# Regular ideals, ideal intersections, and quotients

Jonathan H. Brown  Department of Mathematics
University of Dayton
Dayton
OH 45469-2316 U.S.A<EMAIL_ADDRESS>,  Adam H. Fuller
Department of Mathematics
Ohio University
Athens
OH 45701 U.S.A<EMAIL_ADDRESS>,  David R. Pitts  Department of Mathematics
University of Nebraska-Lincoln
Lincoln
NE 68588 U.S.A<EMAIL_ADDRESS>and  Sarah A. Reznikoff  Department of
Mathematics
Virginia Tech
Blacksburg, VA 24061-1026 U.S.A<EMAIL_ADDRESS>
###### Abstract.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. We study the
relationship between the regular ideals of $B$ and regular ideals of $A$. We
show that if $B\subseteq A$ is a regular $C^{*}$-inclusion and there is a
faithful invariant conditional expectation from $A$ onto $B$, then there is an
isomorphism between the lattice of regular ideals of $A$ and invariant regular
ideals of $B$. We study properties of inclusions preserved under quotients by
regular ideals. This includes showing that if $D\subseteq A$ is a Cartan
inclusion and $J$ is a regular ideal in $A$, then $D/(J\cap D)$ is a Cartan
subalgebra of $A/J$. We provide a description of regular ideals in reduced
crossed products $A\rtimes_{r}\Gamma$.

This work was supported by a grants from the Simons Foundation (DRP #316952,
SAR #36563); and by the American Institute of Mathematics SQuaREs Program. The
authors have no relevant financial or non-financial interests to disclose.

##  1\. Introduction

An inclusion $B\subseteq A$ of $C^{*}$-algebras has the ideal intersection
property if every non-trivial ideal of $A$ has non-trivial intersection with
$B$. The ideal intersection property is useful for obtaining structural
results for the inclusion. Here are a few of the many examples.

  1. (i)

When $B$ is a maximal abelian subalgebra of $A$, the ideal intersection
property is a key ingredient for establishing the Cuntz-Kreiger Uniqueness
theorems for graph $C^{*}$-algebras (see for example [35, Theorem 3.13]) and
for groupoids [11, Theorem 3.1]. Further work on the ideal intersection
property in groupoid C∗-algebras has been carried out in [7] and [26].

  2. (ii)

When there is an action of ${\mathbb{T}}$ on $A$, and $B$ is the fixed point
algebra, the ideal intersection property of $A$ is used to establish the
gauge-invariant uniqueness theorems for graph algebras [1, Theorem 2.3] and
Cuntz-Pimsner algebras [23, Theorem 6.2] among others.

  3. (iii)

If $G$ is a discrete group acting topologically freely on a locally compact
Hausdorff space $X$, the ideal intersection property for $C(X)\subseteq
C(X)\rtimes_{r}G$ is used to obtain the simplicity result of [3, Theorem 2].

  4. (iv)

The ideal intersection property has been used to characterize inclusions for
which every pseudo-expectation is faithful [40, Theorem 3.5].

  5. (v)

More recently, Pitts [38, Theorem 5.2] used the ideal intersection property to
characterize the existence of Cartan envelopes.

The ideal intersection property holds for certain crossed products. Necessary
and sufficient conditions for $C(X)\subseteq C(X)\rtimes_{r}G$ to have the
ideal intersection property are given in [24]. Sierakowski [46] shows that
$A\subseteq A\rtimes_{r}G$ has the ideal intersection property for an exact
action of discrete group $G$ on $C^{*}$-algebra $A$ whose induced action on
$\operatorname{Prim}(A)$ is essentially free. Kennedy and Schafhauser [27]
further characterize when $A\subseteq A\rtimes_{r}G$ has the ideal
intersection property in terms of outerness and a cohomological obstruction.

Unfortunately, the ideal intersection property does not pass, in general, to
quotients (see [41, Example 4.8] or Example 4.6 below). In the setting of
amenable groupoid $C^{*}$-algebras, that is the inclusion
$C_{0}(G^{(0)})\subseteq C^{*}(G)$, the ideal intersection property passes to
quotients if and only if there is a correspondence between the lattice of
ideals of $C_{0}(G^{(0)})$ and $C^{*}(G)$ [8, Corollary 5.9] (for one
direction see [43, Corollary 4.9]). Restricting attention to a class of
inclusions where the ideal intersection property passes to quotients has been
quite productive in the study of ideals of these algebras. For example,
Kumjian et al. [29] introduce condition (K) so they could use [43, Corollary
4.9] to study the ideals for this restricted class of graph $C^{*}$-algebras.
Indeed, for algebras of row-finite directed graphs $C^{*}(E)$ whose underlying
graph $E$ satisfies condition (K), the inclusion $D\subseteq C^{*}(E)$, where
$D$ is the (abelian) algebra generated by the range projections of partial
isometries corresponding to finite paths in $E$, has the property that the
ideal intersection property passes to quotients.

Rather than restricting the types of inclusions considered, the paper [10]
takes a different tack, instead limiting the types of ideals to regular
ideals: a main result of [10] shows that the ideal intersection property
passes to quotients of graph algebras by regular ideals.

For a commutative $C^{*}$-algebra $B$, regular ideals in $B$ correspond
exactly to regular open sets in $\widehat{B}$. More generally, for an
arbitrary $C^{*}$-algebra $B$, let $\overline{B}$ be the monotone completion
of $B$ and let $Z(\overline{B})$ be the center of $\overline{B}$. Hamana shows
that the lattice of regular open sets in $\operatorname{Prim}(B)$ is
isomorphic to the projection lattice of $Z(\overline{B})$, which in turn is
isomorphic to the lattice of all regular ideals of $B$, see [21, Lemma 1.4 and
Theorem 1.5]. Hamana concludes in [21, p. 526, Remark (b)] that if $I(B)$ is
the injective envelope of $B$, and $A$ is a $C^{*}$-algebra with $B\subseteq
A\subseteq I(B)$, then the map $J\mapsto J\cap B$ gives an isomorphism of the
Boolean algebra of regular ideals of $A$ onto the Boolean algebra of regular
ideals of $B$. This conclusion is similar to our Theorem 3.24 below, but we
consider inclusions of the form $B\subseteq A$ having certain regularity
properties and a faithful invariant conditional expectation of $A$ onto $B$
instead of assuming that $A$ lies between $B$ and its injective envelope.

One of the main results of the present paper is Theorem 4.2, which shows that
if $B\subseteq A$ is an inclusion having a faithful invariant conditional
expectation of $A$ onto $B$, then the ideal intersection property is preserved
under quotients by regular ideals. This is a far reaching generalization of
the main result in [10], as many inclusions of interest, such as reduced
crossed products by discrete groups, come equipped with a faithful invariant
conditional expectation. Along the way we give a detailed analysis of regular
ideals, culminating in Theorem 3.24, which shows that if $B\subseteq A$ is a
regular inclusion with the ideal intersection property and $E:A\to B$ is a
faithful invariant conditional expectation, then $J\to J\cap B$ is a one-to-
one map from the regular ideals of $A$ to the regular ideals of $B$.

Establishing that an inclusion is a Cartan inclusion has become important from
the lens of classification. Indeed, Barlak and Li [5, Theorem 1.1] show that
when $A$ is a separable and nuclear $C^{*}$-algebra containing the Cartan MASA
$B$, then $A$ satisfies the UCT. We show in Theorem 4.8 that quotients of
Cartan embeddings by regular ideals remain Cartan.

One of the interesting aspects of our work is that the key to many of our
results is the regular ideal intersection property: $J\cap B=\\{0\\}\implies
J=\\{0\\}$ for all _regular_ ideals $J$ in $A$. In Section 7, we show that for
a very large class of examples, the regular ideal intersection property and
the ideal intersection property are equivalent.

After we circulated an earlier version of this paper, Exel [14] proved a
generalization of Theorem 3.24 using weaker hypotheses than are assumed here.
In particular, Exel does not assume the existence of an invariant faithful
conditional expectation. Instead Exel assumes that $B\subseteq A$ satisfies an
invariance axiom; see [14] for full details.

This paper is organized as follows. We begin with a short section of
preliminaries where we introduce regular ideals. In Section 3 we analyze the
relationship between regular ideals of $B$ and $A$ where $B\subseteq A$ is a
regular inclusion. This culminates in Theorem 3.24; this result gives settings
in which the Boolean algebras of regular ideals in $A$ are isomorphic to the
Boolean algebra of regular and invariant ideals in $B$. In Section 4, we prove
our main theorem, Theorem 4.2, which shows that in the presence of a faithful,
invariant conditional expectation, the ideal intersection property is
inherited by quotients of regular ideals. We use this result to show that the
Cartan property passes to quotients by regular ideals. We then specialize to
$C^{*}$-algebras of exact groupoids in Section 5. In particular, Theorem 5.6
gives an explicit description of the regular ideals of the (reduced)
$C^{*}$-algebra of a twisted exact groupoid $G$ when $C_{0}(G^{(0)})$ has the
ideal intersection property. Section 6 explores some applications of our work,
including applications to graph $C^{*}$-algebras and reduced crossed products
by discrete groups. In Section 7, we give classes of regular inclusions for
which the regular ideal intersection property implies the ideal intersection
property.

### Acknowledgments

We would like to thank the anonymous referee for their helpful feedback on an
earlier draft of this work. We also wish to thank Kang Li for pointing out an
error in our original proof of Proposition 3.22.

##  2\. Preliminaries on regular ideals

The focus of this paper is the family of regular ideals in a $C^{*}$-algebra.
We will recall the definition from [21] after we introduce a piece of
notation. Let $A$ be a $C^{*}$-algebra and let $X\subseteq A$. Denote by
$X^{\perp}$ the set

$X^{\perp}=\\{a\in A\colon ax=xa=0\text{ for all }x\in X\\}.$

If $B$ is a $C^{*}$-algebra such that $X\subseteq B\subseteq A$ we will write
$X^{\perp_{B}}$ to denote the set determined by the operation restricted to
$B$:

$X^{\perp_{B}}=X^{\perp}\cap B=\\{b\in B\colon bx=xb=0\text{ for all }x\in
X\\}.$

We can now define what it means for an ideal to be _regular_.

######  Definition 2.1.

Let $A$ be a C∗-algebra. If $J\subseteq A$ is a subalgebra satisfying $aJ\cup
Ja\subseteq A$ for all $a\in A$, we will call $J$ an _algebraic ideal_ of $A$.
If $J\subseteq A$ is a closed algebraic ideal we will call $J$ an _ideal_ of
$A$.

We use the notation $J\unlhd A$ to say that $J\subseteq A$ is an ideal of $A$.

######  Definition 2.2.

An ideal $J\unlhd A$ is a _regular ideal_ in $A$ if $J=J^{\perp\perp}$.

Hamana [21] gives two characterizations of the regular ideals in a
$C^{*}$-algebra $A$: one in terms of the topology on the primitive ideal space
$\operatorname{Prim}(A)$, and one in terms the monotone closure $\overline{A}$
of $A$. We end this section with his characterization involving the primitive
ideal space, as we require it in the sequel.

Let $X$ be a topological space. For an open set $U\subseteq X$, define

$U^{\perp}={(X\backslash U)}^{\circ}.$

That is, $U^{\perp}$ is the interior of the complement of $U$. Recall that an
open set $U\subseteq X$ is _regular_ if $U={(\overline{U})}^{\circ}$, i.e. $U$
is the interior of its closure. Equivalently, an open set $U$ is regular if
$U=U^{\perp\perp}$. The regular open sets of $X$ form a complete Boolean
algebra with operations:

  1. (i)

$U\wedge V=U\cap V$;

  2. (ii)

$U\vee V=(U\cup V)^{\perp\perp}$; and

  3. (iii)

$\lnot U=U^{\perp};$

see, e.g. [20] or [17].

######  Remark 2.3.

The ideal $C_{0}(U)\subseteq C_{0}(X)$ is regular if and only if $U$ is a
regular open set in $X$.

