A})$. In
order to show this, however, we first need to define a certain conditional
expectation and describe $\mathcal{J}_{\mathcal{A}}$ in a different way.

######  Definition 2.23.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ over $S$, the _cross-
sectional algebra of $\mathcal{A}$_, denoted by
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$, is the quotient
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)\coloneqq
C_{c}(\mathcal{A})/\mathcal{J}_{\mathcal{A}}$.

Since $\mathcal{J}_{\mathcal{A}}$ contains $\mathcal{I}_{\mathcal{A}}$, we get
surjective ∗-homomorphisms (compare with Proposition 2.27 below)

$C_{c}(\mathcal{A})\twoheadrightarrow\mathcal{Q}_{c}(\mathcal{A})\twoheadrightarrow\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right).$

Moreover, it is observed in [Buss-Exel-Meyer:Reduced] that every
representation of $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ vanishes on
$\mathcal{J}_{\mathcal{A}}$, so we get also a homomorphism with dense range

$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)\to
C_{\text{max}}^{*}\left(\mathcal{A}\right).$

In order to understand all this better, we need to recall the construction of
the canonical (weak) conditional expectation of $\mathcal{A}$. For this, first
notice that as a particular case of Equation 2.21, for each $s\in S$ we get an
isomorphism $\theta_{s,1}\colon A_{s}\cdot I_{s,1}\xrightarrow{\sim}I_{s,1}$
extending the identity map $A_{s}\cdot A_{e}=A_{e}\to A_{e}$ for $e\in E$ with
$e\leq s$. This gives rise to a canonical map

$\Theta_{s,1}\colon
A_{s}\rightarrow\mathcal{M}\left(I_{s,1}\right),\;\;\text{where}\;\,\Theta_{s,1}\left(a\right)x\coloneqq\theta_{s,1}\left(a\cdot
x\right).$

Here we write $\mathcal{M}(B)$ for the multiplier of a $C^{*}$-algebra $B$.
Viewing $\mathcal{M}(I_{s,1})\subseteq I_{s,1}^{\prime\prime}\subseteq
A^{\prime\prime}$ in the standard way, the above map can also be represented
as

$\Theta_{s,1}(a)=\lim_{i}\theta_{s,1}(au_{i}),$

where $\\{u_{i}\\}_{i}\subseteq I_{s,1}$ is an approximate unit and the limit
is with respect to the weak topology of the von Neumann algebra
$A_{1}^{\prime\prime}$. Actually, the limit above can also be understood in
the strict topology of $\mathcal{M}(I_{s,1})$, see [Buss-Exel-
Meyer:Reduced]*Lemma 4.5.

It is fruitful to look at the above structure from the point of view of the
von Neumann enveloping Fell bundle $\mathcal{A}^{\prime\prime}$ (and this is
the way this is done in [Buss-Exel-Meyer:Reduced]). The isomorphism
$\theta_{s,1}$ extends to a weakly continuous isomorphism (of Hilbert
$W^{*}$-bimodules):

$\theta_{s,1}^{\prime\prime}\colon A_{s}^{\prime\prime}\cdot
I_{s,1}^{\prime\prime}\xrightarrow{\sim}I_{s,1}^{\prime\prime}$

and so we get an embedding $A_{s}^{\prime\prime}\cdot
I_{s,1}^{\prime\prime}\hookrightarrow A_{1}^{\prime\prime}$ and a map
$\Theta_{s,1}^{\prime\prime}\colon A_{s}^{\prime\prime}\to
A_{1}^{\prime\prime}$ that extends $\Theta_{s,1}$ when we view
$\mathcal{M}(I_{s,1})\subseteq I_{s,1}^{\prime\prime}\subseteq
A_{1}^{\prime\prime}$. Moreover, notice that $A_{s}^{\prime\prime}\cdot
I_{s,1}^{\prime\prime}=A_{s}^{\prime\prime}\cdot 1_{s}$ so that
$A_{s}^{\prime\prime}$ decomposes as a direct sum

$A_{s}^{\prime\prime}=A_{s}^{\prime\prime}\cdot 1_{s}\oplus
A_{s}^{\prime\prime}\cdot 1_{s}^{\perp},$

where $1_{s}^{\perp}$ denotes the unit of the complement ideal
$(I_{s,1}^{\prime\prime})^{\perp}\mathrel{\triangleleft}A_{1}^{\prime\prime}$.
In this way, $\Theta_{s,1}^{\prime\prime}$ (and hence also its restriction
$\Theta_{s,1}$) is just the composition of the projection map
$A_{s}^{\prime\prime}\twoheadrightarrow A_{s}^{\prime\prime}\cdot 1_{s}$,
$x\mapsto x\cdot 1_{s}$ with the embedding $A_{s}^{\prime\prime}\cdot
1_{s}\xrightarrow{\sim}I_{s,1}^{\prime\prime}\subseteq A_{1}^{\prime\prime}$.
As with Theorem 2.8, the following has only been proved in the saturated case,
but the same ideas carry over to the non-saturated setting.

######  Proposition 2.24 ([Buss-Exel-Meyer:Reduced]).

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$, consider the map

$\displaystyle P\colon C_{c}\left(\mathcal{A}\right)$ $\displaystyle\to
A_{1}^{\prime\prime},\quad
P(a_{s}\delta_{s})=\theta_{s,1}^{\prime\prime}(a_{s}1_{s})$

defined as the linear extension of the maps $\Theta_{s,1}\colon
A_{s}\to\mathcal{M}(I_{s,1})\subseteq A_{1}^{\prime\prime}$. Then $P$ vanishes
on $\mathcal{J}_{\mathcal{A}}$ and factors through a faithful (weak)
conditional expectation

$\tilde{P}\colon\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)\to
A_{1}^{\prime\prime}.$

It is clear from the construction above that $P$ extends to
$C_{c}(\mathcal{A}^{\prime\prime})$, that is, we get from the above
construction also a conditional expectation $P^{\prime\prime}\colon
C_{c}(\mathcal{A}^{\prime\prime})\to A_{1}^{\prime\prime}$ which factors
faithfully to
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}^{\prime\prime}\right)\to
A_{1}^{\prime\prime}$.

######  Definition 2.25.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$, we call the map $P\colon
C_{c}(\mathcal{A})\to A_{1}^{\prime\prime}$ defined above the _canonical
conditional expectation_ of $\mathcal{A}$.

Since $C_{c}(\mathcal{A})$ and
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ are only ∗-algebras, it
might be not clear what we mean by a (weak) conditional expectation on these
algebras. This only means $P$ is a linear map which is positive in the sense
that $P(x^{*}x)\geq 0$ for all $x\in C_{c}(\mathcal{A})$; it preserves
adjoints, meaning that $P(x^{*})=P(x)^{*}$, and that $P$ restricts to the
identity on $A_{1}$, that is, $P(a\delta_{1})=a$ for all $a\in A_{1}$. Similar
properties hold for $\tilde{P}$. And the fact that $\tilde{P}$ is faithful
means that $P(x^{*}x)=0$ happens only if $x=0$ for any
$x\in\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$. This is proved in
[Buss-Exel-Meyer:Reduced]*Proposition 3.6. The fact that $\tilde{P}$ is
faithful also implies that $P$ is symmetric in the sense that $P(x^{*}x)=0$ if
and only if $P(xx^{*})=0$, for all $x\in C_{c}(\mathcal{A})$. This means that

(2.26) $\mathcal{N}_{P}\coloneqq\left\\{x\in
C_{c}\left(\mathcal{A}\right):P\left(x^{*}x\right)=0\right\\}$

is a ∗-ideal of $C_{c}(\mathcal{A})$. In fact, since $\tilde{P}$ is faithful,
it also follows that $\mathcal{N}_{P}$ coincides with the kernel of the
quotient homomorphism
$C_{c}(\mathcal{A})\twoheadrightarrow\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$,
that is,

$\mathcal{N}_{P}=\mathcal{J}_{\mathcal{A}}=\operatorname{{span}}\left\\{\theta_{u,t}(a)\delta_{u}-a\delta_{t}\colon
a\in I_{u,t}\right\\}.$

The following result from [Buss-Exel-Meyer:Reduced] indicates that
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ yields the “correct”
definition of the algebraic crossed product of a Fell bundle.

######  Proposition 2.27.

[Buss-Exel-Meyer:Reduced]*Proposition 4.3 Given a Fell bundle
$\mathcal{A}=(A_{s})_{s\in S}$, the ∗-algebra
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ embeds densely in
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$.

For the following lemma recall that we canonically embed $A_{s}$ into
$C_{c}(\mathcal{A})$ as the set $A_{s}\delta_{s}$ of sections
$a_{s}\delta_{s}$ taking value $a_{s}$ at $s$ and $0$ otherwise.

######  Lemma 2.28.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$,
$A_{s}\delta_{s}\cap\mathcal{N}_{P}=0$ for every $s\in S$.

###### Proof.

For any $a_{s}\delta_{s}\in A_{s}\delta_{s}$ we have

$P\left(\left(a_{s}\delta_{s}\right)^{*}\left(a_{s}\delta_{s}\right)\right)=P\left(a_{s}^{*}a_{s}\delta_{s^{*}s}\right)=\theta_{s^{*}s,1}^{\prime\prime}\left(a_{s}^{*}a_{s}1_{s^{*}s}\right)=a_{s}^{*}a_{s}1_{s^{*}s}$

since $s^{*}s\leq 1$. Therefore if $a_{s}\delta_{s}\in\mathcal{N}_{P}$ then
$a_{s}=0$, as claimed. ∎

Observe that, in general, the target of the canonical conditional expectation
$P$ in Proposition 2.24 is only the von Neumann algebra
$A_{1}^{\prime\prime}$, as opposed to $A_{1}$ itself. We end the section with
a condition that guarantees that the image of $P$ lands in $A_{1}$.

######  Corollary 2.29.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle. Then $P(x)\in A_{1}$ for
all $x\in C_{c}(\mathcal{A})$ if and only if $I_{s,1}$ is complemented in
$\operatorname{\overline{span}}A_{s}^{*}A_{s}\subseteq A_{1}$ for all $s\in
S$.

###### Proof.

The proof is given in [Buss-Exel-Meyer:Reduced]*Proposition 6.3, and it does
not assume that the Fell bundle is saturated. However, we now give a sketch of
the proof of the forward implication and refer the reader to [Buss-Exel-
Meyer:Reduced]*Proposition 6.3 for the (more complicated) proof of the
converse. Observe that if $I_{s,1}$ is a complemented ideal of $A_{s^{*}s}$
then $A_{s}$ decomposes as $A_{s}=A_{s}\cdot I_{s,1}\oplus A_{s}\cdot
I_{s,1}^{\perp}$. And in this case $P$ acts on $A_{s}$ by projecting first to
$A_{s}\cdot I_{s,1}$ and then composes this with the isomorphism
$\theta_{s,1}\colon A_{s}\cdot I_{s,1}\xrightarrow{\sim}I_{s,1}\subseteq
A_{1}$. Hence $P(A_{s}\delta_{s})\subseteq A_{1}$ for all $s$ and therefore
also $P(C_{c}(\mathcal{A}))\subseteq A_{1}$. ∎

The following definition is from [Kwaniewski2019EssentialCP]. We will use it
in the sequel, albeit not predominantly.

######  Definition 2.30.

We say a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ is _closed_ if the image
of the canonical conditional expectation $P$ is contained in $A_{1}$.

######  Remark 2.31.

Observe that for $\mathcal{A}$ to be closed it is _not_ enough that the ideals
$A_{e}$ be complemented or even unital. Indeed, this is not even true in the
basic case of a Fell bundle coming from just an ordinary inverse semigroup.
For instance, take $S\coloneqq\\{1,z\\}\cup\\{e_{n}\\}_{n\in\mathbb{N}}$,
where $1$ is a unit, $z^{2}=1$, $z\geq e_{n}$ for all $n\in\mathbb{N}$, and
$e_{n}e_{m}=e_{m}e_{n}=e_{\min\\{n,m\\}}$. Consider the Fell bundle associated
to the action of $S$ on its spectrum
$X=\widehat{E}\cong\mathbb{N}\sqcup\\{\infty\\}$ (the one-point
compactification of $\mathbb{N}$). Notice that the groupoid of germs of this
action is the universal (Paterson) groupoid of $S$ (see [Paterson1999]). Then
the ideals $A_{e}$ are all complemented (as they are all unital).
Nevertheless, $P$ does not take values in $A_{1}$ as the universal groupoid of
$S$ is not Hausdorff (see [Buss-Exel-Meyer:Reduced]*Proposition 5.1 and
[Paterson1999]*Theorem 4.4.1).

###  2.4. Representations of Fell bundles

In this section we introduce the central notion of a _representation of a Fell
bundle_. As expected, it preserves all the inherent structure of the bundle.

######  Definition 2.32.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$.

  1. (i)

A representation of $\mathcal{A}$ on the Hilbert space $\mathcal{H}$ is a
family $\pi=(\pi_{s})_{s\in S}$ of linear maps $\pi_{s}\colon
A_{s}\to\mathcal{L}(\mathcal{H})$ such that

$\pi_{s}\left(a_{s}\right)\pi_{t}\left(b_{t}\right)=\pi_{st}\left(a_{s}b_{t}\right),\;\;\pi_{s}\left(a_{s}\right)^{*}=\pi_{s^{*}}\left(a_{s}^{*}\right)\;\;\text{and}\;\;\pi_{s}\left(a_{s}\right)=\pi_{u}\left(a_{s}\right),$

whenever $s,t,u\in S$ with $s\leq u$ and $a_{s}\in A_{s}$ and $b_{t}\in
A_{t}$.

