#  $C^{*}$-irreducibility for reduced twisted group $C^{*}$-algebras

Erik Bédos  Department of Mathematics
University of Oslo
NO-0316 Oslo
Norway<EMAIL_ADDRESS>and  Tron Omland  Norwegian National Security
Authority<EMAIL_ADDRESS>
###### Abstract.

We study $C^{*}$-irreducibility of inclusions of reduced twisted group
$C^{*}$-algebras and of reduced group $C^{*}$-algebras. We characterize
$C^{*}$-irreducibility in the case of an inclusion arising from a normal
subgroup, and exhibit many new examples of $C^{*}$-irreducible inclusions.

##  1\. introduction

Let $A$ be a unital $C^{*}$-algebra and $B\subseteq A$ be a unital inclusion,
i.e., $B$ is a $C^{*}$-subalgebra of $A$ containing the unit $1$ of $A$. A
$C^{*}$-algebra $C$ such that $B\subseteq C\subseteq A$ is called an
intermediate $C^{*}$-algebra of $B\subseteq A$. Inspired by several previous
works, including [23], [2] and [3], Rørdam recently introduced [42] the notion
of $C^{*}$-irreducibility for such inclusions of $C^{*}$-algebras: $B\subseteq
A$ is said to be _$C^{*}$ -irreducible_ if every intermediate $C^{*}$-algebra
of $B\subseteq A$ is simple. Besides giving an intrinsic characterization of
$C^{*}$-irreducibility, Rørdam presents in [42] several examples of
$C^{*}$-irreducible inclusions arising in various settings, such as groups,
dynamical systems, inductive limits and tensor products. New examples have
since appeared in [15, 29, 26]. The related problem of determining all
intermediate $C^{*}$-algebras of a given inclusion has attracted a lot of
attention over the years, also in its von Neumann algebraic version. As a
sample, we refer to [1, 11, 12, 13, 23, 44, 46, 48].

Some other properties of unital inclusions of $C^{*}$-algebras have also been
studied. In [45], Ursu says that $B\subseteq A$ is _relatively simple_ if any
unital completely positive map on $A$ which is a $*$-homomorphism on $B$ is
faithful on $A$. Every relatively simple inclusion is $C^{*}$-irreducible [45,
Proposition 3.6]. On the other hand, inspired by the terminology used for von
Neumann algebras, a unital inclusion $B\subseteq A$ of $C^{*}$-algebras is
said to be _irreducible_ if $B^{\prime}\cap A=\mathbb{C}1$. Equivalently, as
we will see later, this amounts to require that every intermediate
$C^{*}$-algebra of $B\subseteq A$ has trivial center. Every
$C^{*}$-irreducible inclusion is irreducible, as explained in [42, Remark
3.8], but the converse does not hold in general. Another property of a unital
inclusion $B\subseteq A$ that has been of interest in the past is the
_relative Dixmier property_ , which says that for any $a\in A$ the norm
closure of the convex hull of $\\{uau^{*}:u\text{ unitary in }B\\}$ contains a
scalar (see [40, 42] and references therein). It is stronger than
$C^{*}$-irreducibility when $A$ has a faithful tracial state, and always
stronger than irreducibility, cf. [42, Proposition 3.12].

Now, let $G$ be a discrete group and $\sigma$ a (circle-valued) two-cocycle on
$G$. One may then form the associated reduced twisted group $C^{*}$-algebra
$C_{r}^{*}(G,\sigma)$ and the twisted group von Neumann algebra
$\mathcal{L}(G,\sigma)$. Our main focus in this article is to study
$C^{*}$-irreducibility of inclusions of the form $C^{*}_{r}(H,\sigma)\subseteq
C^{*}_{r}(G,\sigma)$, where $H$ is a subgroup of $G$. A necessary condition
for this to happen is clearly that both $C^{*}_{r}(H,\sigma)$ and
$C^{*}_{r}(G,\sigma)$ are simple. When $\sigma$ is trivial, this means that
$H$ and $G$ have to be $C^{*}$-simple. However, despite the recent
breakthroughs [8, 21, 28] in the theory of $C^{*}$-simple groups, the problem
of determining when a reduced twisted group $C^{*}$-algebra is simple is more
complicated, see for example our discussion in [7]. This indicates that
obtaining an intrinsic characterization of the $C^{*}$-irreducibility of
$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ is probably also a
challenging problem in general. Anyhow, when $H$ is normal in $G$, we are able
to show in Theorem 6.2 that $C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$
is $C^{*}$-irreducible if and only if $C_{r}^{*}(H,\sigma)$ is simple and
$(H\leq G,\sigma)$ satisfies the so-called _relative Kleppner condition_.
Moreover, if $H$ is FC-hypercentral or $C^{*}$-simple, then we get that
$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ is $C^{*}$-irreducible if
and only if $(H\leq G,\sigma)$ satisfies the relative Kleppner condition, if
and only if $C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ satisfies the
relative Dixmier property. The relative Kleppner condition, which is of purely
combinatorial nature, has its origin from Kleppner’s work [30] on factoriality
of twisted group von Neumann algebras. As we show in Proposition 4.3, it is
equivalent to the irreducibility of $C_{r}^{*}(H,\sigma)\subseteq
C_{r}^{*}(G,\sigma)$ (resp.
$\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$). When $\sigma$ is
trivial, this condition just says that $G$ is icc relatively to $H$, and we
obtain in Theorem 6.4 that if $H$ is normal in $G$, then
$C^{*}_{r}(H)\subseteq C^{*}_{r}(G)$ is $C^{*}$-irreducible if and only if $H$
is $C^{*}$-simple and the centralizer $C_{G}(H)$ of $H$ in $G$ is trivial. As
shown by Ursu in [45], this is in turn equivalent to $C_{r}^{*}(H)\subseteq
C_{r}^{*}(G)$ being relatively simple. Our Theorem 6.4 should be seen in light
of [8, Theorem 1.4], which says that $G$ is $C^{*}$-simple if and only if both
$H$ and $C_{G}(H)$ are $C^{*}$-simple.

We pay some special attention to the case where $H$ is a normal subgroup which
is _prime_ , in the sense that the FC-center of $H$ is torsion-free, cf.
Definition 3.12. Under this assumption, we characterize when $(H\leq
G,\sigma)$ satisfies Kleppner’s condition in Theorem 3.15, and show in
Corollary 6.9 that $C_{r}^{*}(H,\sigma)\subseteq C_{r}^{*}(G,\sigma)$ is
$C^{*}$-irreducible if and only if $C_{r}^{*}(H,\sigma)$ is simple and the
_twisted_ centralizer $C^{\sigma}_{G}(H)$ of $H$ in $G$ is trivial. This last
result takes an even simpler form if $H$ is also assumed to be FC-
hypercentral, or if $H$ is $C^{*}$-simple, cf. Corollary 6.10.

When $H$ is a normal subgroup of $G$, our approach is to decompose
$C_{r}^{*}(G,\sigma)$ as the reduced twisted $C^{*}$-crossed product of
$C_{r}^{*}(H,\sigma)$ by a twisted action of the quotient group $G/H$, and
combine this fact with a study of $C^{*}$-irreducibility for reduced twisted
$C^{*}$-crossed products. As a bonus, the intermediate $C^{*}$-algebras of the
inclusion $C_{r}^{*}(H,\sigma)\subseteq C_{r}^{*}(G,\sigma)$ can be described
by making use of Cameron and Smith’s result [12, Theorem 4.4] for simple
reduced twisted $C^{*}$-crossed products. When $H$ is not normal in $G$, such
a decomposition is not available, and we point out in Remark 6.8 that Theorem
6.2 does not necessarily hold in this case, even if $\sigma$ is trivial.

Our paper is organized as follows. Section 2 is devoted to some preliminary
material on reduced twisted group $C^{*}$-algebras and twisted group von
Neumann algebras. In the next section we discuss when a group $G$ is icc
relatively to a subgroup $H$ and introduce the relative Kleppner condition for
$(H\leq G,\sigma)$, where $\sigma$ is a two-cocycle on $G$. The main goal of
Section 4 is to show the equivalence of the relative Kleppner condition being
satisfied and the irreducibility of the associated inclusions of twisted group
algebras, as mentioned above. We also show that this is equivalent to the
primeness of every intermediate $C^{*}$-algebra of
$C_{r}^{*}(H,\sigma)\subseteq C_{r}^{*}(G,\sigma)$. In Section 5 we first
point out that Rørdam’s result [42, Theorem 5.8], which characterizes the
$C^{*}$-irreducibility of the inclusion $A\subseteq A\rtimes_{r}G$, where
$A\rtimes_{r}G$ is the reduced $C^{*}$-crossed product associated to an action
of $G$ on a unital $C^{*}$-algebra $A$, is still valid in the case of a
twisted action. Assuming that $H$ is a normal subgroup of $G$, we obtain in
Theorem 5.3 a characterization of the $C^{*}$-irreducibility of
$A\rtimes_{r}H\subseteq A\rtimes_{r}G$ for a twisted action. Section 6
contains our characterization of the $C^{*}$-irreducibility of
$C_{r}^{*}(H,\sigma)\subseteq C_{r}^{*}(G,\sigma)$, hence of
$C_{r}^{*}(H)\subseteq C_{r}^{*}(G)$, and some of the consequences that may be
drawn from it. We illustrate our findings in Section 7, where we present
various new examples of $C^{*}$-irreducible inclusions, involving
noncommutative tori, the discrete Heisenberg group, the braid group on
infinitely many strands, and wreath products. In Section 8, we apply our
results to produce $C^{*}$-irreducible inclusions associated to groups acting
on trees, e.g., amalgamated free products and HNN-extensions. Our final
section is a sequel where we have included a related result on the simplicity
of $C_{r}^{*}(G,\sigma)$ in the presence of a normal subgroup.

##  2\. Preliminaries

Let $G$ be a discrete group with identity $e$. By a (normalized) _two-cocycle
on $G$_ we will mean in this article a map $\sigma\colon G\times
G\to\mathbb{T}$ satisfying

$\begin{gathered}\sigma(g,h)\sigma(gh,k)=\sigma(g,hk)\sigma(h,k),\\\
\sigma(g,e)=\sigma(e,g)=1\end{gathered}$

for all $g,h,k\in G$. We assume throughout this paper that $G$ and $\sigma$
are given. We recall that if $\sigma^{\prime}$ is also a two-cocycle on $G$,
then $\sigma^{\prime}$ is said to be _similar to $\sigma$_ if there exists a
function $\beta\colon G\to\mathbb{T}$ such that $\beta(e)=1$ and

$\sigma^{\prime}(r,s)=\beta(r)\beta(s)\overline{\beta(rs)}\sigma(r,s)\quad\text{
for all $r,s\in G$}.$

The (left) _regular $\sigma$-projective representation_
$\lambda_{\sigma}\colon G\to B(\ell^{2}(G))$ is defined by

$\lambda_{\sigma}(g)\xi(h)=\sigma(g,g^{-1}h)\xi(g^{-1}h)$

for $g,h\in G$ and $\xi\in\ell^{2}(G)$. The _reduced twisted group
$C^{*}$-algebra $C^{*}_{r}(G,\sigma)$_ is the $C^{*}$-subalgebra of
$B(\ell^{2}(G))$ generated by $\lambda_{\sigma}(G)$, and the _twisted group
von Neumann algebra $\mathcal{L}(G,\sigma)$_ is the von Neumann subalgebra of
$B(\ell^{2}(G))$ generated by $\lambda_{\sigma}(G)$. The unit in both these
algebras is the identity operator on $\ell^{2}(G)$. If $\sigma^{\prime}$ is a
two-cocycle on $G$ which is similar to $\sigma$ via a map $\beta\colon
G\to\mathbb{T}$, then, as is well-known and easy to check,
$C_{r}^{*}(G,\sigma^{\prime})$ (resp. $\mathcal{L}(G,\sigma^{\prime})$) is
$*$-isomorphic to $C_{r}^{*}(G,\sigma)$ (resp. $\mathcal{L}(G,\sigma)$) via a
map $\Phi$ sending $\lambda_{\sigma^{\prime}}(g)$ to
$\beta(g)\lambda_{\sigma}(g)$ for each $g\in G$. There is a canonical faithful
tracial state $\tau$ on $\mathcal{L}(G,\sigma)$, hence also on
$C^{*}_{r}(G,\sigma)$, namely the restriction of the vector state on
$B(\ell^{2}(G))$ associated to the characteristic function $\delta_{e}$ of
$\\{e\\}$ in $G$.