Let $A$ be a $C^{*}$-algebra and let $\operatorname{Prim}(A)$ be the primitive
ideal space. For $R\subseteq A$ define

$\operatorname{hull}(R)=\\{P\in\operatorname{Prim}(A)\colon R\subseteq P\\},$

and for $S\subseteq\operatorname{Prim}(A)$ define

$\ker(S)=\bigcap_{P\in S}P.$

Endow $\operatorname{Prim}(A)$ with the usual hull-kernel topology. That is,
if $S\subseteq\operatorname{Prim}(A)$ then the closure $\overline{S}$ is given
by $\overline{S}=\operatorname{hull}\ker(S).$ The map
$I\mapsto\operatorname{Prim}(A)\backslash\operatorname{hull}(I)$ gives a one-
to-one correspondence between ideals $I\unlhd A$ and open subsets of
$\operatorname{Prim}(A)$ [34, Theorem 5.2.7]. When this map is restricted to
the regular ideals of $A$, Hamana establishes the following result.

######  Proposition 2.4 (c.f. [21, Lemma 1.4]).

Let $A$ be a $C^{*}$-algebra. Then the collection of all regular ideals in $A$
forms a complete Boolean algebra with meet, join and negation given by

  1. (i)

$J\wedge K:=J\cap K$;

  2. (ii)

$J\vee K:=\langle J\cup K\rangle^{\perp\perp}$;

  3. (iii)

$\lnot J:=J^{\perp}$.

Furthermore, the map
$I\mapsto\operatorname{Prim}(A)\backslash\operatorname{hull}(I)$ on regular
ideals of a $C^{*}$-algebra $A$ gives a complete Boolean algebra isomorphism
from the regular ideals of $A$ to the regular open sets of
$\operatorname{Prim}(A)$.

##  3\. Regular ideals in regular $C^{*}$-inclusions

######  Definition 3.1.

Let $A$ and $B$ be $C^{*}$-algebras such that $B\subseteq A$. If $B$ contains
an approximate unit for $A$, we say that $B\subseteq A$ is an _inclusion of
$C^{*}$-algebras._ We will sometimes use the notation $(A,B)$ for an inclusion
of $C^{*}$-algebras (we write the larger algebra first).

######  Remark 3.2.

The condition that $B$ contain an approximate unit for $A$ is sometimes
automatic. For example, when $B$ is maximal abelian and
$\operatorname{span}\\{n\in A:nBn^{*}\cup n^{*}Bn\subseteq B\\}$ is dense in
$A$, [39, Theorem 2.6] shows that every approximate unit for $B$ is an
approximate unit for $A$.

If $B\subseteq A$ is an inclusion of $C^{*}$-algebras, we want to know to what
extent the regular ideals of $A$ determine the regular ideals of $B$, and vice
versa. In general, one should not expect any strong relationship. However,
under some additional hypotheses, we show in Theorem 3.24 that there is an
injective map from the regular ideals of $A$ to the invariant regular ideals
of $B$. We will use this map in our main results on quotients by regular
ideals, Theorem 4.2 and Theorem 4.8.

We use the following notation extensively. For $I\subseteq A$ we denote the
ideal generated by $I$ in $A$ by $\langle I\rangle_{A}$. If there is no chance
of confusion as to where the ideal should lie we will simply write $\langle
I\rangle$ in place of $\langle I\rangle_{A}$.

The following elementary facts will be useful.

######  Lemma 3.3.

If $J\subseteq A$ is an algebraic ideal, then $J^{\perp}\unlhd A$ is a regular
ideal.

######  Lemma 3.4.

Let $A$ be a $C^{*}$-algebra. Suppose $I,\ J\subseteq A$ are algebraic ideals
satisfying $J^{\perp}\subseteq I.$ Then $J^{\perp}=I$ or $I\cap J\neq\\{0\\}.$

######  Proof..

Suppose $J^{\perp}\neq I$. Take $x\in I\backslash J^{\perp}$. Since $x\notin
J^{\perp}$, we have $xJ\neq\\{0\\}$ or $Jx\neq\\{0\\}$. Hence $I\cap
J\neq\\{0\\}$. ∎

We now impose some natural extra hypotheses on the inclusions we study. We
first show that if there is a faithful conditional expectation from $A$ onto
$B$ then $J\cap B$ is a regular ideal of $B$ whenever $J$ is a regular ideal
of $A$.

######  Lemma 3.5.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. Suppose $E\colon
A\rightarrow B$ is a faithful conditional expectation. Then for any $J\unlhd
A$ we have

$J^{\perp}\cap B=E(J)^{\perp_{B}}.$

######  Proof..

Take $d\in E(J)^{\perp_{B}}$ and $a\in J$. Then

$E(d^{*}a^{*}ad)=d^{*}E(a^{*}a)d=0.$

Since $E$ is faithful, it follows that $ad=0$. Similarly $da=0$. Hence $d\in
J^{\perp}$.

Suppose now that $d\in J^{\perp}\cap B$ and $a\in J$. Then $dE(a)=E(da)=0$,
and $E(a)d=E(ad)=0$. Hence $d\in E(J)^{\perp_{B}}.$ ∎

######  Remark 3.6.

While Lemma 3.5 is stated for an ideal $J\unlhd A$, the conclusion holds for
any subset $J\subseteq A$.

######  Proposition 3.7.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. Suppose $E\colon
A\rightarrow B$ is a faithful conditional expectation. If $J\unlhd A$ is a
regular ideal of $A$, then $J\cap B$ is a regular ideal of $B$.

######  Proof..

Let $J\unlhd A$ be a regular ideal. Then, by regularity of $J$ and Lemma 3.5,
we get

$J\cap B=J^{\perp\perp}\cap B=E(J^{\perp})^{\perp_{B}}.$

It follows now by Lemma 3.3 that $J\cap B$ is a regular ideal in $B$. ∎

We recall the following definitions.

######  Definition 3.8.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. An element $n\in A$ is
a _normalizer of $B$_ if $nBn^{*}\subseteq B$ and $n^{*}Bn\subseteq B$. We
denote the set of all normalizers of $B$ in $A$ by ${\mathcal{N}}(A,B)$. If

$A=\overline{\operatorname{span}}{\mathcal{N}}(A,B)$

then we say $B$ is _regular in $A$_, or $B\subseteq A$ is a _regular inclusion
of $C^{*}$-algebras_.

######  Remark 3.9.

We are now using the term regular in two different senses: one refers to
ideals and the other to inclusions. The context will make clear which sense we
intend. This unfortunate development is the result of historical naming
conventions. In the first sense, regular ideal, the term goes back to at least
Hamana [21], and seems to be derived from the notion of regular open sets in
topology. The second, regular inclusion, goes back to at least to both
Vershik’s and Feldman and Moore’s early work on Cartan inclusions of von
Neumann algebras, see [51] and [16].

######  Remark 3.10.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. Note that if
$n\in{\mathcal{N}}(A,B)$, then $n^{*}n\in B$. This follows from the assumption
that there is an approximate unit for $B$ that also an approximate unit for
$A$.

######  Definition 3.11.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras. Let
$N\subseteq{\mathcal{N}}(A,B)$ be a $*$-semigroup with dense span in $A$. If
$K\unlhd B$ we say that $K$ is an _$N$ -invariant ideal_ if $nKn^{*}\subseteq
K$ for all $n\in N$. When $N={\mathcal{N}}(A,B)$ we will simply say that $K$
is an _invariant ideal_.

######  Remark 3.12.

If $K=I\cap B$ for some ideal $I$ in $A$ then $K$ is invariant: indeed, for a
normalizer $n$, $nIn^{*}\subseteq I$ since $I$ is an ideal and
$nBn^{*}\subseteq B$ since $n$ is a normalizer.

######  Definition 3.13.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras. Let
$N\subseteq{\mathcal{N}}(A,B)$ be a $*$-semigroup with dense span in $A$. If
$E\colon A\rightarrow B$ is a conditional expectation, we say that $E$ is an
_$N$ -invariant conditional expectation_ if $nE(a)n^{*}=E(nan^{*})$ for all
$a\in A$ and $n\in N$. When $E$ is ${\mathcal{N}}(A,B)$-invariant we will
simply say that $E$ is an _invariant conditional expectation_.

We have introduced $N$-invariance for ideals and conditional expectations
because in some settings it is unclear whether a given ideal or conditional
expectation is invariant under the entire $*$-semigroup of normalizers.
Nevertheless, in many situations, there is a natural $*$-semigroup $N$ of
normalizers such that $\operatorname{span}N$ is dense in $A$, under which an
ideal or conditional expectation is invariant. Corollary 3.20 will show that
when there is an $N$-invariant faithful conditional expectation, the
$N$-invariant ideals will necessarily be invariant. This occurs in the crossed
product setting.

######  Example 3.14.

Let $\Gamma$ be a discrete group acting via $\alpha$ on a $C^{*}$-algebra $B$
and let $A=B\rtimes_{r}\Gamma$. Let $N=\\{b\delta_{s}\colon b\in B,\
s\in\Gamma\\}$. Then $N$ is a $*$-semigroup of normalizers with dense span in
$B\rtimes_{r}\Gamma$. An ideal $K$ in $B$ is $N$-invariant in the sense of
Definition 3.11 if and only if $\alpha_{s}(K)=K$ for all $s\in\Gamma$.

The map $E\colon\sum_{i=1}^{k}b_{s_{i}}\delta_{s_{i}}\mapsto b_{e}\delta_{e}$
extends to a faithful conditional expectation from $B\rtimes_{r}\Gamma$ to
$B$, [12, Proposition 4.1.9]. The conditional expectation $E$ is
$N$-invariant.

Let us examine the notion of faithful, invariant conditional expectations for
some more classes of examples.

######  Examples 3.15.

  1. (i)

Suppose $A$ is a unital $C^{*}$-algebra and $B={\mathbb{C}}I$. Then
${\mathcal{N}}(A,B)$ is the collection of all scalar multiples of unitary
operators in $A$, so $(A,B)$ is a regular inclusion. Here, the collection of
all conditional expectations may be identified with the state space of $A$:
for a state $\tau$ on $A$, the map $E_{\tau}$ given by $E_{\tau}(a)=\tau(a)I$
is a conditional expectation. Note that $E_{\tau}$ is faithful precisely when
$\tau$ is a faithful; and $E_{\tau}$ is invariant precisely when $\tau$ is
tracial. Thus the existence of a faithful or invariant expectation is not
guaranteed. In this setting, there are many conditional expectations, and
there is no connection between faithfulness and invariance.

  2. (ii)

Suppose $B\subseteq A$ is a regular inclusion with $B$ maximal abelian in $A$.

    1. (a)

The existence of a conditional expectation is not automatic [37, Example 1.2].

    2. (b)

When a conditional expectation $E:A\rightarrow B$, exists, it is always unique
and invariant. If $A$ is unital, uniqueness follows from [37, Theorem 3.5] and
invariance from [37, Proposition 3.14]. When $A$ is not unital, consider the
unitizations, $\tilde{B}\subseteq\tilde{A}$. By [39, Corollary 2.4(i)],
${\mathcal{N}}(A,B)\subseteq{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{B})$.
Thus, $\tilde{B}\subseteq\tilde{A}$ is a regular inclusion, and $\tilde{B}$ is
maximal abelian in $\tilde{A}$. The unitization $\tilde{E}$ of $E$ (that is,
$A\times{\mathbb{C}}\ni(a,\lambda)\mapsto(E(a),\lambda)\in\tilde{B}$) is
therefore an conditional expectation which is invariant and unique. Since
${\mathcal{N}}(A,B)\subseteq{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{B})$, it
follows that $E$ is also invariant and unique.

    3. (c)

It is possible to construct a discrete dynamical system $(\Gamma,X,\alpha)$
with $\Gamma$ acting freely on a compact Hausdorff space $X$ where the full
and reduced crossed products of $C(X)$ by $\Gamma$ differ; in this case,
$C(X)$ is maximal abelian in the full crossed product $C(X)\rtimes_{f}\Gamma$,
yet the conditional expectation onto $C(X)$ is not faithful, [15, Theorem 3.1
and Remark 3.2].

######  Definition 3.16.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. We say that $B$ has
the _ideal intersection property in $A$_ if whenever $J\unlhd A$ is a non-zero
ideal, $J\cap B\neq\\{0\\}$. We say that $B$ has the _regular ideal
intersection property in $A$_ if whenever $J\unlhd A$ is a non-zero regular
ideal, $J\cap B\neq\\{0\\}$. We will also say that the inclusion $(A,B)$ has
the ideal intersection property or the regular ideal intersection property.