  2. (ii)

We say the representation $\pi=(\pi_{s})_{s\in S}$ is _nondegenerate_ if
$\pi_{1}$ is a nondegenerate representation of $A=A_{1}$.

By definition a representation of $\mathcal{A}$ always respects the inclusion
maps $\iota_{t,s}\colon A_{s}\hookrightarrow A_{t}$ for every $s\leq t$ in
$S$. It turns out that they also automatically respect the extended inclusion
maps $\theta_{t,u}\colon A_{u}\cdot I_{t,u}\hookrightarrow A_{t}\cdot I_{t,u}$
for every $t,u\in S$, as the following result shows. Actually, the following
was only proved in the saturated case, but the same ideas work in the non-
saturated setting as well (recall Remark 2.6).

######  Proposition 2.33 ([Buss-Exel-Meyer:Reduced]*Proposition 2.9).

Every representation $\pi$ of $\mathcal{A}$ satisfies

$\pi_{t}\circ\theta_{t,u}\left(a_{u}\right)=\pi_{u}\left(a_{u}\right)$

for all $t,u\in S$ and $a_{u}\in A_{u}\cdot I_{t,u}$.

The following notation will be useful throughout the paper, particularly when
discussing Fell’s absorption principle in the context of Fell bundles over
inverse semigroups.

######  Definition 2.34.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ and a representation
$\pi=(\pi_{s})_{s\in S}$, we define
$\mathcal{H}_{s}:=\operatorname{\overline{span}}\pi_{s}(A_{s})\mathcal{H}\subseteq\mathcal{H}$.

Observe that, by [Buss-Exel-Meyer:Reduced]*Lemma 2.8, we have
$\mathcal{H}_{s}=\mathcal{H}_{ss^{*}}=\pi_{1}(A_{ss^{*}})\mathcal{H}\subseteq\mathcal{H}$
for all $s\in S$. Likewise, notice that $\mathcal{H}=\mathcal{H}_{1}$ if and
only if the representation $\pi$ is nondegenerate.

######  Remark 2.35.

Notice that every representation $\pi$ of $\mathcal{A}$ extends uniquely to a
weakly continuous representation
$\pi^{\prime\prime}=(\pi_{s}^{\prime\prime})_{s\in S}$ of its von Neumann
enveloping Fell bundle $\mathcal{A}^{\prime\prime}$. Moreover, since $A_{s}$
is weakly dense in $A_{s}^{\prime\prime}$, for all $s\in S$ we have

$\mathcal{H}_{s}=\operatorname{\overline{span}}\pi_{s}\left(A_{s}\right)\mathcal{H}=\operatorname{\overline{span}}\pi_{s}^{\prime\prime}\left(A_{s}^{\prime\prime}\right)\mathcal{H}=\operatorname{\overline{span}}\pi_{1}^{\prime\prime}\left(A_{ss^{*}}^{\prime\prime}\right)\mathcal{H}.$

We end this section with the expected universal characterization of the full
cross sectional $C^{*}$-algebra.

######  Proposition 2.36.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over the inverse semigroup
$S$. Then $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ is universal for
representations of $\mathcal{A}$, that is, given any representation
$\pi=(\pi_{s})_{s\in S}$ of $\mathcal{A}$ on a Hilbert space $\mathcal{H}$,
there is a (unique) $*$-homomorphism $\rho\colon
C_{\text{max}}^{*}\left(\mathcal{A}\right)\rightarrow\mathcal{L}(\mathcal{H})$
such that $\rho(a_{s}\delta_{s})=\pi_{s}(a_{s})$ for every $a_{s}\in A_{s}$
and $s\in S$.

##  3\. Regular representation and Fell’s absorption principle

Before going into the _approximation property_ (see Section 4), in this
section we prove an analogue of Fell’s absorption principle in the setting of
Fell bundles over inverse semigroups (see Theorem 3.20).

###  3.1. The regular representation

In this section we introduce the _left regular representation_ of a Fell
bundle, and prove some structural results about it. Note that this
representation will be key in order to construct the _reduced cross-sectional
$C^{*}$-algebra_ of a Fell bundle (see Definition 3.3). We refer the reader to
[Buss-Exel-Meyer:Reduced] for a more comprehensive discussion.

We first need to discuss a canonical inner product on $C_{c}(\mathcal{A})$.
Recall we naturally endow $C_{c}(\mathcal{A})$ with a ∗-algebra structure (see
Section 2.3). Likewise, recall the definition of the ideal
$\mathcal{N}_{P}\subseteq C_{c}(\mathcal{A})$ from Equation 2.26.

######  Proposition 3.1.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle. Consider

$\langle\xi\\!\mid\\!\eta\rangle_{A_{1}^{\prime\prime}}\coloneqq
P\left(\xi^{*}\cdot\eta\right),\;\;\text{where}\;\,\xi,\eta\in
C_{c}\left(\mathcal{A}\right).$

Then $\langle\cdot\\!\mid\\!\cdot\rangle_{A_{1}^{\prime\prime}}$ is a _semi-
inner product_ with values in $A_{1}^{\prime\prime}$ whose kernel is exactly
$\mathcal{N}_{P}$, that is,
$\langle\xi\\!\mid\\!\xi\rangle_{A_{1}^{\prime\prime}}=0$ if and only if
$\xi\in\mathcal{N}_{P}$. Thus, it factors through an
$A_{1}^{\prime\prime}$-valued inner product on
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$.

###### Proof.

The proof follows from the properties of $P$ showed in [Buss-Exel-
Meyer:Reduced]*Proposition 3.6 and Remark 4.7. Recall that
$\langle\cdot\\!\mid\\!\cdot\rangle_{A_{1}^{\prime\prime}}$ being a semi-inner
product with values in $A_{1}^{\prime\prime}$ means that it is a sesquilinear
form, $A_{1}$-linear in the second variable, and is positive meaning that
$\langle\xi\\!\mid\\!\xi\rangle_{A_{1}^{\prime\prime}}\geq 0$ in the
$C^{*}$-algebra $A_{1}^{\prime\prime}$ for all $\xi\in C_{c}(\mathcal{A})$. ∎

We could now proceed and take the completion of
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ with the respect to the norm
induced from the above inner product. But there is a technical complication
here as this completion does not necessarily carry a (right)
$A_{1}^{\prime\prime}$-module structure. The easy way out of this is to go
directly to the von Neumann enveloping Fell bundle
$\mathcal{A}^{\prime\prime}$ and consider the similar completion there. This
is the way it is done in [Buss-Exel-Meyer:Reduced].

######  Definition 3.2.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ we denote by
_$\ell^{2}(\mathcal{A}^{\prime\prime})$_ the completion of
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}^{\prime\prime}\right)$ with respect
to the norm induced from the inner product
$\langle\cdot\\!\mid\\!\cdot\rangle_{A_{1}^{\prime\prime}}$ given by
Proposition 3.1 for $\mathcal{A}^{\prime\prime}$.

Note that $\ell^{2}(\mathcal{A}^{\prime\prime})$ is, in particular, a Hilbert
module over $A_{1}^{\prime\prime}$. We can now introduce the left regular
representation of a Fell bundle.

######  Definition 3.3.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$.

  1. (i)

The _(left) regular representation_ of $\mathcal{A}$ is the canonical
∗-homomorphism
$\Lambda\colon\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)\to\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime}))$
defined by (left) multiplication, that is,

$\Lambda\left(a\right)\left(f\right)\coloneqq af$

for every $f\in\ell^{2}(\mathcal{A}^{\prime\prime})$ and
$a\in\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$.

  2. (ii)

The _reduced cross-sectional $C^{*}$-algebra_ of $\mathcal{A}$ is the
completion $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ of
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ with respect to the
$C^{*}$-norm

$\|a\|_{r}\coloneqq\|\Lambda\left(a\right)\|_{\mathcal{L}\left(\ell^{2}\left(\mathcal{A}^{\prime\prime}\right)\right)},\;\;\text{where}\;\,a\in\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right).$

######  Remark 3.4.

By the universal property of $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ (see
also Proposition 2.36) the left regular representation of $\mathcal{A}$ gives
a representation $C_{\text{max}}^{*}\left(\mathcal{A}\right)\to
C_{\text{red}}^{*}\left(\mathcal{A}\right)\subseteq\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime}))$.
By abuse of notation we will denote both by $\Lambda$.

The following proposition gives an explicit formula for the left regular
representation
$\Lambda\colon\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)\rightarrow\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime}))$.
For it, recall that the sections $A_{s}\cong A_{s}\delta_{s}$ canonically
embed into $\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ (see Lemma 2.28).

######  Proposition 3.5.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over $S$. Then:

$\Lambda_{s}\left(a_{s}\delta_{s}\right)\left(f+\mathcal{N}_{P}\right)=\sum_{t\in
S}a_{s}f\left(t\right)\delta_{st}+\mathcal{N}_{P}$

for every $a_{s}\in A_{s}$ and $f\in C_{c}(\mathcal{A}^{\prime\prime})$.

###### Proof.

We have $f=\sum_{r\in S}f(r)\delta_{r}+\mathcal{N}_{P}$, where the sum has, at
most, finitely many non-zero terms. By definition

$\Lambda\left(a_{s}\delta_{s}\right)\left(f\right)=\sum_{t\in
S}a_{s}\delta_{s}f\left(t\right)\delta_{t}+\mathcal{N}_{P}=\sum_{t\in
S}a_{s}f\left(t\right)\delta_{st}+\mathcal{N}_{P},$

as claimed. ∎

######  Remark 3.6.

Note that, by Proposition 3.5, we have a commuting diagram

${C_{c}\left(\mathcal{A}\right)}$${C_{c}\left(\mathcal{A}\right)}$${\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)}$${\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)}$$\scriptstyle{\pi_{P}}$$\scriptstyle{\tilde{\Lambda}\left(a\right)}$$\scriptstyle{\pi_{P}}$$\scriptstyle{\Lambda\left(a+\mathcal{N}_{P}\right)}$

for every $a\in C_{c}(\mathcal{A})$, where $\pi_{P}\colon
C_{c}(A)\rightarrow\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$ is the
canonical quotient map and $\tilde{\Lambda}(a)(b)\coloneqq ab$. In particular,
the map $\tilde{\Lambda}$ descends to a map $\Lambda$ on the quotients.
Several different situations similar to this will happen throughout the paper
(see, e.g., Propositions 3.18 and 5.9).

We isolate the following notion due to its relevance within the paper.

######  Definition 3.7.

We say a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ has the weak containment
property if the left regular representation $\Lambda\colon
C_{\text{max}}^{*}\left(\mathcal{A}\right)\to
C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is a ∗-isomorphism.

We now turn our attention to the construction of new representations. In
particular, we can induce representations of $A_{1}$ to representations of the
Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ using the left regular
representation in the following way.

######  Definition 3.8.

Given a representation $\rho\colon A_{1}\to\mathcal{L}(\mathcal{H})$ on
$\mathcal{H}$, we define:

  1. (i)

$\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\rho^{\prime\prime}}\mathcal{H}$
to be the _balanced tensor product of $\ell^{2}(\mathcal{A}^{\prime\prime})$
and $\mathcal{H}$_, that is, the Hausdorff completion of
$\ell^{2}(\mathcal{A}^{\prime\prime})\odot\mathcal{H}$ by the semi-inner
product given by

$\langle f_{1}\otimes h_{1}\\!\mid\\!f_{2}\otimes
h_{2}\rangle_{\ell^{2}\left(\mathcal{A}^{\prime\prime}\right)\otimes_{\rho^{\prime\prime}}\mathcal{H}}\coloneqq\langle
h_{1}\\!\mid\\!\rho^{\prime\prime}\left(\langle
f_{1}\\!\mid\\!f_{2}\rangle_{A^{\prime\prime}}\right)h_{2}\rangle_{\mathcal{H}},$

where $f_{1},f_{2}\in\ell^{2}(\mathcal{A}^{\prime\prime})$ and
$h_{1},h_{2}\in\mathcal{H}$ and $\rho^{\prime\prime}$ is the weakly continuous
extension of $\rho$ to $A_{1}^{\prime\prime}$;

  2. (ii)

the _induced representation of $\rho$_ to be the representation given by

$\displaystyle\operatorname{Ind}\rho\colon
C_{\text{max}}^{*}\left(\mathcal{A}\right)$
$\displaystyle\rightarrow\mathcal{L}\left(\ell^{2}\left(\mathcal{A}^{\prime\prime}\right)\otimes_{\rho^{\prime\prime}}\mathcal{H}\right),$
$\displaystyle a_{s}\delta_{s}$
$\displaystyle\mapsto\left(\operatorname{Ind}\rho\right)_{s}\left(a_{s}\delta_{s}\right)\coloneqq\Lambda_{s}\left(a_{s}\delta_{s}\right)\otimes_{\rho^{\prime\prime}}\textup{id}.$

Henceforth, we will denote
$\langle\cdot\\!\mid\\!\cdot\rangle_{\mathcal{H}},\langle\cdot\\!\mid\\!\cdot\rangle_{A^{\prime\prime}}$
and
$\langle\cdot\\!\mid\\!\cdot\rangle_{\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\rho^{\prime\prime}}\mathcal{H}}$
in Definition 3.8 all by $\langle\cdot\\!\mid\\!\cdot\rangle$ unless otherwise
confusing. The following remarks are in order.