Let the map $\widetilde{\sigma}\colon G\times G\to\mathbb{T}$ be defined by

$\widetilde{\sigma}(h,g)=\sigma(h,g)\overline{\sigma(hgh^{-1},h)},$

Then a simple computation gives that

$\lambda_{\sigma}(h)\lambda_{\sigma}(g)\lambda_{\sigma}(h)^{*}=\widetilde{\sigma}(h,g)\lambda_{\sigma}(hgh^{-1})$

for all $h,g\in G$. Some further computations using the cocycle identity give
that for all $r,s,t\in G$ we have:

(1)
$\displaystyle\widetilde{\sigma}(rs,t)=\widetilde{\sigma}(r,sts^{-1})\widetilde{\sigma}(s,t)$
(2)
$\displaystyle\widetilde{\sigma}(r,st)=\overline{\sigma(s,t)}\sigma(rsr^{-1},rtr^{-1})\widetilde{\sigma}(r,s)\widetilde{\sigma}(r,t)$

These identities are similar to those stated in [39, (1.1)-(1.2)], but beware
that our definition of $\widetilde{\sigma}$ is different from theirs: if
$\widetilde{\sigma}_{\textup{PR}}$ denotes this function from [39], then
$\widetilde{\sigma}(r,s)=\widetilde{\sigma}_{\textup{PR}}(s,r^{-1})$. From (1)
and (2) we immediately get

(3)
$\displaystyle\widetilde{\sigma}(rs,t)=\widetilde{\sigma}(r,t)\widetilde{\sigma}(s,t)\quad\text{
whenever $st=ts$},$ (4)
$\displaystyle\quad\quad\quad\widetilde{\sigma}(r,st)=\widetilde{\sigma}(r,s)\widetilde{\sigma}(r,t)\quad\text{
whenever $rs=sr$ and $rt=tr$}.$

As usual, when $\sigma$ is trivial (i.e., $\sigma(g,h)=1$ for all $g,h\in G$),
we skip it in our notation and from our terminology. So, for example,
$\mathcal{L}(G)$ denotes the group von Neumann algebra of $G$. As is well-
known, $\mathcal{L}(G)$ is a factor if and only if $G$ is icc, i.e., every
nontrivial conjugacy class in $G$ is infinite. The twisted version of this
fact, which is due to Kleppner [30], is as follows. An element $g\in G$ is
called _$\sigma$ -regular_ if $\sigma(g,h)=\sigma(h,g)$ whenever $h\in G$ and
$gh=hg$. As $\sigma$-regularity is a property of conjugacy classes, it makes
sense to say that $(G,\sigma)$ satisfies _Kleppner’s condition_ if there is no
nontrivial finite $\sigma$-regular conjugacy class in $G$. As shown in [30],
$\mathcal{L}(G,\sigma)$ is a factor if and only if $(G,\sigma)$ satisfies
Kleppner’s condition. It can also be shown that this is equivalent to
$C^{*}_{r}(G,\sigma)$ having a trivial center (see [37, Theorem 2.7]).

The pair $(G,\sigma)$ is said to be _$C^{*}$ -simple_ (resp. to have _the
unique trace property_) when $C^{*}_{r}(G,\sigma)$ is simple (resp. has a
unique tracial state). Any pair satisfying one of these conditions must
necessarily satisfy Kleppner’s condition, but the converse implications do not
always hold. If $G$ is $C^{*}$-simple, then, as shown in [10], $(G,\sigma)$ is
$C^{*}$-simple and has the unique trace property. We refer to [7] for other
examples of pairs satisfying one or both of these conditions.

A large class of amenable groups for which $C^{*}$-simplicity (resp. the
unique trace property) of $(G,\sigma)$ is known to be equivalent to Kleppner’s
condition being satisfied is the class of FC-hypercentral groups, i.e., groups
having no icc quotient group other than the trivial one, cf. [6] and
references therein. This class contains all abelian groups, all FC-groups, all
finitely generated groups having polynomial growth and, more generally, all
virtually nilpotent groups. Countable FC-hypercentral groups have recently
been characterized as groups with the Choquet-Deny property [16], and as
strongly amenable groups [17].

##  3\. The relative Kleppner condition

Let $H$ be a subgroup of $G$. We will often write $H\leq G$ to indicate this.
The _$H$ -conjugacy class_ of $g$ in $G$ is defined as

$g^{H}:=\\{hgh^{-1}:h\in H\\}.$

Thus $g^{H}$ is the orbit of $g$ under the action of $H$ on $G$ by
conjugation, and we have that $\lvert g^{H}\rvert=\big{[}H:C_{H}(g)\big{]}$,
where $C_{H}(g):=\\{h\in H:hg=gh\\}$ is the centralizer of $g$ in $H$.

We recall that $G$ is said to be an _icc group relatively to $H$_ if every
nontrivial $H$-conjugacy class in $G$ is infinite. We will denote the
centralizer of $H$ in $G$ by $C_{G}(H)$. Thus, $C_{G}(H)=\\{g\in G:\lvert
g^{H}\rvert=1\\}$. Similarly, we define the _FC-centralizer of $H$ in $G$_ as
the subgroup of $G$ given by

$FC_{G}(H)=\\{g\in G:\lvert g^{H}\rvert<\infty\\}.$

Clearly, $C_{G}(H)\subseteq FC_{G}(H)$, and saying that $G$ is icc relatively
to $H$ means precisely that $FC_{G}(H)$ is trivial.

######  Example 3.1.

Let $H,K$ be groups and assume that $\alpha\colon K\to\operatorname{Aut}(H)$
is an action of $K$ on $H$ by automorphisms. Let $G=H\rtimes K$ denote the
associated semidirect product. As usual, we consider $H$ as a normal subgroup
of $G$ and $K$ as a subgroup of $G$, so that $H\cap K=\\{e\\}$ and
$\alpha_{k}(h)=khk^{-1}$ for all $h\in H$, $k\in K$. Then we have that $G$ is
icc relatively to $H$ if and only if $H$ is icc and $\alpha$ is outer, i.e.,
$\alpha_{k}$ is an outer automorphism of $H$ for each $k\in
K\setminus\\{e\\}$. Indeed, if $H$ is icc and $\alpha$ is outer, then using
[4, Lemma 3.4] it is not difficult to deduce that $G$ is icc relatively to
$H$. Alternatively, one may check that $C_{G}(H)$ is trivial and apply
Proposition 3.3. The converse implication is straightforward. On the other
hand, $G$ is icc relatively to $K$ if and only if $K$ is icc and
$\\{\alpha_{k}(h):k\in K\\}$ is infinite for every $h\in H\setminus\\{e\\}$.
The reader should have no trouble in verifying this assertion.

######  Lemma 3.2.

Assume $H$ is icc and normal in $G$. Then $FC_{G}(H)=C_{G}(H)$.

###### Proof.

Since $H$ is icc, we have that $FC_{G}(H)\cap H=\\{e\\}$. Let $g\in FC_{G}(H)$
and $k\in g^{H}$. Then $k^{H}=g^{H}$, so $k\in FC_{G}(H)$. Hence $g^{-1}k\in
FC_{G}(H)$. But, since $H$ is normal, $g^{-1}k\in H$. Thus, $g^{-1}k=e$, i.e.,
$k=g$. This means that $\lvert g^{H}\rvert=1$, i.e., $g\in C_{G}(H)$. ∎

Using this lemma, we immediately get:

######  Proposition 3.3.

If $H$ is normal in $G$, then $G$ is icc relatively to $H$ if and only if $H$
is icc and $C_{G}(H)$ is trivial.

The condition $FC_{G}(H)=C_{G}(H)$ is trivially satisfied when $H$ is
contained in the center of $G$. It is also satisfied in some other situations.

######  Example 3.4.

Let $H\leq G$ and assume that $H$ has no nontrivial finite quotient, or that
$H$ is an $R$-group (meaning that if $h_{1},h_{2}\in H$ are such that
$h_{1}^{n}=h_{2}^{n}$ for some $n\in\mathbb{N}$, then $h_{1}=h_{2}$). Then
$FC_{G}(H)=C_{G}(H)$.

Indeed, suppose (for contradiction) that $FC_{G}(H)\neq C_{G}(H)$. Then there
exists $g\in G\setminus\\{e\\}$ such that

(5) $1<\lvert g^{H}\rvert=[H:C_{H}(g)]<\infty.$

Thus, $H$ has a proper subgroup of finite index, and therefore a normal proper
subgroup of finite index (just take the normal core). Hence $H$ has a
nontrivial finite quotient. Moreover, (5) gives that there must exist some
$h\in H\setminus\\{e\\}$ such that $hgh^{-1}\neq g$. It also implies that we
must have $h^{n}gh^{-n}=g$ for some $n\in\mathbb{N}$, that is
$h^{n}=gh^{n}g^{-1}=(ghg^{-1})^{n}$. So $H$ is not an R-group.

All torsion-free, locally nilpotent groups are known to be $R$-groups (see for
example [34, 2.1.2]). It is also known that every group having a bi-invariant
total order is an $R$-group (see e.g. [27, Lemma 7.7]). Combining this with
[27, Theorem 7.8] one deduces that all pure braid groups are also $R$-groups.

The following definition was given in [7] in the case where $H$ is normal, but
it makes sense without this assumption.

######  Definition 3.5.

An element $g\in G$ is said to be _$\sigma$ -regular w.r.t. $H$_ if
$\sigma(g,h)=\sigma(h,g)$ whenever $h\in H$ and $gh=hg$.

In other words, $g\in G$ is $\sigma$-regular w.r.t. $H$ if and only if
$\widetilde{\sigma}(h,g)=1$ (resp. $\widetilde{\sigma}(g,h)=1$) whenever $h\in
H$ and $gh=hg$. It follows from the argument given in the proof of [7, Lemma
4.4] that $\sigma$-regularity w.r.t. $H$ is a property of $H$-conjugacy
classes. Also, if $\sigma^{\prime}$ is a two-cocycle on $G$ which is similar
to $\sigma$, then it is easy to check that $g\in G$ is $\sigma$-regular w.r.t.
$H$ if and only if it is $\sigma^{\prime}$-regular w.r.t. $H$. The twisted
analogue of $G$ being icc relatively to $H$ is as follows.

######  Definition 3.6.

Let $H\leq G$.Then $(H\leq G,\sigma)$ is said to satisfy the _relative
Kleppner condition_ if every nontrivial $H$-conjugacy class in $G$ that is
$\sigma$-regular w.r.t. $H$ is infinite.

Clearly, this property depends on $\sigma$ only up to similarity, and $(H\leq
G,\sigma)$ automatically satisfies the relative Kleppner condition whenever
$G$ is icc relatively to $H$. Also, if $(H\leq G,\sigma)$ satisfies the
relative Kleppner condition, then both $(H,\sigma)$ and $(G,\sigma)$ satisfy
Kleppner’s condition. The converse implication does not necessarily hold, even
if $H$ is normal in $G$ and $\sigma$ is trivial. Indeed, if $G=H\times K$
where both $H$ and $K$ are icc groups and $K$ is nontrivial, then $G$ is icc,
but not icc relatively to $H$.

######  Remark 3.7.

The reader should be aware of the discrepancy between Definition 3.6 and the
relative Kleppner condition introduced in [7, Definition 4.5], where only
$H$-conjugacy classes in $G\setminus H$ are involved; the difference is
whether or not the assumption that $(H,\sigma)$ satisfies Kleppner’s condition
is part of the definition: in this paper, it is.

######  Remark 3.8.

If $H\leq G$ is such that $FC_{G}(H)=C_{G}(H)$, then $(H\leq G,\sigma)$
satisfies the relative Kleppner condition if and only if for every $g\in
C_{G}(H)\setminus\\{e\\}$, there exists some $h\in H$ such that
$\sigma(g,h)\neq\sigma(h,g)$.

When $H$ is a normal subgroup of $G$, one may wonder if the relative Kleppner
property for $(H\leq G,\sigma)$ can be characterized in a way similar to the
one obtained in Proposition 3.3 when $\sigma$ is trivial. This is a nontrivial
problem in general, but we will provide a positive answer for a large class of
normal subgroups in Theorem 3.15.

We define

$\displaystyle C_{G}^{\sigma}(H):$ $\displaystyle=\\{g\in G:\lvert
g^{H}\rvert=1\text{ and $g$ is $\sigma$-regular w.r.t.\leavevmode\nobreak\
$H$}\\}$ $\displaystyle=\\{g\in C_{G}(H):\widetilde{\sigma}(g,h)=1\,\text{ for
all }h\in H\\}.$

Using (3), one readily checks that $C_{G}^{\sigma}(H)$ is a subgroup of
$C_{G}(H)$. Moreover, if $H$ is normal in $G$, then so is $C_{G}^{\sigma}(H)$.
Next, we define

$\displaystyle FC_{G}^{\sigma}(H):$ $\displaystyle=\\{g\in G:\lvert
g^{H}\rvert<\infty\text{ and $g$ is $\sigma$-regular
w.r.t.\leavevmode\nobreak\ $H$}\\}$ $\displaystyle=\\{g\in FC_{G}(H):\text{$g$
is $\sigma$-regular w.r.t.\leavevmode\nobreak\ $H$}\\}.$

Clearly, $C_{G}^{\sigma}(H)\subseteq FC_{G}^{\sigma}(H)$, and $(H\leq
G,\sigma)$ satisfies the relative Kleppner condition if and only if
$FC_{G}^{\sigma}(H)$ is trivial. If $\sigma$ is nontrivial, then
$FC_{G}^{\sigma}(H)$ is not necessarily a subgroup of $G$. However, the
following two properties hold.

######  Lemma 3.9.

$FC_{G}^{\sigma}(H)$ is closed under the inverse operation.

###### Proof.

Let $g\in FC_{G}^{\sigma}(H)$. Then $g\in FC_{G}(H)$, so $g^{-1}\in
FC_{G}(H)$. Assume that $h\in H$ commutes with $g^{-1}$. Then $h$ also
commutes with $g$, so $\widetilde{\sigma}(g,h)=1$. Therefore, using (3), we
obtain

$1=\widetilde{\sigma}(g^{-1}g,h)=\widetilde{\sigma}(g^{-1},h)\widetilde{\sigma}(g,h)=\widetilde{\sigma}(g^{-1},h).$

This shows that $g^{-1}$ is $\sigma$-regular w.r.t. $H$. Thus, $g^{-1}\in
FC_{G}^{\sigma}(H)$. ∎

######  Lemma 3.10.

Let $r\in\mathbb{N}$ and suppose that $g_{1},\dotsc,g_{r}\in
FC_{G}^{\sigma}(H)$. Then there exists an $n\in\mathbb{N}$ such that
$(g_{1}\dotsm g_{r})^{mn}\in FC_{G}^{\sigma}(H)$ for all $m\in\mathbb{N}$.