Other authors use different terminology for the ideal intersection property.
For example, an inclusion with the ideal intersection property is called
$C^{*}$-essential in [40] and is said to detect ideals in [32]. To the best of
our knowledge, the regular ideal intersection property has not appeared
elsewhere.

######  Remark 3.17.

It is clear that if $B$ has the ideal intersection property in $A$ then it has
the regular ideal intersection property. We will show in Section 7 that for a
large class of examples the regular ideal intersection property implies the
ideal intersection property. Example 7.1 gives an example of a regular
inclusion $B\subseteq A$ with the regular ideal intersection property but not
the ideal intersection property.

######  Notation 3.18.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras with a faithful
conditional expectation $E\colon A\rightarrow B$. If $K\unlhd B$ is an ideal,
denote by $J_{K}$ the set

$J_{K}=\\{a\in A\colon E(a^{*}a)\in K\\}.$

Our aim in the remainder of this section is to prove Theorem 3.24, which shows
that when the regular inclusion $(A,B)$ has the regular ideal intersection
property and $E$ is faithful and invariant, then $K\mapsto J_{K}$ defines a
bijection between invariant regular ideals of $B$ and regular ideals of $A$.
The following gives a monomorphism of the invariant regular ideals of $B$ into
the regular ideals of $A$.

######  Proposition 3.19.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras, let $N$ be a
$*$-semigroup of normalizers with dense span in $A$, and let $E\colon
A\rightarrow B$ be a faithful $N$-invariant conditional expectation. The
following statements hold.

  1. (i)

If $K\unlhd B$ is an $N$-invariant ideal, then $J_{K}$ is an ideal in $A$ such
that

$J_{K}\cap B=K\subseteq E(J_{K}).$

  2. (ii)

If $K\unlhd B$ is an invariant regular ideal, then $J_{K}$ is a regular ideal
satisfying

$J_{K}\cap B=K=E(J_{K})\quad\text{and}\quad J_{K}^{\perp}=J_{K^{\perp_{B}}}.$

######  Proof..

(i) Let $K\unlhd B$ be an $N$-invariant ideal. Take $a\in J_{K}$ and $b\in A$.
Then $a^{*}b^{*}ba\leq\|b\|^{2}a^{*}a$ and hence

$E((ba)^{*}ba)=E(a^{*}b^{*}ba)\leq\|b\|^{2}E(a^{*}a)\in K$

since conditional expectations are (completely) positive maps. As ideals in
$C^{*}$-algebras are hereditary it follows that $E((ab)^{*}(ab))\in K$ and
hence $ab\in J_{K}$. Thus $J_{K}$ is a left-ideal of $A$. Now take any $a\in
J_{K}$ and any $n\in N$. Then
$E((an)^{*}(an))=E(n^{*}a^{*}an)=n^{*}E(a^{*}a)n\in K$, as $E$ is an
$N$-invariant conditional expectation and $K$ is an $N$-invariant ideal. Hence
$an\in J_{K}$. As the span of $N$ is dense in $A$, it follows that $J_{K}$ is
also a right ideal. That $J_{K}\cap B=K\subseteq E(J_{K})$ follows from the
definition, so (i) holds.

(ii) Now suppose that $K\unlhd B$ is an $N$-invariant regular ideal. Note that
$E(J_{K})$ is an algebraic ideal in $B$ since $J_{K}$ is an ideal in $A$. Let
us show $K=E(J_{K})$.

If $K\neq E(J_{K})$ then, by Lemma 3.4, $E(J_{K})\cap
K^{\perp_{B}}\neq\\{0\\}$. Hence there is an $0\neq a\in J_{K}$ such that
$E(a)\in K^{\perp_{B}}$. Thus

$K\ni E(a^{*}a)\geq E(a)^{*}E(a)\in K^{\perp_{B}}.$

Since $K$ is hereditary, $E(a)^{*}E(a)\in K\cap K^{\perp_{B}}$ and hence
$E(a)=0$, contrary to choice of $a$. Thus $E(J_{K})=K$.

It remains to show that $J_{K}$ is a regular ideal. Let
$J_{K^{\perp}}:=\\{a\in A:E(a^{*}a)\in K^{\perp_{B}}\\}$. It suffices to show
that $J_{K}=J_{K^{\perp}}^{\perp}$. Take any $a\in J_{K}$ and $b\in
J_{K^{\perp}}$. Since $J_{K}$ is a right ideal and $J_{K^{\perp}}$ is a left
ideal, $E((ab)^{*}ab)\in K\cap K^{\perp_{B}}=\\{0\\}$. Since $E$ is faithful
it follows that $ab=0$. Similarly, $ba=0$. Hence $J_{K}\subseteq
J^{\perp}_{K^{\perp}}$.

Now take any $a\in J_{K^{\perp}}^{\perp}$ and $b\in K^{\perp_{B}}\subseteq
J_{K^{\perp}}$. Then $E(a)b=E(ab)=0$. Thus
$E(J_{K^{\perp}}^{\perp})\subseteq(K^{\perp_{B}})^{\perp_{B}}=K$; that is,
$J_{K^{\perp}}^{\perp}\subseteq J_{K}$. Combining the two inclusions we get
$J_{K}=J_{K^{\perp}}^{\perp}$ and hence $J_{K}$ is a regular ideal. ∎

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras, let $N$ be a
$*$-semigroup of normalizers with dense span in $A$. The following corollary
shows that, in the presence of an $N$-invariant faithful conditional
expectation $E\colon A\rightarrow B$, the $N$-invariant ideals of $B$ and the
${\mathcal{N}}(A,B)$-invariant ideals of $B$ coincide.

######  Corollary 3.20.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras, let $N$ be a
$*$-semigroup of normalizers with dense span in $A$, and let $E\colon
A\rightarrow B$ be a faithful $N$-invariant conditional expectation. An ideal
$K\unlhd B$ is $N$-invariant if and only if it is invariant.

######  Proof..

Any invariant ideal of $B$ is $N$-invariant. Suppose that $K\unlhd B$ is
$N$-invariant. By Proposition 3.19 (i), $K=J_{K}\cap B$. Hence $K$ is an
invariant ideal by Remark 3.12. ∎

We now return to our study of regular ideals.

######  Corollary 3.21.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras, $N$ be a
$*$-semigroup of normalizers with dense span in $A$, and let $E\colon
A\rightarrow B$ be a faithful $N$-invariant conditional expectation. Further
suppose that $B$ has the regular ideal intersection property in $A$. If
$K\unlhd B$ is an invariant regular ideal, then $\langle
K\rangle_{A}^{\perp\perp}=J_{K}$.

######  Proof..

Since $\left\langle K\right\rangle_{A}^{\perp\perp}\unlhd A$ is the smallest
regular ideal containing $K$, Proposition 3.19(ii) gives $\left\langle
K\right\rangle_{A}^{\perp\perp}\subseteq J_{K}$. Let $L:=J_{K}\cap\left\langle
K\right\rangle_{A}^{\perp}$ and let $b\in L\cap B$. By Proposition 3.19,
$J_{K}\cap B=K$, so $b\in K$. But $bK=0$ because $b\in\left\langle
K\right\rangle_{A}^{\perp}$. Thus $b=0$, so $L\cap B=\\{0\\}$. By Proposition
2.4, $L$ is a regular ideal in $A$. Since $(A,B)$ has the regular ideal
intersection property, we conclude that $L=\\{0\\}$, that is, $\left\langle
K\right\rangle_{A}^{\perp\perp}=J_{K}$. ∎

######  Proposition 3.22.

Let $B\subseteq A$ be a regular inclusion, where $B$ has the regular ideal
intersection property in $A$. Let $E\colon A\rightarrow B$ be a faithful
conditional expectation. If $J\unlhd A$ is a regular ideal, then $J\cap
B=E(J)$.

######  Proof..

Let $I_{1}=J\cap B$ and $I_{2}=E(J).$ Then $I_{1}$ is an ideal of $B$, $I_{2}$
is an algebraic ideal of $B$, and $I_{1}\subseteq I_{2}$. The ideal
$I_{1}\neq\\{0\\}$ since $B$ has the regular ideal intersection property in
$A$; the algebraic ideal $I_{2}\neq\\{0\\}$ since $E$ is a faithful
conditional expectation. Further, $I_{1}$ is an invariant ideal, and it is a
regular ideal by Proposition 3.7.

Assume $I_{1}\neq I_{2}$. Let $I_{3}=I_{2}\cap I_{1}^{\perp_{B}}$. Then
$I_{3}$ is an algebraic ideal. Further, $I_{3}\neq\\{0\\}$ by Lemma 3.4. We
show

(3.23) $\langle I_{3}\rangle_{A}\subseteq(\langle I_{1}\rangle_{A})^{\perp}.$

First notice that since $B$ is regular in $A$, for any algebraic ideal
$I\subseteq B$, $\langle I\rangle_{A}$ is the closed linear span of
$\\{n_{1}bn_{2}\colon b\in I,\ n_{i}\text{ normalizers}\\}.$

So let $b\in I_{3}$, $c\in I_{1}$ and
$\\{n_{i}\\}_{i=1}^{4}\subseteq{\mathcal{N}}(A,B)$. Using the facts that
$n_{4}^{*}n_{4}\in B$, $I_{1}\unlhd B$ is invariant and $I_{3}\subseteq
I_{1}^{\perp_{B}}$, we obtain

$((n_{1}bn_{2})(n_{3}cn_{4}))((n_{1}bn_{2})(n_{3}cn_{4}))^{*}=n_{1}b(n_{2}n_{3}cn_{4}n_{4}^{*}c^{*}n_{3}^{*}n_{2}^{*})b^{*}n_{1}^{*}=0,$

so $(n_{1}bn_{2})(n_{3}cn_{4})=0$. Similarly, $(n_{3}cn_{4})(n_{1}bn_{2})=0$.
So

$n_{1}bn_{2}\in\overline{\operatorname{span}}\\{ncm:n,m\in{\mathcal{N}}(A,B),c\in
I_{1}\\}^{\perp}=(\langle I_{1}\rangle_{A})^{\perp}.$

The inclusion (3.23) follows.

Since $I_{3}\subseteq I_{2}=E(J)$ there exists $a\in J$ such that $0\neq
E(a)\in I_{3}$. Then $0\neq E(a)a^{*}\in\langle I_{3}\rangle_{A}\cap J$,
showing $\langle I_{3}\rangle_{A}\cap J\neq\\{0\\}.$ By (3.23), the regular
ideal $L:=J\cap\langle I_{1}\rangle_{A}^{\perp}\neq\\{0\\}.$

Note that $\langle I_{1}\rangle_{A}^{\perp}\cap B\subseteq I_{1}^{\perp_{B}}$.
As $J\cap B=I_{1}$ it follows that $L\cap B=\\{0\\}$. This contradicts the
regular ideal intersection property. Thus $J\cap B=E(J)$. ∎

Combining Proposition 3.19 and Proposition 3.22 we get the following theorem.

######  Theorem 3.24.

Let $B\subseteq A$ be a regular inclusion of $C^{*}$-algebras satisfying the
regular ideal intersection property, and let $N$ be a $*$-semigroup of
normalizers with dense span in $A$. Further assume there is an $N$-invariant
faithful conditional expectation $E\colon A\rightarrow B$. The invariant
regular ideals of $B$ form a Boolean algebra. The map $J\mapsto J\cap B$, with
inverse given by $K\mapsto J_{K}$, is a Boolean algebra isomorphism between
the regular ideals of $A$ and the invariant regular ideals of $B$.

######  Proof..

Recall that by Corollary 3.20, an ideal $K\unlhd B$ is invariant if and only
if it is $N$-invariant. Let $K_{1},K_{2}\unlhd B$ be invariant regular ideals.
Note that $K_{1}\cap K_{2}$ is an invariant ideal, so Proposition 2.4 shows
$K_{1}\wedge K_{2}=K_{1}\cap K_{2}$ is an invariant regular ideal. For a
regular invariant ideal $K\unlhd B$, Proposition 3.19 shows $J_{K}\unlhd A$ is
a regular ideal and $K^{\perp_{B}}=J_{K}^{\perp}\cap B$. But
$J_{K}^{\perp}\cap B$ is an invariant ideal of $B$, whence $\neg
K=K^{\perp_{B}}$ is also a regular invariant ideal in $B$. As $K_{1}\vee
K_{2}=\neg((\neg K_{1})\wedge(\neg K_{2}))$, we conclude that the invariant
regular ideals of $B$ form a Boolean algebra.