######  Remark 3.9.

Notice that $\operatorname{Ind}\rho$ is the induced representation from the
weakly continuous extension $\rho^{\prime\prime}\colon
A_{1}^{\prime\prime}\to\mathcal{L}(\mathcal{H})$ via
$\ell^{2}(\mathcal{A}^{\prime\prime})$ viewed as a correspondence from
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$ to $A_{1}^{\prime\prime}$. Since
the left action of $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ on
$\ell^{2}(\mathcal{A}^{\prime\prime})$ factors through
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ by construction, the induced
representation $\operatorname{Ind}\rho$ always factors through
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$, regardless of the initial
representation $\rho\colon A_{1}\to\mathcal{L}(\mathcal{H})$. Indeed, the
reduced norm can be described via induced representations: take, e.g., the
universal representation of $A_{1}$ and induce it up to
$\mathbb{C}_{\text{alg}}\left(\mathcal{A}\right)$. The associated
representation of $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is faithful
because the universal representation of $A_{1}$ extends faithfully to
$A_{1}^{\prime\prime}$.

On the one hand a general faithful representation of $A_{1}$ need not induce a
faithful representation of $C_{\text{red}}^{*}\left(\mathcal{A}\right)$. And
on the other hand an induced representation can be faithful on
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ without being faithful on
$A_{1}^{\prime\prime}$ (or even on $A_{1}$) as
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ may be, for instance, simple.
General criteria for faithfulness of induced representations on
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ are given in [Buss-Exel-
Meyer:Reduced]*Section 4. For instance, it is proved in [Buss-Exel-
Meyer:Reduced]*Theorem 7.4 that the _atomic_ representation of $A_{1}$, i.e.,
the direct sum $\rho_{a}=\oplus_{[\pi]\in\hat{A}_{1}}\pi$ of all (classes of)
irreducible representations of $A_{1}$, yields a faithful representation
$\operatorname{Ind}\rho_{a}$ of $C_{\text{red}}^{*}\left(\mathcal{A}\right)$.

###  3.2. Fell’s absorption principle for groups

Let us now recall Fell’s absorption principle for the case of Fell bundles
over (discrete) groups (see [Exel:Partial_dynamical]*Chapter 18). Let
$\mathcal{A}=(A_{g})_{g\in G}$ be a Fell bundle over a group $G$ and let
$\pi=(\pi_{g})_{g\in G}$ be a representation of $\mathcal{A}$ on some Hilbert
space $\mathcal{H}$. Then we obtain a new representation
$\pi^{\lambda}=(\pi^{\lambda}_{g})_{g\in G}$ of $\mathcal{A}$ on
$\ell^{2}(G,\mathcal{H})\cong\ell^{2}(G)\otimes\mathcal{H}$ defined by
$\pi^{\lambda}_{g}\coloneqq\lambda_{g}\otimes\pi_{g}$ for all $g\in G$, where
$\lambda\colon G\to\mathcal{L}(\ell^{2}(G))$ denotes the left regular
representation of $G$. Observe that

$\pi^{\lambda}_{g}\left(f\right)\left(h\right)=\pi_{g}\left(f\left(g^{-1}h\right)\right),\;\;\text{for}\;\,g,h\in
G\;\,\text{and}\;\,f\in\ell^{2}\left(G,\mathcal{H}\right).$

Here $\ell^{2}(\mathcal{A})$ denotes the Hilbert $A_{1}$-module obtained as
the completion of $C_{c}(\mathcal{A})$ with respect to the norm induced by the
$A_{1}$-valued inner product

$\langle\xi\\!\mid\\!\eta\rangle_{A_{1}}\coloneqq\sum_{g\in
G}\xi\left(g\right)^{*}\eta\left(g\right).$

The so-called left regular representation of $\mathcal{A}$ is the
representation

$\Lambda\colon\mathcal{A}\to\mathcal{L}(\ell^{2}(\mathcal{A}))\quad\mbox{given
by }(\Lambda(a_{g})\xi)(h)\coloneqq a_{g}\xi(g^{-1}h).$

By definition, the reduced cross-sectional $C^{*}$-algebra
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is

$C_{\text{red}}^{*}\left(\mathcal{A}\right)\coloneqq\Lambda(C_{\text{max}}^{*}\left(\mathcal{A}\right))\subseteq\mathcal{L}(\ell^{2}(\mathcal{A})).$

The following result is an analogue of Fell’s absorption principle in the
context of Fell bundles over discrete groups (see
[Exel:Partial_dynamical]*Proposition 18.4). Before we state and prove the
result, however, we need some notation. Let $\ell^{2}_{\pi}(G,\mathcal{H})$ be
the (closed) subspace of $\ell^{2}(G,\mathcal{H})$ consisting of the functions
$\xi\in\ell^{2}(G,\mathcal{H})$ satisfying
$\xi(g)\in\mathcal{H}_{g}:=\operatorname{\overline{span}}\pi_{g}(\mathcal{A}_{g})\mathcal{H}\subseteq\mathcal{H}$.
A simple computation shows that this is an invariant subspace under
$\pi\otimes\lambda$, hence it restricts to a representation of $\mathcal{A}$
on the Hilbert space $\ell^{2}_{\pi}(G,\mathcal{H})$, which we denote by
$\pi^{\lambda}$.

######  Proposition 3.10.

Let $\mathcal{A}=(A_{g})_{g\in G}$ be a Fell bundle over a discrete group $G$
and let $\pi=(\pi_{g})_{g\in G}$ be a representation of $\mathcal{A}$. Then
the representation

$\pi^{\lambda}\colon
C_{\text{max}}^{*}\left(\mathcal{A}\right)\to\mathcal{L}\left(\ell^{2}_{\pi}\left(G,\mathcal{H}\right)\right)$

is weakly contained in the regular representation $\Lambda\colon
C_{\text{max}}^{*}\left(\mathcal{A}\right)\to
C_{\text{red}}^{*}\left(\mathcal{A}\right)$, and therefore it factors through
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$.

###### Proof.

A simple computation with the inner products shows that the canonical map

$C_{c}\left(\mathcal{A}\right)\odot\mathcal{H}\to
C_{c}\left(G,\mathcal{H}\right),\;\;\text{where}\;\,f\otimes
v\mapsto\left[g\mapsto\pi_{g}\left(f\left(g\right)\right)v\right]$

is an isometry and extends to an isomorphism

$U_{\pi}\colon\ell^{2}(\mathcal{A})\otimes_{\pi_{1}}\mathcal{H}\xrightarrow{\sim}\ell^{2}_{\pi}(G,\mathcal{H})$

intertwining the representations
$\operatorname{Ind}\pi_{1}\coloneqq\Lambda\otimes_{\pi_{1}}\textup{id}$ and
the restriction of $\pi^{\lambda}$ to $\ell^{2}_{\pi}(G,\mathcal{H})$. The
representation $\operatorname{Ind}\pi_{1}$, being induced via
$\ell^{2}(\mathcal{A})$, is always weakly contained in the regular
representation of $\mathcal{A}$. ∎

######  Remark 3.11.

We note in passing that Proposition 3.10 is _not_ actually Fell’s absorption
principle, as employed in [Exel:Partial_dynamical]*Proposition 18.4, unless
the Fell bundle $\mathcal{A}$ is assumed to be saturated (recall Remark 2.6)
and the representation $\pi$ is nondegenerate. Indeed, in such case the space
$\ell^{2}_{\pi}(G,\mathcal{H})=\ell^{2}(G,\mathcal{H})$ because
$\mathcal{H}_{g}=\operatorname{\overline{span}}\pi_{1}(A_{g}A_{g}^{*})\mathcal{H}=\mathcal{H}$
for all $g\in G$, and hence we do recover [Exel:Partial_dynamical]*Proposition
18.4. Otherwise, Proposition 3.10 only proves that a sub-representation of
$\pi\otimes\lambda$, namely $\pi^{\lambda}$, is weakly contained in $\Lambda$.
Nevertheless, we shall stick to the naming of _Fell’s absorption principle_
here and in the sequel because, in particular, the above version recovers the
usual Fell’s absorption principle for ordinary group actions, in special for
group representations, where the name comes from.

###  3.3. Fell’s absorption principle for inverse semigroups

We are going to follow a similar strategy as the one outlined in Section 3.2
and Proposition 3.10, but this time in the case of inverse semigroups,
describing induced representations of a Fell bundle. The main complication in
the more general case of inverse semigroups is the description of the regular
representation $\Lambda$ of a Fell bundle over an inverse semigroup $S$. By
construction, this representation acts not just on a completion of
$C_{c}(\mathcal{A}^{\prime\prime})$, but of a certain quotient of it. Thus it
is natural to expect that an induced representation will act not just on a
completion of a space of sections $S\to\mathcal{H}$, but on a quotient of it;
and this is what will be happening indeed (see Propositions 3.14 and 3.18 and
Theorem 3.20).

The main technical points of Fell’s absorption principle in the setting of
inverse semigroups are the key Propositions 3.14 and 3.18, which allow to
construct a unitary operator that will intertwine two representations of the
Fell bundle $\mathcal{A}$.

######  Definition 3.12.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over $S$, and let
$\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$.

  1. (i)

We define $C_{c}^{\pi}(S,\mathcal{H})\coloneqq\\{f\in C_{c}(S,\mathcal{H})\mid
f(s)\in\mathcal{H}_{s}\;\,\text{for all}\;\,s\in S\\},$ where
$\mathcal{H}_{s}\coloneqq\operatorname{\overline{span}}\pi_{s}(A_{s})\mathcal{H}\subseteq\mathcal{H}$
(recall Definition 2.34).

  2. (ii)

We denote by $U_{\pi,0}\colon C_{c}(\mathcal{A})\odot\mathcal{H}\rightarrow
C_{c}^{\pi}(S,\mathcal{H})$ the linear map defined by

$U_{\pi,0}\left(f\otimes
v\right)\left(s\right)\coloneqq\pi_{s}\left(f\left(s\right)\right)v$

for $f\in C_{c}(\mathcal{A})$ and $v\in\mathcal{H}$.

  3. (iii)

We also denote by $U^{\prime\prime}_{\pi,0}\colon
C_{c}(\mathcal{A}^{\prime\prime})\odot\mathcal{H}\rightarrow
C_{c}^{\pi}(S,\mathcal{H})$ the linear map defined in the same way by

$U_{\pi,0}^{\prime\prime}\left(f\otimes
v\right)\left(s\right)\coloneqq\pi_{s}^{\prime\prime}\left(f\left(s\right)\right)v$

$f\in C_{c}(\mathcal{A}^{\prime\prime})$ and $v\in\mathcal{H}$. Recall that
$\mathcal{H}_{s}=\operatorname{\overline{span}}\pi_{s}^{\prime\prime}(A_{s}^{\prime\prime})\mathcal{H}$,
so the above makes sense.

For notational convenience we shall henceforth denote $U_{\pi,0}\left(f\otimes
v\right)$ simply by $f_{v}$ whenever $f\in C_{c}(\mathcal{A})$ and
$v\in\mathcal{H}$. We shall use the same notation if $f\in
C_{c}(\mathcal{A}^{\prime\prime})$.

Notice that $C_{c}^{\pi}(S,\mathcal{H})$ is, as a vector space, just the
algebraic direct sum $\oplus_{s\in
S}^{\mathrm{alg}}\mathcal{H}_{s}\delta_{s}$. For technical reasons, we also
need a certain special subspace of $C_{c}^{\pi}(S,\mathcal{H})$. For this we
define:

$\displaystyle\mathcal{H}^{0}_{s}$
$\displaystyle\coloneqq\operatorname{{span}}\pi_{s}\left(A_{s}\right)\mathcal{H}\subseteq\mathcal{H}_{s};\;\,\text{and}$
$\displaystyle C_{c}^{\pi}\left(S,\mathcal{H}^{0}\right)$
$\displaystyle\coloneqq\left\\{f\in C_{c}^{\pi}\left(S,\mathcal{H}\right)\mid
f\left(s\right)\in\mathcal{H}_{s}^{0}\mbox{ for all }s\in S\right\\}.$

Observe that, following from the definitions,
$\mathcal{H}^{0}_{s}\subseteq\mathcal{H}_{s}$ is dense for all $s$, and
$C_{c}^{\pi}(S,\mathcal{H}^{0})$ is exactly the image of $U_{\pi,0}$. Hence we
may see $U_{\pi,0}$ as a surjective linear map $U_{\pi,0}\colon
C_{c}(\mathcal{A})\odot\mathcal{H}\rightarrow C_{c}^{\pi}(S,\mathcal{H}^{0})$.
Furthermore, we now wish to construct a unitary $U_{\pi}$ from $U_{\pi,0}$
between certain Hausdorff completions of its domain and target spaces (see
Proposition 3.18). For this, however, we first need to define the adequate
semi-inner product on $C_{c}^{\pi}(S,\mathcal{H})$ (see Proposition 3.14).

######  Lemma 3.13.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle, and let
$\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$ on
$\mathcal{H}$. For any given $s,t\in S$, consider

$\langle x\\!\mid\\!y\rangle_{s,t}\coloneqq\sum_{i,j}\langle
v_{i}\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(a_{i}^{*}b_{j}\delta_{s^{*}t}\right)\right)w_{j}\rangle,$

where $x=\sum_{i}\pi_{s}(a_{i})v_{i}\in\mathcal{H}_{s}^{0}$ and
$y=\sum_{j}\pi_{t}(b_{j})w_{j}\in\mathcal{H}_{t}^{0}$. Then
$\langle\cdot\\!\mid\\!\cdot\rangle_{s,t}$ is a well-defined sesquilinear form
on $C_{c}^{\pi}(S,\mathcal{H}^{0})$.

Furthermore, $\langle x\\!\mid\\!y\rangle_{s,t}=\langle
x\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)y\rangle$ for every
$x\in\mathcal{H}^{0}_{s}$ and $y\in\mathcal{H}^{0}_{t}$.

