# Intermediate crossed product $C^{*}$-algebras

Tattwamasi Amrutam  Ben Gurion University of the Negev. Department of
Mathematics. Be’er Sheva, 8410501, Israel<EMAIL_ADDRESS>,  Ilan
Hirshberg  Ben Gurion University of the Negev. Department of Mathematics.
Be’er Sheva, 8410501, Israel<EMAIL_ADDRESS>and  Apurva Seth  Ben Gurion
University of the Negev. Department of Mathematics. Be’er Sheva, 8410501,
Israel<EMAIL_ADDRESS>
###### Abstract.

Let $B$ be a separable $C^{*}$-algebra, let $\Gamma$ be a discrete countable
group, let $\alpha\colon\Gamma\to\mathrm{Aut}(B)$ be an action, and let $A$ be
an invariant subalgebra. We find certain freeness conditions which guarantee
that any intermediate $C^{*}$-algebra $A\rtimes_{\alpha,r}\Gamma\subseteq
C\subseteq B\rtimes_{\alpha,r}\Gamma$ is a crossed product of an intermediate
invariant subalgebra $A\subseteq C_{0}\subseteq B$ by $\Gamma$. Those are used
to generalize related results by Suzuki.

The first-named author’s research is supported by the European Research
Council (ERC) under the European Union’s Seventh Framework Programme
(FP7-2007-2013) (Grant agreement No. 101078193), and by the Israel Science
Foundation (ISF 1175/18). The third named author is supported by the US-Israel
Binational Science Foundation. The authors also thank the hospitality of the
Fields Institute, Toronto, where the final part of this work was completed.

##  1\. Introduction

Let $\Gamma$ be a discrete group. Let $B$ be a $\Gamma$-$C^{*}$-algebra, that
is, a $C^{*}$-algebra along with an action $\alpha$ of $\Gamma$ on it by
automorphisms. If $A\subseteq B$ is a $\Gamma$-invariant subalgebra, then this
inclusion induces a natural inclusion of the reduced crossed products,
$A\rtimes_{\alpha,r}\Gamma\subseteq B\rtimes_{\alpha,r}\Gamma$ (whereby slight
abuse of notation, we use $\alpha$ to denote the restriction of the action to
the invariant subalgebra $A$). Suppose we have an intermediate $C^{*}$-algebra
$A\rtimes_{\alpha,r}\Gamma\subseteq C\subseteq B\rtimes_{\alpha,r}\Gamma$. Our
goal in this paper is to find conditions that ensure that any such
intermediate algebra $C$ arises from an intermediate $\Gamma$-invariant
subalgebra $A\subseteq C_{0}\subseteq B$. Related questions were studied under
various other hypotheses in [Suz, AK, A19, ryo2021remark,
rordam2021irreducible]. In [A19], the first named author considers the case in
which $A=\mathbb{C}$, the group $\Gamma$ is $C^{*}$-simple and the kernel of
the action of $\Gamma$ on $A$ is sufficiently large. In [AK], the first named
author and Kalantar studied the case in which $A=\mathbb{C}$, $B=C(X)$, the
group $\Gamma$ is $C^{*}$-simple, and the action of $\Gamma$ on $X$ is
minimal, and the result is that intermediate subalgebras are simple. In
[rordam2021irreducible], one considers, among other things, the case in which
$A=\mathbb{C}$, the group $\Gamma$ is $C^{*}$-simple and finds conditions for
the inclusion to be irreducible. In [Suz], Suzuki addresses the case in which
$A=C(X)$ and $\Gamma$ acts freely on it, as well as the von Neumann algebra
analog. He also considers the case in which $A$ and $B$ are noncommutative
von-Neumann algebras, as well certain actions on $C^{*}$-algebras, including
some noncommutative Bernoulli shifts; see also [ryo2021remark] for more on
Suzuki’s conditions. For somewhat related work concerning the structure of
intermediate $C^{*}$-algebras and von-Neumann algebras of the form $A\subseteq
C\subseteq A\rtimes_{\alpha,r}\Gamma$, we refer the reader to
[choda1978galois, izumi1998galois, Cameron2015BimodulesIC,
cameron2016intermediate].

Our main theorem in this paper is the following: the definitions of the
conditions in the theorem are provided in Section 2.

