# Ideal structure and pure infiniteness of
inverse semigroup crossed products

Bartosz Kosma Kwaśniewski<EMAIL_ADDRESS>Faculty of Mathematics
University of Białystok
ul. K. Ciołkowskiego 1M
15-245 Białystok
Poland  and  Ralf Meyer<EMAIL_ADDRESS>Mathematisches Institut
Georg-August-Universität Göttingen
Bunsenstraße 3–5
37073 Göttingen
Germany

###### Abstract.

We give efficient conditions under which a $\textup{C}^{*}$-subalgebra
$A\subseteq B$ separates ideals in a $\textup{C}^{*}$-algebra $B$, and $B$ is
purely infinite if every positive element in $A$ is properly infinite in $B$.
We specialise to the case when $B$ is a crossed product for an inverse
semigroup action by Hilbert bimodules or a section $\textup{C}^{*}$-algebra of
a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory
works provided $B$ is the recently introduced essential crossed product and
the action is essentially exact and residually aperiodic or residually
topologically free. These last notions are developed in the article.

###### Key words and phrases:

crossed product, inverse semigroup, Fell bundle, groupoid, purely infinite,
primitive ideal space

###### 2020 Mathematics Subject Classification:

46L55, 20M18, 22A22

The first named author was supported by the National Science Center (NCN),
Poland, grant no. 2019/35/B/ST1/02684. We thank Alcides Buss, Diego Martínez
and Jonathan Taylor for pointing out an error in the previous version of
Proposition 4.2.

##  1\. Introduction

Many authors have given sufficient criteria for crossed products by discrete
group actions or for $\textup{C}^{*}$-algebras associated to étale locally
compact groupoids to be purely infinite (see, for instance, [Anantharaman-
Delaroche:Purely_infinite, Laca-Spielberg:Purely_infinite, Jolissaint-
Robertson:Simple_purely_infinite, Rordam-Sierakowski:Purely_infinite, Pasnicu-
Phillips:Spectrally_free, Kirchberg-Sierakowski:Strong_pure, Kwasniewski-
Szymanski:Pure_infinite, Boenicke-Li:Ideal, Rainone-Sims:Dichotomy,
Ma:Purely_infinite_groupoids]). These articles mostly deal with the case when
the bigger $\textup{C}^{*}$-algebra $B$ is simple or when the
$\textup{C}^{*}$-subalgebra $A\subseteq B$ on which the action takes place is
commutative and has totally disconnected spectrum. In addition, étale
groupoids are required to be Hausdorff. These pure infiniteness criteria also
imply that $A$ separates ideals in $B$. Then the ideal lattice of $B$ is
isomorphic to the lattice of invariant ideals in $A$. Here we formulate
sufficient conditions for $A$ to separate ideals in $B$ and for $B$ to be
purely infinite, which allow $A$ to be noncommutative and which impose no
Hausdorffness restrictions. In this generality, it is natural to study actions
of inverse semigroups by Hilbert bimodules (see [Buss-
Meyer:Actions_groupoids]) or, equivalently, section algebras of Fell bundles
over inverse semigroups. This contains Fell bundles over discrete groups and
over étale groupoids – possibly non-Hausdorff – as special cases. Another
special case are Exel’s noncommutative Cartan $\textup{C}^{*}$-inclusions (see
[Exel:noncomm.cartan, Kwasniewski-Meyer:Cartan]), which generalise Renault’s
(commutative) Cartan subalgebras.

This article is based on our recent papers, [Kwasniewski-Meyer:Stone_duality,
Kwasniewski-Meyer:Aperiodicity, Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations, Kwasniewski-Meyer:Essential], where
two key concepts are developed. The first one is the _essential crossed
product_ introduced in [Kwasniewski-Meyer:Essential], which is a variation on
the reduced crossed product that “always” has the expected ideal structure –
even for general actions of inverse semigroups and for actions of non-
Hausdorff groupoids. The second concept is _aperiodicity_. It is a strong
regularity property, abstracted from the work of Kishimoto, Olesen–Pedersen,
and others, defined for a general $\textup{C}^{*}$-inclusion $A\subseteq B$ in
[Kwasniewski-Meyer:Essential]. As shown in [Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations], aperiodicity implies that there is a
_unique pseudo-expectation_ – a unique generalised conditional expectation for
$A\subseteq B$ taking values in Hamana’s injective hull of $A$. If in addition
this pseudo-expectation is almost faithful, then $A$ _supports_ $B$ in the
sense that every element in $B^{+}\setminus\\{0\\}$ is supported by an element
in $A^{+}\setminus\\{0\\}$ in the Cuntz preorder of
[Cuntz:Dimension_functions]. This, in turn, implies that $A$ _detects ideals_
in $B$, that is, $J\cap A\neq 0$ for any ideal $0\neq
J\mathrel{\triangleleft}B$. All these properties are closely related. In fact,
when $B$ is an essential crossed product and $A$ is separable or of Type I,
then the following conditions are equivalent: aperiodicity, unique pseudo-
expectation, supporting positive elements in all intermediate
$\textup{C}^{*}$-algebras, detection of ideals in all intermediate
$\textup{C}^{*}$-algebras, and _topological freeness_ of the dual groupoid
(see [Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]).

In the present paper, we study “residual” versions of these conditions, that
is, when they hold for quotient inclusions $A/I\subseteq B/BIB$ for all ideals
$I\mathrel{\triangleleft}A$ that are restricted from $B$. The relationship
between ideals in $B$ and _restricted ideals_ in $A$ is thoroughly studied in
[Kwasniewski-Meyer:Stone_duality]. For a crossed product inclusion, the
restricted ideals in $A$ are exactly those that are _invariant_ under the
action that produces $B$ from $A$. The residual version of detection of ideals
is separation of ideals. We say that $A$ _separates ideals_ in $B$ if $I\cap
A=J\cap A$ for ideals $I,J\mathrel{\triangleleft}B$ implies $I=J$. This
identifies the ideal lattice of $B$ with the lattice of restricted ideals in
$A$. Under some extra assumptions, we may also identify the primitive ideal
space of $B$ with the _quasi-orbit space_ of the induced action on the
primitive ideal space of $A$ (see [Kwasniewski-Meyer:Stone_duality]). A
residual version of supporting is closely related to the formally stronger
condition called _filling_ , which was used in [Kirchberg-
Sierakowski:Strong_pure, Kirchberg-Sierakowski:Filling_families] to prove
strong pure infiniteness. For a large class of $\textup{C}^{*}$-inclusion
$A\subseteq B$, we show that $A$ residually supports $B$ if and only if $A$
fills $B$. Namely, this holds for the _symmetric_ inclusions defined in
[Kwasniewski-Meyer:Stone_duality], which include all sorts of
$\textup{C}^{*}$-inclusions coming from crossed products. For general
residually supporting $\textup{C}^{*}$-inclusions $A\subseteq B$ and a family
$\mathcal{F}\subseteq A^{+}$ of elements in $A$ that are properly infinite in
$B$, we give sufficient conditions for $B$ to be purely infinite (see Theorem
2.36 below).

To ensure that $A^{+}$ residually supports $B$ we assume that the inclusion
$A\subseteq B$ is _residually aperiodic_ , in the sense that for any
restricted ideal $I\mathrel{\triangleleft}A$, the inclusion $A/I\to B/BIB$ is
aperiodic. We also need to assume that the pseudo-expectations for these
quotient inclusions are almost faithful. For actions by discrete groups this
“residual faithfulness of conditional expectations” is also called _exactness_
(see [Sierakowski:IdealStructureCrossedProducts, Abadie-Abadie:Ideals,
Kwasniewski-Szymanski:Pure_infinite]) and for groupoids inner exactness (see
[AnantharamanDelaroch:Weak_containment, Boenicke-Li:Ideal]). We generalise
this concept to inverse semigroup actions by Hilbert bimodules and Fell
bundles over étale groupoids. In particular, we prove that the full crossed
product is an exact functor and the reduced crossed product is an injective
functor, but only when restricted to special homomorphisms between actions
(Propositions 4.2 and 4.15). The essential crossed product is not functorial.
Therefore, exactness for these crossed products, which we call _essential
exactness_ , is more subtle.

As an appetizer, we formulate here a theorem that summarises and illustrates
some of our results. An action of a unital inverse semigroup $S$ by Hilbert
bimodules on a $\textup{C}^{*}$-algebra $A$ is a semigroup
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$, where each fibre $\mathcal{E}_{t}$
is a Hilbert $A$-bimodule and the semigroup product is compatible with the
internal tensor product (see Definition 3.1 below). This induces an $S$-action
by partial homeomorphisms on the spectrum $\widehat{A}$ and the primitive
ideal space $\check{A}$. The corresponding transformation groupoids
$\widehat{A}\rtimes S$ and $\check{A}\rtimes S$ are called _dual groupoids_ of
$\mathcal{E}$. If the unit spaces in these groupoids are closed, we call
$\mathcal{E}$ a _closed action_. Then the essential crossed product
$A\rtimes_{\mathrm{ess}}S$ coincides with the reduced crossed product
$A\rtimes_{\mathrm{r}}S$. The following theorem combines Theorems 5.8, 5.9 and
5.15 and Corollary 5.22:

######  Theorem 1.

Let $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ be an inverse semigroup action by
Hilbert bimodules on a $\textup{C}^{*}$-algebra $A$. Assume that $\mathcal{E}$
is essentially exact (or exact when the action is closed) and residually
aperiodic (this holds when $\widehat{A}\rtimes S$ is residually topologically
free). Then

  1. (1)

$A$ fills the essential crossed product $A\rtimes_{\mathrm{ess}}S$. So
$A\rtimes_{\mathrm{ess}}S$ is strongly purely infinite if and only if every
pair of elements in $A^{+}$ satisfies the matrix factorisation property of
[Kirchberg-Sierakowski:Filling_families];

  2. (2)

the ideal lattice of $A\rtimes_{\mathrm{ess}}S$ is isomorphic to the lattice
$\mathbb{I}^{\mathcal{E}}(A)$ of $\mathcal{E}$-invariant ideals in $A$. If
$\check{A}$ is second countable, the primitive ideal space of
$A\rtimes_{\mathrm{ess}}S$ is homeomorphic to the quasi-orbit space
$\check{A}/{\sim}$ of the dual groupoid $\check{A}\rtimes S$;

  3. (3)

if $\mathcal{F}\subseteq A^{+}$ residually supports $A$ and consists of
residually $\mathcal{E}$-infinite elements (see Definition 5.18), and
$\mathbb{I}^{\mathcal{E}}(A)$ is finite or the projections in $\mathcal{F}$
separate the ideals in $\mathbb{I}^{\mathcal{E}}(A)$, then
$A\rtimes_{\mathrm{ess}}S$ is purely infinite and has the ideal property.

The above theorem directly applies to (twisted) crossed products by discrete
groups. As we explain in Section 5.2, it covers all the pure infiniteness
results in [Laca-Spielberg:Purely_infinite, Jolissaint-
Robertson:Simple_purely_infinite, Rordam-Sierakowski:Purely_infinite, Pasnicu-
Phillips:Spectrally_free, Kirchberg-Sierakowski:Strong_pure, Kwasniewski-
Szymanski:Pure_infinite]. Theorem 1 has an analogue for Fell bundles over
étale groupoids (Corollary 5.10). In particular, it can be used to generalise
the results in [Boenicke-Li:Ideal, Rainone-Sims:Dichotomy,
Ma:Purely_infinite_groupoids] to twisted (not necessarily Hausdorff) étale
groupoids. The twisted version covers all Cartan inclusions of Renault
[Renault:Cartan.Subalgebras] (see Corollary 5.29 and Remark 5.30). In fact,
Theorem 1 may be applied to all regular residually aperiodic
$\textup{C}^{*}$-inclusions $A\subseteq B$ with a residually faithful
conditional expectation (see Proposition 5.12). This includes a large class of
noncommutative Cartan inclusions in the sense of Exel [Exel:noncomm.cartan].
See [Kwasniewski-Meyer:Cartan]*Theorem 4.3 for a number of equivalent
characterisations of such inclusions.

The article is organised as follows. Section 2 reviews general results about
restriction and induction of ideals from [Kwasniewski-Meyer:Stone_duality],
discusses residually supporting and filling families and their relationship,
and presents our general pure infiniteness criteria for
$\textup{C}^{*}$-algebras. In Section 3, we recall actions of inverse
semigroups by Hilbert bimodules and their crossed products. In Section 4 we
discuss restrictions, functoriality and exactness of inverse semigroup crossed
products. Section 5 discusses our general pure infiniteness criteria for
crossed products by inverse semigroup actions.

##  2\. Separation of ideals and pure infiniteness criteria for
$\textup{C}^{*}$-inclusions

###  2.1. Separation and detection of ideals

Let $A\subseteq B$ be a $\textup{C}^{*}$-inclusion. We recall some results
from [Kwasniewski-Meyer:Stone_duality] that relate the ideal structure of the
two $\textup{C}^{*}$-algebras $A$ and $B$. (In [Kwasniewski-
Meyer:Stone_duality], we also consider the more general situation of an
inclusion $A\to\mathcal{M}(B)$ into the multiplier algebra of $B$.)

######  Definition 2.1.

Let $\mathbb{I}(A)$ and $\mathbb{I}(B)$ be the complete lattices of (closed,
two-sided) ideals in $A$ and $B$, respectively. For $I\in\mathbb{I}(A)$, let
$BIB\in\mathbb{I}(B)$ be the ideal in $B$ generated by $I$. We define the
_restriction map $r$_ and the _induction map $i$_ by

$\displaystyle r$ $\displaystyle\colon\mathbb{I}(B)\to\mathbb{I}(A),$
$\displaystyle\qquad J$ $\displaystyle\mapsto J\cap A,$ $\displaystyle i$
$\displaystyle\colon\mathbb{I}(A)\to\mathbb{I}(B),$ $\displaystyle\qquad I$
$\displaystyle\mapsto BIB.$

We call $I\in\mathbb{I}(A)$ _restricted_ if $I=r(J)$ for some
$J\in\mathbb{I}(B)$. We call $J\in\mathbb{I}(B)$ _induced_ if $J=i(I)$ for
some $I\in\mathbb{I}(A)$. Let $\mathbb{I}^{B}(A)\subseteq\mathbb{I}(A)$ and
$\mathbb{I}^{A}(B)\subseteq\mathbb{I}(B)$ be the subsets of restricted and
induced ideals, respectively.

The maps $r$ and $i$ form a (monotone) _Galois connection_ , that is, if
$I\in\mathbb{I}(A)$, $J\in\mathbb{I}(B)$, then $I\subseteq r(J)$ if and only
if $i(I)\subseteq J$. This observation goes back to Green
[Green:Local_twisted]. It has a number of consequences. For instance, the maps
$i$ and $r$ are monotone and satisfy $r\circ i(I)\supseteq I$ and $i\circ
r\circ i(I)=i(I)$ for all $I\in\mathbb{I}(A)$, and $i\circ r(J)\subseteq J$
and $r\circ i\circ r(J)=r(J)$ for all $J\in\mathbb{I}(B)$. The map $i$
preserves joins and $r$ preserves meets. The maps
$r\colon\mathbb{I}(B)\to\mathbb{I}^{B}(A)\subseteq\mathbb{I}(A)$ and
$i\colon\mathbb{I}(A)\to\mathbb{I}^{A}(B)\subseteq\mathbb{I}(B)$ restrict to
mutually inverse isomorphisms of partially ordered sets

$\mathbb{I}^{A}(B)\cong\mathbb{I}^{B}(A).$

Thus $\mathbb{I}^{A}(B)\cong\mathbb{I}^{B}(A)$ are complete lattices with
inclusion as the partial order. The map $r$ is injective if and only if $i$ is
surjective, if and only if $\mathbb{I}(B)=\mathbb{I}^{A}(B)$. Subalgebras with
this property are said to separate ideals.

######  Definition 2.2.

We say that $A$ _separates ideals_ in $B$ if $J_{1}\cap A\neq J_{2}\cap A$ for
all $J_{1},J_{2}\mathrel{\triangleleft}B$ with $J_{1}\neq J_{2}$ or,
equivalently, $r$ is injective. We say that $A$ _detects ideals_ in $B$ if
$J\cap A\neq 0$ for all $J\mathrel{\triangleleft}B$ with $J\neq 0$ or,
equivalently, $r^{-1}(0)=\\{0\\}$.

Detection of ideals is sometimes called the intersection property. Separation
of ideals is a residual version of detection of ideals.

######  Lemma 2.3 ([Kwasniewski-Meyer:Stone_duality]*Proposition 2.12).

A $\textup{C}^{*}$-subalgebra $A\subseteq B$ separates ideals if and only if
$A/I\subseteq B/BIB$ detects ideals for all restricted ideals
$I\in\mathbb{I}^{B}(A)$.

###  2.2. Symmetric and regular $\textup{C}^{*}$-inclusions

We will be mainly interested in regular inclusions, and these are symmetric as
in the following definition.

######  Definition 2.4 ([Kwasniewski-Meyer:Stone_duality]*Definition 5.2).

A $\textup{C}^{*}$-inclusion $A\subseteq B$ is _nondegenerate_ if $AB=B$. It
is _symmetric_ if for all $I\in\mathbb{I}^{B}(A)$ the inclusion $I\to BIB$ is
nondegenerate. (This is equivalent to $IB=BI$ by [Kwasniewski-
Meyer:Stone_duality]*Lemma 5.1.)

Let $\check{A}$ and $\check{B}$ denote the primitive ideal spaces of $A$ and
$B$, respectively. Let $\operatorname{Prime}^{B}(A)$ denote the space of prime
ideals in the lattice $\mathbb{I}^{B}(A)$ of restricted ideals. For
$I\in\mathbb{I}^{B}(A)$, let
$U_{I}\mathrel{\vcentcolon=}\\{\mathfrak{p}\in\operatorname{Prime}^{B}(A)\,{:}\,\mathopen{}I\not\subseteq\mathfrak{p}\\}$.
We equip $\operatorname{Prime}^{B}(A)$ with the topology
$\\{U_{I}\\}_{I\in\mathbb{I}^{B}(A)}$. The next theorem summarises several
desirable properties of symmetric inclusions.

######  Theorem 2.5 ([Kwasniewski-Meyer:Stone_duality]).

Let $A\subseteq B$ be a symmetric $\textup{C}^{*}$-inclusion.

  1. (1)

If $\mathfrak{p}$ is a primitive ideal in $B$, then
$r(\mathfrak{p})\in\mathbb{I}^{B}(A)$ is prime, and this defines a continuous
map $r\colon\check{B}\to\operatorname{Prime}^{B}(A)$.

  2. (2)

If $\mathfrak{p}$ is a primitive ideal in $A$, then there is a largest
restricted ideal in $A$ that is contained in $\mathfrak{p}$, which we denote
by $\pi(\mathfrak{p})$. This element of $\mathbb{I}^{B}(A)$ is prime, and the
resulting map $\pi\colon\check{A}\to\operatorname{Prime}^{B}(A)$ is
continuous. Define an equivalence relation on $\check{A}$ by
$\mathfrak{p}\sim\mathfrak{q}$ if and only if
$\pi(\mathfrak{p})=\pi(\mathfrak{q})$. So $\pi$ descends to a continuous map
$\widetilde{\pi}\colon\check{A}/{\sim}\to\operatorname{Prime}^{B}(A)$.

  3. (3)

If $\operatorname{Prime}^{B}(A)$ is first countable – this always holds when
$\check{A}$ is second countable – then $\pi$ is open and surjective and
$\widetilde{\pi}$ is a homeomorphism. Then there is a continuous map

$\varrho\colon\check{B}\to\check{A}/{\sim},\qquad\mathfrak{p}\mapsto\widetilde{\pi}^{-1}(r(\mathfrak{p})).$

It is a homeomorphism if and only if $A$ separates ideals in $B$.

###### Proof.

By [Kwasniewski-Meyer:Stone_duality]*Corollary 5.5 we may apply [Kwasniewski-
Meyer:Stone_duality]*Lemmas 4.8 and 4.1 to get (1) and (2), and [Kwasniewski-
Meyer:Stone_duality]*Theorem 4.5 gives (3), see also [Kwasniewski-
Meyer:Stone_duality]*Corollary 4.7.∎

######  Definition 2.6 ([Kwasniewski-Meyer:Stone_duality]*Definitions 4.4,
4.10).

The space $\check{A}/{\sim}$ in Theorem 2.5 is called the _quasi-orbit space_
and the map $\varrho\colon\check{B}\to\check{A}/{\sim}$ is called the _quasi-
orbit map_ of $A\subseteq B$.

######  Remark 2.7.

If $B$ is the full or reduced crossed product – or an exotic crossed product,
for an action of a discrete group $G$ on a $\textup{C}^{*}$-algebra $A$, then
$\check{A}/{\sim}$ coincides with the usual quasi-orbit space of the dual
action of $G$ on $\check{A}$. In [Kwasniewski-Meyer:Stone_duality], the quasi-
orbit space is also described in several other cases.