###### Proof.

Let $g,k\in FC_{G}^{\sigma}(H)$. Since $FC_{G}(H)$ is a group, we have that
$(gk)^{n}\in FC_{G}(H)$ for every $n\geq 1$. Set $R:=C_{H}(g)\cap
C_{H}(k)=C_{H}(\\{g,k\\})$, and for each $n\geq 1$, set
$Q_{n}:=C_{H}((gk)^{n})$.

Clearly, $R$ is contained in $Q_{n}$ for every $n$, and it has finite index in
$H$ since it is an intersection of finite index subgroups. Moreover, if $m$
divides $n$, then $Q_{m}\subseteq Q_{n}$, so

$[Q_{1}:R]\leq[Q_{m}:R]\leq[Q_{n}:R]\leq[H:R]<\infty.$

It follows from (3) that

$\widetilde{\sigma}(xy,(gk)^{n})=\widetilde{\sigma}(x,(gk)^{n})\,\widetilde{\sigma}(y,(gk)^{n})$

for all $x\in G$ and $y\in Q_{n}$. In particular, for every $n\geq 1$, the map
$\nu_{n}\colon Q_{n}\to\mathbb{T}$ given by
$x\mapsto\widetilde{\sigma}(x,(gk)^{n})$ is a homomorphism. Define

$\begin{split}\kappa&:=\max\\{[Q_{n}:R]:n\geq 1\\},\\\
\ell&:=\min\\{n:[Q_{n}:R]=\kappa\\}.\end{split}$

We note that $[Q_{n}:R]=\kappa$ and $Q_{n}=Q_{\ell}$ whenever $n$ is a
multiple of $\ell$, since we then have that

$~\kappa\geq[Q_{n}:R]=[Q_{n}:Q_{\ell}]\,[Q_{\ell}:R]=[Q_{n}:Q_{\ell}]\,\kappa\,\geq\,\kappa.$

By applying (4) repeatedly, we get that the homomorphism $\nu_{\ell}\colon
Q_{\ell}\to\mathbb{T}$ maps $R$ to $\\{1\\}$. Indeed, for every $y\in R$, we
have $\widetilde{\sigma}(y,g)=\widetilde{\sigma}(y,k)=1$, hence

$\displaystyle\widetilde{\sigma}(y,gk)$
$\displaystyle=\widetilde{\sigma}(y,g)\widetilde{\sigma}(y,k)=1,$

and therefore

$\displaystyle\nu_{\ell}(y)=\widetilde{\sigma}(y,(gk)^{\ell})$
$\displaystyle=(\widetilde{\sigma}(y,gk))^{\ell}=1.$

Since $[Q_{\ell}:R]=\kappa$, this means that the image of $\nu_{\ell}$ is a
finite subgroup of $\mathbb{T}$ with at most $\kappa$ elements. Hence, for
each $m\geq 1$ and $x\in Q_{m\kappa\ell}=Q_{\ell}$, applying again (4)
repeatedly, we get that

$\widetilde{\sigma}(x,(gk)^{m\kappa\ell})=\widetilde{\sigma}(x,(gk)^{\ell})^{m\kappa}=(\nu_{\ell}(x))^{m\kappa}=(\nu_{\ell}(x)^{\kappa})^{m}=1.$

In other words, $\sigma(x,(gk)^{m\kappa\ell})=\sigma((gk)^{m\kappa\ell},x)$
for every $x\in H$ which commutes with $(gk)^{m\kappa\ell}$, i.e.,
$(gk)^{m\kappa\ell}$ is $\sigma$-regular w.r.t $H$. Thus,
$(gk)^{m\kappa\ell}\in FC^{\sigma}_{G}(H)$ for all $m\geq 1$.

This shows that the conclusion holds when $r=2$. It also holds if $r=1$ (by
setting $g_{2}:=e$). The proof when $r>2$ proceeds in essentially the same
way, now with $R:=\cap_{j=1}^{r}C_{H}(g_{j})$ and $Q_{n}=C_{H}((g_{1}\cdots
g_{r})^{n})$. ∎

We recall that the FC-center $FC(G)$ of $G$ consists of the elements of $G$
having a finite $G$-conjugacy class. In other words, $FC(G)=FC_{G}(G)$. The
equivalence between conditions (i) and (ii) in the following proposition is
pointed out by Connell in [14, p. 675].

######  Proposition 3.11.

The following conditions are equivalent:

  * (i)

$G$ has no finite normal subgroup except $\\{e\\}$.

  * (ii)

The FC-center of $G$ is a torsion-free group.

  * (iii)

The FC-center of $G$ is a torsion-free abelian group.

###### Proof.

The equivalences betweeen (i), (ii), and (iii) can be deduced from e.g. [41,
Theorem 4.32]. Indeed, assume that there exists some $r\in
FC(G)\setminus\\{e\\}$ having torsion. Then the normal subgroup $N$ of $G$
generated by the $G$-conjugacy class $r^{G}$ is contained in $FC(G)$. Hence,
$N$ is locally finite by [41, Theorem 4.32, ii)]. Since $N$ is finitely
generated, this means that it is finite. This shows that (i) $\Rightarrow$
(ii). Moreover, if $r,s\in FC(G)$, then [41, Theorem 4.32, ii)] also gives
that $rsr^{-1}s^{-1}$ is a torsion element of $FC(G)$. So if (ii) holds, then
we get that $rs=sr$. Thus, (ii) $\Rightarrow$ (iii), and the converse
implication is trivial. Finally, assume that $G$ has a nontrivial finite
normal subgroup $N$. Then $N$ is contained $FC(G)$. Since $N$ is finite, we
get that $FC(G)$ contains nontrivial torsion elements. This shows that (ii)
$\Rightarrow$ (i). ∎

Connell calls the group $G$ _prime_ whenever condition (i) (or (ii)) holds,
and shows in [14, Theorem 8] that $G$ is prime if and only if the group
algebra $\mathbb{C}[G]$ is prime. We will adopt his terminology.

######  Definition 3.12.

The group $G$ is said to be _prime_ when it satisfies any of the equivalent
conditions in Proposition 3.11.

Clearly, every icc group and every torsion-free group is prime. An $R$-group
(cf. Example 3.4) is prime if and only if its center is torsion-free. As shown
by Gelander and Glasner [19, Theorem 1.15], a countable non-elementary
convergence group (e.g., a subgroup of a Gromov hyperbolic group, or a
Kleinian group) is prime if and only if it admits a faithful primitive action
on a set.

Our next result is a twisted analogue of Lemma 3.2 in the case where $H$ is
prime.

######  Lemma 3.13.

Let $H$ be a normal subgroup of $G$. Assume that $H$ is prime and that
$(H,\sigma)$ satisfies Kleppner’s condition. Then

$C_{G}^{\sigma}(H)=FC_{G}^{\sigma}(H).$

###### Proof.

Suppose that $g\in FC_{G}^{\sigma}(H)$ and $k\in g^{H}$. Then $k\in
FC_{G}^{\sigma}(H)$, and $g^{-1}\in FC_{G}^{\sigma}(H)$ by Lemma 3.9. So Lemma
3.10 gives that $(g^{-1}k)^{n}$ is $\sigma$-regular w.r.t. $H$ for some $n\geq
1$. Moreover, $g^{-1}k\in H$ (since $H$ is normal and $k=hgh^{-1}$ for some
$h\in H$). As $(g^{-1}k)^{H}$ is finite, we get that $g^{-1}k\in FC(H)$, hence
that $(g^{-1}k)^{n}\in FC(H)$. Since $(H,\sigma)$ satisfies Kleppner’s
condition, this means that $(g^{-1}k)^{n}=e$. As $H$ is assumed to be prime,
$FC(H)$ is torsion-free, so we must have $g^{-1}k=e$, that is, $k=g$. This
shows that $\lvert g^{H}\rvert=1$. Hence,
$FC_{G}^{\sigma}(H)=C_{G}^{\sigma}(H)$. ∎

######  Remark 3.14.

When $C_{G}^{\sigma}(H)=FC_{G}^{\sigma}(H)$, e.g., when $H$ satisfies the
assumptions in Lemma 3.13, then by modifying the argument of Kleppner in [30,
Lemma 4], one may deduce that $C^{*}_{r}(H,\sigma)^{\prime}\cap
C^{*}_{r}(G,\sigma)=C^{*}_{r}(C_{G}^{\sigma}(H),\sigma)$ and
$\mathcal{L}(H,\sigma)^{\prime}\cap\mathcal{L}(G,\sigma)=\mathcal{L}(C_{G}^{\sigma}(H),\sigma)$.

As an immediate consequence of Lemma 3.13, we get:

######  Theorem 3.15.

Let $H$ be a normal subgroup of $G$ and assume that $H$ is prime. Then $(H\leq
G,\sigma)$ satisfies the relative Kleppner condition if and only if
$(H,\sigma)$ satisfies Kleppner’s condition and $C_{G}^{\sigma}(H)$ is
trivial.

##  4\. Irreducibility and primeness

Let $H$ be a subgroup of $G$. We will denote the restriction of $\sigma$ to
$H\times H$ by the same symbol $\sigma$. It is well-known that
$C^{*}_{r}(H,\sigma)$ (resp. $\mathcal{L}(H,\sigma)$) may be identified with
the $C^{*}$-subalgebra of $C_{r}^{*}(G,\sigma)$ (resp. the von Neumann
subalgebra of $\mathcal{L}(G,\sigma)$) generated by $\lambda_{\sigma}(H)$, cf.
[47, Subsection 6.46].

The following two lemmas are straightforward generalizations of [37, Lemmas
2.2 and 2.3] (the assumption that $H$ is normal is not used in the proof of
these two results).

######  Lemma 4.1.

Let $H\leq G$, $T\in\mathcal{L}(G,\sigma)$ and set
$f_{T}=T\delta_{e}\in\ell^{2}(G)$. Then the following conditions are
equivalent:

  * (i)

$T$ belongs to $\mathcal{L}(H,\sigma)^{\prime}\cap\mathcal{L}(G,\sigma)$.

  * (ii)

$f_{T}(hgh^{-1})=\widetilde{\sigma}(h,g)f_{T}(g)$ for all $h\in H$ and $g\in
G$.

Moreover, $f_{T}$ can be nonzero only on the finite $H$-conjugacy classes.

######  Lemma 4.2.

Let $H\leq G$ and $C$ be an $H$-conjugacy class in $G$. Then the following
conditions are equivalent:

  * (i)

$C$ is $\sigma$-regular w.r.t. $H$.

  * (ii)

There is a function $f\colon G\to\mathbb{C}$ satisfying:

    * 1.

$f(g)\neq 0$ for all $g\in C$.

    * 2.

$f(hgh^{-1})=\widetilde{\sigma}(h,g)f(g)$ for all $g\in C$ and all $h\in H$.

Moreover, $f$ can be chosen in $\ell^{2}(G)$ if and only if $C$ is finite.

The twisted analogue of [42, Proposition 5.1] is the following:

######  Proposition 4.3.

Let $H\leq G$. Then the following assertions are equivalent:

  * (i)

$(H\leq G,\sigma)$ satisfies the relative Kleppner condition.

  * (ii)

$\mathcal{L}(H,\sigma)^{\prime}\cap\,\mathcal{L}(G,\sigma)=\mathbb{C}1$, i.e.,
the inclusion $\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is
irreducible.

  * (iii)

$C^{*}_{r}(H,\sigma)^{\prime}\cap C^{*}_{r}(G,\sigma)=\mathbb{C}1$, i.e., the
inclusion $C_{r}^{*}(H,\sigma)\subseteq C_{r}^{*}(G,\sigma)$ is irreducible.

Moreover, if $G$ is countable, then
$\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is $C^{*}$-irreducible
if and only if $(H\leq G,\sigma)$ satisfies the relative Kleppner condition
and $[G:H]<\infty$.

###### Proof.

(i) $\Longrightarrow$ (ii) follows readily from Lemmas 4.1 and 4.2.

(ii) $\Longrightarrow$ (iii) is obvious.

(iii) $\Longrightarrow$ (i): If there is a finite nontrivial $H$-conjugacy
class $C$ in $G$ that is $\sigma$-regular w.r.t. $H$, then Lemma 4.2 ensures
that there exists a function $f\colon C\to\mathbb{C}$ such that $\sum_{g\in
C}f(g)\lambda_{\sigma}(g)$ is nonscalar and belongs to
$C^{*}_{r}(H,\sigma)^{\prime}\cap C^{*}_{r}(G,\sigma)$.

Finally, assume that $G$ is countable. In view of the equivalence of (i) and
(ii), we have to show that
$\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is $C^{*}$-irreducible
if and only if $\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is
irreducible and $[G:H]<\infty$. We may then assume that both
$\mathcal{L}(H,\sigma)$ and $\mathcal{L}(G,\sigma)$ are separable II1-factors.
Then the desired equivalence follows from [42, Theorem 4.4] (which Rørdam
attributes to Pop and Popa), taking into account that it is well-known that
the Jones index $[\mathcal{L}(G,\sigma):\mathcal{L}(H,\sigma)]$ is equal to
$[G:H]$. ∎

For regular irreducible inclusions of twisted group von Neumann algebras,
intermediate von Neumann algebras have a particularly simple description:

######  Proposition 4.4.

Assume that $H$ is normal in $G$ and $(H\leq G,\sigma)$ satisfies the relative
Kleppner condition, i.e., the inclusion
$\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is irreducible. Then the
map $\Gamma\mapsto\mathcal{L}(\Gamma,\sigma)$ gives a bijective correspondence
between groups $\Gamma$ satisfying $H\leq\Gamma\leq G$, and von Neumann
algebras $N$ satisfying $\mathcal{L}(H,\sigma)\subseteq
N\subseteq\mathcal{L}(G,\sigma)$.