######  Theorem 1.1.

mainintmresult Let $\Gamma$ be a countable discrete group with property AP.
Let $B$ be a separable $C^{*}$-algebra, and let
$\alpha\colon\Gamma\to\mathrm{Aut}(B)$ be an action. Let $A$ be a
$\Gamma$-invariant subalgebra of $B$. Let $\omega$ be a free ultrafilter.
Assume that one of the following conditions holds:

  1. (1)

$A$ contains an approximate identity of $B$ and for every central
$\alpha^{**}$-invariant projection $P\in B^{**}$ such that $PB^{**}P$ is
separable, we have

$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$

and $\alpha^{**}:\Gamma\to\mathrm{Aut}(PA^{**}P)$ is centrally free.

  2. (2)

The action of $\Gamma$ on the inclusion $A\subseteq B$ has pointwise finite
relative Rokhlin dimension.

Then any intermediate $C^{*}$-algebra $C$ with
$A\rtimes_{\alpha,r}\Gamma\subset C\subset B\rtimes_{\alpha,r}\Gamma$ is of
the form $B_{1}\rtimes_{\alpha,r}\Gamma$, where $B_{1}$ is a
$\Gamma$-invariant $C^{*}$-subalgebra of $B$.

We show that the condition on central freeness holds if $A$ is a type I
$C^{*}$-algebra which admits a $\Gamma$-invariant composition series and the
action on $\hat{A}$ is free; this generalizes the freeness condition from
[Suz] where $A$ is assumed to be commutative.

The assumption of property AP on the group $\Gamma$ cannot generally be
relaxed, as shown in [Suz]. Property AP (the _approximation property_) was
introduced by Haagerup-Kraus in [haagerup1994approximation] where it was shown
that every weakly amenable group has property AP (see
[haagerup1994approximation, Theorem 1.12]). In particular, every amenable
group, being weakly amenable, has the approximation property but also
hyperbolic group also has the approximation property [ozawa2008weak]. Roughly
speaking, property AP allows us to recover enough information on crossed-
product elements by considering Fourier coefficients. We refer the reader to
[BroOza08, Chapter 12] for further details on the approximation property.

We present now a brief outline of the proof. Let $\mathbb{E}\colon
B\rtimes_{\alpha,r}\Gamma\to B$ be the canonical conditional expectation
associated with the crossed product. To show that $C$ is a crossed product,
using property AP, it is sufficient to establish that $\mathbb{E}(C)=C$. To
this end, we consider the set $V$ of completely contractive maps $\psi\colon
B\rtimes_{\alpha,r}\Gamma\to B\rtimes_{\alpha,r}\Gamma$ such that
$\psi(C)\subseteq C$, and it suffices to show that for any $x\in C$, the
element $\mathbb{E}(x)$ is in the norm closure of $V_{x}=\\{\psi(x)\mid\psi\in
V\\}$. As the set $V_{x}$ is convex, using the Hahn-Banach theorem, it
suffices to show, in fact, that $\mathbb{E}(x)$ is in the weak closure of
$V_{x}$. In the the first case, we construct a net of completely positive maps
in $V_{x}$ which converge weakly to $\mathbb{E}(x)$ by considering the induced
action on $A^{**}$, and successively using strengthened versions of Rokhlin-
type theorems due to Connes [connes1975outer] and Ocneanu
[ocneanu2006actions], which we derive in this paper, applied to actions by
various single elements. Though we apply the strengthened version of Ocneanu’s
theorem just for finite cyclic groups, we prove it in greater generality, as
it may be of independent interest. Under the assumption of pointwise finite
Rokhlin dimension, the idea is similar, although somewhat more
straightforward, as we can construct the approximations directly in the norm.

Once we find the Rokhlin towers in $A^{**}\subset B^{**}$, we use a modified
version of the Kaplansky density theorem to obtain a net of orthogonal
elements in $A$. We then proceed to piece together these elements to form a
net of uniformly bounded completely positive contractive maps $\Psi_{\nu}$
which converge to $\mathbb{E}$ $\sigma$-weakly (also see correctiveepsilon and
the remark following it). Finally, we use the Hahn-Banach separation theorem
to conclude that the canonical conditional expectation ${\mathbb{E}}$ lies in
the closure of a convex combination of such $\Psi_{v}$’s.

The paper is organized as follows. We review some preliminary notions. In
Section 3 we prove the versions we need of the noncommutative Rokhlin Theorems
of Connes (thm:relativizedconnes) and of Ocneanu (thm:relativizedocneanu). In
Section 4, we prove our main result, mainintmresult. Finally, in Section 5, we
give some natural examples of inclusions and actions which fit into the realm
of mainintmresult.

##  2\. Preliminaries

Throughout this paper, $\Gamma$ will denote a discrete countable group. We
denote the cardinality of a set $S$ by $|S|$. If $S\subset\Gamma$ is finite,
we sometimes write $S\subset\subset\Gamma$.