Next we turn to regular inclusions. To link them to crossed products for
inverse semigroup actions, we describe them through gradings by inverse
semigroups.

######  Definition 2.8 ([Kwasniewski-Meyer:Stone_duality]*Definition 6.15).

Let $S$ be an inverse semigroup with unit $1\in S$. An _$S$ -graded
$\textup{C}^{*}$-algebra_ is a $\textup{C}^{*}$-algebra $B$ with a family of
closed linear subspaces $(B_{t})_{t\in S}$ such that $B_{g}^{*}=B_{g^{*}}$,
$B_{g}\cdot B_{h}\subseteq B_{gh}$ for all $g,h\in S$ and $B_{g}\subseteq
B_{h}$ if $g\leq h$ in $S$ (that is, $g=hg^{*}g$), and $\sum B_{t}$ is dense
in $B$. We call $A\mathrel{\vcentcolon=}B_{1}\subseteq B$ the _unit fibre_ of
the $S$-grading. The grading is _saturated_ if $B_{g}\cdot B_{h}=B_{gh}$ for
all $g,h\in S$.

######  Definition 2.9 ([Kumjian:Diagonals, Renault:Cartan.Subalgebras]).

Let $A\subseteq B$ be a $\textup{C}^{*}$-subalgebra. We call $b\in B$ a
_normaliser_ of $A$ in $B$ if $bAb^{*}\subseteq A$ and $b^{*}Ab\subseteq A$.
The inclusion $A\subseteq B$ is _regular_ if it is nondegenerate and $B$ is
the closed linear span of the normalisers of $A$ in $B$.

######  Proposition 2.10.

Let $A\subseteq B$ be a $\textup{C}^{*}$-inclusion. The following are
equivalent:

  1. (1)

$A$ is a regular subalgebra of $B$;

  2. (2)

$A$ is the unit fibre for some $S$-grading on $B$;

  3. (3)

$A$ is the unit fibre for some saturated $S$-grading on $B$.

If the inclusion is regular, then it is symmetric, and the set
$\mathcal{S}(A,B)$ of all closed linear $A$-subbimodules $M\subseteq B$ that
consist entirely of normalisers is an inverse semigroup with the operations
$M\cdot
N\mathrel{\vcentcolon=}\overline{\operatorname{span}}{}\\{mn\,{:}\,\mathopen{}m\in
M,\ n\in N\\}$ and $M^{*}\mathrel{\vcentcolon=}\\{m^{*}\,{:}\,\mathopen{}m\in
M\\}$, and it gives a saturated grading on $B$.

###### Proof.

Combine [Kwasniewski-Meyer:Stone_duality]*Lemma 6.25 and Proposition 6.26. ∎

######  Definition 2.11.

Let $(B_{t})_{t\in S}$ be an $S$-grading on $B$ with $A=B_{1}$. An ideal
$I\in\mathbb{I}(A)$ is called _$(B_{t})_{t\in S}$ -invariant_ if
$IB_{t}=B_{t}I$ for all $t\in S$. An ideal $J\in\mathbb{I}(B)$ is called _$S$
-graded_ if $J=\overline{\sum_{t\in S}(J\cap B_{t})}$.

######  Proposition 2.12 ([Kwasniewski-Meyer:Stone_duality]*Propositions 6.19
and 6.20).

Let $(B_{t})_{t\in S}$ be an $S$-grading on $B$ with $A=B_{1}$. An ideal
$I\in\mathbb{I}(A)$ is restricted, $I\in\mathbb{I}^{B}(A)$, if and only if it
is $(B_{t})_{t\in S}$-invariant. An ideal $J\in\mathbb{I}(B)$ is induced,
$J\in\mathbb{I}^{A}(B)$, if and only if $J$ is $S$-graded.

The following lemma on quotients of $\textup{C}^{*}$-inclusions is not yet
considered in [Kwasniewski-Meyer:Stone_duality].

######  Lemma 2.13.

Let $A\subseteq B$ be a $\textup{C}^{*}$-subalgebra and $J\in\mathbb{I}(B)$.
Let $I\mathrel{\vcentcolon=}J\cap A$. Let $q\colon B\to B/J$ denote the
quotient map. View $A/I$ as a $\textup{C}^{*}$-subalgebra of $B/J$.

  1. (1)

If $(B_{t})_{t\in S}$ is an $S$-grading of $B$ with $A$ as unit fibre, then
$(q(B_{t}))_{t\in S}$ is an $S$-grading on $B/J$ with unit fibre $A/I$. And
$q(B_{t})\cong B_{t}/B_{t}I$ as Banach spaces – and even as Hilbert
$A/I$-bimodules – for all $t\in S$.

  2. (2)

If $A\subseteq B$ is regular, then $A/I\subseteq B/J$ is regular.

###### Proof.

The canonical map from $A/I$ to $B/J$ is injective because $J\cap A=I$. Thus
we may view $A/I$ as a $\textup{C}^{*}$-subalgebra of $B/J$. We prove (1). It
is easy to see that $(q(B_{t}))_{t\in S}$ is an $S$-grading on $B/J$. Each
$B_{t}$ is naturally a right Hilbert $A$-module with inner product $\langle
a\,|\,\mathopen{}b\rangle\mathrel{\vcentcolon=}a^{*}b\in A$ for $a,b\in B_{t}$
and the right multiplication in $B$. The proof of the Rieffel correspondence
between ideals in $A$ and $\mathbb{K}(B_{t})$ shows that $B_{t}I=\\{b\in
B_{t}\,{:}\,\mathopen{}\langle b\,|\,\mathopen{}b\rangle\in I\\}$. The
quotient Banach space $B_{t}/B_{t}I$ is a right Hilbert $A/I$-module with the
induced multiplication and the inner product $\langle
a+B_{t}I\,|\,\mathopen{}b+B_{t}I\rangle\mathrel{\vcentcolon=}\langle
a\,|\,\mathopen{}b\rangle+I=q(a^{*}b)\in A/I$ for $a,b\in B_{t}$. We claim
that the norm defined by this inner product is equal to the quotient norm on
$B_{t}/B_{t}I$. To show this, let $(u_{n})_{n\in N}$ be an approximate unit
for $I$. Then

$\lVert q(b)\rVert^{2}=\lim{}\lVert
b-bu_{i}\rVert^{2}=\lim{}\lVert(1-u_{i})b^{*}b(1-u_{i})\rVert=\lVert
q(b^{*}b)\rVert=\lVert\langle b+B_{t}I\,|\,\mathopen{}b+B_{t}I\rangle\rVert.$

This finishes the proof of (1). Assertion (2) follows from (1) and Proposition
2.10. ∎

###  2.3. Generalised expectations

######  Definition 2.14.

A _generalised expectation_ for a $\textup{C}^{*}$-inclusion $A\subseteq B$
consists of another $\textup{C}^{*}$-inclusion $A\subseteq\tilde{A}$ and a
completely positive, contractive map $B\to\tilde{A}$ that restricts to the
identity map on $A$. If $\tilde{A}=A$, $\tilde{A}=A^{\prime\prime}$, or
$\tilde{A}=\mathrm{I}(A)$ is Hamana’s injective envelope of $A$, then we speak
of a _conditional expectation_ , a _weak expectation_ , or a _pseudo-
expectation_ , respectively.

Let $E\colon B\to\tilde{A}\supseteq A$ be a generalised expectation. It is
called _faithful_ if $E(b^{*}b)=0$ for some $b\in B$ implies $b=0$, _almost
faithful_ if $E((bc)^{*}bc)=0$ for all $c\in B$ and some $b\in B$ implies
$b=0$, and _symmetric_ if $E(b^{*}b)=0$ for some $b\in B$ implies
$E(bb^{*})=0$. The largest two-sided ideal in $B$ contained in $\ker E$ is
equal to

$\mathcal{N}_{E}\mathrel{\vcentcolon=}\\{b\in
B\,{:}\,\mathopen{}E((bc)^{*}bc)=0\text{ for all }c\in B\\}=\\{b\in
B\,{:}\,\mathopen{}E(xby)=0\text{ for all }x,y\in B\\}$

(see [Kwasniewski-Meyer:Essential]*Proposition 3.6), and $\mathcal{N}_{E}=0$
if and only if $E$ is almost faithful.

Since $E|_{A}=\textup{Id}_{A}$ and $E|_{\mathcal{N}_{E}}=0$, it follows that
$A\cap\mathcal{N}_{E}=0$. Hence the composite map $A\to B\to
B/\mathcal{N}_{E}$ is injective and we may identify $A$ with its image in
$B_{\mathrm{r}}\mathrel{\vcentcolon=}B/\mathcal{N}_{E}$. The map $E$ descends
to a generalised expectation $E_{\mathrm{r}}\colon
B_{\mathrm{r}}\to\tilde{A}\supseteq A$ that we call the _reduced generalised
expectation_ associated to $E$ (see [Kwasniewski-Meyer:Essential]*Definition
3.5). The reduced generalised expectation $E_{\mathrm{r}}$ is always almost
faithful. It is faithful if and only if $E$ is symmetric (see [Kwasniewski-
Meyer:Essential]*Corollary 3.8).

We will mainly work with pseudo-expectations below. The injectivity of
$\mathrm{I}(A)$ implies that any $\textup{C}^{*}$-inclusion has a pseudo-
expectation. The following lemma links detection of ideals to almost
faithfulness of pseudo-expectations:

######  Lemma 2.15.

Let $A\subseteq B$ be a $\textup{C}^{*}$-inclusion. The following are
equivalent:

  1. (1)

$A$ detects ideals in $B$;

  2. (2)

every generalised expectation for the $\textup{C}^{*}$-inclusion $A\subseteq
B$ is almost faithful;

  3. (3)

every pseudo-expectation for $A\subseteq B$ is almost faithful.

###### Proof.

Let $E\colon B\to\widetilde{A}\supseteq A$ be a generalised expectation. Since
$\mathcal{N}_{E}\cap A=0$, we must have $\mathcal{N}_{E}=0$ if $A$ detects
ideals in $B$. That is, (1) implies (2). That (2) implies (3) is obvious. We
prove by contradiction that (3) implies (1). Assume that $A$ does not detect
ideals in $B$. Then there is a nonzero ideal $\mathcal{N}$ in $B$ with
$\mathcal{N}\cap A=0$. The inclusion $A\hookrightarrow B/\mathcal{N}$ has a
pseudo-expectation $E\colon B/\mathcal{N}\to\mathrm{I}(A)$. Let $q\colon B\to
B/\mathcal{N}$ be the quotient map. Then $E\circ q\colon B\to\mathrm{I}(A)$ is
a pseudo-expectation for the inclusion $A\subseteq B$. It is not almost
faithful because $0\neq\mathcal{N}\subseteq\mathcal{N}_{E\circ q}$. ∎

######  Remark 2.16 ([Pitts-Zarikian:Unique_pseudoexpectation]*Theorem 3.5).

Every pseudo-expectation for $A\subseteq B$ is faithful – not only almost
faithful – if and only if $A$ detects ideals in $C$ for each intermediate
$\textup{C}^{*}$-algebra $A\subseteq C\subseteq B$.

Lemma 2.3 says that $A$ separates ideals in $B$ if and only if it “residually
detects” ideals in $B$. The residual version of Lemma 2.15 says that $A$
separates ideals in $B$ if and only if the following happens: if
$I\in\mathbb{I}^{B}(A)$ is a restricted ideal and $E^{I}\colon
B/BIB\to\mathrm{I}(A/I)$ is a pseudo-expectation for the inclusion
$A/I\hookrightarrow B/BIB$, then $E^{I}$ is almost faithful.

In general, it is not practical to check that _all_ pseudo-expectations
$B\to\mathrm{I}(A)$ are faithful or almost faithful. And the residual version
of this statement looks even more hopeless. There are, however, inclusions
with a unique pseudo-expectation. This is the case for aperiodic inclusions by
[Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Theorem 3.6. When the
inclusion is even “residually” aperiodic, then the inclusions
$A/I\hookrightarrow B/BIB$ have a unique pseudo-expectation for all
$I\in\mathbb{I}^{B}(A)$. And if we know pseudo-expectations $E^{I}\colon
B/BIB\to\mathrm{I}(A/I)$ for all $I\in\mathbb{I}^{B}(A)$, then it becomes
possible to check whether they are all (almost) faithful (see Theorem 2.33
below).

The residual version of Lemma 2.15 discussed above uses pseudo-expectations
for the inclusion $A/I\hookrightarrow B/BIB$ for $I\in\mathbb{I}^{B}(A)$. The
following examples show that these need not be closely related to pseudo-
expectations for the original inclusion $A\subseteq B$. In fact, for
inclusions that are not symmetric, even a genuine conditional expectation
$E\colon B\to A$ need not “induce” a conditional expectation $E\colon B/BIB\to
A/I$.

######  Example 2.17.

Let $B=\mathbb{B}(H)$ be the algebra of bounded operators on a separable
Hilbert space $H$, and let $A=\mathbb{K}(H)+1$ be the minimal unitisation of
the compacts. The inclusion $A\subseteq B$ is symmetric,
$\mathbb{I}^{B}(A)=\\{0,\mathbb{K}(H),A\\}$, and
$\mathrm{I}(A)=\mathbb{B}(H)=B$. We may take the identity map as the pseudo-
expectation $E\colon B\to B$ for $A\subseteq B$. Let
$I\mathrel{\vcentcolon=}\mathbb{K}(H)$. Then, on the one hand, $E$ descends to
the identity map $E^{I}\colon B/I\to B/I$ on the Calkin algebra. On the other
hand, pseudo-expectations for $\mathbb{C}\cong A/I\subseteq B/I$ are just
states on the Calkin algebra. It seems that there is no universal way how to
produce a state from the identity map.

######  Example 2.18 (see [Kwasniewski-Meyer:Stone_duality]*Examples 2.16 and
7.9).

Let
$B\mathrel{\vcentcolon=}\mathbb{M}_{2}(\mathbb{C})\oplus\mathbb{M}_{2}(\mathbb{C})$
and consider the commutative $\textup{C}^{*}$-subalgebra $A\subseteq B$
spanned by the orthogonal diagonal projections $(P_{00},0)$, $(0,P_{00})$, and
$(P_{11},P_{11})$. Let $E\colon B\to A\subseteq B$ be any faithful conditional
expectation. For instance,

$E\left(\begin{pmatrix}a_{00}&a_{01}\\\
a_{10}&a_{11}\end{pmatrix}\oplus\begin{pmatrix}b_{00}&b_{01}\\\
b_{10}&b_{11}\end{pmatrix}\right)\mathrel{\vcentcolon=}\frac{1}{2}\begin{pmatrix}2a_{00}&0\\\
0&a_{11}+b_{11}\end{pmatrix}\oplus\frac{1}{2}\begin{pmatrix}2b_{00}&0\\\
0&a_{11}+b_{11}\end{pmatrix}.$

Let $J=\mathbb{M}_{2}(\mathbb{C})\oplus 0$. Then $I\mathrel{\vcentcolon=}J\cap
A=\mathbb{C}\cdot(P_{00},0)$ is a restricted ideal in $A\cong\mathbb{C}^{3}$.
We have $J=BIB$ and
$E(J)=\\{\lambda_{1}(P_{00},0)+\lambda_{2}(P_{11},P_{11})\,{:}\,\mathopen{}\lambda_{1},\lambda_{2}\in\mathbb{C}\\}\not\subseteq
I$. Hence $E\colon B\to A$ does not factor through a map $B/J\to A/I$. Note
that the inclusion $A\subseteq B$ is not symmetric.

######  Lemma 2.19.

Let $E\colon B\to A$ be a conditional expectation for a symmetric
$\textup{C}^{*}$-inclusion $A\subseteq B$ and let $I\in\mathbb{I}^{B}(A)$.
Then $E$ descends to a conditional expectation $E^{I}\colon B/BIB\to A/I$,
$b+J\mapsto E(b)+I$, for the inclusion $A/I\to B/BIB$.

###### Proof.

Let $J\in\mathbb{I}^{A}(B)$ and put $I=J\cap A\in\mathbb{I}^{B}(A)$. Since
$A\subseteq B$ is symmetric we have $J=IBI$. Thus $E(J)=E(IBI)\subseteq
IE(B)I=I$ because $E$ is $A$-bilinear. Since $I\subseteq E(J)$ always holds,
this is equivalent to $I=E(J)$. Then $E^{I}$ is well defined. ∎

Many $\textup{C}^{*}$-algebras, including section algebras for Fell bundles
over Hausdorff étale groupoids, are naturally equipped with a conditional
expectation, which is residually symmetric in the sense described in the
following proposition. Then the residual faithfulness of $E$ is called
exactness of the corresponding action
[Sierakowski:IdealStructureCrossedProducts, Abadie-Abadie:Ideals, Kwasniewski-
Szymanski:Pure_infinite, AnantharamanDelaroch:Weak_containment, Boenicke-
Li:Ideal].

######  Proposition 2.20.

Let $A\subseteq B$ be a symmetric $\textup{C}^{*}$-inclusion with a
conditional expectation $E\colon B\to A$ which is residually symmetric, that
is, for each $I\in\mathbb{I}^{B}(A)$, the conditional expectation $E^{I}$ in
Lemma 2.19 is symmetric. Then $A$ separates ideals in $B$ if and only if $E$
is residually faithful – that is, $E^{I}$ is faithful for each
$I\in\mathbb{I}^{B}(A)$ – and $E$ preserves ideals – that is, $E(J)\subseteq
J$ for each $J\in\mathbb{I}(B)$.

###### Proof.

Assume first that $A$ separates ideals in $B$. Lemmas 2.3 and 2.15 imply that
$E^{I}$ is faithful for all $I\in\mathbb{I}^{B}(A)$. Since we assume that
$\mathbb{I}(B)=\mathbb{I}^{A}(B)$ and $A\subseteq B$ is symmetric, this
implies $E(J)\subseteq J$ by Lemma 2.19. Conversely, assume that $E$ is
residually faithful and preserves ideals.

Let $J\in\mathbb{I}(B)$ and $I\in\mathbb{I}(A)$ satisfy $BIB\subsetneq J$.
Then $E(J)\neq I$ because $E^{I}$ is faithful. Since $E$ preserves ideals,
$E(J)=J\cap A$. Thus $I\subsetneq J\cap A$. This shows that $A/I\subseteq
B/BIB$ detects ideals for any $I\in\mathbb{I}^{B}(A)$. This implies that $A$
separates ideals in $B$ by Lemma 2.3. ∎

###  2.4. Supporting and filling families

Let $B^{+}$ be the set of positive elements in a $\textup{C}^{*}$-algebra $B$.
We equip $B^{+}$ with the _Cuntz preorder_ $\precsim$ introduced in
[Cuntz:Dimension_functions]: for $a,b\in B^{+}$, we write $a\precsim b$ and
say that $a$ _supports_ $b$ if, for every $\varepsilon>0$, there is $x\in B$
with $\lVert a-x^{*}bx\rVert<\varepsilon$. We call $a,b\in B^{+}$ _Cuntz
equivalent_ and write $a\approx b$ if $a\precsim b$ and $b\precsim a$. We call
$a,b\in B^{+}$ _Murray–von Neumann equivalent_ and write $a\sim b$ if there is
$z\in B$ with $a=z^{*}z$ and $b=zz^{*}$. Both $\sim$ and $\approx$ are
equivalence relations, and $a\sim b$ implies $a\approx b$. In the converse
direction, only a weaker result is true (see [Kirchberg-
Rordam:Infinite_absorbing]*Lemma 2.3(iv)). Namely, for $\varepsilon>0$ and
$a\in B^{+}\setminus\\{0\\}$, let $(a-\varepsilon)_{+}\in B$ be the positive
part of $a-\varepsilon\cdot 1\in\mathcal{M}(B)$. Let $bBb$ be the hereditary
subalgebra generated by $b$, that is, the closure of
$\\{bxb\,{:}\,\mathopen{}x\in A\\}$. Then $a\precsim b$ if and only if every
$\varepsilon$-cut-down of $a$ is Murray–von Neumann equivalent to an element
in $bBb$, that is

$a\precsim b\quad\Longleftrightarrow\quad\forall_{\varepsilon>0}\
\exists_{z\in B}\quad(a-\varepsilon)_{+}=z^{*}z\ \text{and}\ zz^{*}\in bBb.$

In particular, $a\in bBb$ implies $a\precsim b$, $aBa=bBb$ implies $a\approx
b$, and $a\precsim b$ implies $a\in BbB$. A $\textup{C}^{*}$-algebra $B$ is
_purely infinite_ [Kirchberg-Rordam:Non-simple_pi] if it admits no characters
and $a,b\in B^{+}\setminus\\{0\\}$ satisfy $a\preceq b$ if and only if $a\in
BbB$.