The result now follows from Proposition 3.19 and Proposition 3.22. ∎

######  Remark 3.25.

Combining Proposition 2.4 with Theorem 3.24 shows that in the setting of a
regular inclusion with the regular ideal intersection property and a faithful,
invariant conditional expectation, the invariant regular ideals in $B$ and the
regular open sets in $\operatorname{Prim}(A)$ are isomorphic Boolean algebras.

##  4\. Quotients by regular ideals

With the results from Section 3, we have all the tools we need to prove our
main results about quotienting by regular ideals. We first note that regular
inclusions are preserved by quotients.

######  Remark 4.1.

Let $(A,B)$ be a regular inclusion and let $J\unlhd A$ be any ideal of $A$.
Then $(A/J,B/(J\cap B))$ is a regular inclusion. Indeed, if $(u_{\lambda})$ is
a net in $B$ which is an approximate unit for $A$, then $(u_{\lambda}+J)$ is a
net in $B/(B\cap J)$ which is an approximate unit for $A/J$, so $(A/J,B/(J\cap
B))$ is an inclusion. Since $\\{n+J:n\in{\mathcal{N}}(A,B)\\}$ has dense span
in $A/J$, the inclusion is regular.

######  Theorem 4.2.

Suppose $(A,B)$ is a regular inclusion with the ideal intersection property.
Let $N$ be a $*$-semigroup of normalizers with dense span in $A$ and suppose
there is an $N$-invariant faithful conditional expectation $E\colon
A\rightarrow B$. Let $J\unlhd A$ be a regular ideal. Then $B/(J\cap B)$ has
the ideal intersection property in $A/J$.

######  Proof..

For notational purposes, use $q$ for the quotient mapping of $A$ onto $A/J$,
let $\dot{A}:=A/J$, and $\dot{B}:=B/(B\cap J)$.

Let $I\unlhd\dot{A}$ satisfy $I\cap\dot{B}=\\{0\\}$ and put
$L:=q^{-1}(I)\unlhd A$. Note that $L$ contains $J$. Our task is to show $J=L$.
Arguing by contradiction, suppose $J\neq L$. Since $J$ is a regular ideal,
Lemma 3.4 shows $L\cap J^{\perp}$ is a non-zero ideal of $A$. Define

$K:=L\cap B$

and note that $K=J\cap B$. Indeed, $J\cap B\subseteq L\cap B$ because
$J\subseteq L$, and the reverse inclusion follows from $I\cap\dot{B}=\\{0\\}$.

By Proposition 3.7, $K$ is a non-zero regular ideal in $B$. By Proposition
3.19 and Theorem 3.24, $J^{\perp}\cap B=K^{\perp_{B}}$. Hence $(L\cap
J^{\perp})\cap B=K\cap K^{\perp_{B}}=\\{0\\}$. This contradicts the ideal
intersection property. Thus $L=J$, and $I$ is the zero ideal in $A/J$. ∎

######  Remark 4.3.

In this remark, we discuss how Theorem 4.2 can be used to give an alternate
proof of [10, Theorem 3.5].

Let $E=\\{E^{0},E^{1},r,s\\}$ be a directed graph. We assume that $E$ is row-
finite with no sources, that is $0<r^{-1}(v)<\infty$ for all $v\in E^{0}$. A
sequence of edges $e_{1}e_{2}\cdots e_{n}$ is a path in $E$, if for all $i$,
$r(e_{i})=s(e_{i-1})$; we say it is a return path if $r(e_{1})=s(e_{n})$. We
say $E$ satisfies condition (L) if for every return path $e_{1}e_{2}\cdots
e_{n}$ there exists an $i$ such that
$r^{-1}(r(e_{i}))\backslash\\{e_{i}\\}\neq\emptyset$.

A set of mutually orthogonal projections $\\{P_{v}\\}_{v\in E^{0}}$ and a set
of partial isometries $\\{S_{e}\\}_{e\in E}$ is a Cuntz-Krieger $E$-family if

$P_{s(e)}=S_{e}^{*}S_{e}=\sum_{r(f)=s(e)}S_{f}S_{f}^{*}.$

The $C^{*}$-algebra of $E$, $C^{*}(E)$, is the unique $C^{*}$-algebra
generated by a universal Cuntz-Krieger $E$-family, $\\{p_{v},s_{e}\\}$ (for
details see [41]). The $C^{*}$-algebra $C^{*}(E)$ comes equipped with a gauge
action. More precisely, for each $t\in{\mathbb{T}}$ the map

$\gamma_{t}(s_{e})=ts_{e}\text{ and }\gamma_{t}(p_{v})=p_{v}$

for $e\in E^{1}$ and $v\in E^{0}$, uniquely defines an automorphism on
C${}^{*}(E)$ so that the map $t\mapsto\gamma_{t}$ is strongly continuous.

Let $D_{E}$ be the $C^{*}$-subalgebra of $C^{*}(E)$ generated by

$\\{(s_{e_{1}}s_{e_{2}}\cdots s_{e_{n}})(s_{e_{1}}s_{e_{2}}\cdots
s_{e_{n}})^{*}:n\in{\mathbb{N}},e_{i}\in E^{1}\\}.$

A consequence of the Cuntz-Kreiger Uniqueness theorem [28, Theorem 3.7] is
that $D_{E}\subseteq C^{*}(E)$ has the ideal intersection property if and only
if $E$ satisfies condition (L).

Now if $J$ is an ideal in $C^{*}(E)$, then by [6, Theorem 4.1], $J$ is
generated by a set of vertex projections $\\{p_{v}:v\in H\subseteq E^{0}\\}$
if and only if $J$ is gauge-invariant and in this case there exists a directed
graph $E/J$ such that $C^{*}(E/J)\cong C^{*}(E)/J$.

Let $E$ be a a directed graph satisfying condition (L). Suppose $J$ is a
gauge-invariant regular ideal. By Theorem 4.2,

$D_{E}/(J\cap D_{E})\subseteq C^{*}(E)/J\cong C^{*}(E/J)$

has the ideal intersection property and thus $E/J$ satisfies condition (L).
Thus [10, Theorem 3.5] follows from Theorem 4.2.

In [10, Proposition 3.7] we go on to show that all regular ideals in
$C^{*}(E)$ are gauge-invariant when the graph $E$ has condition (L). The
analogous result for higher-rank graph $C^{*}$-algebras is proved in [45,
Proposition 6.7]. We generalize these results in Theorem 5.6 below (see
Section 6.1 for details on the relationship to graph algebras).

Theorem 4.2 has implications for quotients of Cartan inclusions by regular
ideals, and to this we now turn our attention. Let us first recall the notion
of a Cartan inclusion from [44].

######  Definition 4.4.

Let $D\subseteq A$ be a regular inclusion of $C^{*}$-algebras. We say that $D$
is a _Cartan subalgebra_ , or _$D$ is Cartan in $A$_ if

  1. (i)

there is a faithful conditional expectation $E\colon A\rightarrow D$;

  2. (ii)

$D$ is maximal abelian in $A$.

We will also say that $(A,D)$ is a Cartan inclusion when these conditions
hold.

######  Remark 4.5.

Recall that if $D$ is a Cartan subalgebra of $A$, then the inclusion
$D\subseteq A$ has the ideal intersection property. This well-known fact can
be found in several places, e.g. [36, Corollary 3.2] and [37, Theorem 6.1].

Thus for a Cartan inclusion $(A,D)$, Theorem 4.2 implies that if $J\unlhd A$
is a regular ideal, then $D/(J\cap D)\subseteq A/J$ has the ideal intersection
property too. It is therefore natural to ask if this quotient inclusion is
also Cartan. We prove this in Theorem 4.8 below. The following example shows
that this does not hold for arbitrary ideals. Another example for graph
algebras is described in [6, Remark 4.5].

######  Example 4.6.

Suppose ${\mathbb{Z}}$ acts on the closed disk $\overline{{\mathbb{D}}}$ by
irrational rotation and

${\mathbb{P}}:=\overline{{\mathbb{D}}}\backslash\\{(0,0)\\}.$

Then $n\cdot x\neq x$ for all $x\in{\mathbb{P}}$. Notice that ${\mathbb{P}}$
is dense in $\overline{{\mathbb{D}}}$ so $C(\overline{{\mathbb{D}}})$ is
Cartan in $C(\overline{{\mathbb{D}}})\rtimes{\mathbb{Z}}$ ( see [44, Example
6.1]).

On the other hand, ${\mathbb{P}}$ is an open invariant subset of
$\overline{{\mathbb{D}}}$ and so by [42, Proposition II.4.6] we have that
$C_{0}({\mathbb{P}})\rtimes{\mathbb{Z}}$ is an ideal in
$C(\overline{{\mathbb{D}}})\rtimes{\mathbb{Z}}$. Moreover the quotient
$C(\overline{{\mathbb{D}}})\rtimes{\mathbb{Z}}/(C_{0}({\mathbb{P}})\rtimes{\mathbb{Z}})$
is isomorphic to $C^{*}({\mathbb{Z}})\cong C({\mathbb{T}})$ and
$C(\overline{{\mathbb{D}}})/C_{0}({\mathbb{P}})\cong{\mathbb{C}}$. As
${\mathbb{C}}$ is not maximal abelian in $C({\mathbb{T}})$ we have
$C(\overline{{\mathbb{D}}})/C_{0}({\mathbb{P}})$ is not Cartan in
$C(\overline{{\mathbb{D}}})\rtimes{\mathbb{Z}}/(C_{0}({\mathbb{P}})\rtimes{\mathbb{Z}})$.
Thus Cartan inclusions are not necessarily preserved by quotients. Notice that
${(\overline{{\mathbb{P}}})}^{\circ}=\overline{{\mathbb{D}}}$ and so
${\mathbb{P}}$ is not regular in $\overline{{\mathbb{D}}}$. Hence, by
Proposition 3.7, we have $C_{0}({\mathbb{P}})\rtimes{\mathbb{Z}}$ is not a
regular ideal in $C(\overline{{\mathbb{D}}})\rtimes{\mathbb{Z}}$.

Before proving Theorem 4.8 which shows the quotient of a Cartan inclusion by a
regular ideal is again a Cartan inclusion, we show that Theorem 4.2 guarantees
the existence of a faithful conditional expectation in the quotient.

######  Lemma 4.7.

Let $(A,B)$ be a regular inclusion with the ideal intersection property, let
$N$ be a $*$-semigroup of normalizers with dense span in $A$, and suppose
$E:A\rightarrow B$ is a faithful $N$-invariant conditional expectation. If
$J\unlhd A$ is a regular ideal, then the map

$\displaystyle E_{A/J}\colon A/J$ $\displaystyle\rightarrow B/(B\cap
J)\quad\text{given by}\quad$ $\displaystyle a+J$ $\displaystyle\mapsto
E(a)+(J\cap B)$

is a well-defined faithful conditional expectation.

######  Proof..

Let $K:=J\cap B$. For $a_{1},a_{2}\in A$, if $a_{1}+J=a_{2}+J$, then
$a_{1}-a_{2}\in J$, whence $E(a_{1}-a_{2})\in E(J)$. By Proposition 3.22,
$E(J)=K$. Thus $E(a_{1})+K=E(a_{2})+K$. Therefore, $E_{A/J}$ is well-defined,
and an argument using [50, Theorem 1] (adjoining a unit if necessary) shows it
is a conditional expectation.

It remains to show that $E_{A/J}$ is faithful. Let

$L:=\\{x\in A/J\colon E_{A/J}(x^{*}x)=0\\}.$

We shall show that $L$ is an ideal in $A/J$. That $L$ is a left ideal follows
as in the proof of Proposition 3.19. To show $L$ is a right ideal, we use the
$N$-invariance of $E$: for $a+J\in L$ and $n\in N$,

$\displaystyle E_{A/J}([(a+J)(n+J)]^{*}(a+J)(n+J))$
$\displaystyle=E(n^{*}a^{*}an)+K$ $\displaystyle=nE(a^{*}a)n+K$
$\displaystyle=0.$

This gives $(a+J)(n+J)\in L$. Therefore for any $y\in\operatorname{span}N$,
$(a+J)(y+J)\in L$. Since $\operatorname{span}N$ is dense in $A$ by assumption,
$L$ is a right ideal.