###  2.1. Monotileable Amenable groups and paving systems

We begin by reviewing the definition of monotileable groups from
[weiss2001monotileable].

######  Definition 2.1.

def:monotileable A _monotile_ $T$ in a discrete group $\Gamma$ is a finite set
such that

  1. (1)

$\exists$ $C$ such that $\\{cT:c\in C\\}$ is a covering of $\Gamma$.

  2. (2)

$cT\cap c^{\prime}T=\emptyset$, for $c\neq c^{\prime}$.

An discrete group $\Gamma$ is said to be _monotileable_ if it admits a Følner
sequence consisting of monotiles. To our knowledge, it is still not known
whether all amenable groups are monotileable. We need a much weaker property.
We say that $\Gamma$ _admits large monotiles_ if for any finite subset
$F\subset\Gamma$ there exists a monotile $T\subset\Gamma$ such that $F\subset
T$. We do not know whether groups exist that do not satisfy this property. For
example, it is easy to see that any residually finite group admits large
monotiles.

We briefly recall some definitions concerning paving systems; see
[ocneanu2006actions, Chapter 3] for more details.

######  Definition 2.2.

For $K\subset\subset\Gamma$ and $\delta>0$, we say that a non empty set $S$ of
$\Gamma$ is $(\delta,K)$-_(right) invariant_ if it is finite and $\lvert
S\cap\bigcap_{g\in K}Sg\rvert>(1-\delta)\lvert S\rvert$.

######  Definition 2.3.

A system $\\{K_{i}\\}_{i\in I}$ of finite sets in $\Gamma$ are said to be
$\varepsilon$-disjoint for $\varepsilon>0$, if there exist subsets
$K_{i}^{\prime}\subseteq K_{i}$ for all $i\in I$ such that $\lvert
K_{i}^{\prime}\rvert\geq(1-\varepsilon)\lvert K_{i}\rvert$ and
$(K_{i}^{\prime})_{i\in I}$ are disjoint.

######  Definition 2.4.

A system $K_{1},K_{2},\ldots,K_{N}$ of finite subsets of $\Gamma$
_$\varepsilon$ -pave_ the finite subset $S$ of $\Gamma$ if there are subsets
$\\{B_{1},B_{2},\ldots,B_{N}\\}$ of $\Gamma$, called the _paving centers_ ,
such that

  1. (1)

${\cup_{i}}K_{i}B_{i}\subset S$

  2. (2)

$\lvert S\setminus{\bigcup_{i\in I}}K_{i}B_{i}\rvert<\varepsilon\lvert
S\rvert$

  3. (3)

$(K_{i}b)_{b\in B_{i},i\in I}$ are $\varepsilon$-disjoint

######  Definition 2.5.

We call $\\{K_{1},K_{2},\ldots,K_{N}\\}$ an _$\varepsilon$ -paving system of
sets_, if there exist $\delta>0$ and $F\subset\subset\Gamma$ such that
$\\{K_{1},K_{2},\ldots,K_{N}\\}$ $\varepsilon$-pave any $(\delta,F)$-invariant
subset $S$ of $\Gamma$.

We conclude this discussion with the following observation concerning paving
systems in amenable groups that admit large monotiles.

######  Lemma 2.6.

prop:monotileableepsilon Let $\Gamma$ be a group, and let $T$ be a monotile in
$G$. Let $\varepsilon>0$. Then $T$ $\varepsilon$-paves any finite
$(\varepsilon,T^{-1}T)$-invariant set $S$ in $G$.

###### Proof.

Since $T$ is a monotile, by def:monotileable, there exists $C\subseteq\Gamma$
such that $\\{cT:c\in C\\}$ forms a disjoint covering of $\Gamma$. Let $S$ be
an $(\varepsilon,TT^{-1})$-invariant subset of $G$. Define
$C_{0}:=\\{c:cT\subseteq S\\}$. Then, $C_{0}T\subseteq S$. Define
$\gamma:\Gamma\to C$ by requiring that $\gamma(g)$ is the unique element in
$\Gamma$ such that $g\in\gamma(g)T$. Then, for $g\in S\setminus C_{0}T$, we
have $g\in S$ and $g\notin C_{0}T$, hence $\gamma(g)T\nsubseteq S$, because if
$\gamma(g)T\subseteq S$, then $\gamma(g)\in C_{0}$ and so
$g=\gamma(g)t\in\gamma(g)T\in C_{0}T$, which is a contradiction. Hence, there
exists some $t^{\prime}\in T$ such that $\gamma(g)t^{\prime}\notin S$. Setting
$t=\gamma(g)^{-1}g\in T$, this implies
$gt^{-1}t^{\prime}=\gamma(g)t^{\prime}\notin S$, so that $g\notin\bigcap_{h\in
T^{-1}T}Sh$. Thus, $S\setminus C_{0}T\subseteq S\setminus\bigcap_{h\in
T^{-1}T}Sh$. Now, since $S$ is taken to be $(\varepsilon,T^{-1}T)$-invariant,
we have $\lvert\bigcap_{h\in TT^{-1}}Sh\rvert>(1-\varepsilon)\lvert S\rvert$,
so that $\lvert S\setminus C_{0}T\rvert\leq\lvert S\setminus\bigcap_{h\in
T^{-1}T}Sh\rvert<\varepsilon\lvert S\rvert$. Thus, $T$ $\varepsilon$-paves any
$(\varepsilon,T^{-1}T)$-invariant subset $S$ of $\Gamma$. ∎