######  Definition 2.21 (compare [Kwasniewski:Crossed_products]*Definition
2.39).

A subset $\mathcal{F}\subseteq B^{+}$ _supports $B$_ if, for each $b\in
B^{+}\setminus\\{0\\}$, there is $a\in\mathcal{F}\setminus\\{0\\}$ with
$a\precsim b$. It _residually supports $B$_ if, for each $J\in\mathbb{I}(B)$,
the image of $\mathcal{F}$ in the quotient $B/J$ supports $B/J$.

######  Lemma 2.22.

Let $B$ be a $\textup{C}^{*}$-algebra and let $\mathcal{F}\subseteq B^{+}$.

  1. (1)

Let $\mathcal{F}$ support $B$. If $J\in\mathbb{I}(B)$, then $J\cap\mathcal{F}$
supports $J$. So $\mathcal{F}$ detects ideals in $B$.

  2. (2)

Let $\mathcal{F}$ residually support $B$. If $J\in\mathbb{I}(B)$, then
$J\cap\mathcal{F}$ residually supports $J$. In addition, $\mathcal{F}$
separates ideals in $B$.

###### Proof.

We first prove (1). Let $J\in\mathbb{I}(B)$ and $b\in J^{+}\setminus\\{0\\}$.
Then $x^{*}bx\in J$ for each $x\in B$. If $a\precsim b$ for some $a\in
B^{+}\setminus\\{0\\}$, then $a=\lim x_{k}^{*}bx_{k}$ for some sequence
$(x_{k})_{k\in\mathbb{N}}$ in $B$. Therefore, any
$a\in\mathcal{F}^{+}\setminus\\{0\\}$ with $a\preceq b$ and $b\in
J^{+}\setminus\\{0\\}$ belongs to $J$. So if $\mathcal{F}$ supports $B$ and
$J\neq\\{0\\}$, then $J\cap\mathcal{F}\neq\\{0\\}$.

Next we prove (2). Let $J\in\mathbb{I}(B)$ and $I\in\mathbb{I}(J)$. Let
$q\colon B\to B/I$ be the quotient map. For any $b\in
q(J)^{+}\setminus\\{0\\}$, there is $a\in\mathcal{F}$ with $0\neq q(a)\precsim
b$. The proof above shows that $q(a)\in q(J)$. Since $q(a)\neq 0$, we have
$a\in J\setminus I$. Thus $J\cap\mathcal{F}$ residually supports $J$. If
$I,J\in\mathbb{I}(B)$ and $I\neq J$, then $I\cap J$ is a proper ideal of $I$
or $J$. Assume, say, that $I\cap J\subsetneq J$. As we have seen above, there
is $a\in(J\cap\mathcal{F})\setminus I$. Hence $I\cap\mathcal{F}\neq
J\cap\mathcal{F}$. ∎

######  Corollary 2.23.

Let $A\subseteq B$ be a $\textup{C}^{*}$-inclusion. A family
$\mathcal{F}\subseteq A^{+}$ residually supports $B$ if and only if the image
of $\mathcal{F}$ supports $B/J$ for each $J\in\mathbb{I}^{A}(B)$.

###### Proof.

If the image of $\mathcal{F}$ supports the quotient $B/J$ for every
$J\in\mathbb{I}^{A}(B)$, then by Lemma 2.22, $A/(J\cap A)$ detects ideals in
$B/J$ for each $J\in\mathbb{I}^{A}(B)$. Then $A$ separates ideals in $B$ by
Lemma 2.3. Thus $\mathbb{I}(B)=\mathbb{I}^{A}(B)$. So our assumption already
says that $\mathcal{F}$ residually supports $B$. ∎

A $\textup{C}^{*}$-algebra $B$ has the _ideal property_ if projections in $B$
separate ideals in $B$.

######  Corollary 2.24.

If the set of projections in $B$ residually supports $B$, then $B$ has the
ideal property. Conversely, a purely infinite $\textup{C}^{*}$-algebra with
the ideal property is residually supported by its projections.

###### Proof.

Lemma 2.22.(2) gives the first statement. For the second, let
$J\in\mathbb{I}(B)$ and $b\in B^{+}\setminus J^{+}$. Let $q\colon B\to B/J$ be
the quotient map. Since $B$ has the ideal property, the ideal $BbB+J\supsetneq
J$ contains a projection $p\notin J$. Then $0\neq q(p)\in
q(BbB)=q(B)q(b)q(B)$. The quotient $B/J$ is purely infinite by [Kirchberg-
Rordam:Non-simple_pi]*Proposition 4.3. Hence we get $q(p)\preceq q(b)$. ∎

Let $\mathbb{H}(B)$ be the set of all _nonzero hereditary
$\textup{C}^{*}$-subalgebras_ of $B$.

######  Lemma 2.25.

Let $\mathcal{F}\subseteq B^{+}$. The following conditions are equivalent:

  1. (1)

$\mathcal{F}$ supports $B$;

  2. (2)

for each $b\in B^{+}\setminus\\{0\\}$, there is $x\in B$ with
$x^{*}bx\in\mathcal{F}\setminus\\{0\\}$;

  3. (3)

for each $D\in\mathbb{H}(B)$, there is $z\in B$ with $zz^{*}\in D$ and
$z^{*}z\in\mathcal{F}\setminus\\{0\\}$.

###### Proof.

(1)$\Rightarrow$(2): Let $b\in B^{+}\setminus\\{0\\}$. Let $\delta\in(0,\lVert
b\rVert)$. Then (1) gives $a\in\mathcal{F}\setminus\\{0\\}$ with
$a\precsim(b-\delta)_{+}$. And [Kirchberg-Rordam:Infinite_absorbing]*Lemma
2.4(ii) gives $x\in B$ with $x^{*}bx=a\in\mathcal{F}\setminus\\{0\\}$.

(2)$\Rightarrow$(3): Let $b\in D^{+}\setminus\\{0\\}$ and choose $x\in B$ with
$x^{*}bx\in\mathcal{F}\setminus\\{0\\}$ as in (2). Then
$z\mathrel{\vcentcolon=}\sqrt{b}\cdot x$ satisfies $zz^{*}\in D$ and
$z^{*}z\in\mathcal{F}\setminus\\{0\\}$, as required in (3).

(3)$\Rightarrow$(1): Let $b\in B^{+}\setminus\\{0\\}$. Then
$D\mathrel{\vcentcolon=}bBb$ is a nonzero hereditary
$\textup{C}^{*}$-subalgebra of $B$. Condition (3) gives $z\in B$ with
$zz^{*}\in D$ and
$a\mathrel{\vcentcolon=}z^{*}z\in\mathcal{F}\setminus\\{0\\}$. Then $a\sim
zz^{*}$ and hence $a\approx zz^{*}$. And $zz^{*}\precsim b$ by [Kirchberg-
Rordam:Non-simple_pi]*Proposition 2.7(i). So $a\precsim b$. ∎

######  Definition 2.26 ([Kirchberg-Sierakowski:Filling_families]*Definition
4.2).

Let $B$ be a $\textup{C}^{*}$-algebra. A set $\mathcal{F}\subseteq B^{+}$
_fills_ $B$ (is a _filling family_ for $B$) if, for each $D\in\mathbb{H}(B)$
and each $J\in\mathbb{I}(B)$ with $D\not\subseteq J$, there is $z\in
B\setminus J$ with $zz^{*}\in D$ and $z^{*}z\in\mathcal{F}$.

In this definition, we may also require $z\in B$ with $zz^{*}\in D$ and
$z^{*}z\in\mathcal{F}\setminus J$ because $z\in J$ if and only if $z^{*}z\in
J$. For $J=0$, this is the condition in Lemma 2.25.(3). This suggests that
filling and residually supporting families are closely related. We are going
to prove some results to this effect.

######  Proposition 2.27.

If $\mathcal{F}\subseteq B^{+}$ fills $B$, then it residually supports $B$.

###### Proof.

Let $J\in\mathbb{I}(B)$. Let $q\colon B\to B/J$ denote the quotient map. Let
$b\in q(B)^{+}\setminus\\{0\\}$. There is $d\in B^{+}\setminus J$ with
$q(d)=b$. For $D\mathrel{\vcentcolon=}dBd$ we have $q(D)=bq(B)b$ and
$D\not\subseteq J$. Since $\mathcal{F}$ fills $B$, there is $z\in B$ with
$z^{*}z\in D$ and $zz^{*}\in\mathcal{F}\setminus J$. Then $q(z)^{*}q(z)\in
bq(B)b$ and $q(z)q(z)^{*}\in q(\mathcal{F})\setminus\\{0\\}$. Hence
$q(z)q(z)^{*}\approx q(z)^{*}q(z)\precsim b$. So $q(\mathcal{F})$ supports
$q(B)$. ∎

######  Proposition 2.28.

Let $A\subseteq B$ be a symmetric $\textup{C}^{*}$-inclusion. Then $A^{+}$
residually supports $B$ if and only if $A^{+}$ fills $B$.

###### Proof.

If $A^{+}$ fills $B$, then it residually supports $B$ by Proposition 2.27.
Conversely, assume that $A^{+}$ residually supports $B$. We are going to prove
that $A^{+}$ fills $B$. Pick $D\in\mathbb{H}(B)$ and $J\in\mathbb{I}(B)$ with
$D\not\subseteq J$. We need $z\in A$ with $z^{*}z\in D$ and $zz^{*}\in
A^{+}\setminus J$. By Lemma 2.22.(2), $A$ separates ideals in $B$. Hence
$\mathbb{I}^{A}(B)=\mathbb{I}(B)$. So $J=BIB$ with
$I\mathrel{\vcentcolon=}A\cap J$. Let $q\colon B\to B/J$ be the quotient map.
There is $d\in D^{+}\setminus J$. Let
$b\mathrel{\vcentcolon=}q(d)\in(B/J)^{+}\setminus\\{0\\}$. Lemma 2.25 gives
$x\in B/J$ with $a\mathrel{\vcentcolon=}x^{*}bx\in(A/I)^{+}\setminus\\{0\\}$.
There are $c\in A^{+}$ with $q(c)=a$ and $w\in B$ with $q(w)=x$. Then
$q(c)=x^{*}bx=q(w^{*}dw)$. So $c=w^{*}dw+v$ for some $v\in J$.

Let $\varepsilon\mathrel{\vcentcolon=}\lVert a\rVert/2$. By assumption, an
approximate unit in $I$ is also one for $J$. So there is $f\in I^{+}$ with
$\lVert f\rVert\leq 1$ and $\lVert v-fv\rVert<\varepsilon$. Let $1$ denote the
formal unit in the unitisation of $B$ and let
$g\mathrel{\vcentcolon=}1-f\in\mathcal{M}(A)^{+}$. Then $\lVert g\rVert\leq 1$
and

$\lVert gw^{*}dwg-gcg\rVert=\lVert gvg\rVert\leq\lVert
v-fv\rVert<\varepsilon.$

Now [Kirchberg-Rordam:Infinite_absorbing]*Lemma 2.2 gives $h\in B$ with

$h^{*}(gw^{*}dwg)h=(gcg-\varepsilon)_{+}\in A^{+}.$

Let $z\mathrel{\vcentcolon=}d^{\nicefrac{{1}}{{2}}}wgh$. Then $zz^{*}\in D$
because $d\in D$, and $z^{*}z=(gcg-\varepsilon)_{+}\in A^{+}$. Since
$q(gcg)=q(c-cf-fc+fcf)=q(c)=a$, we get

$\lVert q(z^{*}z)\rVert=\lVert
q\big{(}(gcg-\varepsilon)_{+}\big{)}\rVert=\lVert(a-\varepsilon)_{+}\rVert\geq\lVert
a\rVert-\varepsilon=\lVert a\rVert/2>0.$

Hence $z^{*}z\notin J$, which is equivalent to $zz^{*}\notin J$. ∎

######  Example 2.29.

Let $B=\textup{C}_{0}(\Omega)$ be commutative and let $\mathcal{F}\subseteq
B^{+}$. The following conditions are equivalent:

  1. (1)

$\mathcal{F}$ fills $B$;

  2. (2)

$\mathcal{F}$ residually supports $B$;

  3. (3)

the open supports of elements of $\mathcal{F}$ form a basis of the topology of
$\Omega$.

Proposition 2.27 implies (1)$\Rightarrow$(2), and (3)$\Rightarrow$(1) is
straightforward. We show (2)$\Rightarrow$(3). Fix an open subset
$U\subseteq\Omega$ and a point $x_{0}\in U$. There is a function
$b\in\textup{C}_{0}(\Omega)$ with $b(x_{0})=1$ and $b|_{\Omega\setminus
U}\equiv 0$. Let $J$ be the ideal in $B$ consisting of functions vanishing on
$\Omega\setminus U\cup\\{x_{0}\\}$. There is $a\in\mathcal{F}\setminus J$ with
$a+J\precsim b+J$. Then
$V\mathrel{\vcentcolon=}\\{x\in\Omega\,{:}\,\mathopen{}a(x)>0\\}$ is an open
subset of $U$ that contains $x_{0}$. This implies (3).

Example 2.29 suggests to view filling families and residually supporting
subsets as noncommutative analogues of bases for topologies.

###  2.5. Residual aperiodicity and criteria for pure infiniteness

We introduce the residual version of aperiodicity and use it to characterise
when a $\textup{C}^{*}$-inclusion $A\subseteq B$ separates ideals. We also
formulate some criteria for $B$ to be purely infinite.

######  Definition 2.30 ([Kwasniewski-Meyer:Essential]*Definition 5.14).

A $\textup{C}^{*}$-inclusion $A\subseteq B$ is _aperiodic_ if the Banach
$A$-bimodule $B/A$ is aperiodic, that is, if for every $x\in B$,
$D\in\mathbb{H}(A)$ and $\varepsilon>0$, there are $a\in D^{+}$ and $y\in A$
with $\lVert axa-y\rVert<\varepsilon$ and $\lVert a\rVert=1$.

######  Remark 2.31 ([Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations]*Theorem 5.5).

If a $\textup{C}^{*}$-inclusion $A\subseteq B$ has the almost extension
property introduced in [Nagy-Reznikoff:Pseudo-diagonals], then $A\subseteq B$
is aperiodic. The converse implication holds if $B$ is separable.

######  Definition 2.32.

A $\textup{C}^{*}$-inclusion $A\subseteq B$ is _residually aperiodic_ if, for
all $I\in\mathbb{I}^{B}(A)$ and $J\mathrel{\vcentcolon=}BIB$, the
$\textup{C}^{*}$-inclusion $A/I\subseteq B/J$ is aperiodic.

######  Theorem 2.33.

Let $A\subseteq B$ be a residually aperiodic $\textup{C}^{*}$-inclusion. Then
for each $I\in\mathbb{I}^{B}(A)$ there is a unique pseudo-expectation
$E^{I}\colon B/BIB\to\mathrm{I}(A/I)$ for the $\textup{C}^{*}$-inclusion
$A/I\subseteq B/BIB$, and the following are equivalent:

  1. (1)

the unique pseudo-expectation $E^{I}$ is almost faithful for all
$I\in\mathbb{I}^{B}(A)$;

  2. (2)

$A$ separates ideals in $B$;

  3. (3)

$A^{+}$ residually supports $B$.

If, in addition, the inclusion $A\subseteq B$ is symmetric, then (1)–(3) are
equivalent to

  1. (4)

$A^{+}$ fills $B$.

If $A\subseteq B$ is symmetric, $\check{A}$ is second countable, and the above
equivalent conditions hold, then $\check{B}\cong\check{A}/{\sim}$ via the
quasi-orbit map.

###### Proof.

It follows from [Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Theorem
3.6 that the expectation $E^{I}$ is unique and that (1) implies (3). Lemma
2.22 shows that (3) implies (2). Next we prove that (2) implies (1). Indeed,
if $E^{I}$ is not almost faithful, then $\mathcal{N}_{E^{I}}$ witnesses that
$A/I$ does not detect ideals in $B/J$. Then $A$ does not separate ideals in
$B$ by Lemma 2.3. In the symmetric case, Proposition 2.28 shows that (3) is
equivalent to (4). The claims in the last sentence follow from Theorem 2.5. ∎

We end this section with pure infiniteness criteria that use filling and
residually supporting families. Infinite and properly infinite elements in
$B^{+}$ are defined in [Kirchberg-Rordam:Non-simple_pi]. We recall their
equivalent descriptions in [Kwasniewski-Szymanski:Pure_infinite]*Lemma 2.1. We
also recall the notion of a separated pair of elements in $B^{+}$ from
[Kwasniewski-Meyer:Aperiodicity]*Definition 5.1 and relate it to the matrix
diagonalisation property. We write $a\approx_{\varepsilon}b$ if $\lVert
a-b\rVert\leq\varepsilon$ for $a,b\in B$.

######  Definition 2.34.

Let $B$ be a $\textup{C}^{*}$-algebra and $a\in B^{+}\setminus\\{0\\}$.

  1. (1)

We call $a\in B^{+}$ _infinite_ in $B$ if there is $b\in
B^{+}\setminus\\{0\\}$ such that for all $\varepsilon>0$ there are $x,y\in aB$
with $x^{*}x\approx_{\varepsilon}a$, $y^{*}y\approx_{\varepsilon}b$ and
$x^{*}y\approx_{\varepsilon}0$.

  2. (2)

We call $a\in B^{+}\setminus\\{0\\}$ _properly infinite_ if for all
$\varepsilon>0$ there are $x,y\in aB$ with $x^{*}x\approx_{\varepsilon}a$,
$y^{*}y\approx_{\varepsilon}a$ and $x^{*}y\approx_{\varepsilon}0$.

  3. (3)

We call $a,b\in B^{+}$ _separated in $B$_ if for all $\varepsilon>0$ there are
$x\in aB$ and $y\in bB$ with $x^{*}x\approx_{\varepsilon}a$,
$y^{*}y\approx_{\varepsilon}b$ and $x^{*}y\approx_{\varepsilon}0$.

By [Kirchberg-Rordam:Non-simple_pi]*Theorem 4.16, a $\textup{C}^{*}$-algebra
is purely infinite if and only if each element $a\in B^{+}\setminus\\{0\\}$ is
properly infinite. By [Kirchberg-Rordam:Infinite_absorbing]*Remark 5.10, $B$
is _strongly purely infinite_ if and only if each pair of elements $a,b\in
B^{+}\setminus\\{0\\}$ is separated in $B$.

We say that a pair of elements $a,b\in B^{+}$ _has the matrix diagonalisation
property in_ $B$, if for each $x\in B$ with
$\left(\begin{smallmatrix}a&x^{*}\\\ x&b\end{smallmatrix}\right)\in
M_{2}(B)^{+}$ and each $\varepsilon>0$ there are $d_{1}\in B$ and $d_{2}\in B$
such that

$d_{1}^{*}ad_{1}\approx_{\varepsilon}a,\qquad
d_{2}^{*}bd_{2}\approx_{\varepsilon}b,\qquad
d_{1}^{*}xd_{2}\approx_{\varepsilon}0.$

We say that a subset $\mathcal{F}\subseteq B^{+}$ is _invariant under
$\varepsilon$-cut-downs_ if $(a-\varepsilon)_{+}\in\mathcal{F}$ for all
$a\in\mathcal{F}$ and arbitrarily small $\varepsilon>0$.

######  Theorem 2.35 ([Kirchberg-Sierakowski:Filling_families]*Theorem 1.1).

Let $\mathcal{F}$ fill $B$ and be invariant under $\varepsilon$-cut-downs.
Then $B$ is strongly purely infinite if and only if each pair of elements
$a,b\in\mathcal{F}$ has the matrix diagonalisation property in $B$.

The following theorem improves upon [Kwasniewski-
Meyer:Aperiodicity]*Proposition 5.4.

######  Theorem 2.36.

Let $A\subseteq B$ be a $\textup{C}^{*}$-subalgebra for which $A^{+}$
residually supports $B$; this is the case, for instance, if $A\subseteq B$ is
residually aperiodic and for each $I\in\mathbb{I}^{B}(A)$ the unique pseudo-
expectation $E^{I}\colon B/BIB\to\mathrm{I}(A/I)$ is almost faithful. Let
$\mathcal{F}\subseteq A^{+}$ residually support $A$. Assume that
$\mathbb{I}^{B}(A)$ is finite or that the projections in $\mathcal{F}$
separate the ideals in $\mathbb{I}^{B}(A)$ (this is automatic when
$\mathcal{F}$ consists of projections). Then the following statements are
equivalent:

  1. (1)

$\mathcal{F}\setminus\\{0\\}$ consists of elements that are properly infinite
in $B$;

  2. (2)

$B$ is purely infinite;

  3. (3)

$B$ is purely infinite and $\operatorname{Prim}B$ has topological dimension
zero;

  4. (4)

$B$ is purely infinite and has the ideal property;

  5. (5)

$B$ is strongly purely infinite.