By definition, $L\cap(B/K)=\\{0\\}$, and an application of Theorem 4.2 shows
$L=\\{0\\}$. Thus $E_{A/J}$ is a faithful conditional expectation. ∎

Here is the promised result concerning quotients of Cartan inclusions by
regular ideals.

######  Theorem 4.8.

If $D$ is a Cartan subalgebra of a $C^{*}$-algebra $A$ and $J\unlhd A$ is a
regular ideal, then $D/(J\cap D)$ is a Cartan subalgebra of $A/J$.

######  Proof..

Let $K=D\cap J$. Remark 4.1 shows $(A/J,D/(D\cap J))$ is a regular inclusion.

Since $(A,D)$ is a Cartan inclusion, it has the ideal intersection property
(see Remark 4.5). Further, by Example 3.15(ii)(b) the faithful conditional
expectation $E\colon A\rightarrow D$ is invariant. We can thus apply Lemma 4.7
to see that there is a faithful conditional expectation $E_{A/J}\colon
A/J\rightarrow D/K$.

It remains to show that $D/K$ is a maximal abelian subalgebra of $A/J$. To
this end, take $a\in A$ such that $ad-da\in J$ for all $d\in D$. To complete
the proof, we will show

(4.9) $a+J=E(a)+J.$

Note that $J^{\perp}\cap D=K^{\perp_{D}}$ by Proposition 3.19(ii). If $e\in
J^{\perp}\cap D$ then $ae-ea\in J^{\perp}\cap J$. Thus $ae-ea=0$. This shows
that

$ae=ea\text{ for all }e\in K^{\perp_{D}}.$

Next we show $ea$ commutes with $D$. For $d\in D$ and $e\in K^{\perp_{D}}$,
the facts that $ad-da\in J$, $D$ is abelian, and $ae=ea$ yield,

$0=e(ad-da)=(ea)d-d(ea).$

Hence $ea$ commutes with $D$.

As $D$ is maximal abelian in $A$ we have $ea=ae\in D$, whence
$ae=E(ae)=E(a)e$. These considerations are independent of the choice of $e\in
K^{\perp_{D}}$, so

$(a-E(a))e=0\text{ for all }e\in K^{\perp_{D}}.$

Let $n_{1}$ and $n_{2}$ be normalizers of $D$ and take $e\in K^{\perp_{D}}$.
Then

$((a-E(a))n_{1}en_{2})((a-E(a))n_{1}en_{2})^{*}=(a-E(a))n_{1}en_{2}n_{2}^{*}e^{*}n_{1}^{*}(a-E(a))^{*}=0,$

since $n_{1}en_{2}n_{2}^{*}e^{*}n_{1}^{*}\in K^{\perp_{B}}$. Thus
$(a-E(a))n_{1}en_{2}=0$. Therefore

$a-E(a)\in\langle K^{\perp_{B}}\rangle^{\perp}.$

However, $\langle K^{\perp_{B}}\rangle^{\perp}=J$ by Proposition 3.19. Hence
$a+J=E(a)+J$, establishing (4.9). ∎

######  Remark 4.10.

The converse of Theorem 4.8 does not hold: it is possible to find a Cartan
inclusion $(A,D)$ and a non-regular ideal $J\unlhd A$ so that $D/(J\cap D)$ is
Cartan in $A/J$, see e.g. [10, Example 3.9].

##  5\. $C^{*}$-algebras of Twisted Étale Groupoids

We now specialize our study to the $C^{*}$-algebras of twisted, Hausdorff,
étale groupoids. As a consequence of exactness we will get an explicit
description of the regular ideals of such $C^{*}$-algebras.

To begin our discussion, we recall a few facts and some notation concerning
$C^{*}$-algebras from twists. We refer the reader to [9, Section 2] for
details.

For a groupoid $G$ we will use $G^{(0)}$ to denote its unit space. We identify
$G^{(0)}$ with the objects in $G$ and use $r,s:G\to G^{(0)}$ to denote the
range and source maps.

For each subset $\Delta\subseteq G^{(0)}$ there is a subgroupoid

$G_{\Delta}:=\\{\gamma\in G:r(\gamma),s(\gamma)\in\Delta\\}.$

When $G_{\Delta}=\\{\gamma\in G:s(\gamma)\in\Delta\\}$, that is,
$s(\gamma)\in\Delta\Rightarrow r(\gamma)\in\Delta$, $\Delta$ is said to be an
invariant subset of $G^{(0)}$; we will also refer to such as set as
_invariant_.

We assume groupoids are endowed with a locally compact Hausdorff topology in
which composition and inversion are continuous. An open set $B\subseteq G$ is
a _bisection_ if $r|_{B}$ and $s|_{B}$ are homeomorphisms onto their images.
We say $G$ is _étale_ if it has a basis consisting of bisections. A _twist_ is
a central extension of a groupoid $G$

$G^{(0)}\times{\mathbb{T}}\overset{\iota}{\to}\Sigma\overset{q}{\twoheadrightarrow}G.$

We usually suppress writing the maps $\iota$ and $q$ and simply denote a twist
by $\Sigma\rightarrow G$ or $(\Sigma;G)$. The map $q$ restricts to a
homeomorphism between the unit spaces $\Sigma^{(0)}$ and $G^{(0)}$. We will
thus identify $\Sigma^{(0)}$ with $G^{(0)}$ without mention of $q$. A well-
known construction produces the reduced $C^{*}$-algebra, $C^{*}_{r}(\Sigma;G)$
of $(\Sigma;G)$; it is the completion of a convolution algebra using the
reduced norm. Details may be found in a number of sources; see for example,
[44], [47] or [9]. If $\Sigma=G\times{\mathbb{T}}$ we drop $\Sigma$ from the
notation and refer to this algebra as $C^{*}_{r}(G)$.

If $H$ is an open subgroupoid of $G$, the twist $\Sigma\rightarrow G$
determines a central extension

$H^{(0)}\times{\mathbb{T}}\to q^{-1}(H)\twoheadrightarrow H$

and $C_{r}^{*}(q^{-1}(H);H)$ embeds in $C^{*}_{r}(\Sigma;G)$ [9, Lemma 2.7].
We will use this embedding in the following instances.

  1. (i)

If $U$ is an invariant open subset of $G^{(0)}$, then $G_{U}$ is open in $G$
and $q^{-1}(G_{U})=\Sigma_{U}$. The C∗-algebra $C^{*}_{r}(\Sigma_{U};G_{U})$
embeds in $C_{r}^{*}(\Sigma;G)$ as a closed ideal. In fact,
$C_{r}^{*}(\Sigma_{U};G_{U})=\langle C_{0}(U)\rangle_{C_{r}^{*}(\Sigma;G)}$ by
[9, Lemma 2.7].

  2. (ii)

Since $G$ is étale, the unit space $G^{(0)}$ is clopen in $G$ and
$C^{*}_{r}(q^{-1}(G^{(0)});G^{(0)})$ embeds in $C^{*}_{r}(\Sigma;G)$. As
$C^{*}_{r}(q^{-1}(G^{(0)});G^{(0)})\cong C_{0}(G^{(0)})$, we identify
$C_{0}(G^{(0)})$ with $C^{*}_{r}(q^{-1}(G^{(0)});G^{(0)})$. In this sense, we
regard $C_{0}(G^{(0)})$ as a subalgebra of $C^{*}_{r}(\Sigma;G)$; furthermore,
$(C^{*}_{r}(\Sigma;G),C_{0}(G^{(0)}))$ is a regular inclusion.

In the last instance, there exists a canonical conditional expectation
$E:C^{*}_{r}(\Sigma;G)\to C_{0}(G^{(0)})$. This conditional expectation is
invariant for the $*$-semigroup $N_{\operatorname{bi}}$ of normalizers that
are supported on bisections. By Corollary 3.20 an ideal $C_{0}(U)$ in
$C_{0}(G^{(0)})$ is invariant in the sense of Definition 3.11 if and only if
it is $N_{\mathrm{bi}}$-invariant. Moreover, $C_{0}(U)$ is an invariant ideal
of $C_{0}(G^{(0)})$ if and only if $U$ is an invariant open set.

Suppose $U$ is an open invariant set in $G^{(0)}$ and $F=G^{(0)}\setminus U$.
Since $F$ is a closed invariant subset, $G_{F}$ is a groupoid with unit space
$F$. Further, $q^{-1}(G_{F})=\Sigma_{F}$. We thus get a twist,

$F\times{\mathbb{T}}\rightarrow\Sigma_{F}\twoheadrightarrow G_{F}.$

It is not difficult to show that the restriction map, $C_{c}(\Sigma;G)\ni
g\mapsto g|_{F}\in C_{c}(\Sigma_{F};G_{F})$, extends to a $*$-homomorphism,

(5.1) $\varphi:C_{r}^{*}(\Sigma;G)\to C^{*}(\Sigma_{F};G_{F}).$

Thus, with $\theta:C_{0}(G^{(0)})\rightarrow C_{0}(F)$ denoting the quotient
map $\theta(f)=f|_{F}$, we obtain the commutative diagram (where the vertical
maps are the conditional expectations described above),

(5.2)
$\textstyle{C_{r}^{*}(\Sigma_{U};G_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{U}}$$\textstyle{C_{r}^{*}(\Sigma;G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\scriptstyle{\varphi}$$\textstyle{C_{r}^{*}(\Sigma_{F};G_{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{F}}$$\textstyle{C_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}(G^{(0)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{C_{0}(F).}$

While the bottom row is exact in the middle, it is possible that the top row
is not exact [43, Appendix by Skandalis].

######  Definition 5.3.

If the top row of (5.2) is also exact in the middle, we say that $G$ is inner
exact at $U$. When $G$ is inner exact for every open invariant set $U\subseteq
G^{(0)}$, we say $G$ is inner exact, or more simply, exact.

By [33, Theorem 3.5], inner exactness is a property of $G$ and is independent
of the twist over $G$. Our next goal is to obtain a description of certain
regular ideals in the reduced $C^{*}$-algebra of a twist over an exact
groupoid, see Theorem 5.6 below.

Let $U\subseteq G^{(0)}$ be an invariant open set. Set

$J_{U}:=\\{x\in C_{r}^{*}(\Sigma;G)\colon E(x^{*}x)\in C_{0}(U)\\}.$

(This is simplified notation for the ideal $J_{C_{0}(U)}$ defined in Notation
3.18.)

######  Proposition 5.4.

Suppose $\Sigma\to G$ is a twist, $U\subseteq G^{(0)}$ is an invariant open
set, and $\varphi$ is defined as in (5.1). Then $\ker\varphi=J_{U}$.

######  Proof..

Suppose $x\in\ker\varphi$. Then $\theta(E(x^{*}x))=E_{F}(\varphi(x^{*}x))=0$
so $E(x^{*}x)\in\ker\theta$. Since the bottom row of (5.2) is exact,
$E(x^{*}x)\in C_{0}(U)$. Thus, $\ker\varphi\subseteq J_{U}$.

Now suppose $x\in J_{U}$. Then $E_{F}(\varphi(x^{*}x))=\theta(E(x^{*}x))=0$,
so faithfulness of $E_{F}$ gives $\varphi(x)=0$. Hence $x\in\ker\varphi$, and
so $\ker\varphi=J_{U}$ as desired. ∎

This new description of $J_{U}$ as the kernel of a quotient map allows us to
strengthen Corollary 3.21 for twisted groupoid C∗-algebras. In the groupoid
setting, Corollary 3.21 says that if $C_{0}(G^{(0)})$ has the regular ideal
intersection property in $C_{r}^{*}(\Sigma;G)$, and $U\subseteq G^{(0)}$ is an
open regular invariant set, then
$C_{r}^{*}(\Sigma_{U};G_{U})^{\perp\perp}=J_{U}$. We now show that for reduced
C∗-algebras of twists we can drop the regular ideal intersection property.