###### Proof.

This result could be proven by using results from [23] or [11], or by
decomposing $\mathcal{L}(G,\sigma)$ as a twisted crossed product and applying
[24, Lemma 3.3]. For the ease of the reader, we sketch a direct proof, close
in spirit to the proof of [13, Corollary 4].

Clearly, if $H\leq\Gamma\leq G$, then $N_{\Gamma}:=\mathcal{L}(\Gamma,\sigma)$
is a subfactor of $\mathcal{L}(G,\sigma)$ containing $\mathcal{L}(H,\sigma)$.
Conversely, let $N$ be a von Neumann algebra satisfying
$\mathcal{L}(H,\sigma)\subseteq N\subseteq\mathcal{L}(G,\sigma)$, and let
$E_{\tau}^{N}$ denote the faithful normal conditional expectation from
$\mathcal{L}(G,\sigma)$ onto $N$ associated to $\tau$. We note that for any
$g\in G$, we have that
$\lambda_{\sigma}(g)^{*}E_{\tau}^{N}(\lambda_{\sigma}(g))\in\mathbb{C}1$.

Indeed, let $x\in\mathcal{L}(H,\sigma)$. Then
$\alpha_{g}(x):=\lambda_{\sigma}(g)x\lambda_{\sigma}(g)^{*}\in\mathcal{L}(H,\sigma)$
(because $H$ is normal in G), so we get

$\alpha_{g}(x)E_{N}^{\tau}(\lambda_{\sigma}(g))=E_{N}^{\tau}(\alpha_{g}(x)\lambda_{\sigma}(g))=E_{N}^{\tau}(\lambda_{\sigma}(g)x)=E_{N}^{\tau}(\lambda_{\sigma}(g))x.$

This implies that

$x\,\lambda_{\sigma}(g)^{*}E_{N}^{\tau}(\lambda_{\sigma}(g))=\lambda_{\sigma}(g)^{*}E_{N}^{\tau}(\lambda_{\sigma}(g))\,x.$

Thus we get that
$\lambda_{\sigma}(g)^{*}E_{N}^{\tau}(\lambda_{\sigma}(g))\in\mathcal{L}(H,\sigma)^{\prime}\cap\,\mathcal{L}(G,\sigma)=\mathbb{C}1$,
as asserted.

Now, set $\Gamma_{N}:=\\{g\in G:\lambda_{\sigma}(g)\in N\\}$. Then
$\Gamma_{N}$ is easily seen to be a subgroup of $G$ containing $H$, and the
observation above gives that $E_{\tau}^{N}(\lambda_{\sigma}(g))=0$ whenever
$g\not\in\Gamma_{N}$. This readily implies that $E_{\tau}^{N}$ maps
$\mathcal{L}(G,\sigma)$ into $\mathcal{L}(\Gamma_{N},\sigma)$, hence that

$N=E_{N}^{\tau}(\mathcal{L}(G,\sigma))\subseteq\mathcal{L}(\Gamma_{N},\sigma)\subseteq
N.$

Thus $N=\mathcal{L}(\Gamma_{N},\sigma)=N_{\Gamma_{N}}$. This shows that the
map $\Gamma\mapsto N_{\Gamma}$ is surjective. To see that it is injective, it
suffices to show that $\Gamma=\Gamma_{N_{\Gamma}}$ whenever $H\leq\Gamma\leq
G$. By definition, the inclusion $\subseteq$ holds. On the other hand, let
$g\in\Gamma_{N_{\Gamma}}$, i.e., $g\in G$ and $\lambda_{\sigma}(g)\in
N_{\Gamma}=\mathcal{L}(\Gamma,\sigma)$. Then we have

$\delta_{g}=\lambda_{\sigma}(g)\delta_{e}\in\mathcal{L}(\Gamma,\sigma)\delta_{e}\subseteq\\{\xi\in\ell^{2}(G):\xi(s)=0\text{
for all }s\in G\setminus\Gamma\\},$

which is possible only if $g\in\Gamma$. ∎

######  Remark 4.5.

The assumption in Proposition 4.4 that $H$ is normal can not be removed. This
can be seen by using an observation due to Jiang in [24, Section 4]. Let $K$
be an infinite simple group, and set $G:=K\times K$ and $H=\\{(k,k):k\in
K\\}$. Then $H$ is a maximal subgroup of $G$, and as shown by Jiang,
$\mathcal{L}(H)$ is not maximal in $\mathcal{L}(G)$, i.e., there exists a von
Neumann subalgebra $N$ of $\mathcal{L}(G)$ containing $\mathcal{L}(H)$ such
that $\mathcal{L}(H)\neq N\neq\mathcal{L}(G)$. Since $K$ is icc, it is easy to
verify that $G$ is icc relatively to $H$.

Recall that a $C^{*}$-algebra is said to be _prime_ if nonzero ideals have
nonzero intersection. As shown in [37, Theorem 2.7], $(G,\sigma)$ satisfies
Kleppner’s condition if and only if $C^{*}_{r}(G,\sigma)$ is prime, if and
only if $C^{*}_{r}(G,\sigma)$ has trivial center. In this connection, we note
that if $B\subseteq A$ is a unital inclusion of $C^{*}$-algebras, then the
following conditions are equivalent:

  * (i)

The inclusion $B\subseteq A$ is irreducible.

  * (ii)

Every $C^{*}$-algebra $C$ satisfying $B\subseteq C\subseteq A$ has trivial
center.

Indeed, if (i) holds and $C$ is an intermediate $C^{*}$-algebra, then
$C^{\prime}\cap C\subseteq B^{\prime}\cap A=\mathbb{C}1$. Conversely, if (i)
does not hold, and we choose a nonscalar $a\in B^{\prime}\cap A$, then the
$C^{*}$-algebra $C$ generated by $B$ and $a$ has a nontrivial center.

Using Proposition 4.3 we get the following generalization:

######  Corollary 4.6.

The following conditions are equivalent:

  * (i)

$(H\leq G,\sigma)$ satisfies the relative Kleppner condition.

  * (ii)

Every $C^{*}$-algebra $A$ satisfying $C^{*}_{r}(H,\sigma)\subseteq A\subseteq
C^{*}_{r}(G,\sigma)$ is prime.

  * (iii)

Every $C^{*}$-algebra $A$ satisfying $C^{*}_{r}(H,\sigma)\subseteq A\subseteq
C^{*}_{r}(G,\sigma)$ has trivial center.

###### Proof.

(i) $\Longrightarrow$ (ii): Assume (i) holds. Then Proposition 4.3 gives that
$\mathcal{L}(H,\sigma)\subseteq\mathcal{L}(G,\sigma)$ is irreducible. Now, if
$A$ is a $C^{*}$-algebra satisfying $C^{*}_{r}(H,\sigma)\subseteq A\subseteq
C^{*}_{r}(G,\sigma)$, then we get $\mathcal{L}(H,\sigma)\subseteq
A^{\prime\prime}\subseteq\mathcal{L}(G,\sigma)$, so $A^{\prime\prime}$ is a
factor; as is well-known, this implies that $A$ is prime. Hence, (ii) holds.

(ii) $\Longrightarrow$ (iii): Since every prime $C^{*}$-algebra has trivial
center, this is clear.

(iii) $\Longrightarrow$ (i): Assume (iii) holds. Then, as observed above,
$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ is irreducible. Hence, (i)
holds by Proposition 4.3. ∎

##  5\. $C^{*}$-irreducibility and reduced twisted $C^{*}$-crossed products

In this section, we assume that $(A,G,\alpha,\sigma)$ is a unital, discrete,
twisted C∗-dynamical system, i.e., $A$ is a $C^{*}$-algebra with unit $1$, $G$
is a discrete group with identity $e$ and $(\alpha,\sigma)$ is a twisted
action of $G$ on $A$, that is, $\alpha$ is a map from $G$ into the group of
$*$-automorphisms of $A$, and $\sigma$ is a map from $G\times G$ into the
unitary group of $A$, satisfying

$\begin{gathered}\alpha_{g}\circ\alpha_{h}=\operatorname{Ad}(\sigma(g,h))\circ\alpha_{gh},\\\
\sigma(g,h)\sigma(gh,k)=\alpha_{g}(\sigma(h,k))\sigma(g,hk),\\\
\sigma(g,e)=\sigma(e,g)=1,\end{gathered}$

for all $g,h,k\in G$, where $\operatorname{Ad}(v)$ denotes the (inner)
$*$-automorphism of $A$ implemented by a unitary $v$ in $A$.

For simplicity, we will denote the reduced crossed product associated to
$(A,G,\alpha,\sigma)$ by $A\rtimes_{r}G$. (If confusion may arise, we will
denote it by $A\rtimes_{(\alpha,\sigma),r}G$.) As a $C^{*}$-algebra,
$A\rtimes_{r}G$ is generated by (a copy of) $A$ and a family $\\{u(g):g\in
G\\}$ of unitaries satisfying

$\alpha_{g}(a)=u(g)au(g)^{*}\quad\text{ and }\quad u(g)u(h)=\sigma(g,h)u(gh)$

for all $g,h\in G$ and $a\in A$. Moreover, there exists a faithful conditional
expectation $E\colon A\rtimes_{r}G\to A$ such that $E(u(g))=0$ for all $g\in
G\,,\,g\neq e\,.$

We recall that $(\alpha,\sigma)$ is said to be _outer_ if $\alpha_{g}$ is
outer for each $g\in G\setminus\\{e\\}$. The twisted version of [42, Theorem
5.8] is as follows:

######  Theorem 5.1.

The following conditions are equivalent:

  * (i)

$A\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible.

  * (ii)

$A$ is simple and $(\alpha,\sigma)$ is outer.

  * (iii)

$A$ is simple and $A^{\prime}\cap(A\rtimes_{r}G)=\mathbb{C}1$.

Moreover, if $A$ has the Dixmier property, then any of the above conditions
implies that $A\subseteq A\rtimes_{r}G$ has the relative Dixmier property
$($as defined in [40]$)$.

###### Proof.

(i) $\Longrightarrow$ (iii): Follows from [42, Remark 3.8].

(iii) $\Longrightarrow$ (ii): Same argument as in the proof of [42, Theorem
5.8].

(ii) $\Longrightarrow$ (i): Assume that (ii) holds. As shown in [4, Theorem
3.2], Kishimoto’s result that $A\rtimes_{r}G$ is simple remains true for
twisted actions. In fact, arguing as in the proof of [42, Theorem 5.8], one
may show that $A\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible. Indeed, let
$B_{0}$ denote the $*$-algebra generated by $A$ and $\\{u(g):g\in G\\}$, i.e.,
$B_{0}=\operatorname{Span}\\{au(g):a\in A,g\in G\\}$. Using [31, Lemma 3.2]
(or [35, Lemma 7.1] if $A$ is separable), and the denseness of $B_{0}$ in
$A\rtimes_{r}G$, one deduces that $A\subseteq A\rtimes_{r}G$ has the pinching
property with respect to $E$ (cf. [42, Definition 3.13]). The conclusion
follows then from [42, Proposition 3.15].

The last statement follows from [40, Corollary 4.1]. ∎

Cameron and Smith’s result about intermediate $C^{*}$-algebras in simple
reduced twisted $C^{*}$-crossed products [12, Theorem 4.4] may now be
reformulated as follows.

######  Theorem 5.2.

Assume that $A\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible. Then there is a
bijective correspondence between subgroups $\Gamma$ of $G$ and
$C^{*}$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes_{r}G$, given
by

$\Gamma\mapsto A\rtimes_{r}\Gamma.$

###### Proof.

Since $A$ is simple and $(\alpha,\sigma)$ is outer by Theorem 5.1, the
conclusion follows from [12, Theorem 4.4]. ∎

Assume that $H$ is a normal subgroup of $G$ and set $K:=G/H$. By restricting
$\alpha$ to $H$ and $\sigma$ to $H\times H$, we get a twisted
$C^{*}$-dynamical system $(A,H,\alpha,\sigma)$. The associated reduced crossed
product $A\rtimes_{(\alpha,\sigma),r}H$ may then be identified with the
$C^{*}$-subalgebra of $A\rtimes_{(\alpha,\sigma),r}G$ generated by $A$ and
$\\{u(h):h\in H\\}$. We recall from [4, Theorem 2.1] that there exists an
induced twisted action $(\beta,\omega)$ of $K$ on
$A\rtimes_{(\alpha,\sigma),r}H$ such that

$(A\rtimes_{(\alpha,\sigma),r}H)\rtimes_{(\beta,\omega),r}K\,\simeq\,A\rtimes_{(\alpha,\sigma),r}G$

under a $*$-isomorphism which maps $A\rtimes_{(\alpha,\sigma),r}H$ onto its
canonical copy in $A\rtimes_{(\alpha,\sigma),r}G$.

Combining this decomposition result with Theorem 5.1 and Theorem 5.2, we
obtain:

######  Theorem 5.3.

Let $H$ be a normal subgroup of $G$. Then the following conditions are
equivalent:

  * (i)

$A\rtimes_{r}H\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible.

  * (ii)

$A\rtimes_{r}H$ is simple and the induced twisted action of $G/H$ on
$A\rtimes_{r}H$ is outer.

  * (iii)

$A\rtimes_{r}H$ is simple and
$(A\rtimes_{r}H)^{\prime}\cap(A\rtimes_{r}G)=\mathbb{C}1$.

Moreover, if $A\rtimes_{r}H\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible,
then there is a bijective correspondence between groups $\Gamma$ satisfying
$H\leq\Gamma\leq G$, and $C^{*}$-algebras $B$ satisfying
$A\rtimes_{r}H\subseteq B\subseteq A\rtimes_{r}G$, given by

$\Gamma\mapsto A\rtimes_{r}\Gamma.$

In turn, combining this result with [4, Lemmas 3.3 and 3.4], we obtain the
following strengthening of [4, Theorem 3.5] (which only asserts the simplicity
of $A\rtimes_{r}G$):

######  Theorem 5.4.

Let $H$ be a normal subgroup of $G$, and assume that the following conditions
are satisfied:

  * •

$A\rtimes_{r}H$ is simple.