###### Proof.

The implications (3)$\Leftarrow$(4)$\Rightarrow$(5) are general facts (see
[Pasnicu-Rordam:Purely_infinite_rr0]*Propositions 2.11 and 2.14). The
implications (3)$\Rightarrow$(2)$\Rightarrow$(1)$\Leftarrow$(5) are clear. To
close the cycles of implications, it suffices to show that
(1)$\Rightarrow$(4). We have
$\mathbb{I}(B)=\mathbb{I}^{A}(B)\cong\mathbb{I}^{B}(A)$ by Lemma 2.22. And
$\mathcal{F}$ residually supports $B$ because $\preceq$ is transitive. In
particular, $\mathcal{F}$ separates ideals in $\mathbb{I}^{B}(A)$. Under our
assumptions, the implication (1)$\Rightarrow$(4) follows from the proof of
[Kwasniewski-Meyer:Aperiodicity]*Proposition 5.4. We considered there the case
$\mathcal{F}=A^{+}$. The proof still works, however, for any $\mathcal{F}$
that residually supports $A$. ∎

When $A$ separates ideals in $B$, the assumption that $A^{+}$ residually
supports $B$ seems necessary for Theorem 2.36 to hold. Evidence for this is
the following result abstracted from [Rordam-
Sierakowski:Purely_infinite]*Proposition 2.1 (see also [Boenicke-
Li:Ideal]*Proposition 4.1).

######  Proposition 2.37.

Let $A\subseteq B$ be a symmetric $\textup{C}^{*}$-inclusion with a residually
symmetric conditional expectation $E\colon B\to A$ as in Proposition 2.20. If
$A$ separates ideals in $B$, then $B$ is purely infinite if and only if every
$a\in A^{+}\setminus\\{0\\}$ is properly infinite in $B$ and $E(b)\precsim b$
for every $b\in B^{+}$. If $B$ is purely infinite and $A$ separates ideals in
$B$, then $A^{+}$ fills $B$.

###### Proof.

Assume that $A$ separates ideals in $B$. Suppose first that $B$ is purely
infinite. By Proposition 2.20, for any $b\in B^{+}\setminus\\{0\\}$, $E(b)$ is
in the ideal in $B$ generated by $b$. Then $E(b)\precsim b$ because $b$ is
properly infinite (see [Kirchberg-Rordam:Non-simple_pi]*Theorem 4.16). Now
suppose that every $a\in A^{+}\setminus\\{0\\}$ is properly infinite in $B$
and $E(b)\precsim b$ for every $b\in B^{+}$. Let $J$ be the ideal in $B$
generated by $b\in B^{+}\setminus\\{0\\}$. Proposition 2.20 implies $J\cap
A=E(J)$. Since every ideal in $B$ is induced, this implies $J=BE(J)B$. So $b$
is in the ideal generated by $E(b)$. This implies that $b\precsim E(b)$ using
that $E(b)$ is properly infinite (see [Kirchberg-Rordam:Non-
simple_pi]*Proposition 3.5(ii)). Since $b\oplus b\precsim E(b)\oplus
E(b)\precsim E(b)\precsim b$ in $B$, we conclude that $b$ is properly
infinite. This shows that $B$ is purely infinite.

If $A$ separates ideals in $B$ and $B$ is purely infinite, the same holds for
all the quotient inclusions $A/I\subseteq B/J$, $I=J\cap A$,
$J\in\mathbb{I}(B)$ (see [Kirchberg-Rordam:Non-simple_pi]*Proposition 4.3). By
Proposition 2.20, the conditional expectation $E^{I}\colon B/J\to A/I$ is
faithful. Hence the first part of the assertion shows that
$b\in(B/J)^{+}\setminus\\{0\\}$ implies $0\neq E^{I}(b)\precsim b$. Thus
$A^{+}$ residually supports $B$. Then $A^{+}$ fills $B$ by Proposition 2.28. ∎

##  3\. Inverse semigroup actions and their crossed products

In this section, we briefly recall inverse semigroup actions by Hilbert
bimodules and their crossed products, referring to [Buss-Exel-Meyer:Reduced,
Kwasniewski-Meyer:Essential] for more details.

###  3.1. Inverse semigroup actions by Hilbert bimodules

Throughout this paper, $S$ is an inverse semigroup with unit $1\in S$.

######  Definition 3.1 ([Buss-Meyer:Actions_groupoids]).

An _action_ of $S$ on a $\textup{C}^{*}$-algebra $A$ (by Hilbert bimodules)
consists of Hilbert $A$-bimodules $\mathcal{E}_{t}$ for $t\in S$ and Hilbert
bimodule isomorphisms
$\mu_{t,u}\colon\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}\xrightarrow{\sim}\mathcal{E}_{tu}$
for $t,u\in S$, such that

  1. (A1)

for all $t,u,v\in S$, the following diagram commutes (associativity):

$(\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u})\otimes_{A}\mathcal{E}_{v}$$\mathcal{E}_{t}\otimes_{A}(\mathcal{E}_{u}\otimes_{A}\mathcal{E}_{v})$$\mathcal{E}_{tu}\otimes_{A}\mathcal{E}_{v}$$\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{uv}$$\mathcal{E}_{tuv}$ass$\mu_{t,u}\otimes_{A}\textup{Id}_{\mathcal{E}_{v}}$$\textup{Id}_{\mathcal{E}_{t}}\otimes_{A}\mu_{u,v}$$\mu_{t,uv}$$\mu_{tu,v}$

  2. (A2)

$\mathcal{E}_{1}$ is the identity Hilbert $A,A$-bimodule $A$;

  3. (A3)

$\mu_{t,1}\colon\mathcal{E}_{t}\otimes_{A}A\xrightarrow{\sim}\mathcal{E}_{t}$
and $\mu_{1,t}\colon
A\otimes_{A}\mathcal{E}_{t}\xrightarrow{\sim}\mathcal{E}_{t}$ for $t\in S$ are
the maps defined by $\mu_{1,t}(a\otimes\xi)=a\cdot\xi$ and
$\mu_{t,1}(\xi\otimes a)=\xi\cdot a$ for $a\in A$, $\xi\in\mathcal{E}_{t}$.

Any $S$-action by Hilbert bimodules comes with canonical involutions
$J_{t}\colon\mathcal{E}_{t}^{*}\to\mathcal{E}_{t^{*}}$ and inclusion maps
$j_{u,t}\colon\mathcal{E}_{t}\to\mathcal{E}_{u}$ for $t\leq u$ that satisfy
the conditions required for a saturated Fell bundle in [Exel:noncomm.cartan]
(see [Buss-Meyer:Actions_groupoids]*Theorem 4.8). Thus $S$-actions by Hilbert
bimodules are equivalent to saturated Fell bundles over $S$. A nonsaturated
Fell bundle over $S$ is turned into a saturated Fell bundle over another
inverse semigroup in [BussExel:InverseSemigroupExpansions], such that the full
and reduced section $\textup{C}^{*}$-algebras stay the same. Therefore, we
usually restrict attention to saturated Fell bundles, which we may replace by
inverse semigroup actions as in Definition 3.1. Definition 3.1 contains
(twisted) actions by partial automorphisms.

######  Example 3.2 (Twisted actions of inverse semigroups, see
[BussExel:Regular.Fell.Bundle]*Definition 4.1).

A _twisted action of an inverse semigroup_ $S$ _by partial automorphisms_ on a
$\textup{C}^{*}$-algebra $A$ consists of partial automorphisms
$\alpha_{t}\colon D_{t^{*}}\to D_{t}$ of $A$ for $t\in S$ – that is, $D_{t}$
is an ideal in $A$ and $\alpha_{t}$ is a ∗-isomorphism – and unitary
multipliers $\omega(t,u)\in\mathcal{UM}(D_{tu})$ for $t,u\in S$, such that
$D_{1}=A$ and the following conditions hold for $r,t,u\in S$ and $e,f\in
E(S)\mathrel{\vcentcolon=}\\{s\in S\,{:}\,\mathopen{}s^{2}=s\\}$:

  1. (1)

$\alpha_{r}\circ\alpha_{t}=\textup{Ad}_{\omega(r,t)}\alpha_{rt}$;

  2. (2)

$\alpha_{r}\big{(}a\omega(t,u)\big{)}\omega(r,tu)=\alpha_{r}(a)\omega(r,t)\omega(rt,u)$
for $a\in D_{r^{*}}\cap D_{tu}$;

  3. (3)

$\omega(e,f)=1_{ef}$ and $\omega(r,r^{*}r)=\omega(rr^{*},r)=1_{r}$, where
$1_{r}$ is the unit of $\mathcal{M}(D_{r})$;

  4. (4)

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

Let $((\alpha_{t})_{t\in S},\omega(t,u))_{t,u\in S}$ be a twisted action as
above. For $t\in S$, let $\mathcal{E}_{t}$ be the Hilbert $A$-bimodule
associated to the partial homeomorphism $\alpha_{t}$; this is $D_{t}$ as a
Banach space, and we denote elements by $b\delta_{t}$ to highlight the $t\in
S$ in which we view $b\in D_{t}\subseteq A$ as an element; the Hilbert
$A$-bimodule structure is defined by

$\displaystyle a\cdot(b\delta_{t})$
$\displaystyle\mathrel{\vcentcolon=}(ab)\delta_{t},$
$\displaystyle\qquad(b\delta_{t})\cdot a$
$\displaystyle\mathrel{\vcentcolon=}\alpha_{t}(\alpha_{t}^{-1}(b)a)\delta_{t},$
$\displaystyle\langle\\!\langle
c\delta_{t}\,|\,\mathopen{}b\delta_{t}\rangle\\!\rangle$
$\displaystyle\mathrel{\vcentcolon=}cb^{*},$ $\displaystyle\qquad\langle
c\delta_{t}\,|\,\mathopen{}b\delta_{t}\rangle$
$\displaystyle\mathrel{\vcentcolon=}\alpha_{t}^{-1}(c^{*}b)$

for $a\in A$, $b,c\in D_{t}$. The formula

$\mu_{t,u}(a\delta_{t}\otimes
b\delta_{u})=\alpha_{t}(\alpha_{t}^{-1}(b)a)\omega(t,u)\delta_{t}$

well defines a Hilbert bimodule isomorphism
$\mu_{t,u}\colon\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}\xrightarrow{\sim}\mathcal{E}_{tu}$
for $t,u\in S$ by [BussExel:Regular.Fell.Bundle]*Theorem 4.12. And
$(\mathcal{E}_{t},\mu_{t,u})_{t,u\in S}$ is an action of $S$ on $A$ by Hilbert
bimodules. Fell bundles over $S$ that come from twisted partial actions are
characterised in [BussExel:Regular.Fell.Bundle]*Corollary 4.16, where they are
called “regular.”

######  Example 3.3 (Inverse semigroup gradings).

An $S$-grading $(B_{t})_{t\in S}$ of a $\textup{C}^{*}$-algebra $B$ as in
Definition 2.8 gives a Fell bundle over $S$, using the multiplication and
involution in $B$. This bundle is saturated if and only the grading is
saturated. Then it is an action of $S$ by Hilbert bimodules on
$A\mathrel{\vcentcolon=}B_{1}\subseteq B$. Thus inverse semigroup actions by
Hilbert bimodules are inevitable in the study of regular inclusions (see also
Proposition 2.10). Any inverse semigroup action
$\mathcal{E}=\bigl{(}(\mathcal{E}_{t})_{t\in S},(\mu_{t,u})_{t,u\in
S}\bigr{)}$ on a $\textup{C}^{*}$-algebra $A$ comes from an $S$-graded
$\textup{C}^{*}$-algebra. Namely, embed the spaces $\mathcal{E}_{t}$ for $t\in
S$ into, say, the full crossed product. They form an $S$-grading of the full
crossed product.

Fell bundles over étale groupoids may be described through $S$-actions.

######  Example 3.4 (Fell bundles over groupoids).

Let $G$ be an _étale groupoid_ with locally compact and Hausdorff unit space
$X$. So the range and source maps $r,s\colon G\rightrightarrows X$ are local
homeomorphisms. A _Fell bundle over $G$_ is defined, for instance, in
[BussExel:Fell.Bundle.and.Twisted.Groupoids]*Section 2. It is an upper
semicontinous bundle $\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ of complex
Banach spaces equipped with a continuous involution
${}^{*}\colon\mathcal{A}\to\mathcal{A}$ and a continuous multiplication
${\cdot}\colon\\{(a,b)\in\mathcal{A}\times\mathcal{A}\,{:}\,\mathopen{}a\in
A_{\gamma_{1}},\ b\in A_{\gamma_{2}},\ (\gamma_{1},\gamma_{2})\in
G^{(2)}\\}\to\mathcal{A}$, which satisfy some natural properties. The set of
(open) bisections

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

is a unital inverse semigroup with $U\cdot
V\mathrel{\vcentcolon=}\\{\gamma\cdot\eta\,{:}\,\mathopen{}\gamma\in U,\
\eta\in V\\}$ for $U,V\in\operatorname{Bis}(G)$. Namely,
$X\in\operatorname{Bis}(G)$ is the unit element and
$U^{*}\mathrel{\vcentcolon=}\\{\gamma^{-1}\,{:}\,\mathopen{}\gamma\in U\\}$
for $U\in\operatorname{Bis}(G)$. Let $A_{U}$ for $U\in\operatorname{Bis}(G)$
be the space of continuous sections of $(A_{\gamma})_{\gamma\in G}$ vanishing
outside $U$. The spaces $A_{r(U)}$ and $A_{s(U)}$ are closed two-sided ideals
in $A=A_{X}$, and $A_{U}$ becomes a Hilbert $A_{r(U)}$-$A_{s(U)}$-bimodule
with the bimodule structure $(a\cdot\xi\cdot
b)(\gamma)\mathrel{\vcentcolon=}a(r(\gamma))\xi(\gamma)b(s(\gamma))$ and the
right and left inner products
$\langle\xi\,|\,\mathopen{}\eta\rangle(x)\mathrel{\vcentcolon=}\xi(s|_{U}^{-1}(x))^{*}\eta(s|_{U}^{-1}(x))$,
$\langle\\!\langle\xi\,|\,\mathopen{}\eta\rangle\\!\rangle(x)\mathrel{\vcentcolon=}\xi(r|_{U}^{-1}(x))\eta(r|_{U}^{-1}(x))^{*}$.
For $U,V\in S$, the formula

$\mu_{U,V}(\xi\otimes\eta)(\gamma)\mathrel{\vcentcolon=}\xi\Bigl{(}r|_{U}^{-1}\bigl{(}r(\gamma)\bigr{)}\Bigr{)}\eta\Bigl{(}s|_{V}^{-1}(s\bigl{(}\gamma)\bigr{)}\Bigr{)},\qquad\gamma\in
UV,$

defines a Hilbert bimodule map $\mu_{U,V}\colon A_{U}\otimes_{A}A_{V}\to
A_{UV}$. This data defines a Fell bundle over $\operatorname{Bis}(G)$, which
is saturated if $\mathcal{A}$ is (see
[BussExel:Fell.Bundle.and.Twisted.Groupoids, Kwasniewski-Meyer:Essential] for
details). If $\mathcal{A}$ is not saturated, this may be naturally turned into
a saturated Fell bundle over another inverse semigroup in a number of ways.
For instance, for any inverse subsemigroup $S\subseteq\operatorname{Bis}(G)$
we may let $\tilde{S}$ be the family of all Hilbert subbimodules of $A_{U}$
for $U\in S$. Equivalently, elements of $\tilde{S}$ are of the form
$A_{U}\cdot I$ for $U\in S$ and $I\triangleleft A$. Then $\tilde{S}$, with
operations defined as above, forms an inverse semigroup that acts by Hilbert
bimodules on $A$ (see [Kwasniewski-Meyer:Essential]*Lemma 7.3).

Let $A$ be a $\textup{C}^{*}$-algebra with an action $\mathcal{E}$ of a unital
inverse semigroup $S$. Let $\widehat{A}$ and
$\widecheck{A}=\operatorname{Prim}(A)$ be the space of irreducible
representations and the primitive ideal space of $A$, respectively. The action
of $S$ on $A$ induces actions
$\widehat{\mathcal{E}}=(\widehat{\mathcal{E}}_{t})_{t\in S}$ and
$\widecheck{\mathcal{E}}=(\widecheck{\mathcal{E}}_{t})_{t\in S}$ of $S$ by
partial homeomorphisms on $\widehat{A}$ and $\widecheck{A}$, respectively (see
[Buss-Meyer:Actions_groupoids]*Lemma 6.12, [Kwasniewski-
Meyer:Essential]*Section 2.3). The homeomorphisms

$\widecheck{\mathcal{E}}_{t}\colon\widecheck{s(\mathcal{E})}_{t}\xrightarrow{\sim}\widecheck{r(\mathcal{E})}_{t},\qquad\widehat{\mathcal{E}}_{t}\colon\widehat{s(\mathcal{E})}_{t}\xrightarrow{\sim}\widehat{r(\mathcal{E})}_{t}$

are given by Rieffel’s correspondence and induction of representations,
respectively. Any action by partial homeomorphisms has a transformation
groupoid, which is étale (see [Exel:Inverse_combinatorial]*Section 4 or
[Kwasniewski-Meyer:Essential]*Section 2.1 for details).

######  Definition 3.5 ([Kwasniewski-Meyer:Stone_duality, Kwasniewski-
Meyer:Essential]).

We call $\widehat{\mathcal{E}}$ and $\widecheck{\mathcal{E}}$ _dual actions_
to the action $\mathcal{E}$ of $S$ on $A$. The transformation groupoids
$\widehat{A}\rtimes S$ and $\widecheck{A}\rtimes S$ are called _dual
groupoids_ of $\mathcal{E}$.

######  Example 3.6 (Dual groupoids to Fell bundles).

Let $\mathcal{A}$ be a Fell bundle over an étale groupoid $G$ with locally
compact Hausdorff object space $X$. Then $G$ acts naturally both on the
primitive ideal space $\check{A}$ and the spectrum $\widehat{A}$ of the
$\textup{C}^{*}$-algebra
$A\mathrel{\vcentcolon=}\textup{C}_{0}(X,\mathcal{A})$. More specifically,
every irreducible representation of $A$ factors through the evaluation map
$A\to A_{x}$ for some $x\in X$, and this defines a continuous map
$\psi\colon\widehat{A}\to X$, which is the anchor map of the $G$-action on
$\widehat{A}$ given by the partial homeomorphisms
$\psi_{\gamma}\colon\widehat{A_{s(\gamma)}}\to\widehat{A_{r(\gamma)}}$ induced
by the Hilbert $A_{r(\gamma)}$-$A_{s(\gamma)}$-bimodules $A_{\gamma}$ for
$\gamma\in G$ (see, for instance, [Ionescu-
Williams:Remarks_ideal_structure]*Section 2). The corresponding transformation
groupoid is

$\widehat{A}\rtimes
G\mathrel{\vcentcolon=}\\{([\pi],\gamma)\in\widehat{A}\times
G\,{:}\,\mathopen{}\psi([\pi])=s(\gamma)\\}.$

Two elements $([\rho],\eta)$ and $([\pi],\gamma)$ are composable if and only
if $[\rho]=\psi_{\gamma}([\pi])$, and then their composite is
$([\pi],\eta\gamma)$. The inverse is
$([\pi],\gamma)^{-1}=(\psi_{\gamma}([\pi]),\gamma^{-1})$. The maps
$\psi\colon\widehat{A}\to X$ and
$\psi_{\gamma}\colon\widehat{A_{s(\gamma)}}\to\widehat{A_{r(\gamma)}}$ for
$\gamma\in G$, factor through to a $G$-action on $\check{A}$, which defines a
transformation groupoid $\check{A}\rtimes G$ (see [Ionescu-
Williams:Remarks_ideal_structure]). These actions give rise to transformation
groupoids $\check{A}\rtimes G$ and $\widehat{A}\rtimes G$. Let
$S\subseteq\operatorname{Bis}(G)$ be a unital, inverse subsemigroup of
bisections of $G$ which is _wide_ in the sense that $\bigcup S=G$ and $U\cap
V$ is a union of bisections in $S$ for all $U,V\in S$. Then turning this into
the inverse semigroup action $\tilde{S}$ on $A$ described in Example 3.4, we
have natural isomorphisms of groupoids (see [Kwasniewski-
Meyer:Essential]*Remark 7.4)

$\widehat{A}\rtimes G\cong\widehat{A}\rtimes\tilde{S},\qquad\widehat{A}\rtimes
G\cong\widehat{A}\rtimes\tilde{S}.$

We call $\check{A}\rtimes G$ and $\widehat{A}\rtimes G$ _dual groupoids_ for
$\mathcal{A}$.