###### Proof.

First, recall that $1_{s^{*}t}$ is the unit of $I_{s^{*}t,1}^{\prime\prime}$
(see Definition 2.12) and note that this is a central projection of
$A_{1}^{\prime\prime}$. By Proposition 2.24 we have

$P\left(a_{s}^{*}b_{t}\delta_{s^{*}t}\right)=\theta_{s^{*}t,1}^{\prime\prime}\left(a_{s}^{*}b_{t}1_{s^{*}t}\right)\;\;\mbox{for
all }a_{s}\in A_{s}\;\,\mbox{and }b_{t}\in A_{t},$

where $\theta_{s^{*}t,1}^{\prime\prime}$ denotes the canonical embedding
$A_{s^{*}t}^{\prime\prime}\cdot I_{s^{*}t,1}^{\prime\prime}\hookrightarrow
A_{1}^{\prime\prime}$. Since these embeddings are compatible with
representations (see [Buss-Exel-Meyer:Reduced]*Proposition 2.9), we have

$\pi_{1}^{\prime\prime}\left(\theta_{s^{*}t,1}^{\prime\prime}\left(a_{s}^{*}b_{t}1_{s^{*}t}\right)\right)=\pi_{s^{*}t}^{\prime\prime}\left(a_{s}^{*}b_{t}1_{s^{*}t}\right)=\pi_{s}\left(a_{s}\right)^{*}\pi_{t}\left(b_{t}\right)\pi_{1}^{\prime\prime}\left(1_{s^{*}t}\right).$

Thus, by the commutation relations given in Proposition 2.17 (iii),

$\displaystyle\langle x\\!\mid\\!y\rangle_{s,t}$
$\displaystyle=\sum_{i,j}\langle
v_{i}\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(a_{i}^{*}b_{j}\delta_{s^{*}t}\right)\right)w_{j}\rangle=\sum_{i,j}\langle
v_{i}\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{s^{*}t}\right)\pi_{s}\left(a_{i}\right)^{*}\pi_{t}\left(b_{j}\right)w_{j}\rangle$
$\displaystyle=\sum_{i,j}\langle\pi_{s}\left(a_{i}\right)\pi_{1}^{\prime\prime}\left(1_{s^{*}t}\right)v_{i}\\!\mid\\!\pi_{t}\left(b_{j}\right)w_{j}\rangle=\sum_{i}\langle\pi_{s}\left(a_{i}\right)\pi_{1}^{\prime\prime}\left(1_{s^{*}t}\right)v_{i}\\!\mid\\!y\rangle$
$\displaystyle=\sum_{i}\langle\pi_{s}\left(a_{i}\right)v_{i}\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)y\rangle=\langle
x\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)y\rangle,$

as desired. The fact that $\langle\cdot\\!\mid\\!\cdot\rangle_{s,t}$ is a
sesquilinear form is now immediate. ∎

We now assemble the sesquilinear forms from Lemma 3.13 in order to get a semi-
inner product $\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$ on
$C_{c}^{\pi}(S,\mathcal{H})$, as the following key proposition shows.

######  Proposition 3.14.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle, and let
$\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$ on
$\mathcal{H}$. Then

$\langle x\\!\mid\\!y\rangle_{\pi}\coloneqq\sum_{s,t\in S}\langle
x\left(s\right)\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)y\left(t\right)\rangle,$

where $x,y\in C_{c}^{\pi}(S,\mathcal{H})$, defines a semi-inner product on
$C_{c}^{\pi}(S,\mathcal{H})$. Furthermore,

$\langle f_{v}\\!\mid\\!g_{w}\rangle_{\pi}=\langle
v\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(f^{*}\cdot
g\right)\right)w\rangle$

for all $f,g\in C_{c}(\mathcal{A})$ and $v,w\in\mathcal{H}$.

###### Proof.

By Lemma 3.13 it follows that $\langle x\\!\mid\\!y\rangle_{\pi}=\sum_{s,t\in
S}\langle x(s)\\!\mid\\!y(t)\rangle_{s,t}$. Observe that, by construction and
Lemma 3.13 again, for $f,g\in C_{c}(\mathcal{A})$ and $v,w\in\mathcal{H}$, we
have

$\displaystyle\langle f_{v}\\!\mid\\!g_{w}\rangle_{\pi}$
$\displaystyle=\sum_{s,t\in S}\langle
f_{v}\left(s\right)\\!\mid\\!g_{w}\left(t\right)\rangle_{s,t}=\sum_{s,t\in
S}\langle\pi_{s}\left(f\left(s\right)\right)v\\!\mid\\!\pi_{t}\left(g\left(t\right)\right)w\rangle_{s,t}$
$\displaystyle=\sum_{s,t\in S}\langle
v\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(f\left(s\right)^{*}g\left(t\right)\delta_{s^{*}t}\right)\right)w\rangle=\langle
v\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(f^{*}\cdot
g\right)\right)w\rangle,$

as stated in the claim. The form $\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$ is
clearly symmetric and linear, so in order to end the proof it is enough to
show it is also positive semi-definite. For that, fix any $x\in
C_{c}^{\pi}(S,\mathcal{H})$ and observe that

$\langle x\\!\mid\\!x\rangle_{\pi}=\sum_{s,t\in S}\langle
x\left(s\right)\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)x\left(t\right)\rangle=\sum_{s,t\in
S}\langle\pi_{1}^{\prime\prime}\left(1_{ts^{*}}\right)x\left(s\right)\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{st^{*}}\right)x\left(t\right)\rangle\geq
0$

by Proposition 2.17 (i). ∎

In light of Proposition 3.14, the following definition is only natural
(compare with Definition 3.2).

######  Definition 3.15.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ we denote by
_$\ell^{2}_{\pi}(S,\mathcal{H})$_ the Hausdorff completion of
$C_{c}^{\pi}(S,\mathcal{H})$ with respect to the semi-inner product
$\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$.

We briefly observe in passing that $\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$
is a semi-inner product, that is, there may be non-zero elements $x\in
C_{c}^{\pi}(S,\mathcal{H})$ such that $\langle x\\!\mid\\!x\rangle_{\pi}=0$.
Thus, $\ell^{2}_{\pi}(S,\mathcal{H})$ is only the completion of a _quotient_
of $C_{c}^{\pi}(S,\mathcal{H})$. However, the fibers $\mathcal{H}_{s}$ embed
isometrically into $\ell^{2}(S,\mathcal{H})$ (compare the following with Lemma
2.28):

######  Lemma 3.16.

For any Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ and representation $\pi$,
the canonical map $\mathcal{H}_{s}^{0}\to
C_{c}^{\pi}(S,\mathcal{H}^{0})\to\ell^{2}_{\pi}(S,\mathcal{H})$ is isometric.
In particular, it extends to an isometry
$\mathcal{H}_{s}\cong\mathcal{H}_{s}\delta_{s}\hookrightarrow\ell^{2}_{\pi}(S,\mathcal{H})$.

###### Proof.

The proof boils down to the fact that $s^{*}s$ is an idempotent, and hence $P$
acts as the identity on $A_{s^{*}s}\subseteq A_{1}$. In fact, we get

$\langle v\delta_{s}\\!\mid\\!v\delta_{s}\rangle_{\pi}=\sum_{t,r\in
S}\langle\left(v\delta_{s}\right)\left(t\right)\\!\mid\\!\pi_{1}^{\prime\prime}\left(1_{tr^{*}}\right)\left(v\delta_{s}\right)\left(r\right)\rangle=\langle
v\\!\mid\\!v\rangle$

for every $v\in\mathcal{H}_{s}$ and $s\in S$. ∎

The following is a technical point that needs to be resolved for later
reference.

######  Lemma 3.17.

The Hilbert space $\ell^{2}(S,\mathcal{H})$ is the Hausdorff completion of
$C_{c}^{\pi}(S,\mathcal{H}^{0})$ with respect to the semi-inner product
$\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$.

###### Proof.

By Definition 3.15, the Hilbert space $\ell^{2}_{\pi}(S,\mathcal{H})$ is the
Hausdorff completion of $C_{c}^{\pi}(S,\mathcal{H})$ with respect to the semi-
inner product $\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$. Thus, in order to
prove the claim it is enough to show the canonical map
$C_{c}^{\pi}(S,\mathcal{H}^{0})\subseteq
C_{c}^{\pi}(S,\mathcal{H})\to\ell^{2}_{\pi}(S,\mathcal{H})$ has dense image.
But $\mathcal{H}^{0}_{s}\subseteq\mathcal{H}_{s}$ is dense for every $s\in S$,
and by Lemma 3.16 $\mathcal{H}_{s}$ embeds isometrically into
$\ell^{2}_{\pi}(S,\mathcal{H})$, and hence so does $\mathcal{H}_{s}^{0}$.
Therefore the image of $C_{c}^{\pi}(S,\mathcal{H}^{0})$ in
$\ell^{2}_{\pi}(S,\mathcal{H})$ is dense in that of
$C_{c}^{\pi}(S,\mathcal{H})$, as desired. ∎

######  Proposition 3.18.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle, and let
$\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$ on
$\mathcal{H}$. Then the map $U_{\pi,0}^{\prime\prime}$ from Definition 3.12
induces a unitary operator
$U_{\pi}\colon\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{{}^{\prime\prime}}}\mathcal{H}\to\ell^{2}_{\pi}(S,\mathcal{H})$
making the diagram

${C_{c}\left(\mathcal{A}^{\prime\prime}\right)\odot\mathcal{H}}$${C_{c}^{\pi}\left(S,\mathcal{H}\right)}$${\ell^{2}\left(\mathcal{A}^{\prime\prime}\right)\otimes_{\pi_{1}^{{}^{\prime\prime}}}\mathcal{H}}$${\ell^{2}_{\pi}\left(S,\mathcal{H}\right),}$$\scriptstyle{U_{\pi,0}^{\prime\prime}}$$\scriptstyle{U_{\pi}}$

commute.

###### Proof.

By Proposition 3.14, notice that

$\langle f_{v}\\!\mid\\!g_{w}\rangle_{\pi}=\langle
v\\!\mid\\!\pi_{1}^{\prime\prime}\left(P\left(f^{*}\cdot
g\right)\right)w\rangle=\langle
f\otimes_{\pi_{1}^{\prime\prime}}v\\!\mid\\!g\otimes_{\pi_{1}^{\prime\prime}}w\rangle,$

that is, the semi-inner product $\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$ is
compatible with $U_{\pi,0}^{\prime\prime}\colon
C_{c}(\mathcal{A}^{\prime\prime})\odot\mathcal{H}\to
C_{c}^{\pi}(S,\mathcal{H}^{0})$. By sesquilinearity the above equation extends
to arbitrary elements of the tensor product, i.e., we get

$\langle
U_{\pi,0}^{\prime\prime}\left(x\right)\\!\mid\\!U_{\pi,0}^{\prime\prime}\left(y\right)\rangle_{\pi}=\langle
x\\!\mid\\!y\rangle\;\;\mbox{for all }x,y\in
C_{c}\left(\mathcal{A}\right)\odot\mathcal{H}.$

The above equation implies the statement. Indeed, recall (see Definitions 3.8
and 3.17) that
$\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H}$
and $\ell^{2}_{\pi}(S,\mathcal{H})$ are, respectively, the Hausdorff
completions of $C_{c}(\mathcal{A}^{\prime\prime})\odot\mathcal{H}$ w.r.t.
$\langle\cdot\\!\mid\\!\cdot\rangle_{\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H}}$
and $C_{c}^{\pi}(S,\mathcal{H})$ w.r.t.
$\langle\cdot\\!\mid\\!\cdot\rangle_{\pi}$. ∎

######  Remark 3.19.

It follows from Proposition 3.18 that the canonical map
$C_{c}(\mathcal{A})\odot\mathcal{H}\to\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H}$
has dense image because the latter is isomorphic to
$\ell^{2}_{\pi}(S,\mathcal{H})$ and the image of $U_{\pi,0}\colon
C_{c}(\mathcal{A})\odot\mathcal{H}\to C_{c}^{\pi}(S,\mathcal{H})$ is dense as
a subspace of $\ell^{2}_{\pi}(S,\mathcal{H})$.

We are now ready to provide a version of Fell’s absorption principle for the
context of inverse semigroups; this is the main result of the section. We
start with a representation $\pi$ of $\mathcal{A}$, restrict it to the
representation $\pi_{1}$ of $A_{1}=A$ and then induce it back to $\mathcal{A}$
via the regular representation $\Lambda$ on
$\ell^{2}(\mathcal{A}^{\prime\prime})$. This results in a description of the
induced representation $\Lambda\otimes_{\pi_{1}^{\prime\prime}}\textup{id}$ on
$\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H}$
in a more explicit way, identifying it with a representation on
$\ell^{2}_{\pi}(S,\mathcal{H})$, which is a quotient of a certain space of
sections $S\to\mathcal{H}$.