  * •

$A$ has a faithful $G$-invariant state.

  * •

$H$ is icc and $C_{G}(H)$ is trivial (i.e., $G$ is icc relatively to $H$ by
Proposition 3.3).

Then $A\rtimes_{r}H\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible.

######  Remark 5.5.

The assumption in Theorem 5.4 that $A$ has a faithful $G$-invariant state is
of a technical nature for the proof of [4, Lemmas 3.3] to go through, and it
might be redundant. Anyhow, it is for example satisfied when $A$ has a unique
faithful tracial state.

A noteworthy consequence of Theorem 5.4 is:

######  Corollary 5.6.

Let $H$ be a normal subgroup of $G$ and assume that the following conditions
are satisfied:

  * •

$A$ is $H$-simple (i.e., the only $H$-invariant ideals of $A$ are $\\{0\\}$
and $A$).

  * •

$A$ has a faithful $G$-invariant state.

  * •

$H$ is $C^{*}$-simple and $C_{G}(H)$ is trivial.

Then $A\rtimes_{r}H\subseteq A\rtimes_{r}G$ is $C^{*}$-irreducible.

###### Proof.

As $A$ is $H$-simple and $H$ is $C^{*}$-simple, we get from [10, Corollary
4.4] that $A\rtimes_{r}H$ is simple. As any $C^{*}$-simple group is icc, we
can apply Theorem 5.4. ∎

As will be seen in Theorem 6.4, if $H$ is normal in $G$, then $H$ is
$C^{*}$-simple and $C_{G}(H)$ is trivial if and only if $H\leq G$ is
$C^{*}$-irreducible.

##  6\. $C^{*}$-irreducibility and reduced twisted group $C^{*}$-algebras

We recall that $G$ is a discrete group, $H$ is a subgroup of $G$, and $\sigma$
is a two-cocycle on $G$.

######  Definition 6.1.

We will say that _$(H\leq G,\sigma)$ is $C^{*}$-irreducible_ if
$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ is $C^{*}$-irreducible.

If $\sigma^{\prime}$ is a two-cocycle on $G$ which is similar to $\sigma$,
then it is clear that the canonical $*$-isomorphism $\Phi\colon
C_{r}^{*}(G,\sigma^{\prime})\to C_{r}^{*}(G,\sigma)$ maps
$C_{r}^{*}(H,\sigma^{\prime})$ onto $C_{r}^{*}(H,\sigma)$, and it therefore
follows that $(H\leq G,\sigma^{\prime})$ is $C^{*}$-irreducible if and only if
$(H\leq G,\sigma)$ is $C^{*}$-irreducible.

When $H$ is normal in $G$, $C^{*}$-irreducibility of $(H\leq G,\sigma)$ can be
characterized as follows:

######  Theorem 6.2.

Assume $H$ is normal in $G$. Then the following conditions are equivalent:

  * (i)

$(H\leq G,\sigma)$ is $C^{*}$-irreducible.

  * (ii)

$(H,\sigma)$ is $C^{*}$-simple and $(H\leq G,\sigma)$ satisfies the relative
Kleppner condition.

Both these conditions are satisfied if the following holds:

  * (iii)

$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ has the relative Dixmier
property.

Moreover, if $(H,\sigma)$ has the unique trace property, then all three
conditions are equivalent.

Finally, if $(H\leq G,\sigma)$ is $C^{*}$-irreducible, then the map
$\Gamma\mapsto C_{r}^{*}(\Gamma,\sigma)$ gives a bijective correspondence
between groups $\Gamma$ satisfying $H\leq\Gamma\leq G$, and $C^{*}$-algebras
$B$ satisfying $C_{r}^{*}(H,\sigma)\subseteq B\subseteq C_{r}^{*}(G,\sigma)$.

###### Proof.

The equivalence between (i) and (ii) follows from Theorem 5.3 in the case
where $A=\mathbb{C}$, in combination with Proposition 4.3. The implication
(iii) $\Longrightarrow$ (i) is a consequence of [42, Proposition 3.12]. If
$(H,\sigma)$ has the unique trace property, then (ii) $\Longrightarrow$ (iii)
follows from [40, Examples 4.3]. Indeed, the observation made there also works
in the twisted case, by decomposing $C_{r}^{*}(G,\sigma)$ as a twisted crossed
product $C^{*}_{r}(H,\sigma)\rtimes_{(\beta,\omega),r}(G/H)$ and noticing that
the twisted action $(\beta,\omega)$ of $G/H$ is then outer by Theorem 5.3. The
last assertion follows from Theorem 5.3 ∎

######  Corollary 6.3.

Let $H$ be a normal subgroup of $G$, and assume $H$ is FC-hypercentral or
$C^{*}$-simple. Then the following conditions are equivalent:

  * (i)

$(H\leq G,\sigma)$ is $C^{*}$-irreducible.

  * (ii)

$(H\leq G,\sigma)$ satisfies the relative Kleppner condition.

  * (iii)

$C^{*}_{r}(H,\sigma)\subseteq C^{*}_{r}(G,\sigma)$ has the relative Dixmier
property.

###### Proof.

As shown in [6, Theorem 3.1], if $H$ is FC-hypercentral, then $(H,\sigma)$ is
$C^{*}$-simple if and only if $(H,\sigma)$ has the unique trace property, if
and only if $(H,\sigma)$ satisfies Kleppner’s condition. On the other hand, if
$H$ is $C^{*}$-simple, then $(H,\sigma)$ is $C^{*}$-simple and has the unique
trace property, cf. [10, Corollaries 4.5 and 5.3]. Hence, Theorem 6.2 gives
the assertion. ∎

We will obtain a related result involving the twisted centralizer
$C_{G}^{\sigma}(H)$ in Corollary 6.10. Breuillard, Kalantar, Kennedy and Ozawa
have shown in [8, Theorem 1.4] that if $H$ is normal in $G$, then $G$ is
$C^{*}$-simple if and only if both $H$ and $C_{G}(H)$ are $C^{*}$-simple. In
general, if $H$ is normal in $G$ and $G$ is $C^{*}$-simple, then the inclusion
$H\leq G$ is not necessarily $C^{*}$-irreducible (consider for example
$G=H\times K$ with $H$ and $K$ both $C^{*}$-simple). In fact, we have:

######  Theorem 6.4.

Assume $H$ is normal in $G$. Then the following conditions are equivalent:

  * (i)

$H\leq G$ is $C^{*}$-irreducible.

  * (ii)

$H$ is $C^{*}$-simple and $G$ is icc relatively to $H$.

  * (iii)

$H$ is $C^{*}$-simple and $C_{G}(H)$ is trivial.

  * (iv)

$C^{*}_{r}(H)\subseteq C^{*}_{r}(G)$ has the relative Dixmier property.

  * (v)

$H\leq G$ is relatively $C^{*}$-simple.

###### Proof.

Theorem 6.2 gives that (i) is equivalent to (ii). Since any $C^{*}$-simple
group is icc, the equivalence (ii) $\Longleftrightarrow$ (iii) follows from
Proposition 3.3. Taking into account that $H$ has the unique trace property
whenever $H$ is $C^{*}$-simple, cf. [8, Theorem 1.3], the equivalence between
(i) and (iv) follows also from Theorem 6.2. Finally, the equivalence (iii)
$\Longleftrightarrow$ (v) is a consequence of [45, Theorem 1.3 and 1.6]. ∎

######  Remark 6.5.

A subgroup $H$ of $G$ is said to be _plump_ in $G$ if a relative version of
Powers’ averaging property holds (see [1] and [45, Definition 1.1]). In
general, plumpness of $H$ in $G$ implies $C^{*}$-irreducibility of $H\leq G$
[42, Theorem 5.3], and is equivalent to relative $C^{*}$-simplicity of $H\leq
G$ when $H$ is normal [45, Theorem 1.6]. It therefore follows from Theorem 6.4
that plumpness of $H$ in $G$ is equivalent to $C^{*}$-irreducibility of $H\leq
G$ when $H$ is normal.

Let $\Gamma$ be a $C^{*}$-simple group, and let $K=\operatorname{Aut}(\Gamma)$
denote the group of all automorphisms of $\Gamma$. It is shown in [4,
Corollary 3.7] that $K$ is $C^{*}$-simple too. As $\Gamma$ has trivial center,
we may identify $\Gamma$ with the normal subgroup of $K$ consisting of all
inner automorphisms of $\Gamma$, and $C_{K}(\Gamma)$ is then trivial. Theorem
6.4 therefore gives:

######  Corollary 6.6.

Let $\Gamma$ be a $C^{*}$-simple group. Then
$\Gamma\leq\operatorname{Aut}(\Gamma)$ is $C^{*}$-irreducible.

Similarly, if $\Gamma$ is $C^{*}$-simple and we let it act on itself by inner
automorphisms, then we have that $\Gamma\rtimes\Gamma\simeq\Gamma\times\Gamma$
is $C^{*}$-simple, so we get that
$\Gamma\times\Gamma\leq\Gamma\rtimes\operatorname{Aut}(\Gamma)$ is
$C^{*}$-irreducible (cf. the proof of [4, Corollary 3.8]).

######  Corollary 6.7.

Assume $H$ is normal in $G$ and $H\leq G$ is $C^{*}$-irreducible. Then $(H\leq
G,\sigma)$ is $C^{*}$-irreducible.

###### Proof.

By Theorem 6.4, $H$ is $C^{*}$-simple and $C_{G}(H)$ is trivial. Using
Corollary 5.6 with $A=\mathbb{C}$, we get that $(H\leq G,\sigma)$ is
$C^{*}$-irreducible. ∎

######  Remark 6.8.

When $H$ is not normal in $G$, the equivalence between (i) and (ii) in Theorem
6.2 does not hold in general, even if $\sigma$ is trivial. Here is an example
where $H$ and $G$ are both $C^{*}$-simple, $G$ is icc relatively to $H$, but
$H\leq G$ is not $C^{*}$-irreducible. Consider

$H=\left\langle\begin{bmatrix}1&2\\\ 0&1\end{bmatrix},\begin{bmatrix}1&0\\\
2&1\end{bmatrix}\right\rangle\leq\operatorname{SL}(2,\mathbb{Z}).$

It is well-known that $H\simeq\mathbb{F}_{2}$ ($H$ is sometimes called the
Sanov subgroup of $\operatorname{SL}(2,\mathbb{Z})$).

Furthermore, set

$K=\mathbb{Z}^{2}\rtimes H\quad\text{ and }\quad G=K*\mathbb{Z},$

where $H$ acts on $\mathbb{Z}^{2}$ in the natural way. Then $H\leq K\leq G$,
both $H$ and $G$ are $C^{*}$-simple, but $K$ is not $C^{*}$-simple since it
contains a nontrivial normal abelian subgroup.

We now check that $G$ is icc relatively to $H$. Let
$x=(x_{1},x_{2})\in\mathbb{Z}^{2}$ and let $u$ and $v$ be the generators of
$H$ inside $K$. Then

$u^{n}xu^{-n}=(x_{1}+2nx_{2},x_{2})\quad\text{ and }\quad
v^{n}xv^{-n}=(x_{1},x_{2}+2nx_{1}),$

and it follows that $x^{H}=\\{yxy^{-1}:y\in H\\}$ is infinite for all
$x\in\mathbb{Z}^{2}\setminus\\{0\\}$. Next, consider $xy\in K$ where $y\in
H\setminus\\{e\\}$. We can always pick $w$ to be one of $u,u^{-1},v,v^{-1}$
such that $w^{n}yw^{-n}$ has no cancellation for $n\geq 1$. Then

$\lvert(xy)^{H}\rvert\geq\lvert\\{w^{n}xyw^{-n}:n\geq
1\\}\rvert=\lvert\\{w^{n}xw^{-n}\cdot w^{n}yw^{-n}:n\geq 1\\}\rvert=\infty.$

One checks in a similar way that any word in $K$ and $\mathbb{Z}$, with at
least one letter from $\mathbb{Z}$, has an infinite conjugacy class w.r.t.
$H$. Thus, $g^{H}$ is infinite for all $g\in G\setminus\\{e\\}$.

######  Corollary 6.9.

Let $H$ be a normal subgroup of $G$ and assume that $H$ is prime. Then $(H\leq
G,\sigma)$ is $C^{*}$-irreducible if and only if $(H,\sigma)$ is
$C^{*}$-simple and $C_{G}^{\sigma}(H)$ is trivial.

###### Proof.

The assertion follows by combining Theorem 6.2 with Theorem 3.15. ∎

######  Corollary 6.10.

Let $H$ be a normal subgroup of $G$.

If $H$ is prime and $FC$-hypercentral, then $(H\leq G,\sigma)$ is
$C^{*}$-irreducible if and only if $(H,\sigma)$ satisfies Kleppner’s condition
and $C_{G}^{\sigma}(H)$ is trivial.

If $H$ is $C^{*}$-simple, then $(H\leq G,\sigma)$ is $C^{*}$-irreducible if
and only if $C_{G}^{\sigma}(H)$ is trivial.