###  3.2. Crossed products

Fix an action $\mathcal{E}=\bigl{(}(\mathcal{E}_{t})_{t\in
S},(\mu_{t,u})_{t,u\in S}\bigr{)}$ of $S$ on $A$. For any $t\in S$, let
$r(\mathcal{E}_{t})$ and $s(\mathcal{E}_{t})$ be the ideals in $A$ generated
by the left and right inner products of vectors in $\mathcal{E}_{t}$,
respectively. Thus $\mathcal{E}_{t}$ is an
$r(\mathcal{E}_{t})$-$s(\mathcal{E}_{t})$-imprimitivity bimodule. If $v\leq
t$, then the inclusion map $j_{t,v}$ restricts to a Hilbert bimodule
isomorphism
$\mathcal{E}_{v}\xrightarrow{\sim}r(\mathcal{E}_{v})\cdot\mathcal{E}_{t}=\mathcal{E}_{t}\cdot
s(\mathcal{E}_{v})$. For $t,u\in S$ and for all $v\leq t,u$ this gives Hilbert
bimodule isomorphisms $\vartheta^{v}_{u,t}\colon\mathcal{E}_{t}\cdot
s(\mathcal{E}_{v})\xleftarrow[\cong]{j_{t,v}}\mathcal{E}_{v}\xrightarrow[\cong]{j_{u,v}}\mathcal{E}_{u}\cdot
s(\mathcal{E}_{v})$. Let

(3.7) $I_{t,u}\mathrel{\vcentcolon=}\overline{\sum_{v\leq
t,u}s(\mathcal{E}_{v})}$

be the closed ideal generated by $s(\mathcal{E}_{v})$ for $v\leq t,u$. It is
contained in $s(\mathcal{E}_{t})\cap s(\mathcal{E}_{u})$, and the inclusion
may be strict. There is a unique Hilbert bimodule isomorphism

(3.8) $\vartheta_{u,t}\colon\mathcal{E}_{t}\cdot
I_{t,u}\xrightarrow{\sim}\mathcal{E}_{u}\cdot I_{t,u}$

that restricts to $\vartheta_{u,t}^{v}$ on $\mathcal{E}_{t}\cdot
s(\mathcal{E}_{v})$ for all $v\leq t,u$ by [Buss-Exel-Meyer:Reduced]*Lemma
2.4. The _∗ -algebra_ $A\rtimes_{\mathrm{alg}}S$ of the action $\mathcal{E}$
is defined as the quotient vector space of $\bigoplus_{t\in S}\mathcal{E}_{t}$
by the linear span of $\vartheta_{u,t}(\xi)\delta_{u}-\xi\delta_{t}$ for all
$t,u\in S$ and $\xi\in\mathcal{E}_{t}\cdot I_{t,u}$. The algebraic structure
on $A\rtimes_{\mathrm{alg}}S$ is given by the multiplication maps $\mu_{t,u}$
and the involutions $J_{t}$.

######  Definition 3.9.

The _( full) crossed product_ $A\rtimes S$ of the action $\mathcal{E}$ is the
maximal $\textup{C}^{*}$-completion of the ∗-algebra
$A\rtimes_{\mathrm{alg}}S$ described above.

######  Definition 3.10.

A _representation_ of $\mathcal{E}$ in a $\textup{C}^{*}$-algebra $B$ is a
family of linear maps $\pi_{t}\colon\mathcal{E}_{t}\to B$ for $t\in S$ such
that $\pi_{tu}(\mu_{t,u}(\xi\otimes\eta))=\pi_{t}(\xi)\pi_{u}(\eta)$,
$\pi_{t}(\xi_{1})^{*}\pi_{t}(\xi_{2})=\pi_{1}(\langle\xi_{1}\,|\,\mathopen{}\xi_{2}\rangle)$
and
$\pi_{t}(\xi_{1})\pi_{t}(\xi_{2})^{*}=\pi_{1}(\langle\\!\langle\xi_{1}\,|\,\mathopen{}\xi_{2}\rangle\\!\rangle)$
for all $t,u\in S$, $\xi,\xi_{1},\xi_{2}\in\mathcal{E}_{t}$,
$\eta\in\mathcal{E}_{u}$. The representation is called _injective_ if
$\pi_{1}$ is injective; then all the maps $\pi_{t}$ for $t\in S$ are
isometric.

Any representation $\pi$ of $\mathcal{E}$ in $B$ induces a ∗-homomorphism
$\pi\rtimes S\colon A\rtimes S\to B$. Conversely, every ∗-homomorphism
$A\rtimes S\to B$ is equal to $\pi\rtimes S$ for a unique representation $\pi$
(compare [Buss-Exel-Meyer:Reduced]*Proposition 2.9). This universal property
determines $A\rtimes S$ uniquely up to isomorphism.

The $\textup{C}^{*}$-algebra $A\rtimes S$ is canonically isomorphic to the
full section $\textup{C}^{*}$-algebra of the Fell bundle over $S$
corresponding to $\mathcal{E}$. The reduced section $\textup{C}^{*}$-algebra
of a Fell bundle over $S$ was first defined using inducing pure states, see
[Exel:noncomm.cartan]. An equivalent definition appears in [Buss-Exel-
Meyer:Reduced], where this is called the reduced crossed product
$A\rtimes_{\mathrm{r}}S$ of the action $\mathcal{E}$ (the reduced
$\textup{C}^{*}$-algebra obtained in [MR3690239], using a regular
representation, is in general different). The main ingredient in the
construction in [Buss-Exel-Meyer:Reduced] is the weak conditional expectation
for $A\rtimes S\to A^{\prime\prime}$ described in [Buss-Exel-
Meyer:Reduced]*Lemma 4.5 through the formula

(3.11) $E(\xi\delta_{t})=\operatorname*{s-lim}_{i}\vartheta_{1,t}(\xi\cdot
u_{i}),$

where $\xi\in\mathcal{E}_{t}$, $t\in S$, $(u_{i})$ is an approximate unit for
$I_{1,t}$ and $\operatorname*{s-lim}$ denotes the limit in the strict topology
on $\mathcal{M}(I_{1,t})\subseteq A^{\prime\prime}$. This weak expectation is
symmetric by [Kwasniewski-Meyer:Essential]*Theorem 3.22.

######  Definition 3.12.

The _reduced crossed product_ is the quotient $\textup{C}^{*}$-algebra
$A\rtimes_{\mathrm{r}}S\mathrel{\vcentcolon=}(A\rtimes S)/\mathcal{N}_{E}$.
Hence there is a canonical surjection $\Lambda\colon A\rtimes S\to
A\rtimes_{\mathrm{r}}S$, and

(3.13) $\ker\Lambda=\mathcal{N}_{E}=\\{b\in A\rtimes
S\,{:}\,\mathopen{}E(b^{*}b)=0\\}.$

So the induced weak expectation $E_{\mathrm{r}}$ on $A\rtimes_{\mathrm{r}}S$
is faithful.

######  Remark 3.14.

The canonical maps from $A\rtimes_{\mathrm{alg}}S$ to $A\rtimes S$ and to
$A\rtimes_{\mathrm{r}}S$ are injective by [Buss-Exel-
Meyer:Reduced]*Proposition 4.3. In particular, both $A\rtimes S$ and
$A\rtimes_{\mathrm{r}}S$ are naturally $S$-graded with the same Fell bundle
$(\mathcal{E}_{t})_{t\in S}$ over $S$.

######  Definition 3.15.

The action $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ is called _closed_ if the
weak expectation $E\colon A\rtimes S\to A^{\prime\prime}$ given by (3.11) is
$A$-valued. So it is a genuine conditional expectation $A\rtimes S\to
A\subseteq A\rtimes S$.

######  Remark 3.16.

The action $\mathcal{E}$ is closed if and only if the unit space $\widehat{A}$
is closed in the dual groupoid $\widehat{A}\rtimes S$ or, equivalently,
$\check{A}$ is closed in $\check{A}\rtimes S$ (see [Buss-Exel-
Meyer:Reduced]*Theorem 6.5 and [Kwasniewski-Meyer:Essential]*Proposition
3.20). This explains the name. By [Buss-Exel-Meyer:Reduced]*Proposition 6.3,
$\mathcal{E}$ is closed if and only if the ideal $I_{t,1}$ defined in (3.7) is
complemented in $s(\mathcal{E}_{t})$ for each $t\in S$.

######  Example 3.17 ($\textup{C}^{*}$-algebras of Fell bundles over
groupoids).

Retain the notation from Example 3.4. The ∗-algebra associated to the Fell
bundle $\mathcal{A}$ over the étale groupoid $G$ is denoted by
$\mathfrak{S}(G,\mathcal{A})$. It is the linear span of compactly supported
continuous sections $A_{U}=\textup{C}_{\textup{c}}(U,\mathcal{A})$ for all
bisections $U\in\operatorname{Bis}(G)$ with a convolution and involution given
by

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

for all $f,g\in\mathfrak{S}(G,\mathcal{A})$, $\gamma\in G$. The _full section
$\textup{C}^{*}$-algebra_ $\textup{C}^{*}(G,\mathcal{A})$ is defined as the
maximal $\textup{C}^{*}$-completion of the ∗-algebra
$\mathfrak{S}(G,\mathcal{A})$. This $\textup{C}^{*}$-algebra contains
$A\mathrel{\vcentcolon=}\textup{C}^{*}(X,\mathcal{A})$ as a
$\textup{C}^{*}$-subalgebra. It is equipped with a generalised expectation
$E\colon\textup{C}^{*}(G,\mathcal{A})\to\mathfrak{B}(X,\mathcal{A})$, where
$\mathfrak{B}(X,\mathcal{A})$ is the $\textup{C}^{*}$-algebra of bounded Borel
sections of the $\textup{C}^{*}$-bundle $\mathcal{A}|_{X}$ and $E$ on
$\mathfrak{S}(G,\mathcal{A})$ restricts sections to $X$. The _reduced section
$\textup{C}^{*}$-algebra_ can be defined as the quotient

$\textup{C}^{*}_{\mathrm{r}}(G,\mathcal{A})\mathrel{\vcentcolon=}\textup{C}^{*}(G,\mathcal{A})/\mathcal{N}_{E}.$

All this follows from [Kwasniewski-Meyer:Essential]*Proposition 7.10. Let
$S\subseteq\operatorname{Bis}(G)$ be a unital, inverse subsemigroup of
bisections of $G$ which is _wide_ in the sense that $\bigcup S=G$ and $U\cap
V$ is a union of bisections in $S$ for all $U,V\in S$. If $\mathcal{A}$ is
saturated, then the spaces $(A_{U})_{U\in S}$, with operations inherited from
$\mathfrak{S}(G,\mathcal{A})$, form an action of $S$ on $A$ by Hilbert
bimodules, and the associated crossed products are isomorphic to the
corresponding section $\textup{C}^{*}$-algebras. In general, we modify the
construction as follows. The convolution and multiplication from
$\mathfrak{S}(G,\mathcal{A})$ make $(A_{U}\cdot I)_{{U\in S,I\triangleleft
A}}$ an action by Hilbert bimodules on $A$, and there are natural isomorphisms

$\textup{C}^{*}(G,\mathcal{A})\cong
A\rtimes\tilde{S},\qquad\textup{C}^{*}_{\mathrm{r}}(G,\mathcal{A})\cong
A\rtimes_{\mathrm{r}}\tilde{S}$

(see [Kwasniewski-Meyer:Essential]*Propositions 7.6 and 7.9). If the groupoid
$G$ is Hausdorff, that is, the unit space $X$ is closed in $G$, then the
inverse semigroup action $\tilde{S}$ is closed. The converse implication holds
if $\mathcal{A}$ is a Fell line bundle. Fell line bundles over $G$ are
equivalent to twists of $G$. If $(G,\Sigma)$ is a twisted groupoid and
$\mathcal{L}$ the corresponding line bundle, then
$\textup{C}^{*}(G,\Sigma)\cong\textup{C}^{*}(G,\mathcal{L})\cong A\rtimes S$
and
$\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)\cong\textup{C}^{*}_{\mathrm{r}}(G,\mathcal{L})\cong
A\rtimes_{\mathrm{r}}S$. That is, twisted groupoid $\textup{C}^{*}$-algebras
are also modelled by inverse semigroup actions.

###  3.3. Essential crossed products

The _local multiplier algebra_ $\mathcal{M}_{\mathrm{loc}}(A)$ of $A$ is the
inductive limit of the multiplier algebras $\mathcal{M}(J)$, where $J$ runs
through the directed set of essential ideals in $A$ (see [Ara-
Mathieu:Local_multipliers]). A key idea in [Kwasniewski-Meyer:Essential] is a
natural generalised expectation $EL\colon A\rtimes
S\to\mathcal{M}_{\mathrm{loc}}(A)$ with values in
$\mathcal{M}_{\mathrm{loc}}(A)$. It is defined as follows: for each
$\xi\in\mathcal{E}_{t}$, $t\in S$, the element
$EL(\xi)\in\mathcal{M}(I_{1,t}\oplus
I_{1,t}^{\bot})\subseteq\mathcal{M}_{\mathrm{loc}}(A)$ is given by

$EL(\xi)(u+v)\mathrel{\vcentcolon=}\vartheta_{1,t}(\xi u)\in A$

for $u\in I_{1,t}$, $v\in I_{1,t}^{\bot}$. This generalised expectation is
symmetric by [Kwasniewski-Meyer:Essential]*Theorem 4.11. Hence $EL$ factors
through a faithful pseudo-expectation on the quotient

$A\rtimes_{\mathrm{ess}}S\mathrel{\vcentcolon=}(A\rtimes S)/\mathcal{N}_{EL}.$

There is a canonical embedding
$\iota\colon\mathcal{M}_{\mathrm{loc}}(A)\hookrightarrow\mathrm{I}(A)$
compatible with the inclusions $A\subseteq\mathcal{M}_{\mathrm{loc}}(A)$ and
$A\subseteq\mathrm{I}(A)$ (see [Frank:Injective_local_multiplier]*Theorem 1).
Thus the _canonical $\mathcal{M}_{\mathrm{loc}}$-expectation_ $EL$ may be
viewed as a pseudo-expectation.

######  Definition 3.18 ([Kwasniewski-Meyer:Essential]*Definition 4.4).

We call $A\rtimes_{\mathrm{ess}}S$ the _essential crossed product_.

For a closed action, both $E$ and $EL$ take values in $A$ and then
$A\rtimes_{\mathrm{ess}}S=A\rtimes_{\mathrm{r}}S$. In general,
$\mathcal{N}_{E}\subseteq\mathcal{N}_{EL}$ and there are surjective maps

$A\rtimes S\twoheadrightarrow A\rtimes_{\mathrm{r}}S\twoheadrightarrow
A\rtimes_{\mathrm{ess}}S.$

A $\textup{C}^{*}$-algebra $B$ with ∗-epimorphisms $A\rtimes
S\twoheadrightarrow B\twoheadrightarrow A\rtimes_{\mathrm{ess}}S$ that compose
to the canonical quotient map $A\rtimes S\to A\rtimes_{\mathrm{ess}}S$ is
called an _exotic crossed product_ (see [Kwasniewski-Meyer:Essential]). The
following proposition characterises when the reduced and essential crossed
products coincide:

######  Proposition 3.19 ([Kwasniewski-Meyer:Essential]*Corollary 4.17).

$A\rtimes_{\mathrm{ess}}S=A\rtimes_{\mathrm{r}}S$ if and only if for every
$b\in(A\rtimes S)^{+}\setminus\\{0\\}$ there is $\varepsilon>0$ such that
$\\{\pi\in\widehat{A}\,{:}\,\mathopen{}\lVert\pi^{\prime\prime}(E(b))\rVert>\varepsilon\\}$
has nonempty interior, if and only if for every $b\in(A\rtimes
S)^{+}\setminus\\{0\\}$ the set
$\\{\pi\in\widehat{A}\,{:}\,\mathopen{}\lVert\pi^{\prime\prime}(E(b))\rVert\neq
0\\}$ is not meagre in $\widehat{A}$.

######  Example 3.20 (Essential section $\textup{C}^{*}$-algebras).

Let $\mathcal{A}$ be a Fell bundle over an étale groupoid $G$ with locally
compact Hausdorff unit space $X$. There is a canonical generalised expectation
$EL\colon\textup{C}^{*}(G,\mathcal{A})\to\mathcal{M}_{\mathrm{loc}}(A)$ (see
[Kwasniewski-Meyer:Essential]*Section 7.4). Let

$\mathfrak{M}(X,\mathcal{A})\mathrel{\vcentcolon=}\\{f\in\mathfrak{B}(X,\mathcal{A})\,{:}\,\mathopen{}f\text{
vanishes on a comeagre set}\\}.$

If the bundle $\mathcal{A}|_{X}$ is continuous, then there is a natural
embedding
$\mathcal{M}_{\mathrm{loc}}(A)\hookrightarrow\mathfrak{B}(X,\mathcal{A})/\mathfrak{M}(X,\mathcal{A})$,
and $EL$ is the composite of $E$ and the quotient map
$\mathfrak{B}(X,\mathcal{A})\twoheadrightarrow\mathfrak{B}(X,\mathcal{A})/\mathfrak{M}(X,\mathcal{A})$.
If the bundle is discontinuous, we define $EL$ using the isomorphism
$\textup{C}^{*}(G,\mathcal{A})\cong A\rtimes\tilde{S}$ from Example 3.17. The
_essential section $\textup{C}^{*}$-algebra_ is defined in [Kwasniewski-
Meyer:Essential]*Definition 7.12 as the quotient

$\textup{C}^{*}_{\mathrm{ess}}(G,\mathcal{A})\mathrel{\vcentcolon=}\textup{C}^{*}(G,\mathcal{A})/\mathcal{N}_{EL}.$

######  Example 3.21 (Essential twisted groupoid $\textup{C}^{*}$-algebras).

We define the _essential groupoid $\textup{C}^{*}$-algebra_
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ of a twisted groupoid $(G,\Sigma)$
as $\textup{C}^{*}_{\mathrm{ess}}(G,\mathcal{L})$ for the corresponding Fell
line bundle $\mathcal{L}$. Denoting by $X$ the unit space of $G$,
[Kwasniewski-Meyer:Essential]*Proposition 7.18 implies that the following are
equivalent:

  1. (1)

$\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)=\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$;

  2. (2)

$\left\\{x\in X\,{:}\,\mathopen{}E_{\mathrm{r}}(f)(x)\neq 0\right\\}$ is not
meagre for every
$f\in\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)^{+}\setminus\\{0\\}$;

  3. (3)

if $f\in\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)^{+}\setminus\\{0\\}$, then
$\left\\{x\in X\,{:}\,\mathopen{}\lVert
E_{\mathrm{r}}(f)(x)\rVert>\varepsilon\right\\}$ has nonempty interior for
some $\varepsilon>0$.

Here
$E_{\mathrm{r}}\colon\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)\to\mathfrak{B}(X)$
is the canonical generalised expectation that restricts sections of the
corresponding Fell line bundle to $X$.

######  Example 3.22 (Twisted crossed products by partial automorphisms).

Let $(\alpha,\omega)$ be a twisted action of an inverse semigroup $S$ by
partial automorphisms on a $\textup{C}^{*}$-algebra $A$ as in Example 3.2. By
[BussExel:Regular.Fell.Bundle]*Definition 6.2, a _covariant representation_ of
$(\alpha,\omega)$ on a Hilbert space $\mathcal{H}$ is a pair $(\rho,v)$
consisting of a ∗-homomorphism $\rho\colon A\to\mathbb{B}(\mathcal{H})$ and a
family $v=(v_{t})_{t\in S}$ of partial isometries in $\mathbb{B}(\mathcal{H})$
such that

$\rho(\alpha_{t}(b))=v_{t}\rho(b)v_{t}^{*},\quad\overline{\rho}(\omega(t,u))=v_{t}v_{u}v_{tu}^{*},\quad
v_{t}v_{t}^{*}=\overline{\rho}(1_{tt^{*}}),\quad
v_{t}^{*}v_{t}=\overline{\rho}(1_{t^{*}t}),$

for all $b\in D_{t}$, $t,u\in S$. Here $\overline{\rho}$ is the extension of
$\rho$ to the enveloping von Neumann algebra of $A$, so that
$\overline{\rho}(\omega(t,u))$ and $\overline{\rho}(1_{e})$ make sense. By
definition, the _full crossed product_ for $(\alpha,\omega)$ is the universal
$\textup{C}^{*}$-algebra for covariant representations, and by
[BussExel:Regular.Fell.Bundle]*Theorem 6.3 it is naturally isomorphic to the
full crossed product $A\rtimes S$ for the associated inverse semigroup action
$\mathcal{E}$ by Hilbert bimodules (see Example 3.2). The _reduced crossed
product_ for $(\alpha,\omega)$ may be identified with the reduced crossed
product $A\rtimes_{\mathrm{r}}S$ by [BussExel:Regular.Fell.Bundle]*Definition
6.6. We define the _essential crossed product_ for $(\alpha,\omega)$ as
$A\rtimes_{\mathrm{ess}}S$. By [BussExel:Regular.Fell.Bundle]*Theorem 7.2, for
any twisted groupoid $(G,\Sigma)$ the bisections $S$ of $G$ that trivialise
the twist $\Sigma$ give rise to a twisted inverse semigroup action
$(\alpha,\omega)$ by partial automorphisms of
$A\mathrel{\vcentcolon=}\textup{C}_{0}(G^{0})$ such that
$\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)\cong A\rtimes_{\mathrm{r}}S$. By
definition, this descends to an isomorphism
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)\cong A\rtimes_{\mathrm{ess}}S$.