######  Theorem 3.20.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$, and let $\pi=(\pi_{s})_{s\in S}$ be a nondegenerate representation of
$\mathcal{A}$. Consider the induced representation
$\Lambda\otimes_{\pi_{1}^{{}^{\prime\prime}}}\textup{id}=((\Lambda\otimes_{\pi_{1}^{{}^{\prime\prime}}}\textup{id})_{s})_{s\in
S}$ given by

(3.21)
$\displaystyle(\Lambda\otimes_{\pi_{1}^{{}^{\prime\prime}}}\textup{id})_{s}\colon
A_{s}$
$\displaystyle\rightarrow\mathcal{L}\left(\ell^{2}\left(\mathcal{A}^{\prime\prime}\right)\otimes_{\pi_{1}^{{}^{\prime\prime}}}\mathcal{H}\right),$
$\displaystyle(\Lambda\otimes_{\pi_{1}^{{}^{\prime\prime}}}\textup{id})_{s}$
$\displaystyle\left(a_{s}\right)\left(f\otimes
v\right)\coloneqq\Lambda_{s}\left(a_{s}\right)\left(f\right)\otimes v,$

where $a_{s}\in A_{s},f\in\ell^{2}(\mathcal{A}^{\prime\prime})$ and
$v\in\mathcal{H}$. Then the unitary isomorphism $U_{\pi}$ from Proposition
3.18 conjugates $\Lambda\otimes_{\pi_{1}^{\prime\prime}}\textup{id}$ into the
representation $\pi^{\Lambda}=(\pi^{\Lambda}_{s})_{s\in S}$ given by

(3.22) $\displaystyle\pi^{\Lambda}_{s}\colon A_{s}$
$\displaystyle\rightarrow\mathcal{L}\left(\ell^{2}_{\pi}\left(S,\mathcal{H}\right)\right),$
$\displaystyle\pi^{\Lambda}_{s}$
$\displaystyle\left(a_{s}\delta_{s}\right)\left(v_{t}\delta_{t}\right)\coloneqq\pi_{s}\left(a_{s}\right)v_{t}\delta_{st}.$

In other words, we have
$\pi^{\Lambda}_{s}(a_{s}\delta_{s})U_{\pi}=U_{\pi}(\Lambda\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}})(a_{s}\delta_{s})$
for all $a_{s}\in A_{s}$ and $s\in S$, that is, $U_{\pi}$ intertwines
$\pi^{\Lambda}$ and
$\Lambda\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}}$.

###### Proof.

Given the machinery of Propositions 3.14 and 3.18 the proof of this result
boils down to a computation. Indeed, note that for all basic tensors
$a_{t}\delta_{t}\otimes
v\in\ell^{2}(\mathcal{A}^{\prime\prime})\otimes_{\pi_{1}^{\prime\prime}}\mathcal{H}$
(recall Lemmas 2.28 and 3.16), where $t\in S$, $a_{t}\in A_{t}$ and
$v\in\mathcal{H}$, we have

$U_{\pi}\left(\Lambda_{s}\left(a_{s}\delta_{s}\right)\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}}\right)\left(a_{t}\delta_{t}\otimes
v\right)=U_{\pi}\left(a_{s}a_{t}\delta_{st}\otimes
v\right)=\pi_{st}\left(a_{s}a_{t}\right)v\delta_{st}.$

Likewise,

$\pi^{\Lambda}_{s}\left(a_{s}\delta_{s}\right)\left(U_{\pi}\left(a_{t}\delta_{t}\otimes
v\right)\right)=\pi^{\Lambda}_{s}\left(a_{s}\delta_{s}\right)\left(\pi_{t}\left(a_{t}\right)v\delta_{t}\right)=\pi_{s}\left(a_{s}\right)\pi_{t}\left(a_{t}\right)v\delta_{st}.$

Using Remark 3.19 it follows that
$\pi^{\Lambda}_{s}(a_{s}\delta_{s})=U_{\pi}(\Lambda_{s}(a_{s}\delta_{s})\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}})U_{\pi}^{*}$,
in particular this shows that $\pi^{\Lambda}$ defined as in the statement
gives a well-defined representation of $\mathcal{A}$ since
$\Lambda\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}}$ is,
itself, a representation of $\mathcal{A}$. ∎

Our main use of Fell’s absorption principle comes from the following
consequence of Theorem 3.20.

######  Corollary 3.23.

Let $\mathcal{A}=(A_{s})_{s\in S}$ be a Fell bundle over an inverse semigroup
$S$, and let $\pi=(\pi_{s})_{s\in S}$ be a representation of $\mathcal{A}$.
Suppose that $\pi_{1}$ extends to a faithful representation of
$A^{\prime\prime}$. Then the reduced cross-sectional $C^{*}$-algebra
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is isomorphic to the
$C^{*}$-algebra

$C^{*}\left(\left\\{\pi^{\Lambda}_{s}\left(a_{s}\right)\;\mid\;a_{s}\in
A_{s}\;\text{and}\;s\in
S\right\\}\right)\subseteq\mathcal{L}\left(\ell^{2}_{\pi}\left(S,\mathcal{H}\right)\right),$

generated by the image of the representation
$\pi^{\Lambda}=(\pi^{\Lambda}_{s})_{s\in S}$ given by Theorem 3.20.

###### Proof.

By definition, $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ is the
$C^{*}$-algebra generated by the image of the regular representation
$\Lambda\colon\mathcal{A}\to\mathcal{L}(\ell^{2}(\mathcal{A}^{\prime\prime}))$,
so that $\ell^{2}(\mathcal{A}^{\prime\prime})$ may be viewed as a faithful
correspondence from $C_{\text{red}}^{*}\left(\mathcal{A}\right)$ to
$A_{1}^{\prime\prime}$, that is, it is a Hilbert module over
$A_{1}^{\prime\prime}$ with a faithful representation of
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$. Since $\pi_{1}^{\prime\prime}$
is assumed to be faithful, the induced representation
$\operatorname{Ind}\pi_{1}^{\prime\prime}=\Lambda\otimes_{\pi_{1}^{\prime\prime}}\operatorname{\mathrm{id}}$
is also faithful as a representation of
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$. And therefore so is the
equivalent representation $\pi^{\Lambda}$ by Theorem 3.20, from which the
result follows. ∎

We end the discussion about Fell’s absorption principle with the following
remarks.

######  Remark 3.24.

Notice that, in Corollary 3.23, with the same proof we may actually only
assume that $\pi_{1}$ extends to a faithful representation of the
$C^{*}$-algebra generated by $P(C_{c}(\mathcal{A}))\subseteq
A_{1}^{\prime\prime}$.

######  Remark 3.25.

Observe that if $S$ is a group then all the discussion above reduces to Fell’s
absorption principle for discrete groups in the sense of Section 3.2. In
particular, in such case Theorem 3.20 reduces to Proposition 3.10.

##  4\. The approximation property

The present section introduces the central notion of the paper, namely the
_approximation property_ of a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$. The
approach here taken generalizes the approximation property for Fell bundles
over discrete groups, which was first defined by Exel in [Exel:Amenability] as
follows.

######  Definition 4.1.

A Fell bundle $\mathcal{A}=(A_{s})_{s\in G}$ over a discrete group $G$ has the
approximation property if there is a net $\\{\xi_{i}\colon G\to A_{1}\\}_{i\in
I}$ of finitely supported sections such that

  1. (i)

$\sup_{i\in I}\|\sum_{s\in G}\xi_{i}(s)^{*}\xi_{i}(s)\|<\infty$;

  2. (ii)

and for every $s\in G$ and $a_{s}\in A_{s}$

$\sum_{t\in G}\xi_{i}(st)^{*}a_{s}\xi_{i}(t)\xrightarrow{i}a_{s}$

in norm.

######  Remark 4.2.

Observe that assertion Definition 4.1 (i) is actually stating that
$\\{\xi_{i}\\}_{i\in I}$ is a bounded net in the Hilbert $A_{1}$-module
$\ell^{2}(G,A_{1})$, since $\|\xi_{i}\|_{2}^{2}=\|\sum_{s\in
G}\xi_{i}(s)^{*}\xi_{i}(s)\|$.

We use the word _section_ in Definition 4.1 to refer to functions $\xi\colon
G\to A_{1}$. We do this because, in fact, in the setting of inverse semigroups
the functions $\xi\colon S\to A_{1}$ will be honest sections of a certain
bundle (see Definition 4.4). The following notation will, hence, be useful
throughout the rest of the text.

######  Definition 4.3.

Given a Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ over $S$, we denote by
$\Gamma_{c}(S,\mathcal{A})$ the space of finitely supported sections of the
bundle $(A_{ss^{*}})_{s\in S}$, that is, functions $\xi\colon S\to A_{1}$ such
that $\xi(s)\in A_{ss^{*}}$ for every $s\in S$.

Working, as we are, in the setting of Fell bundles over inverse semigroups
makes the definition of the approximation property a little more subtle. For
it, recall that $I_{s,1}\mathrel{\triangleleft}A_{1}$ is a two-sided closed
ideal, and that $1_{s}$ denotes the unit of the von Neumann algebra
$I_{s,1}^{\prime\prime}$ (recall Definitions 2.12 and 2.17).

######  Definition 4.4.

A Fell bundle $\mathcal{A}=(A_{s})_{s\in S}$ over an inverse semigroup with
unit $1\in S$ has the approximation property if there is a net
$\\{\xi_{i}\colon S\rightarrow A_{1}\\}_{i\in
I}\subseteq\Gamma_{c}(S,\mathcal{A})$ of finitely supported sections such that

  1. (i)

$\sup_{i\in I}\|\sum_{p,t\in S}\xi_{i}(p)^{*}\xi_{i}(t)1_{pt^{*}}\|<\infty$;

  2. (ii)

and for every $s\in S$ and $a_{s}\in A_{s}$

$\sum_{p,t\in
S}1_{p\left(st\right)^{*}}\xi_{i}\left(p\right)^{*}a_{s}\xi_{i}\left(t\right)\to
a_{s}$

in norm.

We note in passing that assertion (i) in Definition 4.4 states that
$\\{\xi_{i}\\}_{i\in I}$ is bounded in _some_ $\ell^{2}$-space of sections
$\ell^{2}_{\Gamma}(S,\mathcal{A})$ (as is the case for groups, recall Remark
4.2). Nevertheless, we will not introduce such a space as we do not need it in
this paper. The following remarks are in order.

######  Remark 4.5.

Note that, by Proposition 2.17, the element $1_{s}\in I_{s,1}^{\prime\prime}$
is central in $A_{1}^{\prime\prime}$, and hence, as $\xi(s)\in
A_{ss^{*}}\subseteq A_{1}^{\prime\prime}$, we have
$1_{p(st)^{*}}\xi(p)=\xi(p)1_{p(st)^{*}}$. This fact will be used virtually
without mention in the sequel.

######  Remark 4.6.

Observe that Definition 4.4 does, in fact, generalize Definition 4.1. Indeed,
if $S=G$ is a group then it is routine to show that $1_{s^{-1}t}=0$ unless
$s=t$, in which case $1_{s^{-1}t}=1\in A_{1}^{\prime\prime}$. Therefore the
double sums in Definition 4.4 simplify to simple sums, and we recover
Definition 4.1 (recall that we are considering non-saturated Fell bundles, see
Remark 2.6).

######  Remark 4.7.

Definition 4.4 indicates the expected behaviour of the inverse semigroup as
_bisections_ over a groupoid. Indeed, the _local_ behaviour of the groupoid
might be implemented by different bisections, and hence the nuclearity of
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ might be witnessed over possibly
different choices of elements $s\in S$. Definition 4.4 fixes these choices, by
summing over all possible labels $p,t\in S$ and then restricting to the
appropriate ideals, namely $I_{p,t}^{\prime\prime}$ and
$I_{st,p}^{\prime\prime}$ in (i) and (ii) respectively.

We end the section with the following simple examples.

######  Example 4.8.

Let $\alpha\colon G\curvearrowright X$ be an action of a discrete group $G$ on
a compact Hausdorff space $X$. Then we may construct a canonical Fell bundle
$\mathcal{A}_{\alpha}=(\mathcal{A}_{\alpha,g})_{g\in G}$ by setting
$\mathcal{A}_{\alpha,g}\coloneqq C(X)\times\\{g\\}$. The multiplication and
involution maps of $\mathcal{A}_{\alpha}$ are given by

$\left(f_{1},g_{1}\right)\cdot\left(f_{2},g_{2}\right)\coloneqq\left(f_{1}\alpha_{g_{1}}\left(f_{2}\right),g_{1}g_{2}\right),\quad\left(f_{1},g_{1}\right)^{*}\coloneqq\left(\alpha_{g_{1}^{-1}}(f_{1}^{*}),g_{1}^{-1}\right).$

It is not hard to see that $\mathcal{A}_{\alpha}$ is then a Fell bundle over
$G$, and that $C_{\text{red}}^{*}\left(\mathcal{A}_{\alpha}\right)\cong
C(X)\rtimes_{\text{red}}G$ and
$C_{\text{max}}^{*}\left(\mathcal{A}_{\alpha}\right)\cong
C(X)\rtimes_{\text{max}}G$. With this notation, it is also routine to show
that $\mathcal{A}_{\alpha}$ has the approximation property, in the sense of
Definition 4.4, if, and only if, $\alpha$ is amenable (as an action, see
[Brown-Ozawa:Approximations]*Definition 4.3.5). In fact, if $m_{i}\colon
X\to\text{Prob}\left(G\right)$ witnesses the amenability of $\alpha$ in the
sense of [Brown-Ozawa:Approximations]*Definition 4.3.5, then
$\xi_{i}(g)\coloneqq[x\mapsto(m_{i}^{x}(g))^{1/2}]$ witnesses the
approximation property of $\mathcal{A}_{\alpha}$ in the sense of Definition
4.4. Likewise, given $\xi_{i}$ witnessing the approximation property, the maps
$m_{i}^{x}(g)\coloneqq\xi_{i}(g)(x)^{2}$ witness the amenability of $\alpha$
(see Theorem 4.16 below for a similar argument).