######  Proposition 5.5.

Let $\Sigma\to G$ be a twist. Let $U\subseteq G^{(0)}$ be a regular invariant
open set and $F=G^{(0)}\setminus U$. Let $\varphi:C_{r}^{*}(\Sigma;G)\to
C_{r}^{*}(\Sigma_{F};G_{F})$ be the $*$-homomorphism that extends restriction.
Then $\ker\varphi$ is a regular ideal and
$C_{r}^{*}(\Sigma_{U};G_{U})^{\perp\perp}=\ker\varphi$.

######  Proof..

By Proposition 5.4, $\ker\varphi=J_{U}$. Since $U$ is regular and invariant,
Proposition 3.19 shows that $\ker\varphi$ is regular. Since
$C_{r}^{*}(\Sigma_{U};G_{U})\subseteq\ker\varphi$, we obtain
$C_{r}^{*}(\Sigma_{U};G_{U})^{\perp\perp}\subseteq\ker\varphi$.

Aiming at a contradiction, assume
$C_{r}^{*}(\Sigma_{U};G_{U})^{\perp\perp}\neq\ker\varphi$. By Lemma 3.4,

$\ker\varphi\cap C_{r}^{*}(\Sigma_{U};G_{U})^{\perp}\neq\\{0\\}.$

Fix a non-zero positive element $a\in\ker\varphi\cap
C_{r}^{*}(\Sigma_{U};G_{U})^{\perp}$. For $e\in C_{0}(U)$, $ea=0$ because
$a\in C_{r}^{*}(\Sigma_{U};G_{U})^{\perp}$. On the other hand, if $f\in
C_{0}(U^{\perp})$, then $fE(a)\in C_{0}(U^{\perp})\subseteq C_{0}(F)$, so

$fE(a)=\theta(fE(a))=\theta(E(fa))=E_{F}(\varphi(f)\varphi(a))=0.$

Thus for any $h\in C_{0}(U\cup U^{\perp})$

$hE(a)=0.$

But $U\cup U^{\perp}$ is dense in $G^{(0)}$, so $E(a)=0$. Faithfulness of $E$
gives $a=0$, contrary to hypothesis. Thus
$\ker\varphi=C_{r}^{*}(\Sigma_{U};G_{U})^{\perp\perp}$, as desired. ∎

Let $\Sigma\to G$ be a twist and assume that $G$ is exact. We now show that if
$(C_{r}^{*}(\Sigma;G),C_{0}(G^{(0)}))$ has the ideal intersection property
then the regular ideals of $C^{*}_{r}(\Sigma;G)$ have an explicit description
in terms of the dynamics of $G$.

######  Theorem 5.6.

Let $\Sigma\rightarrow G$ be a twist. Suppose that $G$ is exact and
$U\subseteq G^{(0)}$ is a regular invariant open set. Then
$C^{*}_{r}(\Sigma_{U};G_{U})$ is a regular ideal in $C^{*}_{r}(\Sigma;G)$.

If, in addition, $C_{0}(G^{(0)})$ has the regular ideal intersection property
in $C_{r}^{*}(\Sigma;G)$, every regular ideal of $C^{*}_{r}(\Sigma;G)$ is of
this form.

######  Proof..

Since $G$ is exact, $\ker\varphi=C^{*}_{r}(\Sigma_{U};G_{U})=J_{U}$. By
Proposition 5.5, $C^{*}_{r}(\Sigma_{U};G_{U})$ is a regular ideal in
$C^{*}_{r}(\Sigma;G)$.

Theorem 3.24 shows that in the presence of the regular ideal intersection
property, all regular ideals are of the form $C^{*}_{r}(\Sigma_{U};G_{U})$. ∎

The following corollary to (non-twisted) groupoid C∗-algebras is a special
case of Theorem 5.6. We record it, however, as many of the examples which
motivated this study are in this setting.

######  Corollary 5.7.

Let $G$ be an exact locally compact Hausdorff étale groupoid. If $U\subseteq
G^{(0)}$ is a regular open invariant set, then $C^{*}_{r}(G_{U})$ is a regular
ideal in $C_{r}^{*}(G)$.

If $C_{0}(G^{(0)})$ has the ideal intersection property inside $C_{r}^{*}(G)$
then all regular ideals of $C_{r}^{*}(G)$ are of this form.

In Section 6.1, we discuss how Corollary 5.7 applies to graph C*-algebras.
More generally, Proposition 6.3 gives an application of Theorem 5.6 for Cartan
inclusions in nuclear C∗-algebras. In Theorem 6.5, a result analogous to
Theorem 5.6 and Corollary 5.7 for reduced crossed products of discrete group
actions on (not necessarily abelian) C∗-algebras is given. If a discrete group
$\Gamma$ acts on a locally compact Hausdorff space $X$, then the reduced
crossed product $C_{0}(X)\rtimes_{r}\Gamma$ is both an example of a reduced
groupoid C∗-algebra and a reduced crossed-product C∗-algebra. Corollary 6.6
gives an application of both Corollary 5.7 and Theorem 6.5 in this setting.

######  Remark 5.8.

A careful reader will note that we did not need that $G$ was exact for Theorem
5.6 or Corollary 5.7, only that Diagram (5.2) is exact for regular open sets
$U\subseteq G^{(0)}$.

Our aim in the remainder of this section is to present Example 5.13. This is
an example of a groupoid $G$ that is not inner exact but which is inner exact
at all open regular invariant sets $U\subseteq G^{(0)}$.

######  Definition 5.9.

Let $\Gamma$ be a discrete group. A sequence of subgroups $(K_{n})$ is an
_approximating sequence_ for $\Gamma$ if

  1. (i)

each $K_{n}$ is a normal, finite index subgroup of $\Gamma$;

  2. (ii)

$K_{n}\supseteq K_{n+1}$ for all $n$; and

  3. (iii)

$\bigcap_{n}K_{n}=\\{e\\}$.

Given a discrete group $\Gamma$ with an approximating sequence $(K_{n})$ one
can construct what is known as a HLS groupoid. These groupoids were introduced
(and named for) Higson, Lafforgue and Skandalis [22]. Higson, Lafforgue and
Skandalis use HLS groupoids to give a counter-example to the groupoid version
of the Baum-Connes conjecture. In so doing, they give an example of a HLS
groupoid which is not exact. We give now the definition of a HLS groupoid, as
presented by Willett [52].

######  Definition 5.10.

Let $\Gamma$ be a discrete group with an approximating sequence $(K_{n})$. For
each $n$ let $\Gamma_{n}=\Gamma/K_{n}$, and let
$\pi_{n}\colon\Gamma\rightarrow\Gamma_{n}$ be the quotient map. Let
$\Gamma_{\infty}=\Gamma$ and $\pi_{\infty}$ be the identity map on $\Gamma$.
Let ${\mathbb{N}}^{*}={\mathbb{N}}\cup\\{\infty\\}$ be the one-point
compatification of ${\mathbb{N}}$ and let $G$ be the disjoint union of
$\\{n\\}\times\Gamma_{n}$ for all $n\in{\mathbb{N}}^{*}$. That is,

$G=\bigsqcup_{n\in{\mathbb{N}}^{*}}\\{n\\}\times\Gamma_{n}.$

Equip $G$ with the topology generated by the sets

  1. (i)

$\\{(n,g)\\}$ for $n\in{\mathbb{N}}$ and $g\in\Gamma_{n}$; and

  2. (ii)

for each $\gamma\in\Gamma$ and $N\in{\mathbb{N}}$,
$\\{(n,\pi_{n}(\gamma))\colon n\in{\mathbb{N}}\cup\\{\infty\\},\ n\geq N\\}$.

With this topology, $\\{(n,\pi_{n}(e)):n\in{\mathbb{N}}^{*}\\}$ is a clopen
set in $G$, and with its relative topology, it is homeomorphic to
${\mathbb{N}}^{*}$. We identify $\\{(n,\pi_{n}(e)):n\in{\mathbb{N}}^{*}\\}$
with ${\mathbb{N}}^{*}$.

Endow $G$ with a partial product

$(n,g_{1})(m,g_{2})=\begin{cases}(n,g_{1}g_{2})&\text{if }n=m\\\
\text{undefined}&\text{when }n\neq m.\end{cases}$

The inverse operation is given by $(n,g)^{-1}=(n,g^{-1})$. With these
operations, $G$ becomes a Hausdorff étale topological groupoid and is called
the _HLS groupoid_ associated to $\Gamma$ and $(K_{n})$. The unit space of $G$
is ${\mathbb{N}}^{*}$.

We will need the following proposition for Example 5.13.

######  Proposition 5.11.

Let $\Gamma$ be a discrete group with an approximating sequence $(K_{n})$ and
let $G$ be the associated HLS groupoid. Assume that $C_{r}^{*}(G)=C^{*}(G)$.
Then

$C_{r}^{*}(G_{U})\rightarrow C_{r}^{*}(G)\rightarrow
C_{r}^{*}(G_{{\mathbb{N}}^{*}\backslash U})$

is exact for all regular open sets $U\subseteq{\mathbb{N}}^{*}.$

######  Proof..

This result hinges on two claims.

Claim 1. If $U\subseteq{\mathbb{N}}$, then $C_{r}^{*}(G_{U})=C^{*}(G_{U})$.

To prove this claim note that if $U$ is finite then the groupoid $G_{U}$ is
finite, and the result is clear. If $U$ is infinite, then the result follows
from [52, Lemma 2.6]. Thus Claim 1 is proved.

Claim 2. If $U\subseteq{\mathbb{N}}^{*}$ with $\infty\in U$ and
$U\cap{\mathbb{N}}$ infinite, then $C_{r}^{*}(G_{U})=C^{*}(G_{U})$.

To prove this claim, note that $G_{U}$ is the HLS groupoid associated to
$\Gamma$ and the approximating sequence $(K_{n})_{n\in U\cap{\mathbb{N}}}$. By
assumption $C_{r}^{*}(G)=C^{*}(G)$, so it follows from [52, Lemma 2.7] that we
also have $C_{r}^{*}(G_{U})=C^{*}(G_{U})$. Thus the second claim holds.

Now, let $U\subseteq{\mathbb{N}}^{*}$ be a regular open set. Then $U$ must
take one of the following forms:

  1. (i)

$U\subseteq{\mathbb{N}}$ and $U$ is finite;

  2. (ii)

$\infty\in U$ and ${\mathbb{N}}^{*}\backslash U$ is finite;

  3. (iii)

$U\subseteq{\mathbb{N}}$ and both $U$ and ${\mathbb{N}}\backslash U$ are
infinite.

In all cases, Claim 1 and Claim 2 show that we have the commutative diagram

(5.12)
$\begin{CD}C^{*}(G_{U})@>{}>{}>C^{*}(G)@>{}>{}>C^{*}(G_{{\mathbb{N}}^{*}\backslash
U})\\\ \Big{\|}\Big{\|}\Big{\|}\\\
C_{r}^{*}(G_{U})@>{}>{}>C_{r}^{*}(G)@>{}>{}>C_{r}^{*}(G_{{\mathbb{N}}^{*}\backslash
U}).\end{CD}$

As the top line of (5.12) is exact, the bottom line is also exact. ∎

######  Example 5.13.

Let ${\mathbb{F}}_{2}$ be the free group over two generators. Let $(K_{n})$ be
the approximating sequence for ${\mathbb{F}}_{2}$ given in [52, Lemma 2.8],
and let $G$ be the associated groupoid. By [2, Proposition 3.5] $G$ is not
exact, since ${\mathbb{F}}_{2}$ is not amenable. However, by [52, Lemma 2.7
and Lemma 2.8], $C_{r}^{*}(G)=C^{*}(G)$. Hence, by Proposition 5.11 the
failure of exactness for $G$ does not happen for open regular invariant sets
$U\subseteq G^{(0)}$.

##  6\. Applications

In this section, we give a number of applications.

###  6.1. Directed graphs

Let $E$ be a directed graph. In Remark 4.3, we showed that if $J$ is a gauge-
invariant regular ideal in $C^{*}(E)$, then the graph $E/J$ satisfies
condition (L) if $E$ does. In this subsection we use Corollary 5.7 to show
that if $E$ satisfies condition (L) then all regular ideals in $C^{*}(E)$ are
gauge-invariant recovering [10, Proposition 3.7].