The reduced section $\textup{C}^{*}$-algebra
$\textup{C}^{*}_{\mathrm{r}}(G,\mathcal{A})$ is usually defined through the
regular representation, which is the direct sum of representations

$\lambda_{x}\colon\textup{C}^{*}(G,\mathcal{A})\to\mathbb{B}\bigl{(}\ell^{2}(G_{x},\mathcal{A})\bigr{)}.$

Here $\ell^{2}(G_{x},\mathcal{A})$ is the Hilbert $A_{x}$-module completion of
$\bigoplus_{s(\gamma)=x}A_{\gamma}$ with the obvious right multiplication and
the standard inner product $\langle
f\,|\,\mathopen{}g\rangle\mathrel{\vcentcolon=}\sum_{s(\gamma)=x}f(\gamma)^{*}g(\gamma)$.
For $f\in\mathfrak{S}(H,\mathcal{A})$ and
$g\in\bigoplus_{s(\gamma)=x}A_{\gamma}$, define
$\lambda_{x}(f)(g)(\gamma)\mathrel{\vcentcolon=}\sum_{r(\eta)=r(\gamma)}f(\eta)g(\eta^{-1}\gamma)$.
The kernel of $\bigoplus_{x\in X}\lambda_{x}$ is $\mathcal{N}_{E}$. So the
reduced $\textup{C}^{*}$-algebra $\textup{C}^{*}_{\mathrm{r}}(H,\mathcal{A})$
is isomorphic to the completion of $\mathfrak{S}(H,\mathcal{A})$ in the
reduced norm $\lVert f\rVert_{\mathrm{r}}\mathrel{\vcentcolon=}\sup_{x\in
X}{}\lVert\lambda_{x}(f)\rVert$. We now describe essential algebras in a
similar fashion:

######  Definition 3.23 ([Kwasniewski-Meyer:Essential]*Definition 7.14).

Call $x\in X$ _dangerous_ if there is a net $(\gamma_{n})$ in $G$ that
converges towards two different points $\gamma\neq\gamma^{\prime}$ in $G$ with
$s(\gamma)=s(\gamma^{\prime})=x$.

######  Proposition 3.24.

Let $\mathcal{A}$ be a continuous Fell bundle over an étale groupoid $G$ with
locally compact and Hausdorff unit space $X$. Assume $G$ is covered by
countably many bisections and let $D\subseteq X$ be the set of dangerous
points. Then

$\ker\bigoplus_{x\in X\setminus D}\lambda_{x}=\mathcal{N}_{EL}.$

That is, $\textup{C}^{*}_{\mathrm{ess}}(G,\mathcal{A})$ is isomorphic to the
Hausdorff completion of $\mathfrak{S}(G,\mathcal{A})$ in the seminorm $\lVert
f\rVert_{\mathrm{ess}}\mathrel{\vcentcolon=}\sup_{x\in X\setminus
D}{}\lVert\lambda_{x}(f)\rVert$.

###### Proof.

For each $x\in X$, $\gamma\in G_{x}=s^{-1}(x)$ and
$f\in\textup{C}^{*}(G,\mathcal{A})$,

$\lVert\lambda_{\gamma}(f)1_{\gamma}\rVert^{2}=\langle
1_{\gamma}\,|\,\mathopen{}\lambda_{x}(f^{*}f)1_{\gamma}\rangle=E(f^{*}f)(r(\gamma)).$

Thus
$\ker\lambda_{x}=\\{f\in\textup{C}^{*}(G,\mathcal{A})\,{:}\,\mathopen{}E(f^{*}f)(r(G_{x}))=0\\}$.
We claim that The set of dangerous points is $G$-invariant. Indeed, if
$\eta\in s^{-1}(x)$ and there is a net $(\gamma_{n})$ that converges towards
two different $\gamma\neq\gamma^{\prime}\in H$ with
$s(\gamma)=s(\gamma^{\prime})=x$, then the net $(\gamma_{n}\eta^{-1})$
converges to $\gamma\eta^{-1}\neq\gamma^{\prime}\eta^{-1}\in H$ with
$s(\gamma\eta^{-1})=s(\gamma^{\prime}\eta^{-1})=r(\eta)$. Hence $x\in D$
implies that $r(G_{x})\subseteq D$. Then

$\ker\bigoplus_{x\in X\setminus D}\lambda_{x}=\bigcap_{x\in X\setminus
D}\ker\lambda_{x}=\\{f\in\textup{C}^{*}(G,\mathcal{A})\,{:}\,\mathopen{}E(f^{*}f)(x)=0\text{
for all }x\in X\setminus D\\}.$

The set on the right hand side is equal to $\mathcal{N}_{EL}$ by [Kwasniewski-
Meyer:Essential]*Proposition 7.18. ∎

##  4\. Exactness of inverse semigroup actions

###  4.1. Functoriality

For group actions, the full and reduced crossed products are functors, and the
reduced one preserves injective homomorphisms. We extend this to inverse
semigroup actions by Hilbert bimodules.

######  Definition 4.1.

Let $\mathcal{E}=\bigl{(}(\mathcal{E}_{t})_{t\in S},(\mu_{t,u})_{t,u\in
S}\bigr{)}$ and $\mathcal{F}=\bigl{(}(\mathcal{F}_{t})_{t\in
S},(\nu_{t,u})_{t,u\in S}\bigr{)}$ be two actions of $S$ by Hilbert bimodules
on $A$ and $B$, respectively. A _homomorphism_ $\psi$ from $\mathcal{E}$ to
$\mathcal{F}$ is a family of linear maps
$\psi_{t}\colon\mathcal{E}_{t}\to\mathcal{F}_{t}$ for $t\in S$ such that for
all $t,u\in S$, $\xi\in\mathcal{E}_{t}$, $\eta\in\mathcal{E}_{u}$ we have

$\psi_{tu}(\mu_{t,u}(\xi\otimes\eta))=\nu_{t,u}(\psi_{t}(\xi)\otimes\psi_{u}(\eta)),\,\,\langle\psi_{t}(\xi)\,|\,\mathopen{}\psi_{t}(\eta)\rangle=\psi_{1}(\langle\xi\,|\,\mathopen{}\eta\rangle),\,\,\langle\\!\langle\psi_{t}(\xi)\,|\,\mathopen{}\psi_{t}(\eta)\rangle\\!\rangle=\psi_{1}(\langle\\!\langle\xi\,|\,\mathopen{}\eta\rangle\\!\rangle).$

The maps $\psi_{t}$ are always contractive. We call $\psi$ _injective_ if
$\psi_{1}$ is injective; then the maps $\psi_{t}$ are isometric for all $t\in
S$. We call $\psi$ an _isomorphism_ if all $\psi_{t}$ are isomorphisms.

We use the superscripts E and F to distinguish between objects defined for the
actions ${\mathcal{E}}$ and $\mathcal{F}$.

######  Proposition 4.2.

Let $\psi$ be a homomorphism from an action
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ on $A$ to an action
$\mathcal{F}=(\mathcal{F}_{t})_{t\in S}$ on $B$. It induces a ∗-homomorphism
$\psi\rtimes S\colon A\rtimes S\to B\rtimes S$ where $(\psi\rtimes
S)(\xi)=\psi_{t}(\xi)$ for $\xi\in\mathcal{E}_{t}$, $t\in S$. In particular,
$\psi$ respects the involution and inclusions maps on $\mathcal{E}$ and
$\mathcal{F}$, and $\psi\rtimes S$ restricts to a ∗-homomorphism
$\psi\rtimes_{\mathrm{alg}}S\colon A\rtimes_{\mathrm{alg}}S\to
B\rtimes_{\mathrm{alg}}S$. Moreover the following conditions are equivalent:

  1. (1)

$\psi\rtimes S$ descends to a ∗-homomorphism $\psi\rtimes_{\mathrm{r}}S\colon
A\rtimes_{\mathrm{r}}S\to B\rtimes_{\mathrm{r}}S$ that respects the canonical
weak expectations, that is the following diagram comutes

${A\rtimes_{\mathrm{alg}}S}$${A\rtimes
S}$${A\rtimes_{\mathrm{r}}S}$${A^{\prime\prime}}$${B\rtimes_{\mathrm{alg}}S}$${B\rtimes
S}$${B\rtimes_{\mathrm{r}}S}$${B^{\prime\prime}}$$\scriptstyle{\subseteq}$$\scriptstyle{\psi\rtimes_{\mathrm{alg}}S}$$\scriptstyle{\Lambda^{\mathcal{E}}}$$\scriptstyle{\psi\rtimes
S}$$\scriptstyle{E^{\mathcal{E}}}$$\scriptstyle{\psi\rtimes_{\mathrm{r}}S}$$\scriptstyle{\psi_{1}^{\prime\prime}}$$\scriptstyle{\subseteq}$$\scriptstyle{\Lambda^{\mathcal{F}}}$$\scriptstyle{E^{\mathcal{F}}}$

where $\Lambda^{\mathcal{E}}$ (resp. $\Lambda^{\mathcal{F}}$) is the regular
representation and $E^{\mathcal{E}}$ (resp. $E^{\mathcal{F}}$) is the
canonical weak conditional expectation associated to the action $\mathcal{E}$
(resp. $\mathcal{F}$).

  2. (2)

$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])[I_{1,t}^{\mathcal{F}}]$,
for all $t\in S$, where $[I_{1,t}^{\mathcal{E}}],[s(\mathcal{E}_{t})]\in
A^{\prime\prime}$ and $[I_{1,t}^{\mathcal{F}}]\in B^{\prime\prime}$ are the
support projections of the ideals
$I_{1,t}^{\mathcal{E}},s(\mathcal{E}_{t})\subseteq A$ and
$I_{1,t}^{\mathcal{F}}\subseteq B$, respectively.

If the above equivalent conditions hold and $\psi$ is injective, then so are
$\psi\rtimes_{\mathrm{r}}S$, $\psi\rtimes_{\mathrm{alg}}S$, and
$\psi_{1}^{\prime\prime}$.

###### Proof.

The Hilbert bimodules $\mathcal{F}_{t}$ for $t\in S$ embed into $B\rtimes S$.
Hence we may treat the maps $\psi_{t}\colon\mathcal{E}_{t}\to\mathcal{F}_{t}$
as taking values in $B\rtimes S$. Then $\psi$ is a representation of
$\mathcal{E}$ in $B\rtimes S$. It integrates to a ∗-homomorphism $\psi\rtimes
S\colon A\rtimes S\to B\rtimes S$ by [Buss-Exel-Meyer:Reduced]*Proposition
2.9. This restricts to a ∗-homomorphism $\psi\rtimes_{\mathrm{alg}}S\colon
A\rtimes_{\mathrm{alg}}S\to B\rtimes_{\mathrm{alg}}S$ and therefore $\psi$
respects the induced involutions and inclusions maps on $\mathcal{E}$ and
$\mathcal{F}$. Our first goal is to show that (1) is equivalent to

(4.3) $E^{\mathcal{F}}\circ\Lambda^{\mathcal{F}}\circ(\psi\rtimes
S)=\psi_{1}^{\prime\prime}\circ E^{\mathcal{E}}\circ\Lambda^{\mathcal{E}}.$

It is clear that (1) implies (4.3). Conversely, assume (4.3). For any
$a\in(A\rtimes S)^{+}$, we get

(4.4) $a\in\ker\Lambda^{\mathcal{E}}\iff
E\bigl{(}\Lambda^{\mathcal{E}}(a)\bigr{)}=0\Longrightarrow\psi_{1}^{\prime\prime}\Bigl{(}E^{\mathcal{E}}\bigl{(}\Lambda^{\mathcal{E}}(a)\bigr{)}\Bigr{)}=0\\\
\iff E^{\mathcal{F}}\Bigl{(}\Lambda^{\mathcal{F}}\bigl{(}(\psi\rtimes
S)(a)\bigr{)}\Bigr{)}=0\iff(\psi\rtimes S)(a)\in\ker\Lambda^{\mathcal{F}}.$

Hence $(\psi\rtimes
S)(\ker\Lambda^{\mathcal{E}})\subseteq\ker(\Lambda^{\mathcal{F}})$. So
$\psi\rtimes S$ descends to a ∗-homomorphism $\psi\rtimes_{\mathrm{r}}S$ as in
(1). If, in addition, $\psi$ is injective, then so is
$\psi_{1}^{\prime\prime}$, and then the only one-sided implication in (4.4)
may be reversed. Thus $(\psi\rtimes
S)(\ker\Lambda^{\mathcal{E}})=\ker(\Lambda^{\mathcal{F}})$, so that
$\psi\rtimes_{\mathrm{r}}S$ is injective. Since the canonical map
$A\rtimes_{\mathrm{alg}}S\to A\rtimes_{\mathrm{r}}S$ is still injective, it
also follows that $\psi\rtimes_{\mathrm{alg}}S$ is injective. So we get all
assertions in (1).

Next, we prove that (2) is equivalent to (4.3). By passing to biduals, we get
a weakly continuous ∗-homomorphism $(\psi\rtimes
S)^{\prime\prime}\colon(A\rtimes S)^{\prime\prime}\to(B\rtimes
S)^{\prime\prime}$. The $\textup{C}^{*}$-algebra $A^{\prime\prime}$ and the
Hilbert bimodules $\mathcal{E}_{t}^{\prime\prime}$ for $t\in S$ embed
naturally into $(A\rtimes S)^{\prime\prime}$. Similarly,
$\mathcal{F}_{t}^{\prime\prime}$ for $t\in S$ are embedded into $(B\rtimes
S)^{\prime\prime}$. Then $(\psi\rtimes
S)^{\prime\prime}|_{\mathcal{E}_{t}^{\prime\prime}}=\psi_{t}^{\prime\prime}$
for $t\in S$. Moreover, if $\xi\in\mathcal{E}_{t}\subseteq A\rtimes S$, then
$E^{\mathcal{E}}(\Lambda^{\mathcal{E}}(\xi))=\xi\cdot[I_{1,t}]\in
A^{\prime\prime}$, where the product is taken in $(A\rtimes S)^{\prime\prime}$
(see (3.11) or the proof of [Buss-Exel-Meyer:Reduced]*Lemma 4.5). Hence (4.3)
is equivalent to

(4.5)
$\psi_{t}(\xi)\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{t}(\xi)[I_{1,t}^{\mathcal{F}}],\qquad\text{
for every }\xi\in\mathcal{E}_{t},t\in S,$

Since $\xi=\xi\cdot[s(\mathcal{E}_{t})]$ holds inside $(A\rtimes
S)^{\prime\prime}$, (2) implies (4.5):

$\psi_{t}(\xi)\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{t}(\xi)\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{t}(\xi)\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])[I_{1,t}^{\mathcal{F}}]=\psi_{t}(\xi)[I_{1,t}^{\mathcal{F}}].$

Conversely, (4.5) implies
$\psi_{t}(\xi_{1})^{*}\psi_{t}(\xi_{2})\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{t}(\xi_{1})^{*}\psi_{t}(\xi_{2})[I_{1,t}^{\mathcal{F}}]$
for all $\xi_{1},\xi_{2}\in\mathcal{E}_{t}$. Taking linear combinations, we
may then replace $\psi_{t}(\xi_{1})^{*}\psi_{t}(\xi_{2})$ by an approximate
unit for the ideal $s(\mathcal{E}_{t})$. And then we may take a strong limit
over this approximate unit to arrive at
$\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])=\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])[I_{1,t}^{\mathcal{F}}]$.
The left hand side then simplifies to
$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])$ because
$I_{1,t}^{\mathcal{E}}\subseteq s(\mathcal{E}_{t})$ implies
$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])\leq\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])$.
Hence (4.5) implies (2). ∎

######  Remark 4.6.

Retain the notation from Proposition 4.2. Since $\psi$ respects inclusions, we
have $\psi_{1}(I_{1,t}^{\mathcal{E}})\subseteq I_{1,t}^{\mathcal{F}}$ and
therefore
$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])\leq[I_{1,t}^{\mathcal{F}}]$.
We also have
$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])\leq\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])$
because $I_{1,t}^{\mathcal{E}}\subseteq s(\mathcal{E}_{t})$ . Thus every
representation satisfies

$\psi_{1}^{\prime\prime}([I_{1,t}^{\mathcal{E}}])\leq\psi_{1}^{\prime\prime}([s(\mathcal{E}_{t})])[I_{1,t}^{\mathcal{F}}]\qquad\text{
for all }t\in S.$

So condition (2) only asserts the inverse inequality. This condition is always
satisfied when $S=G$ is a group, as then $[I_{1,t}^{\mathcal{E}}]=0$ for
$t\neq 1$ (and $[I_{1,1}^{\mathcal{E}}]=1$). For general actions, (2) holds
whenever
$\psi_{1}(I_{1,t}^{\mathcal{E}})=\psi_{1}(s(\mathcal{E}_{t}))I_{1,t}^{\mathcal{F}}$
for all $t\in S$. The latter equality is automatic for inclusion and quotient
homomorphisms in Proposition 4.15 below.

Condition (2) in Proposition 4.2 may fail (and perhaps for some purposes one
might want to include it in the definition of a homomorphism). We thank
Alcides Buss, Diego Martínez and Jonathan Taylor for point this to us.

######  Example 4.7.

Consider the actions whose crossed products are described in [Buss-Exel-
Meyer:Reduced]*Proposition 8.5. Namely, let $S=\\{-1,0,1\\}$ be the inverse
semigroup with the usual number multiplication. Take any
$\textup{C}^{*}$-algebra $A$ and any ideal $I$ in $A$ different from $A$. Let
$\mathcal{E}_{1}=\mathcal{E}_{-1}\mathrel{\vcentcolon=}A$ and
$\mathcal{E}_{0}=I$ be trivial Hilbert bimodules over $A$, and let
$\mu_{t,u}(a\otimes b)=a\cdot b$ for $t,u\in S$ be just the multiplication in
$A$. Then
$A\rtimes^{\mathcal{E}}S=A\rtimes_{\mathrm{r}}^{\mathcal{E}}S=A\rtimes_{\mathrm{alg}}^{\mathcal{E}}S\cong
A\oplus A/I$ (see [Buss-Exel-Meyer:Reduced]*(8.6)). We let $\mathcal{F}$ be
the similar action with $I$ replaced by $A$. Then
$A\rtimes^{\mathcal{F}}S=A\rtimes_{\mathrm{r}}^{\mathcal{F}}S=A\rtimes_{\mathrm{alg}}^{\mathcal{F}}S\cong
A$. The inclusion maps yield a homomorphism $\psi$ from $\mathcal{E}$ to
$\mathcal{F}$ where $\psi\rtimes S=\psi\rtimes_{\mathrm{alg}}S\colon A\oplus
A/I\to A$ is given by $(a\oplus b+I)\mapsto a$. This homomorphism is not
injective, although $\psi$ is. So conditions in Proposition 4.2 are not
satisfied. Indeed, note that $\psi\rtimes_{\mathrm{r}}S=\psi\rtimes S$ exists
in this example, but it does not intertwine the canonical weak expectations,
which are given by
$E^{\mathcal{E}}(a\oplus(b+I))=\frac{a+b}{2}+\frac{a-b}{2}[I]$ and
$E^{\mathcal{F}}(a)=a$, for $a,b\in A$, where $[I]\in A^{\prime\prime}$ is the
support projection of $I$.

###  4.2. Restrictions of actions

We fix an action $\mathcal{E}=\bigl{(}(\mathcal{E}_{t})_{t\in
S},(\mu_{t,u})_{t,u\in S}\bigr{)}$ of a unital inverse semigroup $S$ on a
$\textup{C}^{*}$-algebra $A$ by Hilbert bimodules.

######  Definition 4.8.

Let
$\mathbb{I}^{\mathcal{E}}(A)\mathrel{\vcentcolon=}\\{I\in\mathbb{I}(A)\,{:}\,\mathopen{}I\mathcal{E}_{t}=\mathcal{E}_{t}I\text{
for every }t\in S\\}$ be the set of _$\mathcal{E}$ -invariant_ ideals in $A$.

######  Lemma 4.9.

Let $I\in\mathbb{I}(A)$ and let $B$ be any $S$-graded $\textup{C}^{*}$-algebra
with grading $(\mathcal{E}_{t})_{t\in S}$. The following are equivalent:

  1. (1)

$I$ is $\mathcal{E}$-invariant, that is $I\in\mathbb{I}^{\mathcal{E}}(A)$;

  2. (2)

$I$ is restricted, that is, $I\in\mathbb{I}^{B}(A)$;

  3. (3)

the open subset $\check{I}\subseteq\check{A}$ is invariant in the dual
groupoid $\widecheck{A}\rtimes S$;

  4. (4)

the open subset $\widehat{I}\subseteq\widehat{A}$ is invariant in the dual
groupoid $\widehat{A}\rtimes S$.