######  Example 4.9.

Let $G$ be a discrete group, and let $S\coloneqq G\sqcup\\{0\\}$ be the
inverse semigroup obtained from $G$ by adjoining a $0$ element. Consider the
Banach spaces $A_{g}\coloneqq C[0,1]$ whenever $g\in G$ and $A_{0}\coloneqq
C(0,1]$. It is shown in [Kwaniewski2019EssentialCP]*Example 4.7 that
$\mathcal{A}=(A_{s})_{s\in S}$ forms a Fell bundle over $S$. Moreover, it is
also mentioned that both $C_{\text{max}}^{*}\left(\mathcal{A}\right)$ and
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ are $C[0,1]$-algebras with
trivial fibers at $x\in(0,1]$. Nevertheless, their fiber at $0$ may differ.
Indeed, if $G$ is not amenable then
$C_{\text{max}}^{*}\left(\mathcal{A}\right)$ has fiber
$C_{\text{max}}^{*}\left(G\right)$ at $0$, whereas
$C_{\text{red}}^{*}\left(\mathcal{A}\right)$ has fiber $\mathbb{C}\oplus
C_{\text{red}}^{*}\left(G\right)$. On the contrary, if $G$ is amenable then
both have $C_{\text{red}}^{*}\left(G\right)$ as fiber at $0$, and hence
$C_{\text{max}}^{*}\left(\mathcal{A}\right)\cong
C_{\text{red}}^{*}\left(\mathcal{A}\right)$ via the left regular
representation, that is, $\mathcal{A}$ has the weak containment property. In
fact, it is not hard to show that $\mathcal{A}$ has the approximation property
if, and only if, $G$ is amenable (compare with Theorems 6.2 and 6.12 below).
Indeed, if $\\{F_{n}\\}_{n}$ is a Følner net for $G$ and $\\{u_{i}\\}_{i\in
I}\subseteq C_{0}(0,1]$ is any approximate unit, then we can define

$\xi_{i,n}\left(0\right)\coloneqq
u_{i},\;\;\text{and}\;\;\xi_{i,n}\left(g\right)\coloneqq\left\\{\begin{array}[]{rl}\frac{1}{\left|F_{n}\right|^{1/2}}1_{C\left[0,1\right]}&\text{if}\;g\in
F_{n},\\\ 0&\text{otherwise.}\end{array}\right.$

It then follows from a computation that the net $\\{\xi_{i,n}\\}_{i,n}$
witnesses the approximation property for $\mathcal{A}$. Likewise, by
[Kwaniewski2019EssentialCP]*Example 4.7, if $G$ is non-amenable then
$C_{\text{max}}^{*}\left(\mathcal{A}\right)\not\cong
C_{\text{red}}^{*}\left(\mathcal{A}\right)$. By Theorem 6.2 below it follows
that $\mathcal{A}$ does not have the approximation property.

###  4.1. The approximation property for groupoids and inverse semigroups

This section builds upon Examples 4.8 and 4.9 above, and deals with the case
of Fell bundles over inverse semigroups that arise from Fell bundles over
groupoids. Indeed, it is well known, and discussed below, that Fell bundles
$\mathcal{A}=(A_{s})_{s\in S}$ over inverse semigroups generalize in a
canonical way Fell bundles over (possibly non-Hausdorff) locally compact,
étale groupoids. Recently, Kranz [Kranz:AP-groupooids-Fell-bundles] defined
the _approximation property_ for Fell bundles over (second countable)
_Hausdorff_ étale groupoids as in Definition 4.10. This section has the goal
of proving that our approach generalizes that of [Kranz:AP-groupooids-Fell-
bundles] in a canonical way (see Theorem 4.16). In fact, we will not need our
groupoids to be Hausdorff or second countable, nor will we need the Fell
bundle to be separable or saturated, as is assumed in the main results from
[Kranz:AP-groupooids-Fell-bundles].

We first recall that, given a locally compact groupoid $G$, a set $u\subseteq
G$ is a _bisection_ if the range and source maps $r,s\colon u\to G^{0}$ are
homeomorphisms. Likewise, recall that $G$ is called _étale_ if its topology
admits a basis formed solely by open bisections. Moreover, the set
$\operatorname{\mathrm{Bis}}(G)$ of open bisections is well known to be an
inverse semigroup. For instance, given $u,v\in\operatorname{\mathrm{Bis}}(G)$
their product is defined to be

$u\cdot v\coloneqq\left\\{gh\;\mid\;g\in u,\,h\in
v\;\text{and}\;s\left(g\right)=r\left(h\right)\right\\}.$

For the sequel, we refer the reader to [Kranz:AP-groupooids-Fell-bundles] for
the appropriate definition of a Fell bundle over a groupoid, along with
several examples of Fell bundles over locally compact Hausdorff étale
groupoids. Moreover, recall that in the non-Hausdorff case
$C_{c}(\mathcal{A})$ denotes the space spanned by sections $G\to\mathcal{A}$
that are compactly supported and continuous in a bisection of $G$. Moreover,
one may define $C_{c}(G,r^{*}(\mathcal{A}^{0}))$ in a similar vein. If $G$ is
not Hausdorff, these sections are generally not globally continuous, but they
are always Borel measurable functions. In any case, it makes sense to talk
about pointwise or uniform convergence of these sections, as is done in the
following.

######  Definition 4.10.

Let $\mathcal{A}=(A_{g})_{g\in G}$ be a Fell bundle over a locally compact
étale groupoid $G$ with Hausdorff unit space $G^{0}$. We say that
$\mathcal{A}$ has the _approximation property_ if there is a net
$\\{\xi_{i}\\}_{i\in I}\subseteq C_{c}(G,r^{*}(\mathcal{A}^{0}))$ such that

  1. (i)

$\sup_{i\in I}\sup_{x\in G^{0}}\|\sum_{g\in
G^{x}}\xi_{i}(g)^{*}\xi_{i}(g)\|<\infty$;

  2. (ii)

and for every $f\in C_{c}(\mathcal{A})$ the net

$f_{i}\colon G\to\mathcal{A},\;\;g\mapsto\sum_{h\in
G^{s(g)}}\xi_{i}\left(gh\right)^{*}f\left(g\right)\xi_{i}\left(h\right)=\sum_{h\in
G^{r(g)}}\xi_{i}\left(h\right)^{*}f\left(g\right)\xi_{i}\left(g^{-1}h\right)$

converges to $f$ uniformly over compact sets of $G$.

The following definition first appeared in
[BussExel:Fell.Bundle.and.Twisted.Groupoids]*Definition 2.12, but was based on
the previous [Exel:Inverse_combinatorial]*Proposition 5.4. It precisely
captures those inverse subsemigroups
$S\subseteq\operatorname{\mathrm{Bis}}(G)$ that retain all the topological
information of $G$.

######  Definition 4.11.

We say an inverse semigroup $S\subseteq\operatorname{\mathrm{Bis}}(G)$ is
_wide_ if

  1. (i)

$S$ covers $G$, that is, $\cup_{u\in S}u=G$; and

  2. (ii)

for any $g\in G$, $u,v\in S$, if $g\in u\cap v$ then there is some $w\in S$
such that $g\in w\subseteq u\cap v$.

Note that if $S\subseteq\operatorname{\mathrm{Bis}}(G)$ then $S$ naturally
acts on $G^{0}$. Indeed, given $u\in S$ and $x\in G^{0}$, the action of $u$ on
$x$ is only defined if $s^{-1}(x)\cap u\neq\emptyset$, i.e., there is some
$g\in u$ with $s(g)=x$. In such a case, $ux=r(g)$, as $g\in u$ is then
uniquely determined, since $u$ is assumed to be a bisection. Therefore, if
$S\subseteq\operatorname{\mathrm{Bis}}(G)$ we may consider the _groupoid of
germs $G^{0}\rtimes S$_ of the natural action of $S$ on $G^{0}$, that is, the
set of equivalence classes $[s,x]$, where $s\in S$ and $s$ is defined at $x$,
where $[s,x]=[t,y]$ if, and only if, $x=y$ and there is some idempotent $e\in
E$ such that $x\in e$ and $se=te$. The groupoid structure of $X\rtimes S$
comes from the multiplication and involution maps

$\left[s,ty\right]\cdot\left[t,y\right]\coloneqq\left[st,y\right]\;\;\text{and}\;\;\left[s,x\right]^{-1}\coloneqq\left[s^{*},sx\right].$

We then call $[s,x]\in X\rtimes S$ the _germ of $s$ at $x$_. Note that
$G^{0}\rtimes S$ is an étale locally compact groupoid, and that $s$ defines a
bisection of $G^{0}\rtimes S$. It follows that $[s,x]$ is the unique arrow in
the usual groupoid of germs $G^{0}\rtimes S$ contained in the bisection
defined by $s$ and with source $x$.

######  Proposition 4.12.

Let $G$ be a locally compact étale groupoid, and let
$S\subseteq\operatorname{\mathrm{Bis}}(G)$. The following conditions are
equivalent:

  1. (i)

$G^{0}\rtimes S\cong G$ as topological groupoids;

  2. (ii)

$S$ is wide.

In particular, $G^{0}\rtimes\operatorname{\mathrm{Bis}}(G)\cong G$.

###### Proof.

The proof this is mostly given in [Exel:Inverse_combinatorial]*Proposition
5.4, see [Kwaniewski2019EssentialCP]*Proposition 2.2 for a complete proof. ∎

We now turn our attention to Theorem 4.16, which shows that our approximation
property for Fell bundles over inverse semigroups is, at least, as general as
their counterparts over étale groupoids. In order to prove it, however, we
first need to set some notation and prove some preliminary results. Let $G$ be
a locally compact étale groupoid, and fix any wide
$S\subseteq\operatorname{\mathrm{Bis}}(G)$. Then, given a Fell bundle
$\mathcal{A}=(A_{g})_{g\in G}$ over $G$ we can construct a Fell bundle
$\mathcal{B}=(B_{u})_{u\in S}$, where $B_{u}$ is the Banach space of
$C_{0}$-sections $u\to\mathcal{A}$, i.e., continuous functions $f\colon
u\to\mathcal{A}$ that vanish at infinity and such that $f(g)\in A_{g}$ for
every $g\in u$; see [BussExel:Fell.Bundle.and.Twisted.Groupoids]*Example 2.11
for further details on this construction.

######  Lemma 4.13.

Let $G$ be a locally compact étale groupoid, and let
$S\subseteq\operatorname{\mathrm{Bis}}(G)$ be wide. With notations as above,
given any $\eta\in C_{c}(G,r^{*}(\mathcal{A}^{0}))$ there are finitely many
bisections $\\{u_{j}\\}_{j=1}^{n}\subseteq S$ and
$\\{\eta_{j}\\}_{j=1}^{n}\subseteq C_{c}(u_{j},r^{*}(\mathcal{A}^{0}))$ such
that $\eta=\eta_{1}+\dots+\eta_{n}$.

###### Proof.

Recall that $C_{c}(G,r^{*}(\mathcal{A}^{0}))$ is, by definition, the set of
sections $G\rightarrow\mathcal{A}^{0}$ that can be written as linear
combinations of functions that are compactly supported in a bisection.
Therefore, there are finitely many bisections
$\\{v_{i}\\}_{i=1}^{m}\subseteq\operatorname{\mathrm{Bis}}(G)$ and
$\\{\eta_{i}\\}_{i=1}^{m}\subseteq C_{c}(v_{i},r^{*}(\mathcal{A}^{0}))$ such
that $\eta=\eta_{1}+\dots+\eta_{m}$. The statement will follow once we prove
that the bisections $v_{i}$ may be taken to be in $S$.

Since $S\subseteq\operatorname{\mathrm{Bis}}(G)$ is wide we have, for every
$i$,

$K_{i}\coloneqq\operatorname{\mathrm{supp}}\left(\eta_{i}\right)\subseteq
v_{i}\subseteq G=\cup_{v\in S}v.$

By hypothesis $K_{i}$ is compact, and hence we may choose a finite subcover,
i.e., finitely many bisections $\\{u_{i,\ell}\\}^{n_{i}}_{\ell=1}\subseteq S$
such that $K_{i}\subseteq\cup_{\ell=1}^{n_{i}}u_{i,\ell}$. Now take any
partition of unity $\\{\varphi_{i,\ell}\\}^{n_{i}}_{\ell=1}$ subject to the
cover $\\{u_{i,\ell}\\}^{n_{i}}_{\ell=1}$, and let
$\eta_{i,\ell}\coloneqq\eta_{i}\varphi_{i,\ell}$. Then
$\\{\eta_{i,\ell}\\}_{i,\ell}$ and $\\{u_{i,\ell}\\}_{i,\ell}$ give the
desired realization of $\eta$ as a linear combination of functions compactly
supported in elements of $S$. ∎

We highlight the following result, which details the (expected) relation
between the projections $1_{s}$ in Definition 4.4 and the germ relation. For
the following and in the sequel, we shall take an étale groupoid $G$, fix a
wide inverse semigroup $S\subseteq\operatorname{\mathrm{Bis}}(G)$ and identify
$G$ with the groupoid of germs $G\cong G^{0}\rtimes S$ as in Proposition 4.12.
We shall also tacitly use the convention that whenever a germ $[s,x]$ is
written it is already implied that $s$ is defined at $x$, so that $s\cdot x$
makes sense.

######  Lemma 4.14.

Let $G$ be a locally compact étale groupoid and let
$S\subseteq\operatorname{\mathrm{Bis}}(G)$ be wide. Given a Fell bundle
$\mathcal{A}=(A_{g})_{g\in G}$ let $\mathcal{B}=(B_{u})_{u\in S}$ be
constructed as above. For any $p,t\in S$ and $x\in G^{0}$, the following
assertions are equivalent:

  1. (i)

$1_{p^{*}t}(x)=1_{A_{x}^{\prime\prime}}\in\mathcal{M}\left(A_{x}\right)\subseteq
A_{x}^{\prime\prime}$.

  2. (ii)

There is an idempotent $e\in E$ such that $x\in e\leq p^{*}pt^{*}t$ and
$pe=te$.

  3. (iii)

$[p,x]=[t,x]$.