To see this, we use the groupoid $G$ constructed from $E$ in [29] which, among
other things, has $C^{*}(E)\cong C_{r}^{*}(G)$ with the isomorphism sending
each vertex projection $p_{v}$ to a characteristic function on a compact open
subset of $G^{(0)}$. Further, the groupoid $G$ is amenable [29, Corollary
5.5], and so $G$ is exact, see e.g. [43, Remark 4.5].

Since the gauge-invariant ideals of $C^{*}(E)$ are precisely the ideals
generated by their vertex projections [6, Theorem 4.1], the gauge-invariant
ideals are ideals of the form $C_{r}^{*}(G_{U})$ for open invariant sets
$U\subseteq G^{(0)}$. As noted in Example 4.3, $D_{E}$ has the intersection
property in $C^{*}(E)$ if and only if $E$ satisfies condition (L). Thus
Corollary 5.7 generalizes [10, Proposition 3.7]. Similarly, for higher-rank
graph $C^{*}$-algebras, Corollary 5.7 recovers [45, Proposition 6.7].

In [18] analogous results to the graph $C^{*}$-algebra results of [10] are
shown in the purely algebraic setting of Leavitt path algebras. In a Leavitt
path algebra the correct analogy of a gauge-invariant ideal is a
${\mathbb{Z}}$-graded ideal. It is shown in [18, Corollary 3.4] that all
regular ideals in a Leavitt path algebra are graded, even when the graph does
not satisfy condition (L). For graph $C^{*}$-algebras, however, condition (L)
is needed. Indeed, if $E$ is the graph of one vertex and one edge, then
$C^{*}(E)=C({\mathbb{T}})$. Here any open regular subset of ${\mathbb{T}}$
gives a regular ideal. None of these, other than $\\{0\\}$ and
$C({\mathbb{T}})$, are gauge-invariant.

###  6.2. Transformation groups

Let $\Gamma$ be a discrete group acting by homeomorphisms on a compact
Hausdorff space $X$. The action of $\Gamma$ on $X$ is _topologically free_ if
the interior of $X_{t}:=\\{x\in X\colon t\cdot x=x\\}$ is empty for all non-
identity elements $t\in\Gamma$. The ideal intersection property for
$(C(X)\rtimes_{r}\Gamma,C(X))$ is closely related to topological freeness. In
fact, when the action of $\Gamma$ on $X$ is topologically free, S. Kawamura
and J. Tomiyama show in [25, Theorem 4.1] that $(C(X)\rtimes_{r}\Gamma,C(X))$
has the ideal intersection property, and the converse holds when $\Gamma$ is
amenable. We do not know whether the ideal intersection property for
$(C(X)\rtimes_{r}\Gamma,C(X))$ is equivalent to topological freeness of the
action of $\Gamma$ on $X$ for discrete groups $\Gamma$ if the hypothesis of
amenability on $\Gamma$ is dropped. However, some relaxation of the
amenability hypothesis is possible; see the equivalence of statements (iv) and
(v) of [40, Theorem 4.6].

Recall that a closed set $F\subseteq X$ is regular if it is the closure of its
interior; it is easy to see that a closed set $F$ is regular if and only if
$X\setminus F$ is a regular open set. We now observe that the restriction of a
topologically free action to an invariant regular closed set is again
topologically free.

######  Proposition 6.1.

Let $\Gamma$ be a discrete group acting topologically freely on a compact
Hausdorff space $X$. If $Y\subseteq X$ is a regular and closed
$\Gamma$-invariant set, then the restricted action of $\Gamma$ to $Y$ is also
topologically free.

######  Proof..

Let $Y$ be an invariant regular closed set. We need to show that the interior
of $Y_{s}$ as a subset of $Y$, ${(Y_{s})}^{\circ_{Y}}$, is empty for all
$e\neq s\in\Gamma$. Suppose $s\in\Gamma$ and
${(Y_{s})}^{\circ_{Y}}\neq\emptyset$. This means there exists an open set
$U\subseteq X$ such that $U\cap Y\subseteq Y_{s}$ with $U\cap Y$ nonempty.
Since $Y$ is a regular closed set, $U\cap{Y}^{\circ}\neq\emptyset$. Thus,

$\emptyset\neq U\cap{Y}^{\circ}\subseteq U\cap Y\subseteq Y_{s}\subseteq
X_{s}.$

This shows the interior of $X_{s}$ is non-empty. Since $\Gamma$ acts
topologically freely on $X$, we conclude $s=e$. ∎

######  Remark 6.2.

Proposition 6.1 can be used to construct an alternate proof of Theorem 4.2 for
inclusions $C(X)\subseteq C(X)\rtimes_{r}\Gamma$.

Compare Proposition 6.1 to Example 4.6. In Example 4.6
$X=\overline{{\mathbb{D}}}$, and $F:=X\backslash{\mathbb{P}}$ is a single
point. The action of $\Gamma={\mathbb{Z}}$ on this point is (necessarily)
trivial, and thus not topologically free. In this case, however, $F$ is not a
regular closed set.

###  6.3. Cartan embeddings

We used Theorem 4.2 to show that the quotient of a Cartan inclusion by a
regular ideal is again a Cartan inclusion, see Theorem 4.8. We have already
discussed in Section 6.1 that Theorem 5.6 recovers [10, Proposition 6.7].
Recall that condition (L) in [10, Proposition 6.7] is precisely the condition
that the subalgebra $D_{E}$ is a Cartan subalgebra of the graph algebra
$C^{*}(E)$.

Suppose $(A,D)$ is a Cartan inclusion and let $K\unlhd D$ be an invariant
regular ideal. If $L\unlhd A$ satisfies $L\cap B=K=E(L)$, then $\langle
K\rangle_{A}\subseteq L\subseteq J_{K}$. This leads to the obvious question:
does $\left\langle K\right\rangle_{A}=J_{K}$? Equivalently, is $\langle
K\rangle_{A}$ a regular ideal of $A$? In the graph algebra setting, [10,
Proposition 6.7] precisely says that we always have $\left\langle
K\right\rangle_{A}=J_{K}$ for invariant regular ideals $K$. We generalize this
to more general Cartan inclusions now, using Theorem 5.6.

######  Proposition 6.3.

Let $(A,D)$ be a Cartan inclusion with $A$ nuclear. The map $J\mapsto J\cap D$
is a bijection between the regular ideals of $A$ and the invariant regular
ideals of $D$. The inverse map is given by $K\mapsto\langle K\rangle_{A}$.

######  Proof..

By [44] (see also [30, Corollary 7.6]), there is a twist $\Sigma\rightarrow
G$, with $G$ Hausdorff, étale, and effective so that $A$ is isomorphic to
$C^{*}_{r}(\Sigma;G)$ via an isomorphism which carries $D$ to
$C_{0}(G^{(0)})$. By [49, Theorem 5.4] $G$ is amenable if and only if $A$ is
nuclear. Further, when $G$ is amenable, $G$ is exact, see, e.g. [43, Remark
4.11].

Recall from Remark 4.5 that $(A,D)$ has the ideal intersection property. Thus
by applying Theorem 3.24 and Theorem 5.6, we obtain the result. ∎

###  6.4. Crossed products of $C^{*}$-algebras

So far, the applications in this section have dealt with inclusions
$B\subseteq A$ where $B$ is abelian. Here we sketch how our arguments can be
applied to reduced crossed products of possibly nonabelian $C^{*}$-algebras by
discrete exact groups to obtain an analogue of Theorem 5.6.

Let $\Gamma$ be a discrete group acting by automorphisms on a $C^{*}$-algebra
$A$. Let $A\rtimes_{r}\Gamma$ denote the reduced crossed product and let
$E_{A}\colon A\rtimes_{r}\Gamma\rightarrow A$ be the usual faithful
conditional expectation [12, Proposition 4.1.9]. Recall from Example 3.14,
that $E_{A}$ is invariant under $N=\\{a\delta_{s}\colon s\in\Gamma\\}$. Note
that, an ideal of $A$ is $N$-invariant if and only if it is invariant under
the action of $\Gamma$. By Corollary 3.20 it then follows that an ideal in $A$
is invariant under the action of $\Gamma$ if and only if it is an invariant
ideal in the sense of Definition 3.11.

If $K\unlhd A$ is a $\Gamma$-invariant ideal we get the following commutative
diagram.

(6.4)
$\textstyle{K\rtimes_{r}\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{K}}$$\textstyle{A\rtimes_{r}\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{E_{A}}$$\textstyle{A/K\rtimes_{r}\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{A/K}}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{A/K}$

The bottom line of (6.4) will always be exact. However, it is possible that
the top line may not be exact. As with algebras associated with groupoids (see
Definition 5.3), if the top line of (6.4) is exact for all invariant ideals
$K\unlhd A$ we say that the action of $\Gamma$ on $A$ is _exact_. In
particular, this will happen if $\Gamma$ is exact. See [46] for a study of
exact actions and the ideal structure of $A\rtimes_{r}\Gamma$.

######  Theorem 6.5.

Let $\Gamma$ be a discrete group that acts by an exact action on a
$C^{*}$-algebra $A$. If $K\unlhd A$ is an invariant regular ideal, then
$K\rtimes_{r}\Gamma$ is a regular ideal in $A\rtimes_{r}\Gamma$.

If in addition $A\subseteq A\rtimes_{r}\Gamma$ has the ideal intersection
property, every regular ideal in $A\rtimes_{r}\Gamma$ has this form.

######  Proof..

Let $K$ be an invariant regular ideal in $A$. Since the action of $\Gamma$ is
exact we get that $\ker\varphi=K\rtimes_{r}\Gamma$. Arguing as in Proposition
5.5 one shows $K\rtimes_{r}\Gamma=\ker\varphi$ is a regular ideal, giving the
first statement.

Additionally, assume that $A\subseteq A\rtimes_{r}\Gamma$ has the ideal
intersection property, and let $J_{K}$ be as in Notation 3.18. Arguing as in
Proposition 5.4 we obtain $J_{K}=\ker\varphi$. Now Theorem 3.24 gives that all
regular ideals are of this form. ∎

Let $\Gamma$ be a discrete group acting by homeomorphisms on a compact space
$X$. The reduced crossed-product $C(X)\rtimes_{r}\Gamma$ is isomorphic to the
reduced groupoid C∗-algebra $C_{r}^{*}(\Gamma\times X)$, where $\Gamma\times
X$ is the transformation group. Thus, crossed products of abelian C∗-algebras
are special cases of groupoid C∗-algebras. We have the following corollary
which is a special case of both Theorem 6.5 and Corollary 5.7.

######  Corollary 6.6.

Let $\Gamma$ be an exact discrete group acting on a compact Hausdorff space
$X$ by homeomorphisms. If $U\subseteq X$ is an open regular $\Gamma$-invariant
subset, then $C_{0}(U)\rtimes_{r}\Gamma$ is a regular ideal in
$C_{0}(X)\rtimes_{r}\Gamma$.

If the action of $\Gamma$ on $X$ is topologically free, then all regular
ideals of $C_{0}(X)\rtimes_{r}\Gamma$ are of this form.

######  Proof..

The groupoid $\Gamma\times X$ is exact since the group $\Gamma$ is exact. This
follows immediately on comparing Definition 5.3 with Definition 1.5 of [46].
If the action of $\Gamma$ is topologically free, [25, Theorem 4.1] shows
$C(X)$ has the ideal intersection property in $C(X)\rtimes\Gamma$. Thus, the
result follows by Corollary 5.7 or Theorem 6.5. ∎

######  Remark 6.7.

We compare Theorem 6.5 to results of Sierakowski [46]. The inclusion
$A\subseteq A\rtimes_{r}\Gamma$ has the _residual intersection property_ if
$A/J\subseteq A/J\rtimes_{r}\Gamma$ has the ideal intersection property for
every invariant ideal $J\unlhd A$. In [46, Theorem 1.3] it is shown that all
ideals of $A\rtimes_{r}\Gamma$ are of the form $J\rtimes_{r}\Gamma$ for an
invariant ideal $J\unlhd A$ if and only if the action of $\Gamma$ is exact on
$A$ and $A\subseteq A\rtimes_{r}\Gamma$ has the residual intersection
property. Theorem 6.5 shows that the ideal intersection property suffices for
regular ideals; the residual ideal intersection property is only needed for
non-regular ideals.