###### Proof.

Proposition 2.12 implies that (1) and (2) are equivalent. It is easy to see
that (3) and (4) are equivalent. If $t\in S$, then
$I\cdot\mathcal{E}_{t}=\mathcal{E}_{t}\cdot I$ is equivalent to
$\widecheck{\mathcal{E}_{t}}(\widecheck{I}\cap\widecheck{D}_{t})=\widecheck{I}\cap\widecheck{D}_{t^{*}}$
(see [Kwasniewski:Topological_freeness]*page 645 or the proof of [Abadie-
Abadie:Ideals]*Proposition 3.10). That is, (1) and (3) are equivalent. ∎

Let $I\in\mathbb{I}^{\mathcal{E}}(A)$ be an $\mathcal{E}$-invariant ideal in
$A$. There are natural induced actions of $S$ on $I$ and $A/I$. Namely, the
family $\mathcal{E}|_{I}\mathrel{\vcentcolon=}(\mathcal{E}_{t}I)_{t\in S}$ of
Hilbert $I$-bimodules, with the restrictions of the isomorphisms $\mu_{t,u}$
to
$\mathcal{E}_{t}I\otimes_{A}\mathcal{E}_{u}I=\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}I\xrightarrow{\sim}\mathcal{E}_{tu}I$
for $t,u\in S$, forms an inverse semigroup action by Hilbert bimodules on the
$\textup{C}^{*}$-algebra $I$. For $t\in S$, the quotient Banach space
$\mathcal{E}_{t}/\mathcal{E}_{t}I$ is a Hilbert $A/I$-bimodule in a natural
way because $\mathcal{E}_{t}I=I\mathcal{E}_{t}$. If $t,u\in S$, then the
isomorphism
$\mu_{t,u}\colon\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}\xrightarrow{\sim}\mathcal{E}_{tu}$
induces an isomorphism

$\widetilde{\mu}_{t,u}\colon(\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u})\bigm{/}(\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}I)\xrightarrow{\sim}\mathcal{E}_{tu}\mathbin{/}\mathcal{E}_{tu}I.$

There are natural isomorphisms of Hilbert bimodules

$q_{t,u}\colon(\mathcal{E}_{t}/\mathcal{E}_{t}I)\otimes_{A/I}(\mathcal{E}_{u}/\mathcal{E}_{u}I)\xrightarrow{\sim}(\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u})\bigm{/}(\mathcal{E}_{t}\otimes_{A}\mathcal{E}_{u}I)$

because $I\mathcal{E}_{u}=\mathcal{E}_{u}I$ (see, for instance,
[Kwasniewski:Cuntz-Pimsner-Doplicher]*Lemma 1.8). Now
$\mathcal{E}|_{A/I}\mathrel{\vcentcolon=}(\mathcal{E}_{t}/\mathcal{E}_{t}I)_{t\in
S}$ with the isomorphisms $\widetilde{\mu}_{t,u}\circ
q_{t,u}\colon(\mathcal{E}_{t}/\mathcal{E}_{t}I)\otimes_{A}(\mathcal{E}_{u}/\mathcal{E}_{u}I)\xrightarrow{\sim}(\mathcal{E}_{tu}/\mathcal{E}_{tu}I)$
for $t,u\in S$ is an action of $S$ by Hilbert bimodules on $A/I$.

######  Definition 4.10.

We call the actions $\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$ above the
_restrictions_ of $\mathcal{E}$ to $I$ and $A/I$, respectively.

######  Remark 4.11.

Let $I\in\mathbb{I}^{\mathcal{E}}(A)$. The inclusions
$\mathcal{E}_{t}I\subseteq\mathcal{E}_{t}$ for $t\in S$ yield an injective
homomorphism from $\mathcal{E}|_{I}$ to $\mathcal{E}$. The quotient maps
$\mathcal{E}_{t}\to\mathcal{E}_{t}/\mathcal{E}_{t}I$ for $t\in S$ yield a
homomorphism from $\mathcal{E}$ to $\mathcal{E}|_{A/I}$.

######  Remark 4.12.

Let $B$ be an $S$-graded $\textup{C}^{*}$-algebra with grading $(B_{t})_{t\in
S}$ and let $I\in\mathbb{I}^{B}(A)$. The induced ideal $BIB$ carries the
$S$-grading $BIB\cap B_{t}$ by Proposition 2.12. The quotient $B/BIB$ is
$S$-graded by the images of $B_{t}$ by Lemma 2.13.

######  Remark 4.13.

If $I$ is $\mathcal{E}$-invariant, then the dual actions for the restrictions
$\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$ are equal to the restrictions of
the action $(\widehat{\mathcal{E}}_{t})_{t\in S}$ to $\widehat{I}$ and
$\widehat{A}\setminus\widehat{I}$, respectively. Let $\widehat{I}\rtimes S$
and $\widehat{A}\setminus\widehat{I}\rtimes S$ denote the transformation
groupoids dual to $\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$, respectively.
Then

$\widehat{I}\rtimes S=(\widehat{A}\rtimes
S)|_{\widehat{I}},\qquad\widehat{A}\setminus\widehat{I}\rtimes
S=(\widehat{A}\rtimes S)|_{\widehat{A}\setminus\widehat{I}},$

where the right hand sides in the above equalities mean the restrictions of
the transformation groupoid $\widehat{A}\rtimes S$ to the invariant subsets
$\widehat{I}$ and $\widehat{A}\setminus\widehat{I}$ of the unit space
$\widehat{A}$ of $\widehat{A}\rtimes S$. In particular, if the space of units
in $\widehat{A}\rtimes S$ is closed, then the same holds for the
transformation groupoids dual to the restrictions $\mathcal{E}|_{I}$ and
$\mathcal{E}|_{A/I}$.

######  Proposition 4.14.

If the action $\mathcal{E}$ on $A$ is closed, then so are the restrictions
$\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$ for each $\mathcal{E}$-invariant
ideal $I\mathrel{\triangleleft}A$.

###### Proof.

This follows from Remarks 3.16 and 4.13. ∎

###  4.3. Exact actions

######  Proposition 4.15.

Let $\mathcal{E}$ be an action of $S$ by Hilbert bimodules on a
$\textup{C}^{*}$-algebra $A$ and let $I\in\mathbb{I}^{\mathcal{E}}(A)$. Let
$\iota$ be the injective homomorphism from $\mathcal{E}|_{I}$ into
$\mathcal{E}$ and let $\kappa$ be the quotient homomorphism from $\mathcal{E}$
onto $\mathcal{E}|_{A/I}$ in Remark 4.11. They induce an exact sequence

$0\to I\rtimes S\xrightarrow{\iota\rtimes S}A\rtimes
S\xrightarrow{\kappa\rtimes S}A/I\rtimes S\to 0.$

It descends to a sequence

(4.16) $0\to
I\rtimes_{\mathrm{r}}S\xrightarrow{\iota\rtimes_{\mathrm{r}}S}A\rtimes_{\mathrm{r}}S\xrightarrow{\kappa\rtimes_{\mathrm{r}}S}A/I\rtimes_{\mathrm{r}}S\to
0,$

which may fail to be exact only in the middle: $\iota\rtimes_{\mathrm{r}}S$ is
injective, $\kappa\rtimes_{\mathrm{r}}S$ is surjective and the range of
$\iota\rtimes_{\mathrm{r}}S$ is contained in the kernel of
$\kappa\rtimes_{\mathrm{r}}S$.

###### Proof.

Note that $\iota(I_{1,t}^{\mathcal{E}I})=I_{1,t}^{\mathcal{E}}\cdot
I=\iota(s(\mathcal{E}_{t}I))\cdot I_{1,t}^{\mathcal{E}}$ and
$\kappa(I_{1,t}^{\mathcal{E}})=I_{1,t}^{{\mathcal{E}/\mathcal{E}I}}$, for all
$t\in S$. Hence $\iota$ and $\kappa$ satisfy the equivalent conditions in
Proposition 4.2 by the last part of Remark 4.6. Thus, by Proposition 4.2, not
only the ∗-homomorphisms $\iota\rtimes S$, $\kappa\rtimes S$ but also
$\iota\rtimes_{\mathrm{r}}S$, $\kappa\rtimes_{\mathrm{r}}S$ exist, and
$\iota\rtimes_{\mathrm{r}}S$ is injective. The maps $\kappa\rtimes S$ and
$\kappa\rtimes_{\mathrm{r}}S$ are surjective because their images contain the
dense ∗-subalgebra $A/I\rtimes_{\mathrm{alg}}S$.

We prove that $\iota\rtimes S$ is injective. Let $\pi$ be a faithful,
nondegenerate representation of $I\rtimes S$ on a Hilbert space $\mathcal{H}$.
Since $I$ is nondegenerate in $I\rtimes S$, the representation $\pi|_{I}$ is
also nondegenerate. Therefore, for each $\xi_{t}\in\mathcal{E}_{t}$, the
formula

$\widetilde{\pi}(\xi_{t})\pi(a)h\mathrel{\vcentcolon=}\pi(\xi_{t}a)h$

for $a\in I$ and $h\in\mathcal{H}$ defines a bounded operator on
$\mathcal{H}$. Alternatively, $\widetilde{\pi}(\xi_{t})$ could be defined
using an approximate identity $(\mu_{\lambda})$ for $I$, as the limit of the
strongly convergent net $\pi(\xi_{t}\mu_{\lambda})$. A standard proof shows
that $(\widetilde{\pi}_{t})_{t\in S}$ is a representation $\widetilde{\pi}$ of
$\mathcal{E}$. The integrated representation $\widetilde{\pi}\rtimes S\colon
A\rtimes S\to\mathbb{B}(\mathcal{H})$ satisfies $(\widetilde{\pi}\rtimes
S)\circ(\iota\rtimes S)=\pi$. Hence $\iota\rtimes S$ is injective.

The composite maps $(\kappa\rtimes S)\circ(\iota\rtimes S)$ and
$(\kappa\rtimes_{\mathrm{r}}S)\circ(\iota\rtimes_{\mathrm{r}}S)$ vanish. Hence
the range of $\iota\rtimes S$ is contained in $\ker(\kappa\rtimes S)$ and the
range of $\iota\rtimes_{\mathrm{r}}S$ is contained in
$\ker(\kappa\rtimes_{\mathrm{r}}S)$. Conversely, we claim that the range of
$\iota\rtimes S$ contains $\ker(\kappa\rtimes S)$. We identify $I\rtimes S$
with its image in $A\rtimes S$. Let $q\colon A\rtimes S\to(A\rtimes
S)\mathbin{/}(I\rtimes S)$ be the quotient map. For $t\in S$, the restriction
of $q$ to $\mathcal{E}_{t}I$ vanishes. Hence $q$ induces maps

$\psi_{t}\colon\mathcal{E}_{t}/\mathcal{E}_{t}I\to(A\rtimes S)/(I\rtimes S).$

They form a representation of $\mathcal{E}_{A/I}$. It integrates to a
homomorphism

$\psi\rtimes S\colon(A/I)\rtimes S\to(A\rtimes S)/(I\rtimes S).$

We have $(\psi\rtimes S)\circ(\kappa\rtimes S)=q$. Hence $\ker(\kappa\rtimes
S)\subseteq\ker(q)=I\rtimes S$. ∎

######  Definition 4.17.

The action $\mathcal{E}$ is _exact_ if the sequence (4.16) is exact for each
$I\in\mathbb{I}^{\mathcal{E}}(A)$.

######  Example 4.18 (Exactness of twisted groupoids).

An action $\mathcal{E}$ of an inverse semigroup $S$ on a commutative
$\textup{C}^{*}$-algebra $A\cong\textup{C}_{0}(X)$ corresponds to a twisted
étale groupoid $(\Sigma,G)$ with unit space $X$ (see
[BussExel:Fell.Bundle.and.Twisted.Groupoids]). Here $G=X\rtimes S$ is the dual
groupoid of $\mathcal{E}$. The action $\mathcal{E}$ is exact if and only if
the corresponding _twisted groupoid $(G,\Sigma)$ is exact_ in the sense that
for any open invariant subset $U\subseteq X$, the sequence of reduced twisted
groupoid $\textup{C}^{*}$-algebras

$\textup{C}^{*}_{\mathrm{r}}(G|_{U},\Sigma|_{U})\rightarrowtail\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)\twoheadrightarrow\textup{C}^{*}_{\mathrm{r}}(G|_{X\setminus
U},\Sigma|_{X\setminus U})$

is exact. If the twist is trivial, the name _inner exact_ for such groupoids
is introduced in [AnantharamanDelaroch:Weak_containment]. This example
generalises as follows.

######  Example 4.19 (Exactness of Fell bundles over groupoids).

Let $\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ be a Fell bundle over an étale
groupoid $G$ with locally compact Hausdorff unit space $X$. Let
$S\subseteq\operatorname{Bis}(G)$ be a wide inverse semigroup of bisections
and turn the Fell bundle $\mathcal{A}$ into an action $\mathcal{E}$ of
$\tilde{S}$ on the section $\textup{C}^{*}$-algebra $A=A|_{X}$, such that the
associated universal and reduced $\textup{C}^{*}$-algebras remain the same
(see Examples 3.4 and 3.17). This natural correspondence extends to invariant
ideals and the associated algebras. Indeed, by Lemma 4.9 an ideal $I$ in $A$
is $\mathcal{E}$-invariant if and only if it is $G$-invariant in the sense of
[Ionescu-Williams:Remarks_ideal_structure] ($\check{I}$ is invariant under the
dual groupoid $\check{A}\rtimes G\cong\widecheck{A}\rtimes\tilde{S}$). The
restricted actions $\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$ of $\tilde{S}$
correspond to _restricted Fell bundles_
$\mathcal{A}|_{I}=(A_{\gamma}I_{s(\gamma)})_{\gamma\in G}$ and
$\mathcal{A}_{A/I}=(A_{\gamma}/(A_{\gamma}I_{s(\gamma)}))_{\gamma\in G}$ over
$G$ as defined in [Ionescu-Williams:Remarks_ideal_structure]*Propositions 3.3
and 3.4. The authors of [Ionescu-Williams:Remarks_ideal_structure] consider
separable Fell bundles over locally compact Hausdorff groupoids. However,
neither separability nor Hausdorffness are used in the construction of
$\mathcal{A}|_{I}$ and $\mathcal{A}_{A/I}$. Proposition 4.15 implies that the
sequence

$\textup{C}^{*}(\mathcal{A}|_{I})\rightarrowtail\textup{C}^{*}(\mathcal{A})\twoheadrightarrow\textup{C}^{*}(\mathcal{A}|_{A/I})$

is exact. This extends the main result of [Ionescu-
Williams:Remarks_ideal_structure] to étale groupoids that are not separable or
not Hausdorff. We will say that the _Fell bundle $\mathcal{A}$ is exact_ if
for every $G$-invariant ideal $I$ in $A$, the sequence

$\textup{C}^{*}_{\mathrm{r}}(\mathcal{A}|_{I})\rightarrowtail\textup{C}^{*}_{\mathrm{r}}(\mathcal{A})\twoheadrightarrow\textup{C}^{*}_{\mathrm{r}}(\mathcal{A}|_{A/I})$

is exact. Equivalently, the action $\mathcal{E}$ corresponding to
$\mathcal{A}$ is exact.

###  4.4. Exactness for essential crossed products

The essential crossed product is not functorial (see [Kwasniewski-
Meyer:Essential]*Remark 4.8). This complicates the definition of “exactness”
for essential crossed products. Only the quotient maps cause extra problems:

######  Lemma 4.20.

Let $\mathcal{E}$ be an action of $S$ by Hilbert bimodules on a
$\textup{C}^{*}$-algebra $A$ and let $I\in\mathbb{I}^{\mathcal{E}}(A)$. The
injective homomorphism $\iota$ from $\mathcal{E}|_{I}$ into $\mathcal{E}$
induces an injective ∗-homomorphism $\iota\rtimes_{\mathrm{ess}}S\colon
I\rtimes_{\mathrm{ess}}S\to A\rtimes_{\mathrm{ess}}S$. Its image is the ideal
in $A\rtimes_{\mathrm{ess}}S$ generated by $I$.

###### Proof.

If $J$ is an essential ideal in $I$, then $J\oplus I^{\bot}$ is an essential
ideal in $A$. The obvious inclusions $\mathcal{M}(J)\to\mathcal{M}(J\oplus
I^{\bot})$ for the essential ideals $J\subseteq I$ induce a natural
isomorphism from $\mathcal{M}_{\mathrm{loc}}(I)$ onto an ideal in
$\mathcal{M}_{\mathrm{loc}}(A)$. Let $EL_{I}\colon I\rtimes
S\to\mathcal{M}_{\mathrm{loc}}(I)$ be the canonical essential expectation. We
are going to prove below that the following diagram commutes:

(4.21) ${I\rtimes S}$${A\rtimes
S}$${\mathcal{M}_{\mathrm{loc}}(I)}$${\mathcal{M}_{\mathrm{loc}}(A)}$$\scriptstyle{\iota\rtimes
S}$$\scriptstyle{EL_{I}}$$\scriptstyle{EL}$

Then $\mathcal{N}_{EL}\cap\iota\rtimes S(I\rtimes S)=\iota\rtimes
S(\mathcal{N}_{EL_{I}})$ because $\iota\rtimes S(I\rtimes S)$ is an ideal in
$A\rtimes S$. This, in turn, implies that the injective homomorphism
$\iota\rtimes S$ factors through an injective homomorphism
$\iota\rtimes_{\mathrm{ess}}S\colon I\rtimes_{\mathrm{ess}}S=I\rtimes
S/\mathcal{N}_{EL_{I}}\to A\rtimes_{\mathrm{ess}}S=A\rtimes
S/\mathcal{N}_{EL}$.

To check (4.21), let $t\in S$. If $I_{t,1}=\overline{\sum_{v\leq
t,1}s(\mathcal{E}_{v})}$ then $I_{t,1}I=\overline{\sum_{v\leq
t,1}s(\mathcal{E}_{v}I)}$ and the restriction of the map
$\vartheta_{t,1}\colon\mathcal{E}_{t}\cdot I_{t,1}\xrightarrow{\sim}I_{t,1}$
(defined in (3.8)) to $\mathcal{E}_{t}I\cdot I_{t,1}$ coincides with the
corresponding map defined for the restricted action $(\mathcal{E}_{s}I)_{s\in
S}$. Hence for each $\xi\in\mathcal{E}_{t}I$ the element
$EL(\xi)\in\mathcal{M}(I_{1,t}\oplus
I_{1,t}^{\bot})\subseteq\mathcal{M}(I_{1,t}I\oplus I_{1,t}^{\bot}I\oplus
I_{1,t}I^{\bot}\oplus I_{1,t}^{\bot}I^{\bot})$ acts on $u\in I_{1,t}I$ in the
same way as $EL_{I}(\xi)$:

$EL(\xi)u=\vartheta_{1,t}(\xi u)=EL_{I}(\xi)u.$

Since $EL(\xi)(I_{1,t}^{\bot}I\oplus I_{1,t}I^{\bot}\oplus
I_{1,t}^{\bot}I^{\bot})=0$, the embedding
$\mathcal{M}_{\mathrm{loc}}(I)\subseteq\mathcal{M}_{\mathrm{loc}}(A)$ maps
$EL_{I}(\xi)$ to $EL(\xi)$. This proves (4.21). ∎

######  Example 4.22 ([Kwasniewski-Meyer:Essential]*Example 4.7).

Let $S\mathrel{\vcentcolon=}G\cup\\{0\\}$ be the inverse semigroup obtained by
adjoining a zero element to an amenable discrete group $G$. Let $G$ act on
$A=\textup{C}[0,1]$ by $\mathcal{E}_{g}=\textup{C}[0,1]$ for $g\in G$ and
$\mathcal{E}_{0}=\textup{C}_{0}(0,1]$, equipped with the usual involution and
multiplication maps. Then $A\rtimes_{\mathrm{ess}}S=A$. Every ideal $I$ in $A$
is $\mathcal{E}$-invariant and $I\rtimes_{\mathrm{ess}}S=I$, and so
$I\rtimes_{\mathrm{ess}}S\subseteq A\rtimes_{\mathrm{ess}}S$. However, if
$I\mathrel{\vcentcolon=}\textup{C}_{0}(0,1]$, then
$A/I\rtimes_{\mathrm{ess}}S=\mathbb{C}\rtimes G=\textup{C}^{*}(G)$ and the
quotient homomorphism $\kappa$ from $\mathcal{E}$ onto $\mathcal{E}|_{A/I}$
does not induce a map from $A$ to $\textup{C}^{*}(G)$.