###### Proof.

We note that if $H$ is $C^{*}$-simple, then $H$ is icc, so $H$ is prime and
$(H,\sigma)$ satisfies Kleppner’s condition. Both assertions therefore follows
by combining Corollary 6.3 with Theorem 3.15. ∎

##  7\. Some examples

######  Example 7.1 (Noncommutative tori I).

Let $\theta\in\mathbb{R}$ and let $A_{\theta}$ be the associated
noncommutative $2$-torus, i.e., the universal unital $C^{*}$-algebra generated
by two unitaries $U_{1},U_{2}$ satisfying the relation

$U_{1}U_{2}=e^{i2\pi\theta}U_{2}U_{1}$

As is well-known, we may assume that
$A_{\theta}=C_{r}^{*}(\mathbb{Z}^{2},\sigma_{\theta})$, where
$\sigma_{\theta}$ is the two-cocycle on $\mathbb{Z}^{2}$ given by

$\sigma_{\theta}(\mathbf{x},\mathbf{y})=e^{i\pi\,\theta(x_{1}y_{2}-x_{2}y_{1})}$

for all $\mathbf{x}=(x_{1},x_{2}),\mathbf{y}=(y_{1},y_{2})\in\mathbb{Z}^{2}$.
Moreover, $A_{\theta}$ is simple if and only if
$(\mathbb{Z}^{2},\sigma_{\theta})$ satisfies Kleppner’s condition, which
happens if and only if $\theta$ is irrational.

Now, let $p,q\in\mathbb{N}$ and set $H_{p,q}:=p\mathbb{Z}\times
q\mathbb{Z}\leq\mathbb{Z}^{2}$,
$B_{p,q,\theta}:=C_{r}^{*}(H_{p,q},\sigma_{\theta})$. One readily checks that
$B_{p,q,\theta}\simeq A_{pq\theta}$. Further, we have that the inclusion
$B_{p,q,\theta}\subseteq A_{\theta}$ is $C^{*}$-irreducible if (and only if)
$\theta$ is irrational.

Indeed, let $\theta$ be irrational. Since $H_{p,q}$ is abelian, it suffices to
show that $(H_{p,q}\leq\mathbb{Z}^{2},\sigma_{\theta})$ satisfies the relative
Kleppner condition, cf. Corollary 6.3. Let
$\mathbf{x}=(x_{1},x_{2})\in\mathbb{Z}^{2}$. Then a short computation gives
that $\mathbf{x}$ is $\sigma_{\theta}$-regular w.r.t. $H_{p,q}$ if and only if

$\theta\,(qx_{1}m_{2}-pm_{1}x_{2})\in\mathbb{Z}$

for all $\mathbf{y}=(pm_{1},qm_{2})\in H_{p,q}$, which is clearly possible if
and only if $\mathbf{x}=(0,0)$. Thus $(0,0)$ is the only element of
$\mathbb{Z}^{2}$ which is $\sigma_{\theta}$-regular w.r.t. $H_{p,q}$, and the
desired conclusion follows. Using the final assertion of Theorem 6.2, we get
that there is a one-to-one correspondence between intermediate
$C^{*}$-algebras of $B_{p,q,\theta}\subseteq A_{\theta}$ and subgroups of
$\mathbb{Z}_{p}\times\mathbb{Z}_{q}$. In particular, there are no strict
intermediate $C^{*}$-algebras in this inclusion if for example $p=1$ and $q$
is prime.

######  Example 7.2 (Noncommutative tori II).

Let $\mathbf{\Theta}=(\theta_{1},\theta_{2},\theta_{3})\in\mathbb{R}^{3}$ and
let $A=A_{\mathbf{\Theta}}$ be the associated noncommutative $3$-torus, i.e.,
the universal unital $C^{*}$-algebra generated by three unitaries
$U_{1},U_{2},U_{3}$ satisfying the relations

$U_{1}U_{2}=e^{i2\pi\theta_{3}}U_{2}U_{1},\,\,U_{2}U_{3}=e^{i2\pi\theta_{1}}U_{3}U_{2},\,\,U_{3}U_{1}=e^{i2\pi\theta_{2}}U_{1}U_{3}\,.$

Then $A$ is simple if and only if the dimension $d(\mathbf{\Theta})$ of
$\mathbb{Q}+\theta_{1}\mathbb{Q}+\theta_{2}\mathbb{Q}+\theta_{3}\mathbb{Q}$
(as a vector space over $\mathbb{Q}$) is $3$ or $4$ (see e.g. [5]).

Now, let $B$ be the $C^{*}$-subalgebra of $A$ generated by $U_{1}$ and
$U_{2}$. We may then deduce from Theorem 6.2 that the following statements are
equivalent:

  * i)

$B\subseteq A$ is $C^{*}$-irreducible,

  * ii)

$\theta_{3}\not\in\mathbb{Q}$ and $d(\mathbf{\Theta})\in\\{3,4\\}$.

Indeed, we may assume that
$A=C_{r}^{*}(\mathbb{Z}^{3},\sigma_{\mathbf{\Theta}})$, where
$\sigma_{\mathbf{\Theta}}$ is the two-cocycle on $\mathbb{Z}^{3}$ given by

$\sigma_{\mathbf{\Theta}}(\mathbf{x},\mathbf{y})=e^{i\pi\,\mathbf{\Theta}\cdot(\mathbf{x}\times\mathbf{y})}$

for all $\mathbf{x},\mathbf{y}\in\mathbb{Z}^{3}$,
$\mathbf{\Theta}\cdot(\mathbf{x}\times\mathbf{y})$ denoting the scalar triple
product.

Letting $H$ be the subgroup of $\mathbb{Z}^{3}$ given by
$H=\mathbb{Z}^{2}\times\\{0\\}=\\{(y_{1},y_{2},0):y_{1},y_{2}\in\mathbb{Z}\\}$,
we have that $B=C_{r}^{*}(H,\sigma_{\mathbf{\Theta}})$ is isomorphic to the
noncommutative $2$-torus $A_{\theta_{3}}$ associated to $\theta_{3}$. Thus $B$
is simple if and only if it $\theta_{3}\not\in\mathbb{Q}$.

Since $\mathbb{Z}^{3}$ is abelian,
$(H\leq\mathbb{Z}^{3},\sigma_{\mathbf{\Theta}})$ satisfies the relative
Kleppner’s condition if and only if $(0,0,0)$ is the only element of
$\mathbb{Z}^{3}$ which is $\sigma_{\mathbf{\Theta}}$-regular w.r.t. $H$. Now,
$\mathbf{x}=(x_{1},x_{2},x_{3})\in\mathbb{Z}^{3}$ is
$\sigma_{\mathbf{\Theta}}$-regular w.r.t. $H$ if and only if

$\sigma_{\mathbf{\Theta}}(\mathbf{x},\mathbf{y})=\sigma_{\mathbf{\Theta}}(\mathbf{y},\mathbf{x})$

for all $\mathbf{y}\in H$. This is equivalent to
$e^{i2\pi\,\mathbf{\Theta}\cdot(\mathbf{x}\times\mathbf{y})}=1$ for all
$\mathbf{y}\in H$, i.e.,
$\mathbf{\Theta}\cdot(\mathbf{x}\times\mathbf{y})\in\mathbb{Z}$ for all
$\mathbf{y}\in H$. As
$\mathbf{\Theta}\cdot(\mathbf{x}\times\mathbf{y})=\mathbf{y}\cdot(\mathbf{\Theta}\times\mathbf{x})$,
it follows readily that $(H\leq\mathbb{Z}^{3},\sigma_{\mathbf{\Theta}})$
satisfies the relative Kleppner’s condition if and only if the only element
$\mathbf{x}$ of $\mathbb{Z}^{3}$ satisfying

(6) $~\theta_{2}x_{3}-\theta_{3}x_{2}\in\mathbb{Z}\,\text{ and
}\,\theta_{1}x_{3}-\theta_{3}x_{1}\in\mathbb{Z}\,$

is $\mathbf{x}=(0,0,0)$.

Assume now that $(H\leq\mathbb{Z}^{3},\sigma_{\mathbf{\Theta}})$ does not
satisfy the relative Kleppner’s condition. Using what we just have shown, we
can find $\mathbf{x}=(x_{1},x_{2},x_{3})\neq(0,0,0)$ such that (6) holds. If
$x_{3}\neq 0$, then we get that
$\theta_{1},\theta_{2}\in\operatorname{Span}_{\mathbb{Q}}\\{1,\theta_{3}\\}$,
hence that $d(\mathbf{\Theta})\leq 2$. On the other hand, if $x_{3}=0$, then
$x_{1}\neq 0$ or $x_{2}\neq 0$, and we get that $\theta_{3}\in\mathbb{Q}$.
This shows that Condition ii) does not hold.

Combining this with Theorem 6.2, we get the following chain of implications:

Condition ii) holds
$\displaystyle\Rightarrow\,\,(H\leq\mathbb{Z}^{3},\sigma_{\mathbf{\Theta}})\,\text{
satisfies the relative Kleppner's condition}$
$\displaystyle\Rightarrow\,\,B\subseteq A\text{ is $C^{*}$-irreducible}$
$\displaystyle\Rightarrow\,\,B\text{ and }A\,\text{ are simple }$
$\displaystyle\Rightarrow\,\,\text{Condition ii) holds},$

which proves the asserted equivalence.

When Condition ii) holds, then the last assertion in Theorem 6.2 gives that
the intermediate $C^{*}$-algebras of the inclusion $B\subseteq A$ are the
noncommutative tori of the form $C^{*}(U_{1},U_{2},U_{3}^{\,n})$ for
$n\in\mathbb{N}\cup\\{0\\}$.

In the case where $A$ is an higher-dimensional noncommutative $n$-torus
($n\geq 4$), and $B$ is the $C^{*}$-subalgebra of $A$ generated by some of the
unitary generators of $A$, it is not easy to describe explicitly when the
inclusion $B\subseteq A$ will be $C^{*}$-irreducible. We note that in such a
situation, the necessary assumption that both $A$ and $B$ are simple is not
sufficient in general. Indeed, if $\theta$ and $\theta^{\prime}$ are
irrational numbers, and we set $B=A_{\theta}$, $A=A_{\theta}\otimes
A_{\theta^{\prime}}$, then $A$ and $B$ are both simple, but $B\subseteq A$ is
not $C^{*}$-irreducible.

######  Example 7.3 (The Heisenberg group).

Let $G$ denote the discrete Heisenberg group, i.e., $G=\mathbb{Z}^{3}$ as a
set, with multiplication given by

$(a_{1},a_{2},a_{3})(b_{1},b_{2},b_{3})=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3}+a_{1}b_{2}).$

Let $\sigma$ be a two-cocycle on $G$. As shown in [38, Proposition 1.1],
$\sigma$ is (up to similarity) determined by two parameters
$\gamma,\theta\in[0,1)$ such that

$\sigma\big{(}(a_{1},a_{2},a_{3}),(b_{1},b_{2},b_{3})\big{)}=e^{i2\pi\gamma(b_{3}a_{1}+\frac{1}{2}b_{2}a_{1}(a_{1}-1))}e^{i2\pi\theta(a_{2}(b_{3}+a_{1}b_{2})+\frac{1}{2}a_{1}b_{2}(b_{2}-1))}$

The restriction of $\sigma$ to
$H:=\\{(0,a_{2},a_{3}):a_{2},a_{3}\in\mathbb{Z}\\}\simeq\mathbb{Z}^{2}$
depends only on $\theta$. It is straightforward to check that the
$H$-conjugacy class $(a_{1},a_{2},a_{3})^{H}$ is finite if and only if
$a_{1}=0$, i.e., $(a_{1},a_{2},a_{3})\in H$. It follows readily that $(H\leq
G,\sigma)$ satisfies the relative Kleppner condition if and only if $\theta$
is irrational. Since $H$ is abelian, Corollary 6.3 gives that $(H\leq
G,\sigma)$ is $C^{*}$-irreducible if and only if $\theta$ is irrational. In
this case, the intermediate algebras of $C_{r}^{*}(H,\sigma)\subseteq
C_{r}^{*}(G,\sigma)$ are then classified by $\mathbb{N}\cup\\{0\\}$ via the
map $n\mapsto C_{r}(\Gamma_{n},\sigma)$, where
$\Gamma_{n}:=\\{(na_{1},a_{2},a_{3}):a_{1},a_{2},a_{3}\in\mathbb{Z}\\}\leq G$.

Alternatively, we may consider $S^{\sigma}(H):=\\{h\in H:h\text{ is
$\sigma$-regular w.r.t.\leavevmode\nobreak\ $H$}\\}$. Since $H$ is abelian, we
have that $(H,\sigma)$ satisfies Kleppner’s condition if and only if
$S^{\sigma}(H)$ is trivial. On the other hand, one readily checks that
$C_{G}(H)=H$, so we get that $C_{G}^{\sigma}(H)=S^{\sigma}(H)$. As $H$ is
prime and FC-hypercentral, Corollary 6.10 gives that $(H\leq G,\sigma)$ is
$C^{*}$-irreducible if and only if $S^{\sigma}(H)$ is trivial. Now, a short
computation gives that

$S^{\sigma}(H)=\\{(0,a_{2},a_{3})\in
H:e^{i2\pi\theta(a_{2}b_{3}-a_{3}b_{2})}=1\text{ for all
$b_{2},b_{3}\in\mathbb{Z}$}\\},$

which is trivial if and only if $\theta$ is irrational, in accordance with
what we found above.

Note that
$C_{r}^{*}(H,\sigma)=C^{*}(\lambda_{\sigma}(0,1,0),\lambda_{\sigma}(0,0,1))\simeq
A_{\theta}$. If $\gamma=0$, then $C_{r}^{*}(G,\sigma)\simeq
A_{\theta}\rtimes_{r}\mathbb{Z}\simeq A_{\theta}\rtimes\mathbb{Z}$ for the
action of $\mathbb{Z}$ on $A_{\theta}$ implemented by the $*$-automorphism of
$A_{\theta}$ given by conjugation with $\lambda_{\sigma}(1,0,0)$. In the
natural action of $\operatorname{SL}(2,\mathbb{Z})$ on $A_{\theta}$, this
automorphism corresponds to the matrix $\left[\begin{array}[]{cc}1&0\\\
1&1\end{array}\right]$. Actions of $\mathbb{Z}$ on $A_{\theta}$ associated to
other infinite cyclic subgroups of $\operatorname{SL}(2,\mathbb{Z})$ are
studied in [15, Section 5].