###### Proof.

Before we start the proof, we need to explain why the point evaluations in (i)
make sense, that is, we need to explain why it makes sense to evaluate
$1_{s}(x)$ for $s\in S$. Recall that $1_{s}$ is the unit of the multiplier
$C^{*}$-algebra $\mathcal{M}(I_{s,1})\subseteq I_{s,1}^{\prime\prime}$, and
$I_{s,1}$ is a certain ideal of $B_{1}$ which, in turn, is the $C^{*}$-algebra
of $C_{0}$-sections of the upper semi-continuous $C^{*}$-bundle $(A_{x})_{x\in
G^{0}}$. Hence $I_{s,1}$ also admits such a description as an upper semi-
continuous bundle. In fact, one can see that $I_{s,1}\cong
C_{0}(O_{s,1},(A_{x})_{x\in O_{s,1}})$ for the open set $O_{s,1}=\cup_{e\leq
s,1}e\subseteq G^{0}$. Note, however, we do not actually need such description
in this proof. We may then describe the multiplier $C^{*}$-algebra
$\mathcal{M}(I_{1,s})$ as the $C^{*}$-algebra of sections $f\colon
O_{s,1}\to(\mathcal{M}(A_{x}))_{x\in O_{s,1}}$ satisfying $f\cdot g\in
C_{0}(O_{s,1},(A_{x})_{x\in O_{s,1}})$ for all $g\in
C_{0}(O_{s,1},(A_{x})_{x\in O_{s,1}})$.

We now proceed with the actual proof. First note that, by construction, either
$1_{p^{*}t}(x)=1_{A_{x}^{\prime\prime}}$ or $1_{p^{*}t}(x)=0$. Moreover,
observe that if $1_{p^{*}t}(x)\neq 0$ then, by construction there is some
$e\in E$ such that $x\in e$ and $e\leq p^{*}t$. Thus,
$p^{*}pe=p^{*}pp^{*}te=p^{*}te=e$. Likewise, $et^{*}t=p^{*}tt^{*}te=e$, and
hence $x\in e\leq p^{*}pt^{*}t$. Moreover, recall that, in general, $a\leq b$
if, and only if, $a^{*}\leq b^{*}$. Hence $e\leq(p^{*}t)^{*}=t^{*}p$, and

$pe=pp^{*}tt^{*}te=t\left(t^{*}pp^{*}t\right)e=t\left(\left(p^{*}t\right)^{*}p^{*}t\right)e=te,$

which proves that (i) implies (ii). For the reciprocal, let $e\in E$ witness
(ii), and observe that then $p^{*}te=p^{*}pe=e$, and therefore
$1_{p^{*}t}(x)=1_{e}(x)=1_{A_{x}^{\prime\prime}}$, as claimed. The equivalence
between (ii) and (iii) is well known (and routine to check), so we omit its
proof. ∎

######  Remark 4.15.

It follows from Lemma 4.14 that for $1_{p^{*}t}(x)\neq 0$ it is _not_ enough
that $px=tx$ (recall that $S$ canonically acts on $G^{0}$).

We can now prove the approach taken in this paper generalizes that of
[Kranz:AP-groupooids-Fell-bundles] to étale groupoids that may be non-
Hausdorff.

######  Theorem 4.16.

Let $G$ be a locally compact étale groupoid with Hausdorff unit space $G^{0}$,
and let $S\subseteq\operatorname{\mathrm{Bis}}(G)$ be wide. If $\mathcal{A}$
is a Fell bundle over $G$ and $\mathcal{B}$ is the corresponding Fell bundle
over $S$, then $\mathcal{A}$ has the approximation property if, and only if,
$\mathcal{B}$ has the approximation property.

###### Proof.

First, let $\eta_{i}\in C_{c}(G,r^{*}(\mathcal{A}^{0}))$ witness the
approximation property for $\mathcal{A}=(A_{g})_{g\in G}$. By Lemma 4.13 for
every $i\in I$ there are $\\{\eta_{i,j}\\}_{j=1}^{n_{i}}\subseteq
C_{c}(u_{i,j},r^{*}(\mathcal{A}^{0}))$ where $u_{i,j}\in S$ and
$\eta_{i}=\eta_{i,1}+\dots+\eta_{i,n_{i}}$. For convenience, let
$K_{i,j}\coloneqq\operatorname{\mathrm{supp}}(\eta_{i,j})$, and note that
$K_{i,j}$ is compact. In this context, consider

$\xi_{i}\colon S\rightarrow
B_{1},\;\;\text{where}\;\,\xi_{i}\left(v\right)\left(r\left(g\right)\right)\coloneqq\left\\{\begin{array}[]{rl}\eta_{i,j}\left(g\right)&\text{if}\;\;g\in
v=u_{i,j},\\\ 0&\text{otherwise.}\end{array}\right.$

We claim that the net $\\{\xi_{i}\\}_{i\in I}$ witnesses the approximation
property of $\mathcal{B}$. First of all, note that $\xi_{i}$ is finitely
supported as a section $S\rightarrow B_{1}$, as
$\operatorname{\mathrm{supp}}{(\xi)}\subseteq\\{u_{i,1},\dots,u_{i,n_{i}}\\}\subseteq
S$. Furthermore, for every $g\in G$

(4.17) $\left(\sum_{g\in s\in
S}\xi_{i}\left(s\right)\right)\left(r\left(g\right)\right)=\sum_{\begin{subarray}{c}j=1\\\
g\in
u_{i,j}\end{subarray}}^{n_{i}}\xi_{i}\left(u_{i,j}\right)\left(r\left(g\right)\right)=\sum_{j=1}^{n_{i}}\eta_{i,j}\left(g\right)=\eta_{i}\left(g\right)$

by construction. Moreover, fixing $i$ and $j\in\\{1,\dots,n_{i}\\}$, and
putting $v\coloneqq u_{i,j}$, it follows that
$\operatorname{\mathrm{supp}}(\xi_{i}(v))\subseteq vv^{*}$ is compact. Indeed,
the range map $r$ restricted to $v$ is a homeomorphism between

$\displaystyle v\supset\operatorname{\mathrm{supp}}{\left(\eta_{i,j}\right)}=$
$\displaystyle\left\\{g\in v\;\colon\;\eta_{i,j}\left(g\right)\neq
0\right\\}\xrightarrow{r|_{v}}$ $\displaystyle\left\\{r\left(g\right)\in
vv^{*}\;\colon\;\eta_{i,j}\left(g\right)\neq
0\right\\}=\operatorname{\mathrm{supp}}{\left(\xi_{i}\left(v\right)\right)}\subset
vv^{*},$

and therefore $\xi_{i}(u)\in B_{uu^{*}}$ and
$\xi_{i}\in\Gamma_{c}(S,\mathcal{B})$, as desired. We will now check that
conditions (i) and (ii) in Definition 4.4 are also met. For (i), recall that
$S\subseteq\operatorname{\mathrm{Bis}}(G)$ is wide by assumption, and hence
$G^{0}\rtimes S\cong G$ by Proposition 4.12. We will use this fact to identify
a germ $[s,x]\in G^{0}\rtimes S$ as an arrow $[s,x]\in G$. In such fashion,
recall Lemma 4.14 and observe that

$\displaystyle\sup_{i\in I}$ $\displaystyle\Bigg{\|}\sum_{p,t\in
S}\xi_{i}\left(p\right)^{*}\xi_{i}\left(t\right)1_{pt^{*}}\Bigg{\|}=\sup_{i\in
I}\sup_{x\in G^{0}}\Bigg{\|}\sum_{p,t\in
S}\left(\xi_{i}\left(p\right)^{*}\xi_{i}\left(t\right)1_{pt^{*}}\right)\left(x\right)\Bigg{\|}\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in
G^{0}}\Bigg{\|}\sum_{\begin{subarray}{c}p,t\in S\\\
\left[p^{*},x\right]=\left[t^{*},x\right]\end{subarray}}\xi_{i}\left(p\right)\left(x\right)^{*}\xi_{i}\left(t\right)\left(x\right)\Bigg{\|}\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in
G^{0}}\Bigg{\|}\sum_{\begin{subarray}{c}l,j=1\\\
\left[u_{i,j}^{*},x\right]=\left[u_{i,l}^{*},x\right]\end{subarray}}^{n_{i}}\xi_{i}\left(u_{i,j}\right)\left(x\right)^{*}\xi_{i}\left(u_{i,l}\right)\left(x\right)\Bigg{\|}\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in G^{0}}\Bigg{\|}\sum_{g\in
G^{x}}\sum_{\begin{subarray}{c}l,j=1\\\ g\in u_{i,j}\cap
u_{i,l}\end{subarray}}^{n_{i}}\xi_{i}\left(u_{i,j}\right)\left(r\left(g\right)\right)^{*}\xi_{i}\left(u_{i,l}\right)\left(r\left(g\right)\right)\Bigg{\|}\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in G^{0}}\Bigg{\|}\sum_{g\in
G^{x}}\sum_{\begin{subarray}{c}l,j=1\\\ g\in u_{i,j}\cap
u_{i,l}\end{subarray}}^{n_{i}}\eta_{i,j}\left(g\right)^{*}\eta_{i,l}\left(g\right)\Bigg{\|}\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in G^{0}}\Bigg{\|}\sum_{g\in
G^{x}}\eta_{i}\left(g\right)^{*}\eta_{i}\left(g\right)\Bigg{\|}<\infty$

by the assumptions on $\\{\eta_{i}\\}_{i\in I}$ and Equation 4.17 (for the
previous-to-last equality). The invariance condition (ii) is proved similarly.
Indeed, fix any section $b_{s}\in B_{s}$ and arrow $[s,x]\in s\subseteq G$,
and observe that

$\displaystyle\Bigg{(}\sum_{p,t\in S}1_{p\left(st\right)^{*}}$
$\displaystyle\xi_{i}\left(p\right)^{*}b_{s}\xi_{i}\left(t\right)\Bigg{)}\left(\left[s,x\right]\right)=\sum_{\begin{subarray}{c}p,t\in
S\\\
\left[p^{*},sx\right]=\left[(st)^{*},sx\right]\end{subarray}}\xi_{i}\left(p\right)\left(sx\right)^{*}b_{s}\left(\left[s,x\right]\right)\xi_{i}\left(t\right)\left(x\right)\allowdisplaybreaks$
$\displaystyle=\sum_{\begin{subarray}{c}l,j=1\\\
\left[u_{i,j}^{*},sx\right]=\left[(su_{i,l})^{*},sx\right]\end{subarray}}^{n_{i}}\xi_{i}\left(u_{i,j}\right)\left(sx\right)^{*}b_{s}\left(\left[s,x\right]\right)\xi_{i}\left(u_{i,l}\right)\left(x\right)\allowdisplaybreaks$
$\displaystyle=\sum_{\begin{subarray}{c}l,j=1\\\
\left[u_{i,j}^{*},sx\right]=\left[(su_{i,l})^{*},sx\right]\end{subarray}}^{n_{i}}\eta_{i,j}\left(\left[u_{i,j},u_{i,j}^{*}sx\right]\right)^{*}b_{s}\left(\left[s,x\right]\right)\eta_{i,l}\left(\left[u_{i,l},u_{i,l}^{*}x\right]\right)\allowdisplaybreaks$
$\displaystyle=\sum_{h\in
G^{x}}\sum_{l,j=1}^{n_{i}}\eta_{i,j}\left(\left[s,x\right]h\right)^{*}b_{s}\left(\left[s,x\right]\right)\eta_{i,l}\left(h\right)\allowdisplaybreaks$
$\displaystyle=\sum_{h\in
G^{x}}\eta_{i}\left(\left[s,x\right]h\right)^{*}b_{s}\left(\left[s,x\right]\right)\eta_{i}\left(h\right).$

Therefore, by Equation 4.17 and the hypothesis on $\\{\eta_{i}\\}_{i\in I}$,
it follows that for every compact set $K\subseteq G$

$\displaystyle\sup_{\left[s,x\right]\in K}$
$\displaystyle\left\|\Bigg{(}\sum_{p,t\in
S}1_{p\left(st\right)^{*}}\xi_{i}\left(p\right)^{*}b_{s}\xi_{i}\left(t\right)-b_{s}\Bigg{)}\left(\left[s,x\right]\right)\right\|$
$\displaystyle=\sup_{\left[s,x\right]\in K}\left\|\sum_{h\in
G^{x}}\eta_{i}\left(\left[s,x\right]h\right)^{*}b_{s}\left(\left[s,x\right]\right)\eta_{i}\left(h\right)-b_{s}\left(\left[s,x\right]\right)\right\|\xrightarrow{i}0,$

which proves assertion (ii), as desired.