The question of when $A\subseteq A\rtimes_{r}\Gamma$ has the ideal
intersection property has been studied by several authors. The ideal
intersection property is determined by the action of $\Gamma$ on some
injective objects. However, instead of topological freeness, one must instead
look to see if the action is _properly outer_. See [55, Section 2.3] or [27,
Definition 2.8] for definitions of properly outer actions. Let $I(A)$ be the
injective envelope of $A$ and let $I_{\Gamma}(A)$ be the $\Gamma$-injective
envelope of $A$. We summarize some of the results of Zarikian [55] and Kennedy
and Schafhauser [27].

######  Theorem 6.8 (cf. [55] and [27]).

Let $\Gamma$ be a discrete group acting on a $C^{*}$-algebra $A$. Then the
following are equivalent:

  1. (i)

$A\subseteq A\rtimes_{r}\Gamma$ has the ideal intersection property;

  2. (ii)

$I(A)\subseteq I(A)\rtimes_{r}\Gamma$ has the ideal intersection property;

  3. (iii)

$I_{\Gamma}(A)\subseteq I_{\Gamma}(A)\rtimes_{r}\Gamma$ has the ideal
intersection property.

In particular, this will happen if the action of $\Gamma$ on $I(A)$ is
properly outer.

######  Proof..

The equivalence of (i), (ii) and (iii) is [27, Theorem 5.5]. For the second
part see [55, Theorem 6.4] or [27, Theorem 6.4]. ∎

##  7\. Settings where the regular ideal intersection property and the ideal
intersection property coincide

For most of our theorems we require the seemingly weaker regular ideal
intersection property instead of ideal intersection property. The main purpose
of this section is to describe some classes of inclusions for which the ideal
intersection property and regular ideal intersection property coincide. As
noted in the introduction, the ideal intersection property has become an
important tool for obtaining structural results about inclusions, and it seems
to us that the equivalence of the ideal intersection property and the regular
ideal intersection property is also an interesting structural result in its
own right.

We first give an example of an inclusion with the regular ideal intersection
property but without the ideal intersection property.

######  Example 7.1.

Let $H$ be a separable Hilbert space. Let $B\subseteq B(H)$ be any unital,
non-zero $C^{*}$-algebra having trivial intersection with the compact
operators, $K(H)$. The only ideals in $B(H)$ are $\\{0\\},\ K(H),$ and $B(H)$.
Of these, $\\{0\\}$ and $B(H)$ are regular ideals in $B(H)$, while $K(H)$ is
not a regular ideal. As $B(H)\cap B=B\neq\\{0\\}$, the inclusion $(B(H),B)$
has the regular ideal intersection property, but it does not have the ideal
intersection property because $K(H)\cap B=\\{0\\}$.

We can choose $B$ above so that $B$ is abelian and the inclusion is regular.
For a straightforward example, the inclusion $(B(H),B)$ will be regular when
$B={\mathbb{C}}I$ since unitaries normalize $B$ and every $T\in B(H)$ is a
linear combination of four unitary operators.

We recall that a topological space is called _semiregular_ if it has a basis
of regular open sets.

######  Proposition 7.2.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. Assume that
$\operatorname{Prim}(A)$ with the hull kernel topology is semiregular. Then
$B\subseteq A$ has the ideal intersection property if and only if it has the
regular ideal intersection property.

######  Proof..

Since the ideal intersection property implies the regular ideal intersection
property, it suffices to show that for $B\subseteq A$ the regular ideal
intersection property implies the ideal intersection property.

Assume $B\subseteq A$ has the regular ideal intersection property and let
$I\unlhd A$ be a non-trivial ideal. Then
$\operatorname{Prim}(A)\backslash\operatorname{hull}(I)$ is an open subset of
$\operatorname{Prim}(A)$. By the semiregularity of $\operatorname{Prim}(A)$
there is a regular open set
$U\subseteq\operatorname{Prim}(A)\backslash\operatorname{hull}(I)$. By
Proposition 2.4, the ideal $J=\ker(\operatorname{Prim}(A)\backslash U)$ is a
regular ideal $A$, and by [34, Theorem 5.4.7 (3)] $J\subseteq I$. By the
regular ideal intersection property $J\cap B\neq\\{0\\}$, and hence $I\cap
B\neq\\{0\\}$. Hence $B\subseteq A$ has the ideal intersection property. ∎

######  Corollary 7.3.

Let $B\subseteq A$ be an inclusion of $C^{*}$-algebras. Assume that
$\operatorname{Prim}(A)$ with the hull kernel topology is Hausdorff. Then
$B\subseteq A$ has the ideal intersection property if and only if it has the
regular ideal intersection property.

In particular, if $A$ is abelian, then $B\subseteq A$ has the ideal
intersection property if and only if it has the regular ideal intersection
property.

######  Proof..

The primitive ideal space $\operatorname{Prim}(A)$ is always locally compact.
Thus if $\operatorname{Prim}(A)$ is Hausdorff then $\operatorname{Prim}(A)$ is
semiregular. This follows since locally compact Hausdorff spaces are regular,
and regular spaces are semiregular, see [48, pg. 16]. ∎

######  Remark 7.4.

By combining Corollary 7.3 with [40, Corollary 3.21] we obtain several
checkable characterizations of unital inclusions $B\subseteq A$ with $A$
abelian having the regular ideal intersection property or equivalently, the
ideal intersection property.

######  Remark 7.5.

We can apply Corollary 7.3 to show that regular ideal intersection property
and ideal intersection property are equivalent for inclusions in a variety of
crossed products that are known to have Hausdorff spectrum. For example,
Williams characterizes when the crossed product of a transformation group
$(\Gamma,X,\alpha)$ with $\Gamma$ second countable and abelian has Hausdorff
spectrum [54]. If the stability subgroups of $(\Gamma,X,\alpha)$ are all
subgroups of a fixed abelian group, Williams characterizes when the crossed
product is continuous trace (and so by definition has Hausdorff spectrum) [53,
Theorem 5.1]. Echterhoff uses Williams’ result to give conditions that
guarantee crossed products are continuous trace for transformation groups
$(\Gamma,X,\alpha)$ with $\Gamma$ a Lie group [13, Corollary 3] or $\Gamma$ is
a discrete group [13, Theorem 3]. Moreover, Archbold and an Huef [4, Theorem
3.9] characterize when the crossed product has continuous trace when the
action of $\Gamma$ on $\operatorname{Prim}(A)$ is free.111We would like to
thank the anonymous referee for pointing out the above applications.

We state one application using a theorem of Green [19].

######  Corollary 7.6.

Let $\Gamma$ be a locally compact group with with a free, proper action on a
locally compact Hausdorff space $X$. Let $B\subseteq C_{0}(X)\rtimes\Gamma$ be
a $C^{*}$-subalgebra. Then $B\subseteq C_{0}(X)\rtimes\Gamma$ has the ideal
intersection property if and only if it has the regular ideal intersection
property.

######  Proof..

Note that the reduced crossed product $C(X)\rtimes_{r}\Gamma$ and the full
crossed product $C(X)\rtimes\Gamma$ coincide since the action of $\Gamma$ is
free and proper. The quotient space $X/G$ is Hausdorff and
$\operatorname{Prim}(C(X)\rtimes\Gamma)\cong X/G$ by [19, Theorem 14]. The
result thus follows from Corollary 7.3. ∎

The following application of Corollary 7.3 gives a wide variety of examples
where the regular ideal intersection property is equivalent to the ideal
intersection property.

######  Theorem 7.7.

Let $(A,B)$ be a regular inclusion with $B$ abelian and assume that $B^{c}$,
the relative commutant of $B$ in $A$, is abelian. If $(A,B^{c})$ is a Cartan
inclusion, then $(A,B)$ has the ideal intersection property if and only if
$(A,B)$ has the regular ideal intersection property.

######  Remark 7.8.

Suppose $(A,B)$ is a regular inclusion with both $B$ and $B^{c}$ abelian. We
wish to observe that $B^{c}\subseteq A$ is a regular inclusion of
$C^{*}$-algebras. An easy argument using the partial automorphism $\theta_{n}$
of [39, Corollary 2.3] shows
${\mathcal{N}}(A,B)\subseteq{\mathcal{N}}(A,B^{c})$. Therefore, $B^{c}$ is
regular in $A$. Because $B$ contains an approximate unit for $A$, so does
$B^{c}$, whence $(A,B^{c})$ is a regular inclusion. Thus, in the setting of
Theorem 7.7, $(A,B^{c})$ is a Cartan inclusion if and only if there is a
faithful conditional expectation of $A$ onto $B^{c}$.

######  Proof of Theorem 7.7.

To show that the regular ideal intersection property implies the ideal
intersection property, we establish the contrapositive. So assume $B\subseteq
A$ does not have the ideal intersection property and let $\\{0\\}\neq J\unlhd
A$ be such that $J\cap B=\\{0\\}$. By hypothesis, $B^{c}\subseteq A$ has the
ideal intersection property. Thus $J\cap B^{c}\neq\\{0\\}$. Thus the inclusion
$B\subseteq B^{c}$ does not have the ideal intersection property. By Corollary
7.3, $B\subseteq B^{c}$ does not have the regular ideal intersection property.

Therefore, we may find a non-trivial regular ideal $K\unlhd B^{c}$ such that
$K\cap B=\\{0\\}$. By Theorem 3.24, there is a regular ideal $J_{K}\unlhd A$
such that $J_{K}\cap B^{c}=K$. Hence $J_{K}\cap B=K\cap B=\\{0\\}$. That is,
$J_{K}\unlhd A$ is a non-trivial regular ideal with trivial intersection with
$B$. Hence $B\subseteq A$ does not have the regular ideal intersection
property. ∎

We can readily apply Theorem 7.7 to graph C∗-algebras.

######  Corollary 7.9.

Let $E$ be a row-finite graph. Let $D_{E}$ be the abelain subalgebra of the
graph C∗-algebra $C^{*}(E)$ described in Remark 4.3. The following are
equivalent:

  1. (i)

$E$ satisfies condition (L);

  2. (ii)

$D_{E}$ has the ideal intersection property in $C^{*}(E)$; and

  3. (iii)

$D_{E}$ has the regular ideal intersection property in $C^{*}(E)$.

######  Proof..

The commutant of $D_{E}$ in $C^{*}(E)$ is a Cartan subalgebra of $C^{*}(E)$ by
[35, Theorem 3.6 and Theorem 3.7]. Thus the result follows from Theorem 7.7
and the Cuntz-Krieger Uniqueness theorem [28, Theorem 3.7]. ∎

Our final corollary states that the regular ideal intersection property and
the ideal intersection property coincide for many groupoid $C^{*}$-algebras of
interest, not just graph C∗-algebras.

######  Corollary 7.10.

Suppose $G$ is an étale groupoid and let $G^{\prime}$ be the isotropy of $G$.
Assume that $G^{\prime}$ is abelian and the interior of $G^{\prime}$ is closed
in $G$. Then $C_{0}(G^{(0)})\subseteq C_{r}^{*}(G)$ has the ideal intersection
property if and only if it has the regular ideal intersection property.

######  Proof..

Let $(G^{\prime})^{\circ}$ be the interior of the isotropy of $G$. By [11,
Corollary 4.5], $C^{*}({(G^{\prime})}^{\circ})$ is Cartan in $C_{r}^{*}(G)$.
Let $C_{0}(G^{(0)})^{c}$ be the relative commutant of $C_{0}(G^{(0)})$ inside
$C_{r}^{*}(G)$. The result will follow from Theorem 7.7 once we show,

(7.11) $C_{0}(G^{(0)})^{c}=C^{*}_{r}({(G^{\prime})}^{\circ}).$

Now Proposition II.4.7 (i) in [42] shows first that
$C^{*}_{r}({(G^{\prime})}^{\circ})\subseteq C_{0}(G^{(0)})^{c}$ and second
that for any $b\in C_{0}(G^{(0)})^{c}$ we must have $b$ supported in
$G^{\prime}$. But then the open support of $b$ must be contained in
${(G^{\prime})}^{\circ}$ and since ${(G^{\prime})}^{\circ}$ is closed we get
$b\in C^{*}_{r}({(G^{\prime})}^{\circ})$, that is
$C_{0}(G^{(0)})^{c}=C^{*}_{r}({(G^{\prime})}^{\circ})$ as desired.∎

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