######  Definition 4.23.

We call the action $\mathcal{E}$ _essentially exact_ if for each
$I\in\mathbb{I}^{\mathcal{E}}(A)$ there is a ∗-homomorphism
$\kappa\rtimes_{\mathrm{ess}}S\colon A\rtimes_{\mathrm{ess}}S\to
A/I\rtimes_{\mathrm{ess}}S$ whose restriction to each fibre $\mathcal{E}_{t}$
is the quotient map onto $\mathcal{E}_{t}/\mathcal{E}_{t}I$, $t\in S$, and the
kernel of $\kappa\rtimes_{\mathrm{ess}}S$ is
$\iota\rtimes_{\mathrm{ess}}S(I\rtimes_{\mathrm{ess}}S)$.

######  Remark 4.24.

When the action $\mathcal{E}$ is closed, then the reduced and essential
crossed products coincide for all restrictions of $\mathcal{E}$ (see
Proposition 4.14). Thus essential exactness is the same as exactness for
closed actions of inverse semigroups and for Fell bundles over Hausdorff
groupoids.

######  Example 4.25 (Essentially exact Fell bundles).

Consistently with Example 4.19, we will call a Fell bundle
$\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ over an étale groupoid essentially
exact if the corresponding action $\mathcal{E}$ is essentially exact. More
specifically, by Lemma 4.20, for any $G$-invariant ideal $I$ in $A$, the
inclusion
$\textup{C}_{\textup{c}}(\mathcal{A}|_{I})\subseteq\textup{C}_{\textup{c}}(\mathcal{A})$
extends to an injective ∗-homomorphism
$\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A}|_{I})\rightarrowtail\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A})$.
So $\mathcal{A}$ is _essentially exact_ if and only if, for every
$G$-invariant ideal $I$ in $A$, restriction of sections gives a well-defined
∗-homomorphism
$\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A})\twoheadrightarrow\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A}|_{A/I})$
and the following sequence is exact:

$\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A}|_{I})\rightarrowtail\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A})\twoheadrightarrow\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A}|_{A/I}).$

If the $S$-action on $A$ is residually aperiodic, then $A$ separates ideals in
$A\rtimes_{\mathrm{ess}}S$ if and only if the $S$-action on $A$ is essentially
exact (see Theorem 5.9 below). In Example 4.22, however, $A$ separates ideals
in $A\rtimes_{\mathrm{ess}}S$ although the $S$-action on $A$ is not
essentially exact. The following proposition shows that the $S$-action on $A$
must be essentially exact if $A$ separates ideals in $A\rtimes_{\mathrm{r}}S$.

######  Proposition 4.26.

The following are equivalent:

  1. (1)

$A$ separates ideals in the reduced crossed product $A\rtimes_{\mathrm{r}}S$;

  2. (2)

the action is exact and for each $I\in\mathbb{I}^{\mathcal{E}}(A)$, $A/I$
detects ideals in $A/I\rtimes_{\mathrm{r}}S$.

If these equivalent conditions hold, then
$A/I\rtimes_{\mathrm{r}}S=A/I\rtimes_{\mathrm{ess}}S$ for every
$I\in\mathbb{I}^{\mathcal{E}}(A)$ and the action is essentially exact.

###### Proof.

Lemma 2.3 and Proposition 2.12 show that $A$ separates ideals in
$A\rtimes_{\mathrm{r}}S$ if and only if $A/I$ detects ideals in
$A\rtimes_{\mathrm{r}}S/\iota_{\mathrm{r}}\rtimes S(I\rtimes_{\mathrm{r}}S)$
for all $I\in\mathbb{I}^{\mathcal{E}}(A)$. Then the action is exact because
the kernel of $\kappa\rtimes_{\mathrm{r}}S$ has to be
$\iota_{\mathrm{r}}\rtimes S(I\rtimes_{\mathrm{r}}S)$, and then
$A\rtimes_{\mathrm{r}}S/\iota_{\mathrm{r}}\rtimes
S(I\rtimes_{\mathrm{r}}S)\cong A/I\rtimes S$ for all
$I\in\mathbb{I}^{\mathcal{E}}(A)$. Thus (1) and (2) are equivalent. Condition
(2) implies $A/I\rtimes_{\mathrm{r}}S=A/I\rtimes_{\mathrm{ess}}S$ because
otherwise $A/I\rtimes_{\mathrm{ess}}S$ would be a quotient of
$A/I\rtimes_{\mathrm{r}}S$ by a nonzero ideal not detected by $A/I$. ∎

We illustrate by an example what can go wrong with the exactness of essential
crossed products. Our example is closely related to the Reeb foliation or,
more precisely, to its restriction to a transversal.

######  Example 4.27.

Let $\vartheta\colon\mathbb{R}\to\mathbb{R}$ be a homeomorphism with
$\vartheta(t)=t$ for $t\leq 0$ and $\vartheta(t)>t$ for all $t>0$. Let $G$ be
the germ groupoid of the transformation groupoid
$\mathbb{R}\times_{\vartheta}\mathbb{Z}$. We claim that
$\textup{C}^{*}_{\mathrm{r}}(G)\cong\textup{C}^{*}_{\mathrm{ess}}(G)$. To see
this, we use that the restrictions of $G$ to $[0,\infty)$ and $(-\infty,0)$
are Hausdorff. Indeed, $G|_{(0,\infty)}$ is the Hausdorff transformation
groupoid $(0,\infty)\rtimes_{\vartheta}\mathbb{Z}$ because $\mathbb{Z}$ acts
freely (and properly) on $(0,\infty)$. Similarly, $G|_{[0,\infty)}$ is the
Hausdorff transformation groupoid $[0,\infty)\rtimes_{\vartheta}\mathbb{Z}$;
the action of $\mathbb{Z}$ on $[0,\infty)$ fixes $0$, but the germ of any
$n\in\mathbb{Z}$ at $0$ is nontrivial because $\vartheta^{n}$ acts
nontrivially on $(0,\infty)$. Therefore, the support of any nonzero element of
$\textup{C}^{*}_{\mathrm{r}}(G)$ must intersect $[0,\infty)$ and $(-\infty,0)$
in relatively open subsets. Hence the support cannot be meagre, and this
proves our claim. This equality of reduced and essential groupoid
$\textup{C}^{*}$-algebras for $G$ is not inherited by the restriction to the
closed invariant subset $(-\infty,0]$. Indeed, the restriction of $G$ to
$(-\infty,0]$ is the non-Hausdorff group bundle with trivial fibre over
$(-\infty,0)$ and the fibre $\mathbb{Z}$ at $0$ (see [Kwasniewski-
Meyer:Essential]*Example 4.7 and Example 4.22). In this case, the full and
reduced groupoid $\textup{C}^{*}$-algebras are obtained by gluing together
$\textup{C}_{0}((-\infty,0))$ and the fibre $\textup{C}^{*}(\mathbb{Z})$ at
$0$. However,
$\textup{C}^{*}_{\mathrm{ess}}(G|_{(-\infty,0]})=\textup{C}_{0}((-\infty,0])$,
so

$\textup{C}^{*}_{\mathrm{r}}(G|_{(-\infty,0]})\not\cong\textup{C}^{*}_{\mathrm{ess}}(G|_{(-\infty,0]}).$

Since the essential and reduced crossed products coincide for $G$ and
$G|_{(0,\infty)}$, but not for $G|_{(-\infty,0]}$, the following sequence of
essential crossed products exists, but fails to be exact:

$0\to\textup{C}^{*}_{\mathrm{ess}}(G|_{(0,\infty)})\to\textup{C}^{*}_{\mathrm{ess}}(G)\to\textup{C}^{*}_{\mathrm{ess}}(G|_{(-\infty,0]})\to
0.$

This is one way how essential crossed products may fail to be exact. The
restriction $G|_{\\{0\\}}$ is simply the group $\mathbb{Z}$. So
$\textup{C}^{*}_{\mathrm{r}}(G|_{\\{0\\}})=\textup{C}^{*}_{\mathrm{ess}}(G|_{\\{0\\}})\cong\textup{C}^{*}(\mathbb{Z})$.
The restriction ∗-homomorphism
$\textup{C}^{*}(G|_{(-\infty,0]})\to\textup{C}^{*}(G|_{\\{0\\}})$ does not
descend to the essential crossed products. That is, there is no canonical map
from $\textup{C}^{*}_{\mathrm{ess}}(G|_{(-\infty,0]})$ to
$\textup{C}^{*}_{\mathrm{ess}}(G|_{\\{0\\}})$. This is the second way how
essential crossed products may fail to be exact.

Since $G|_{[0,\infty)}\cong[0,\infty)\rtimes_{\vartheta}\mathbb{Z}$, it
follows that
$\textup{C}^{*}(G|_{[0,\infty)})\cong\textup{C}^{*}_{\mathrm{r}}(G|_{[0,\infty)})$.
By a diagram, it follows that the sequence

$0\to\textup{C}^{*}_{\mathrm{r}}(G|_{(-\infty,0)})\to\textup{C}^{*}_{\mathrm{r}}(G)\to\textup{C}^{*}_{\mathrm{r}}(G|_{[0,\infty)})\to
0$

is exact. Since also
$\textup{C}^{*}(G|_{(-\infty,0)})\cong\textup{C}^{*}_{\mathrm{r}}(G|_{(-\infty,0)})$,
the Five Lemma shows that
$\textup{C}^{*}(G)\cong\textup{C}^{*}_{\mathrm{r}}(G)$. A similar argument
shows $\textup{C}^{*}(G|_{U})\cong\textup{C}^{*}_{\mathrm{r}}(G|_{U})$ for all
locally closed $G$-invariant subsets $U\subseteq\mathbb{R}$. Therefore, $G$ is
inner exact.

###  4.5. Amenability vs exactness

######  Definition 4.28.

Let $\mathcal{E}$ be an $S$-action by Hilbert bimodules on a
$\textup{C}^{*}$-algebra $A$. We call the action $\mathcal{E}$ _amenable_ if
the regular representation $\Lambda\colon A\rtimes S\to
A\rtimes_{\mathrm{r}}S$ is an isomorphism.

######  Remark 4.29.

By (3.13), the action $\mathcal{E}$ is amenable if and only if the weak
conditional expectation $E\colon A\rtimes S\to A^{\prime\prime}$ is faithful.

######  Lemma 4.30.

Let $\mathcal{E}$ be an $S$-action by Hilbert bimodules on a
$\textup{C}^{*}$-algebra $A$ and let $I\in\mathbb{I}^{\mathcal{E}}(A)$ be an
invariant ideal. If $\mathcal{E}|_{A/I}$ is amenable, then

$I\rtimes_{\mathrm{r}}S\rightarrowtail
A\rtimes_{\mathrm{r}}S\twoheadrightarrow A/I\rtimes_{\mathrm{r}}S$

is exact. If this sequence is exact, then $\mathcal{E}$ is amenable if and
only if $\mathcal{E}|_{I}$ and $\mathcal{E}|_{A/I}$ are amenable.

###### Proof.

There is a commutative diagram

${I\rtimes S}$${A\rtimes S}$${A/I\rtimes
S}$${I\rtimes_{\mathrm{r}}S}$${A\rtimes_{\mathrm{r}}S}$${A/I\rtimes_{\mathrm{r}}S,}$$\scriptstyle{\iota\rtimes
S}$$\scriptstyle{\Lambda^{I}}$$\scriptstyle{\kappa\rtimes
S}$$\scriptstyle{\Lambda^{A}}$$\scriptstyle{\Lambda^{A/I}}$$\scriptstyle{\iota\rtimes_{\mathrm{r}}S}$$\scriptstyle{\kappa\rtimes_{\mathrm{r}}S}$

where the top horizontal sequence is exact, and $\Lambda^{I}$, $\Lambda^{A}$,
$\Lambda^{A/I}$ denote the respective regular representations. If
$\Lambda^{A/I}$ is injective, then

$\ker(\kappa\rtimes_{\mathrm{r}}S)=\Lambda^{A}\left(\ker(\kappa\rtimes
S)\right)=\Lambda^{A}\left(\iota\rtimes S(I\rtimes
S)\right)=\iota\rtimes_{\mathrm{r}}S(I\rtimes_{\mathrm{r}}S).$

Hence $I\rtimes_{\mathrm{r}}S\rightarrowtail
A\rtimes_{\mathrm{r}}S\twoheadrightarrow A/I\rtimes_{\mathrm{r}}S$ is exact.
In general, if this sequence is exact, the Snake Lemma from homological
algebra yields a short exact sequence

$\ker(\Lambda^{I})\rightarrowtail\ker(\Lambda^{A})\twoheadrightarrow\ker(\Lambda^{A/I}).$

Hence $\ker(\Lambda^{A})=0$ if and only if $\ker(\Lambda^{I})=0$ and
$\ker(\Lambda^{A/I})=0$. ∎

######  Remark 4.31.

It is unclear whether the quotient action $\mathcal{E}|_{A/I}$ for
$I\in\mathbb{I}^{\mathcal{E}}(A)$ is amenable if $\mathcal{E}$ is (the proof
of [Kwasniewski-Szymanski:Pure_infinite]*Lemma 3.9 is incorrect). If so, then
by Lemma 4.30, the amenable action is also exact. Let $S=G$ be a group. If
$\mathcal{E}$ satisfies the approximation property in
[Exel:Amenability]*Definition 4.4, then also every restriction
$\mathcal{E}|_{A/I}$ satisfies the approximation property. Hence the
approximation property of a Fell bundle over a group implies both amenability
and exactness. In particular, if there is an amenable group action
$\mathcal{E}$ such that $\mathcal{E}|_{A/I}$ is not amenable for some
$I\in\mathbb{I}^{\mathcal{E}}(A)$, then the approximation property is strictly
stronger then amenability (see also [Exel:Amenability]*Page 169).

######  Example 4.32.

Let $\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ be a Fell bundle over an étale,
locally compact, Hausdorff groupoid $G$. Suppose that $G$ is second countable
and that each fibre $A_{\gamma}$ is separable. If $G$ is amenable, then the
regular representation
$\textup{C}^{*}(\mathcal{A})\twoheadrightarrow\textup{C}^{*}_{\mathrm{r}}(\mathcal{A})$
is an isomorphism by [Sims-
Williams:Amenability_for_Fell_bundles_over_groupoids]*Theorem 1. This applies
also to every Fell bundle $\mathcal{A}_{A/I}$ over $G$ where $I$ is a
$G$-invariant ideal in $A$. Thus if $G$ is amenable, then $\mathcal{A}$ is
exact in the sense described in Example 4.19.

######  Example 4.33.

If $\mathcal{E}$ is a closed action of a countable inverse semigroup $S$ on a
separable, commutative $\textup{C}^{*}$-algebra $A=\textup{C}_{0}(X)$ such
that $A\rtimes_{\mathrm{r}}S$ is nuclear, then the action $\mathcal{E}$ is
amenable and exact. Indeed, the action $\mathcal{E}$ corresponds to a twisted
étale, locally compact, Hausdorff groupoid $(G,\Sigma)$, where $G=X\rtimes S$
is the dual groupoid of $\mathcal{E}$. Equivalently, this corresponds to a
Fell line bundle over $G$. Then $G$ is amenable by
[Takeishi:Nuclearity]*Theorem 5.4. Hence our claim follows from Example 4.32.
In particular, a twisted étale, locally compact, second countable, Hausdorff
groupoid $(G,\Sigma)$ with nuclear $\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)$ is
exact in the sense of Example 4.18.

##  5\. Ideals and pure infiniteness for inverse semigroup crossed products

In this section we present efficient criteria for separation of ideals and
pure infiniteness in essential crossed products.

###  5.1. Residual aperiodicity and ideal structure

Let $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ be an action of a unital inverse
semigroup $S$ by Hilbert bimodules on a $\textup{C}^{*}$-algebra $A$.

######  Definition 5.1 ([Kwasniewski-Meyer:Essential]*Definition 6.1).

The action $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ is _aperiodic_ if the
Hilbert $A$-bimodules $\mathcal{E}_{t}\cdot I_{1,t}^{\bot}$ are aperiodic for
all $t\in S$, where $I_{1,t}$ is defined in (3.7).

######  Proposition 5.2 ([Kwasniewski-Meyer:Essential]*Proposition 6.3).

Let $B$ be an $S$-graded $\textup{C}^{*}$-algebra with a grading
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$. Let
$A\mathrel{\vcentcolon=}\mathcal{E}_{1}\subseteq B$ and turn $\mathcal{E}$
into an $S$-action on $A$. If this action is aperiodic, then the inclusion
$A\subseteq B$ is aperiodic. The converse holds if the grading on $B$ is
topological, that is, the canonical quotient map $A\rtimes S\twoheadrightarrow
A\rtimes_{\mathrm{ess}}S$ factors through $A\twoheadrightarrow B$.

######  Definition 5.3 ([Kwasniewski-Meyer:Essential]*Definition 2.20).

Let $G$ be an étale groupoid and $X\subseteq G$ its unit space. The _isotropy
group_ of a point $x\in X$ is $G(x)\mathrel{\vcentcolon=}s^{-1}(x)\cap
r^{-1}(x)\subseteq G$. We call $G$ _topologically free_ if, for every open
$U\subseteq G\setminus X$, the set $\\{x\in X\,{:}\,\mathopen{}G(x)\cap
U\neq\emptyset\\}$ has empty interior.

######  Remark 5.4.

A groupoid $G$ is _effective_ if any open subset $U\subseteq G$ with
$r|_{U}=s|_{U}$ is contained in $X$. Effective groupoids are topologically
free. The converse implication holds if $X$ is closed in $G$, but not in
general.

Aperiodicity is related in [Kwasniewski-Meyer:Aperiodicity, Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations, Kwasniewski-Meyer:Essential] to
topological freeness and several other conditions. We now introduce residual
versions of aperiodicity and topological freeness.

######  Definition 5.5.

The action $\mathcal{E}$ is _residually aperiodic_ if, for each
$I\in\mathbb{I}^{\mathcal{E}}(A)$, the restricted action $\mathcal{E}|_{A/I}$
is aperiodic.

######  Definition 5.6.

An étale groupoid $G$ with unit space $X\subseteq G$ is _residually
topologically free_ if, for each nonempty closed $G$-invariant subset
$Y\subseteq X$, the restricted groupoid $r^{-1}(Y)=s^{-1}(Y)$ is topologically
free.

The following lemma may help to show that a transformation groupoid is
(residually) topologically free.

######  Lemma 5.7.

Let $G$ be an étale groupoid. Let $X$ and $Y$ be topological spaces with
$G$-actions and let $f\colon X\to Y$ be a continuous $G$-equivariant map.

  1. (1)

if $Y$ is closed in $G\ltimes Y$, then $X$ is closed in $G\ltimes X$

  2. (2)

if $f\colon X\to Y$ is open and $G\ltimes Y$ is topologically free, then so is
$G\ltimes X$.

  3. (3)

if $f\colon X\to Y$ is open, maps closed $G$-invariant sets to closed sets,
and $G\ltimes Y$ is residually topologically free, then so is $G\ltimes X$.

###### Proof.

The map $f$ induces a continuous groupoid homomorphism $f_{*}\colon G\ltimes
X\to G\ltimes Y$ such that $f_{*}^{-1}(Y)=X$. So (1) follows. The map $f_{*}$
is open if and only if $f$ is open. Assume this. To show (2) assume that
$G\ltimes Y$ is topologically free and let $U\subseteq(G\ltimes X)\setminus
X$. Then $f_{*}(U)$ is open and contained in $(G\ltimes Y)\setminus Y$. Let
$I_{U}\mathrel{\vcentcolon=}\\{x\in X\,{:}\,\mathopen{}G(x)\cap
U\neq\emptyset\\}$ and $I_{f^{*}(U)}\mathrel{\vcentcolon=}\\{y\in
Y\,{:}\,\mathopen{}G(y)\cap f_{*}(U)\neq\emptyset\\}$. We claim that
$f(I_{U})=I_{f^{*}(U)}$. To see this, let $x\in I_{U}$. Then there is an arrow
$(x,g,x)\in U$; here $g\in G$ is such that $\varrho(x)=s(g)$ and $g\cdot x=x$.
Since $f$ is $G$-equivariant, then $(f(x),g,f(x))\in f_{*}(U)$. This witnesses
that $f(x)\in I_{f^{*}(U)}$. Since $G\ltimes Y$ is topologically free,
$I_{f^{*}(U)}$ has empty interior. Since $f$ is open, the preimage of
$I_{f^{*}(U)}$ in $X$ has empty interior as well. It follows that $I_{U}$ has
empty interior. This witnesses that $G\ltimes X$ is topologically free and
proves (2).

Finally, for any $G$-invariant set $D\subseteq X$ the set $f(D)$ is
$G$-invariant. If we assume $f(D)$ is closed in $Y$ and $G\ltimes Y$ is
residually topologically free, then $G\ltimes f(D)$ is topologically free.
Since the restriction of $f_{*}$ to $G\ltimes D$ is a continuous open map onto
$G\ltimes f(D)$, (2) implies that $G\ltimes D$ is topologically free. This
proves (3). ∎

######  Theorem 5.8.

Let $\mathcal{E}$ be an action of a unital inverse semigroup $S$ on a