We now turn to the proof that if $\mathcal{B}$ has the approximation property
then so does $\mathcal{A}$. To this end, for every idempotent $e\in E$ let
$\\{\varphi_{e,j}\\}_{j}\subseteq C_{0}(e,\mathcal{A}^{0})$ be a quasi-central
approximate unit such that $\varphi_{e,j}$ has compact support (in $e\subseteq
G^{0}$) and such that $\|\varphi_{e,j}\|\leq 1$. In addition, fix a net
$\\{\xi_{i}\\}_{i\in I}$ witnessing the approximation property of
$\mathcal{B}$ and positive numbers $\varepsilon_{i}\in(0,1)$ such that
$\delta_{i}\coloneqq\varepsilon_{i}\cdot|\operatorname{\mathrm{supp}}(\xi_{i})|^{2}\leq
1$ and $\delta_{i}\to 0$ when $i$ grows.222 For instance, if $I$ was countable
then
$\varepsilon_{i}\coloneqq(i\cdot|\operatorname{\mathrm{supp}}(\xi_{i})|)^{-2}$
would suffice. In this context, consider the net $\\{\eta_{i,j}\\}_{i,j}$
given by

$\eta_{i,j}\left(g\right)\coloneqq\sum_{\begin{subarray}{c}s\in S\\\ g\in
s\end{subarray}}\xi_{i}\left(s\right)\left(r\left(g\right)\right)\cdot\varphi_{ss^{*},j}\left(r\left(g\right)\right).$

We shall prove that for every $i$ there is a $j=j(i)$ such that the subnet
$\\{\eta_{i,j(i)}\\}_{i\in I}$ witnesses the approximation property of
$\mathcal{A}$. To this end, observe $\eta_{i,j}$ is, by construction, a finite
linear combination of functions that are continuous and have compact support
on a bisection, and thus $\eta_{i,j}\in C_{c}(G,r^{*}(\mathcal{A}^{0}))$.
Moreover, observe that for all $x\in G^{0}$ and $g\in G^{x}$

$\displaystyle\eta_{i,j}\left(g\right)^{*}\eta_{i,j}\left(g\right)$
$\displaystyle=\sum_{\begin{subarray}{c}p,t\in S\\\ g\in p\cap
t\end{subarray}}\left(\varphi_{pp^{*},j}\left(x\right)\cdot\xi_{i}\left(p\right)\left(x\right)^{*}\right)\left(\xi_{i}\left(t\right)\left(x\right)\cdot\varphi_{tt^{*},j}\left(x\right)\right)\allowdisplaybreaks$
(4.18) $\displaystyle\approx_{\varepsilon_{i}}\sum_{\begin{subarray}{c}p,t\in
S\\\ g\in p\cap
t\end{subarray}}\xi_{i}\left(p\right)\left(x\right)^{*}\xi_{i}\left(t\right)\left(x\right)$

for all large $j$. Therefore, since $\delta_{i}\leq 1$ for all large $j$, we
have

$\displaystyle\sup_{i\in I}\sup_{x\in G^{0}}\left\|\sum_{g\in
G^{x}}\eta_{i,j}\left(g\right)^{*}\eta_{i,j}\left(g\right)\right\|$
$\displaystyle\approx_{1}\sup_{i\in I}\sup_{x\in G^{0}}\left\|\sum_{g\in
G^{x}}\sum_{\begin{subarray}{c}p,t\in S\\\ g\in p\cap
t\end{subarray}}\xi_{i}\left(p\right)\left(x\right)^{*}\xi_{i}\left(t\right)\left(x\right)\right\|\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\sup_{x\in G^{0}}\left\|\sum_{g\in
G^{x}}\sum_{\begin{subarray}{c}p,t\in S\\\
\left[p^{*},x\right]=\left[t^{*},x\right]=g\end{subarray}}\xi_{i}\left(p\right)\left(x\right)^{*}\xi_{i}\left(t\right)\left(x\right)\right\|\allowdisplaybreaks$
$\displaystyle=\sup_{i\in I}\left\|\sum_{p,t\in
S}1_{pt^{*}}\xi_{i}\left(p\right)^{*}\xi_{i}\left(t\right)\right\|<\infty.$

For the invariance condition, observe that given any $f\in
C_{c}(\mathcal{A})$, by assumption we may decompose it as a finite sum
$f=\sum_{s\in S}b_{s}$, where $b_{s}$ is compactly supported in the bisection
$s\in S$. By a simple approximation argument we may, without loss of
generality, assume that $f=b_{s}\in B_{s}$ for some $s\in S$. In such case,
observe that for every $g=[s,x]\in s$ and $h\in G$ such that $r(h)=r(g)=sx$,
we have

$\displaystyle\eta_{i,j}\left(h\right)^{*}b_{s}\left(g\right)\eta_{i,j}(g^{-1}h)$
$\displaystyle=\sum_{\begin{subarray}{c}p\in S\\\ h\in
p\end{subarray}}\sum_{\begin{subarray}{c}t\in S\\\ g^{-1}h\in
t\end{subarray}}\varphi_{pp^{*},j}\left(sx\right)\xi_{i}\left(p\right)\left(sx\right)^{*}b_{s}\left(g\right)\xi_{i}\left(t\right)\left(x\right)\varphi_{tt^{*},j}\left(x\right)$
(4.19) $\displaystyle\approx_{\varepsilon_{i}}\sum_{\begin{subarray}{c}p\in
S\\\ h\in p\end{subarray}}\sum_{\begin{subarray}{c}t\in S\\\ g^{-1}h\in
t\end{subarray}}\xi_{i}\left(p\right)\left(sx\right)^{*}b_{s}\left(g\right)\xi_{i}\left(t\right)\left(x\right)$

for all large $j$. Hence, and since
$\delta_{i}=\varepsilon_{i}\cdot|\operatorname{\mathrm{supp}}(\xi_{i})|^{2}\to
0$, we have that for all compact set $K\subseteq G$

$\displaystyle\sup_{[s,x]\in K}$ $\displaystyle\left\|\sum_{h\in
G^{sx},}\eta_{i,j}\left(h\right)^{*}b_{s}\left(\left[s,x\right]\right)\eta_{i,j}\left(\left[s,x\right]^{-1}h\right)-b_{s}\left(\left[s,x\right]\right)\right\|\allowdisplaybreaks$
$\displaystyle\approx_{\delta_{i}}\sup_{\left[s,x\right]\in
K}\left\|\sum_{h\in G^{sx}}\sum_{\begin{subarray}{c}p\in S\\\ h\in
p\end{subarray}}\sum_{\begin{subarray}{c}t\in S\\\ \left[s,x\right]^{-1}h\in
t\end{subarray}}\xi_{i}\left(p\right)\left(sx\right)^{*}b_{s}\left(\left[s,x\right]\right)\xi_{i}\left(t\right)\left(x\right)-b_{s}\left(\left[s,x\right]\right)\right\|\allowdisplaybreaks$
$\displaystyle=\sup_{\left[s,x\right]\in
K}\left\|\sum_{\begin{subarray}{c}p,t\in S\\\
\left[p^{*},sx\right]=\left[\left(st\right)^{*},sx\right]\end{subarray}}\xi_{i}\left(p\right)\left(sx\right)^{*}b_{s}\left(\left[s,x\right]\right)\xi_{i}\left(t\right)\left(x\right)-b_{s}\left(\left[s,x\right]\right)\right\|\allowdisplaybreaks$
$\displaystyle=\sup_{\left[s,x\right]\in K}\left\|\left(\sum_{p,t\in
S}1_{p\left(st\right)^{*}}\xi_{i}\left(p\right)^{*}b_{s}\xi_{i}\left(t\right)-b_{s}\right)\left(\left[s,x\right]\right)\right\|\xrightarrow{i}0$

for all large $j$. Thus, for each $i$ let $j=j(i)$ be large enough so that
Equations 4.18 and 4.19 are both satisfied. Then, by the above computations,
the net $\\{\eta_{i,j(i)}\\}_{i\in I}$ witnesses the approximation property of
$\mathcal{A}$, finishing the proof. ∎

######  Remark 4.20.

Observe that, in Theorem 4.16, if $G$ is assumed to be Hausdorff then the
canonical expectation $P$ lands in $B_{1}$, as opposed to
$B_{1}^{\prime\prime}$, i.e., $\mathcal{B}$ is closed (recall Definition
2.30). In fact, by the arguments in the proof of Theorem 4.16, all the sums in
Definition 4.4 are actually in $B_{1}$, and we recover the definition of the
approximation property for Hausdorff étale groupoids in [Kranz:AP-groupooids-
Fell-bundles].

######  Remark 4.21.

We note in passing that Theorem 4.16 also holds in a slightly more general
setting. Indeed, if $T\subseteq\operatorname{\mathrm{Bis}}(G)$ is wide, and
$\pi\colon S\rightarrow T$ is a homomorphism, then the canonical action of $T$
on $G^{0}$ induces via $\pi$ an action of $S$ on $G^{0}$. Given a Fell bundle
$\mathcal{A}=(A_{g})_{g\in G}$ over $G$, we translate it into a Fell bundle
over $T$ and pull it back to $S$, so this gives a Fell bundle
$\mathcal{B}=(B_{s})_{s\in S}$. Assume that the action of $S$ on $G^{0}$ gives
$G$ back, that is, $G^{0}\rtimes S\cong G$. In such scenario, the conclusions
of Theorem 4.16 also hold.

######  Remark 4.22.

In Theorem 4.16 it is apparent that the net $\\{\xi_{i}\\}_{i\in I}$
witnessing the approximation property for $\mathcal{B}$ depends on the choice
of bisections $\\{u_{i,j}\\}\subseteq S$ and $\eta_{i,j}\in
C_{c}(u_{i,j},r^{*}(\mathcal{A}^{0}))$ (see Lemma 4.13). This is related to
the fact that $C_{c}(G,r^{*}(\mathcal{A}^{0}))$ is a quotient of
$\Gamma_{c}\left(S,\mathcal{B}\right)$. More precisely, we have a canonical
surjective linear map

$\displaystyle\Phi\colon\Gamma_{c}\left(S,\mathcal{B}\right)$
$\displaystyle\twoheadrightarrow
C_{c}\left(G,r^{*}\left(\mathcal{A}^{0}\right)\right),$
$\displaystyle\Phi\left(\xi\right)\left(g\right)$
$\displaystyle\coloneqq\sum_{s\in
S}\xi\left(s\right)\left(r\left(g\right)\right).$

In fact, one may view this as a bornological quotient map and describe its
kernel, as is done in [Buss-Meyer:Actions_groupoids]*Proposition B.2.
Nevertheless, we are not going to need this, but notice that given any
$\eta\in C_{c}(G,r^{*}(\mathcal{A}^{0}))$ the possibly different
$\xi\in\Gamma_{c}(S,\mathcal{B})$ that are used in the proof of Theorem 4.16
represent the same element in the quotient space, that is, $\Phi(\xi)=\eta$.

Let us now compare our approximation property with groupoid amenability. We
would like to thank Julian Kranz for pointing us to
[Renault_AnantharamanDelaroche:Amenable_groupoids]*Proposition 2.2.13, which
we use as the definition of _(topological) amenability_ of an étale groupoid.

######  Definition 4.23.

Let $G$ be a locally compact étale groupoid with Hausdorff unit space $G^{0}$.
We say $G$ is _topologically amenable_ if there is a net $\\{\eta_{i}\\}_{i\in
I}\subseteq C_{c}(G)$ such that the following conditions hold:

  1. (i)

$\|\eta_{i}\|_{2}\coloneqq\sup_{x\in G^{0}}(\sum_{h\in
G^{x}}|\eta_{i}(h)|^{2})^{1/2}\leq 1$ for all $i\in I$; and

  2. (ii)

$\sum_{h\in G^{r(g)}}\overline{\eta_{i}(h)}\eta_{i}(g^{-1}h)\xrightarrow{i}1$
for $g\in G$ uniformly in compact subsets of $G$.

######  Remark 4.24.

Normalizing the nets appropriately, one can actually drop the first
boundedness condition (i) in the above definition (this is part of the
statements in [Renault_AnantharamanDelaroche:Amenable_groupoids]*Proposition
2.2.13). The functions $g\mapsto\varphi_{i}(g):=\sum_{h\in
G^{r\left(g\right)}}\overline{\eta_{i}(h)}\eta_{i}\left(g^{-1}h\right)$
appearing in (ii) are prototypical examples of _positive type functions_ on
$G$, as defined and used to characterize amenability in
[Renault_AnantharamanDelaroche:Amenable_groupoids].

We shall use in the following the terminology that an action of an inverse
semigroup $S$ on a $C^{*}$-algebra $A$ (or on a locally compact space $X$) has
the approximation property if the associated Fell bundle over $S$ has the
approximation property. Observe that in the following we do not require $G$ to
be globally Hausdorff, only its unit space.

######  Corollary 4.25.

Let $G$ be a locally compact étale groupoid with Hausdorff unit space $G^{0}$.
Then the following assertions are equivalent:

  1. (i)

$G$ is amenable.

  2. (ii)

The canonical action
$\alpha\colon\operatorname{\mathrm{Bis}}(G)\curvearrowright G^{0}$ has the
approximation property.

  3. (iii)

The canonical action $\beta_{S}\colon S\curvearrowright G^{0}$ has the
approximation property, where $S\subseteq\operatorname{\mathrm{Bis}}(G)$ is
any wide subsemigroup.

###### Proof.

This is a special case of Theorem 4.16 for the trivial Fell bundle
$\mathcal{A}$ over $G$, that is, the Fell bundle with
$A_{g}\coloneqq\mathbb{C}$ for all $g\in G$. In this case the approximation
property of $\mathcal{A}$ is equivalent to the amenability of $G$: the only
small possible difference lies in the boundedness conditions required in the
first item of Definition 4.10 and Definition 4.23. However in this case these
conditions are redundant by Remark 4.24.

If $S\subseteq\operatorname{\mathrm{Bis}}(G)$ is a wide inverse semigroup of
bisections of $G$, then the associated Fell bundle $\mathcal{B}$ over $S$
coming from $\mathcal{A}$ is the Fell bundle associated to the canonical
action of $S$ on $G^{0}$. ∎

Before we end this section, let us discuss the relationship between our
approximation property with _amenability_ of inverse semigroups in the
following sense.

######  Definition 4.26.

We say that an inverse semigroup $S$ is _$C^{*}$ -amenable_ if its canonical
action on its spectrum $\widehat{E}$ has the approximation property.

Let $G(S)\coloneqq\widehat{E}\rtimes S$ be the universal groupoid of $S$;
which is sometimes also called the _Paterson groupoid_ of $S$ (see
[Paterson1999]*Theorem 4.4.1). Essentially, $G(S)$ is the universal groupoid
for ‘ample’ actions of $S$, that is, actions on locally compact spaces with
clopen domains.333 In the language of Fell bundles, such actions give rise to