######  Example 7.4 (The braid group on infinitely many strands).

The braid group $B_{\infty}$ on infinitely many strands is the group generated
by $\\{s_{i}\\}_{i=1}^{\infty}$ subject to relations

$\begin{gathered}s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}\quad\text{ for all
}i\geq 1,\\\ s_{i}s_{j}=s_{j}s_{i}\quad\text{ when }\lvert i-j\rvert\geq
2.\end{gathered}$

There is a surjection from $B_{\infty}$ onto the infinite symmetric group
$S_{\infty}$, sending the generator $s_{i}$ to the permutation $(i,i+1)$, and
thus $s_{i}^{2}$ to the identity for every $i$. The pure braid group
$P_{\infty}$ is defined as the kernel of the map $B_{\infty}\to S_{\infty}$.

It follows from [36, Proof of Theorem 6.2] that $P_{\infty}$ is $C^{*}$-simple
and $B_{\infty}$ is icc relatively to $P_{\infty}$. Hence, $P_{\infty}\leq
B_{\infty}$ is $C^{*}$-irreducible by Theorem 6.4, and the intermediate
algebras $C^{*}_{r}(P_{\infty})\subseteq B\subseteq C^{*}_{r}(B_{\infty})$ are
classified by the subgroups of $S_{\infty}$.

######  Example 7.5 (Wreath products).

Let $N$ and $K$ be nontrivial groups. Recall that the restricted wreath
product $N\wr K$ is defined as the semidirect product $(\bigoplus_{K}N)\rtimes
K$, where $K$ acts by (left) translation on the index set $K$. The
unrestricted wreath product is defined in a similar way by
$N\,\bar{\wr}\,K=(\prod_{K}N)\rtimes K$. Then the following conditions are
equivalent:

  * (i)

$N$ is $C^{*}$-simple.

  * (ii)

$N\wr K$ is $C^{*}$-simple.

  * (iii)

$\bigoplus_{K}N\,\leq\,N\wr K$ is $C^{*}$-irreducible.

  * (iv)

$\bigoplus_{K}N\,\leq\,N\,\bar{\wr}\,K$ is $C^{*}$-irreducible.

Indeed, the implications (iv) $\Rightarrow$ (iii) $\Rightarrow$ (ii) are
trivial and since $N$ is subnormal in $N\wr K$ we get (ii) $\Rightarrow$ (i).
Now, assume that (i) holds. Then [45, Theorem 5.6] implies that
$\bigoplus_{K}N$ is relatively $C^{*}$-simple in $N\,\bar{\wr}\,K$, so Theorem
6.4 gives that (iv) holds.

This example is also discussed in [7, Proposition 5.5], where (i)
$\Rightarrow$ (ii) is shown.

######  Example 7.6.

Let $G=\mathbb{F}_{2}\times\mathbb{Z}_{2}$ with $\mathbb{Z}_{2}=\\{0,1\\}$,
and let $a,b$ denote the two generators of the free group $\mathbb{F}_{2}$.
Any element of $\mathbb{F}_{2}$ may be written in an unique way in reduced
form as

$x=a^{m_{1}}b^{n_{1}}\cdots a^{m_{k}}b^{n_{k}}$

for some $k\geq 1$, where $m_{1},n_{k}\in\mathbb{Z}$, and
$m_{2},\ldots,m_{k},n_{1},\ldots,n_{k-1}\in\mathbb{Z}\setminus\\{0\\}$ if
$k\geq 2$. Define then

$\theta_{1}(x):=\sum_{j=1}^{k}m_{j},\quad\theta_{2}(x):=\sum_{j=1}^{k}n_{j}\quad\text{and}\quad\theta_{3}(x):=\sum_{j=1}^{k}(m_{j}+n_{j}).$

For $j=1,2,3$ we may then define a two-cocycle $\sigma_{j}$ on $G$ by

$\sigma_{j}\big{(}(x,k),(y,l)\big{)}=\begin{cases}-1\quad\text{if }k=1\text{
and }\,\theta_{j}(y)\,\text{ is odd,}\\\ \,\,\,1\quad\quad\text{
otherwise}\end{cases}$

for $(x,k),(y,k)\in\mathbb{F}_{2}\times\mathbb{Z}_{2}$. (One may show that
every two-cocycle on $G$ which is not similar to $1$ is similar to
$\sigma_{j}$ for some $j\in\\{1,2,3\\}$, but we do not need this.)

Set $H=\mathbb{F}_{2}\times\\{0\\}$. Then $H$ is $C^{*}$-simple, and
$C_{G}(H)=\\{e\\}\times\mathbb{Z}_{2}$. For $y\in\mathbb{F}_{2}$ and
$j\in\\{1,2,3\\}$, we have

$\sigma_{j}\big{(}(e,1),(y,0)\big{)}=\begin{cases}-1\quad\text{if
}\theta_{j}(y)\,\text{ is odd,}\\\ \,\,\,1\quad\quad\text{
otherwise}\end{cases},\,\text{~while
}\,\sigma_{j}\big{(}(y,0),(e,1)\big{)}=1.$

Thus $(e,1)$ is not $\sigma_{j}$-regular w.r.t. $H$. It follows that
$C_{G}^{\sigma_{j}}(H)=\\{(e,0)\\}$ for $j=1,2,3$. Hence, Corollary 6.10 gives
that $(H\leq G,\sigma_{j})$ is $C^{*}$-irreducible for $j=1,2,3$.

##  8\. $C^{*}$-irreducibility and groups acting on trees

Let $T$ be a tree and $\partial T$ its boundary [43]. An automorphism
$g\in\operatorname{Aut}(T)$ is _elliptic_ if it fixes a vertex of $T$, is an
_inversion_ if it does not fix any vertices, but permutes two adjacent
vertices, i.e., inverts an edge, and is called _hyperbolic_ if it is not
elliptic nor an inversion. The fixed point set $T^{g}$ of an elliptic
automorphism $g$ of $T$ is a (possibly finite) subtree of $T$. An hyperbolic
automorphism $h$ does not fix any vertices, but has an axis $L^{h}$, which is
an infinite linear subtree on which $h$ acts by translation. Moreover, $h$
admits exactly two fixed points in $T\cup\partial T$, namely the two points in
$\partial T$ arising from the $h$-invariant axis $L^{h}$. Two hyperbolic
automorphisms are said to be _transverse_ if they have disjoint fixed point
sets. We refer to [43, Proposition 6.4.24] for details.

An action of a discrete group $G$ on a tree $T$ is _minimal_ if $T$ does not
contain any proper $G$-invariant subtrees, and of _general type_ (or _strongly
hyperbolic_ [20]) if its image in $\operatorname{Aut}(T)$ contains two
transverse hyperbolic automorphisms of $T$. The following result can be found
in [32, Proposition 3]; see [18, Lemma 2.10] for a related result.

######  Lemma 8.1.

Let $G$ act faithfully on a tree $T$ and assume that the action is minimal and
of general type. Let $H$ be a nontrivial normal subgroup of $G$. Then the
action of $H$ on $T$ is also minimal and of general type.

The following consequence of Lemma 8.1 is surely part of the folklore, but we
include a proof for completeness.

######  Proposition 8.2.

Let $G$ act faithfully on a tree $T$ and assume that the action is minimal and
of general type. Let $H$ be a nontrivial normal subgroup of $G$. Then the
centralizer $C_{G}(H)$ of $H$ in $G$ is trivial.

###### Proof.

Since the action of $H$ on $T$ is of general type by Lemma 8.1, $H$ contains a
hyperbolic element $h$. Let $g\in C_{G}(H)$. Then $gh=hg$, and the axis
$L^{h}$ is invariant under the action of $g$. Indeed, we have
$gL^{h}=ghL^{h}=hgL^{h}$, so $gL^{h}$ is another infinite $h$-invariant axis,
and so $gL^{h}=L^{h}$. Thus $L^{h}$ is invariant under the action of
$C_{G}(H)$ on $T$. The action of $G$ on $T$ being of general type, we have
$L^{h}\neq T$. Thus the action of $C_{G}(H)$ on $T$ is not minimal. Since $H$
is normal in $G$, $C_{G}(H)$ is normal in $G$ too, so Lemma 8.1 gives that
$C_{G}(H)$ must be trivial. ∎

If $G$ is the free product of two groups not both of order $2$ and $H$ is a
normal subgroup of $G$, then [45, Theorem 5.2] says that $H$ is relatively
$C^{*}$-simple in $G$, hence that $H\leq G$ is $C^{*}$-irreducible by Theorem
6.4. This may be generalized as follows.

######  Theorem 8.3.

Assume $G$ has a faithful, minimal action of general type on a tree $T$, and
let $H$ be a nontrivial normal subgroup of $G$. Then the following conditions
are equivalent:

  * (i)

$G$ is $C^{*}$-simple.

  * (ii)

$H$ is $C^{*}$-simple.

  * (iii)

$H\leq G$ is $C^{*}$-irreducible.

  * (iv)

$(H\leq G,\sigma)$ is $C^{*}$-irreducible for every two-cocycle $\sigma$ on
$G$.

###### Proof.

Since $C_{G}(H)$ is trivial by Proposition 8.2, (i) is equivalent to (ii) by
[8, Theorem 1.4], while Theorem 6.4 gives that (ii) and (iii) are equivalent.
The equivalence between (iii) and (iv) follows from Corollary 6.7. ∎

We recall that an action of a group $G$ on a set $X$ is called _strongly
faithful_ if for any finite subset $F\subseteq G\setminus\\{e\\}$, there
exists $x\in X$ such that $fx\not=x$ for all $f\in F$. Also, an action of $G$
on a topological space $Y$ is said to be _topologically free_ if the set
$Y^{g}:=\\{y\in Y:gy=y\\}$ has empty interior for each $g\in
G\setminus\\{e\\}$. Now, consider a minimal action of $G$ of general type on a
tree $T$. As shown in [9, Proposition 3.8], such an action is strongly
faithful if and only if the induced action of $G$ on $\partial T$ (equipped
with the relative shadow topology) is topologically free, if and only if the
induced action of $G$ on the closure $\overline{\partial T}$ (of $\partial T$
in $T\cup\partial T$ w.r.t. the shadow topology) is topologically free.
Moreover, $G$ is $C^{*}$-simple (in fact a Powers group) when one of these
conditions holds. Hence, Theorem 8.3 gives:

######  Corollary 8.4.

Assume that $G$ has a strongly faithful, minimal action of general type on a
tree $T$, and let $H$ be a nontrivial normal subgroup of $G$.Then $H\leq G$ is
$C^{*}$-irreducible. Moreover, $(H\leq G,\sigma)$ is $C^{*}$-irreducible for
every two-cocycle $\sigma$ on $G$.

######  Example 8.5.

Let $G=A*_{C}B$ be an amalgam of groups which is nondegenerate, i.e., at least
one of the indices $[A:C],[B:C]$ is strictly larger than $2$. Then the action
of $G$ on its Bass-Serre tree $T$ is minimal and of general type, cf. [20,
Proposition 19]. Moreover, [22, Proposition 5.3] gives that this action is
strongly faithful (equivalently, the action of $G$ on $\partial T$ is
topologically free) whenever the so-called interior group of $G$,
$\operatorname{int}(G)$, is trivial. We also note that [22, Proposition 3.2]
characterizes in several ways when this happens, e.g., it is equivalent to
require that for every finite subset $F$ of $C\setminus\\{e\\}$, there exists
$g\in G$ such that $gFg^{-1}\cap C=\emptyset$. When $\operatorname{int}(G)$ is
trivial and $H$ is a nontrivial normal subgroup of $G$, we can apply Corollary
8.4 and deduce that $(H\leq G,\sigma)$ is $C^{*}$-irreducible for any two-
cocycle $\sigma$ on $G$.

We note that if $C$ is trivial, i.e., $G=A*B$ with $\max\\{\lvert
A\rvert,\lvert B\rvert\\}\geq 3$, then every two-cocycle on $G$ is similar to
a two-cocycle of the form $\sigma_{1}*\sigma_{2}$, where $\sigma_{1}$ (resp.
$\sigma_{2}$) is a two-cocycle on $A$ (resp. $B$), see for instance [37,
Section 4] and references therein. It is therefore easy to obtain concrete
examples $(H\leq A*B,\sigma)$ that are $C^{*}$-irreducible. A natural choice
here is to let $H$ be the kernel of the canonical homomorphism from $A*B$ into
$A\times B$.

######  Example 8.6.

Similarly, let $G=\operatorname{HNN}(\Gamma,A,\theta)$ be an HNN-extension
which is nonascending, i.e., we have $A\neq G\neq\theta(A)$. Then the action
of $G$ on its Bass-Serre tree $T$ is minimal and of general type, cf. [20,
Proposition 20] (see also [9, Proposition 4.16]). Moreover, [9, Proposition
4.18] gives that this action is strongly faithful (equivalently, the action of
$G$ on $\partial T$ is topologically free) whenever the interior group of $G$,
$\operatorname{int}(G)$, is trivial. Several characterizations of
$\operatorname{int}(G)$ being trivial are given in [9, Theorem 4.10], e.g.,
this happens if and only if for every finite subset $F$ of
$A\setminus\\{e\\}$, there exists $g\in G$ such that $gFg^{-1}\cap
A=\emptyset$. When $\operatorname{int}(G)$ is trivial, we can apply Corollary
8.4 and get that $(H\leq G,\sigma)$ is $C^{*}$-irreducible whenever $H$ is a
nontrivial normal subgroup of $G$ and $\sigma$ is a two-cocycle on $G$.

To illustrate this, let $m,n\in\mathbb{Z}$ be such that $\min\\{\lvert
m\rvert,\lvert n\rvert\\}\geq 2$ and $\lvert m\rvert\neq\lvert n\rvert$, and
let $G=BS(m,n)=\langle t,b\mid t^{-1}b^{m}t=b^{n}\rangle$ denote the
associated Baumslag-Solitar group, which is an HNN-extension with
$\Gamma=\mathbb{Z}$, $A=m\mathbb{Z}$, and $\theta(mk)=nk$ for
$k\in\mathbb{Z}$. Then it is well-known that $G$ is $C^{*}$-simple (see for
example [20, Theorem 3] and [9, Example 4.21]). Moreover, $G$, as an HNN-
extension, is clearly nonascending, and its interior group is trivial. Indeed,
the final part of [9, Example 4.21] shows that the so-called quasi-kernel
$K_{1}$ of $G$ is trivial, and this is equivalent to $\operatorname{int}(G)$
being trivial by [9, Theorem 4.10]. Let now $\varphi\colon G\to\mathbb{Z}$ be
the homomorphism determined by $\varphi(t)=1$ and $\varphi(b)=0$ and set
$H=SBS(m,n)=\ker\varphi$ (as in [20, Corollary 5]). Then we can conclude from
the previous paragraph that $H\leq G$ is $C^{*}$-irreducible.

If $G$ acts on compact Hausdorff space $X$ in such a way that every orbit is
dense in $X$ (i.e., the action is minimal) and the weak*-closure of every
orbit in the space of probability measures on $X$ contains a point mass (i.e.,
the action is strongly proximal), then the action of $G$ is called a _boundary
action_ and $X$ is called a _$G$ -boundary_. Up to $G$-equivariant
homeomorphism, the _Furstenberg boundary_ $\partial_{F}G$ is the unique
$G$-boundary having the universal property that for any $G$-boundary $X$,
there exists a (unique) $G$-equivariant, continuous surjection
$\partial_{F}G\to X$. As shown in [25, Theorem 6.2] (see also [8, Theorem
3.1]), $G$ is $C^{*}$-simple if and only if there exists a topologically free
$G$-boundary, if and only if the action of $G$ on $\partial_{F}G$ is free
(resp. topologically free).

When a faithful action of $G$ on a tree $T$ is minimal and of general type,
then $\overline{\partial T}$ is a $G$-boundary (cf. [9, Lemma 3.5]; see also
[33, Proposition 4.26]), and thus also an $H$-boundary for any nontrivial
normal subgroup $H$ of $G$ by Lemma 8.1. Corollary 8.4 can therefore also be
deduced from the following result for boundary actions, which relies heavily
on [45, Theorem 1.3] and its proof.

######  Proposition 8.7.

Let $H$ be a nontrivial normal subgroup of $G$ and suppose that there exists a
topologically free boundary action of $G$ on $X$ which restricts to a boundary
action of $H$. Then $(H\leq G,\sigma)$ is $C^{*}$-irreducible for every two-
cocycle $\sigma$ on $G$.

###### Proof.

Since $X$ is an $H$-boundary, there exists an $H$-equivariant, continuous
surjection map $\partial_{F}H\to X$. By [8, Lemma 5.2], the action of $H$ on
$\partial_{F}H$ extends in a unique way to an action of $G$ on
$\partial_{F}H$, so $\partial_{F}H$ is a $G$-boundary. Further, it follows
from [45, Corollary 4.3] that the surjection $\partial_{F}H\to X$ is
$G$-equivariant. Hence, if $g\in G$ is such that $(\partial_{F}H)^{g}$ has
nonempty interior, then $X^{g}$ has nonempty interior. By assumption, this
means that $g=e$. Thus, the action of $G$ on $\partial_{F}H$ is topologically
free, and [45, Theorem 1.3] gives that $H$ is $C^{*}$-simple and $C_{G}(H)$ is
trivial. The conclusion follows from Theorem 6.4 and Corollary 6.7. ∎

##  9\. On $C^{*}$-simplicity and normal subgroups

Let $H$ be a normal subgroup of $G$. As recalled in Section 6, we then have
that $G$ is $C^{*}$-simple if and only if $H$ and $C_{G}(H)$ are both
$C^{*}$-simple, cf. [8, Theorem 1.4]. It would be interesting to know what
kind of assumptions are needed to ensure that a twisted version of this result
holds. Our goal in this section is to prove a result in this direction, cf.
Corollary 9.7.

######  Lemma 9.1.

Let $H$ be a normal subgroup of $G$. Assume that $H$ is prime and that
$(H,\sigma)$ and $(C_{G}^{\sigma}(H),\sigma)$ both satisfy Kleppner’s
condition. Then $(G,\sigma)$ satisfies Kleppner’s condition.

###### Proof.

Suppose that $g\in G$ is $\sigma$-regular w.r.t. $G$ and $\lvert
g^{G}\rvert<\infty$. Then $g\in FC_{G}^{\sigma}(H)$, so Lemma 3.13 gives that
$g\in C_{G}^{\sigma}(H)$. Now, it is clear that $g$ is also $\sigma$-regular
w.r.t. $C_{G}^{\sigma}(H)$ and $\lvert g^{C^{\sigma}_{G}(H)}\rvert<\infty$.
Since $(C_{G}^{\sigma}(H),\sigma)$ satisfies Kleppner’s condition, we get that
$g=e$. This shows that $(G,\sigma)$ satisfies Kleppner’s condition. ∎

######  Lemma 9.2.

Suppose that $H$ is a nontrivial normal subgroup of a $FC$-hypercentral group
$G$. Then $H\cap FC(G)$ is nontrivial.

###### Proof.

We recall, see for example [41, Section 4.3], that $G$ is $FC$-hypercentral if
and only if $G$ is equal to its $FC$-hypercenter $FCH(G)$, which is defined as
follows. Let $\\{F_{\alpha}\\}_{\alpha}$ be the ascending series of normal
subgroups of $G$ indexed by the ordinal numbers, given by $F_{0}=\\{e\\}$,
$F_{\alpha}/F_{\beta}=FC(G/F_{\beta})$ if $\alpha=\beta+1$, and
$F_{\alpha}=\bigcup_{\beta<\alpha}F_{\beta}$ when $\alpha$ is a limit ordinal
number. This series eventually stabilizes and
$FCH(G):=\lim_{\alpha}F_{\alpha}=\bigcup_{\alpha}F_{\alpha}$.

Now, since $G=FCH(G)$, there is some $\alpha$ such that $H\cap
F_{\alpha}=\\{e\\}$ while $H\cap F_{\alpha+1}\neq\\{e\\}$. Pick $h\in H\cap
F_{\alpha+1}$, $h\neq e$. As $H\cap F_{\alpha}=\\{e\\}$, the homomorphism
$h^{\prime}\mapsto h^{\prime}F_{\alpha}$ from $H$ into $G/F_{\alpha}$ is
injective. Also, $hF_{\alpha}$ belongs to $FC(G/F_{\alpha})$ since $h\in
F_{\alpha+1}$. We therefore get that

$\lvert\\{ghg^{-1}:g\in G\\}\rvert=\lvert\\{ghg^{-1}F_{\alpha}:g\in
G\\}\rvert=\lvert\\{(gF_{\alpha})hF_{\alpha}(gF_{\alpha})^{-1}:g\in
G\\}\rvert<\infty.$

Hence, $h\in FC(G)$. Thus we have $e\neq h\in H\cap FC(G)$. ∎

######  Definition 9.3.

A subgroup $H$ of $G$ will be said to be _$\sigma$ -regular in $G$_ if $h\in
H$ is $\sigma$-regular w.r.t. $G$ whenever $h$ is $\sigma$-regular w.r.t. $H$.

######  Example 9.4.

Let $\theta$ be irrational, $p,q\in\mathbb{N}$, $H_{p,q}=p\mathbb{Z}\times
q\mathbb{Z}$ and $\sigma_{\theta}$ be the two-cocycle on $\mathbb{Z}^{2}$
defined in Example 7.1. Then $H_{p,q}$ is $\sigma$-regular in
$\mathbb{Z}^{2}$. Indeed, $\mathbf{y}=(0,0)$ is the only element of $H_{p,q}$
which is $\sigma$-regular w.r.t. $H_{p,q}$ (because
$(H_{p,q},\sigma_{\theta})$ satisfies Kleppner’s condition), and $(0,0)$ is
obviously $\sigma$-regular w.r.t. $\mathbb{Z}^{2}$. The same argument shows
that if $H\leq G$, $H$ is abelian and $(H,\sigma)$ satisfies Kleppner’s
condition, then $H$ is $\sigma$-regular in $G$.

######  Example 9.5.

Let $G=\mathbb{F}_{2}\times\mathbb{Z}_{2}$, $H=\mathbb{F}_{2}\times\\{0\\}$
and $\sigma_{j}$ be as in Example 7.6 for $j\in\\{1,2,3\\}$. Then $H$ is not
$\sigma_{j}$-regular in $G$. Indeed, every $(x,0)\in H$ is
$\sigma_{j}$-regular w.r.t. $H$ (because $\sigma_{j}$ restricts to the trivial
two-cocycle on $H$). But if we pick $x\in\mathbb{F}_{2}$ such that
$\theta_{j}(x)$ is odd, then $(x,1)$ commutes with $(x,0)$, and

$~\sigma_{j}((x,0),(x,1))=1\neq-1=\sigma_{j}((x,1),(x,0)),$

i.e., $(x,0)$ is not $\sigma_{j}$-regular w.r.t. $G$.

It is not difficult to see that if $(G,\sigma)$ satisfies Kleppner’s condition
and $H$ is a $\sigma$-regular subgroup of $G$ having finite index, then
$(H,\sigma)$ satisfies Kleppner’s condition. If $H$ is of infinite index, this
might not be true. However, we have:

######  Proposition 9.6.

Let $G$ be a FC-hypercentral group and $H$ be a normal subgroup of $G$. Assume
that $(G,\sigma)$ satisfies Kleppner’s condition and that $H$ is prime and
$\sigma$-regular in $G$. Then $(H,\sigma)$ satisfies Kleppner’s condition.

###### Proof.

Assume $h\in H$ is $\sigma$-regular w.r.t. $H$ and $h^{H}$ is finite. Then $h$
is $\sigma$-regular w.r.t. $G$ (since $H$ is $\sigma$-regular in $G$). Let $N$
be the normal subgroup of $G$ generated by $h$, i.e., the subgroup generated
by $h^{G}$.

Note that $FC(H)$ is a characteristic subgroup of $H$, so it is normal in $G$
(this can also be checked directly). It follows that $N$ is contained in
$FC(H)$, so $N$ is torsion-free (since $H$ is prime). Moreover, using Lemma
9.2, we get that $N\cap FC(G)$ is nontrivial.

Let $y\in N\cap FC(G)$, $y\neq e$. Then $y$ can be written as a finite product
of elements from $h^{G}$ and their inverses, and all these elements clearly
belong to $FC_{G}^{\sigma}(G)=\\{g\in FC(G):g\text{ is $\sigma$-regular
w.r.t.}\leavevmode\nobreak\ G\\}$. Thus, by Lemma 3.10, there exists some
$n\in\mathbb{N}$ such that $y^{n}\in FC_{G}^{\sigma}(G)$. As $y\in N$, and $N$
is torsion-free, we can conclude that $FC_{G}^{\sigma}(G)$ is nontrivial,
i.e., $(G,\sigma)$ does not satisfy Kleppner’s condition. ∎

######  Corollary 9.7.

Let $G$ be a FC-hypercentral group and $H$ be a normal subgroup. Assume that
$H$ is prime and $\sigma$-regular in $G$. Consider the following two
conditions:

  * (i)

$(H,\sigma)$ and $(C_{G}^{\sigma}(H),\sigma)$ are both $C^{*}$-simple.

  * (ii)

$(G,\sigma)$ is $C^{*}$-simple.

Then (i) $\Rightarrow$ (ii). If we also assume that $C_{G}^{\sigma}(H)$ is
prime and $\sigma$-regular in $G$, then (ii) $\Rightarrow$ (i).

###### Proof.

Assume (i) holds. Then $(H,\sigma)$ and $(C_{G}^{\sigma}(H),\sigma)$ both
satisfy Kleppner’s condition, so Lemma 9.1 gives that $(G,\sigma)$ satisfies
Kleppner’s condition. Since $G$ is $FC$-hypercentral, we get that (ii) holds.

Next, assume also that $C_{G}^{\sigma}(H)$ is prime and $\sigma$-regular in
$G$, and that (ii) holds. Then $(G,\sigma)$ satisfies Kleppner’s condition, so
Proposition 9.6 gives that $(H,\sigma)$ and $(C_{G}^{\sigma}(H),\sigma)$ both
satisfy Kleppner’s condition. Since $H$ and $C_{G}^{\sigma}(H)$ are
$FC$-hypercentral (being subgroups of $G$), we get that $(H,\sigma)$ and
$(C_{G}^{\sigma}(H),\sigma)$ both are $C^{*}$-simple. ∎

######  Question 9.8.

Let $H$ be a normal subgroup of $G$ and consider the following two properties:

  * (i)

$(H,\sigma)$ and $(C_{G}^{\sigma}(H),\sigma)$ are both $C^{*}$-simple.

  * (ii)

$(G,\sigma)$ is $C^{*}$-simple.

Under which assumptions (other than $\sigma=1$ and those imposed in Corollary
9.7) do we have that (i) $\Rightarrow$ (ii), or that (i) $\Leftrightarrow$
(ii) ?

## Acknowledgements

We are very grateful to the referee for carefully reading the manuscript and
for pointing out several places where it could be improved.

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