###  2.2. Ultraproduct von Neumann Algebras and
$C^{*}$-to-$W^{*}$-Ultraproducts

We briefly recall the definitions for ultraproduct von Neumann algebras; see
[ocneanu2006actions, Chapter 5] for more details. Fix a free ultrafilter
$\omega$ on ${\mathbb{N}}$. Let $M$ be a separable von Neumann algebra and let
$M_{*}$ be its predual. For $a\in M$ and $\varphi\in M_{*}$, denote by
$a\varphi,\,\varphi a,\,[a,\varphi]\in M_{*}$ the linear functionals

$a\varphi(x):=\varphi(ax),\,\,\varphi a(x):=\varphi(xa)\text{
and}\,\,[a,\varphi]=\varphi a-a\varphi.$

For $a\in M$ and $\varphi\in M_{*}^{+}$, we define the following seminorms

$\|a\|_{\varphi}:=\varphi(a^{*}a)^{1/2},\,\,\|a\|_{\varphi}^{\\#}:=\varphi(a^{*}a+aa^{*})^{1/2}.$

Consider $\ell^{\infty}({\mathbb{N}},M)$, the $C^{*}$-algebra of norm bounded
sequences in $M$. An element $(x^{\nu})_{\nu}$ of
$\ell^{\infty}({\mathbb{N}},M)$ is said to be

  1. (1)

$\omega$-trivial if $x^{\nu}\underset{{\nu}\to\omega}{\longrightarrow}0$
*-strongly.

  2. (2)

$\omega$-central if for any $\varphi\in M_{*}$ we have
$\|[x^{\nu},\varphi]\|\underset{{\nu}\to\omega}{\longrightarrow}0$.

Let $\mathcal{I}_{\omega}(M)$ and $\mathcal{C}_{\omega}(M)$ be the sets of
$\omega$-trivial and $\omega$-central sequences in $M$, respectively. Those
are unital $C^{*}$-subalgebras of $\ell^{\infty}({\mathbb{N}},M)$. Let
$N_{\omega}(M)$ be the normalizer of $\mathcal{I}_{\omega}(M)$ in
$\ell^{\infty}({\mathbb{N}},M)$. Then the quotient $C^{*}$-algebras defined as
$M^{\omega}:=N_{\omega}(M)/\mathcal{I}_{\omega}(M)$ and
$M_{\omega}:=C_{\omega}(M)/\mathcal{I}_{\omega}(M)$ are von Neumann algebras.
Notice that $M$ and $M_{\omega}$ are von Neumann subalgebras of $M^{\omega}$
(where $M$ is identified with the image of the constant sequences in
$\ell^{\infty}({\mathbb{N}},M)$). For any state $\varphi\in M_{*}$ we define a
state $\varphi^{\omega}$ on $M^{\omega}$ by
$\varphi^{\omega}((x^{\nu})_{\nu})=\lim_{{\nu}\to\omega}\varphi(x^{\omega})$.
Its restriction to $M_{\omega}$ will be denoted by $\varphi_{\omega}$. Note
that $\varphi_{\omega}$ is a normal trace on $M_{\omega}$, regardless of
whether $\varphi$ is a trace or not. Any automorphism
$\alpha\in\mathrm{Aut}(M)$ induces automorphisms $\alpha^{\omega}$ and
$\alpha_{\omega}$ of $M^{\omega}$ and $M_{\omega}$ respectively. This follows
from the fact that for any $\varphi\in M_{*}$ and for any
$(x^{\nu})_{\nu}\in\ell^{\infty}(N,M)$, we have

$\left\|\alpha(x^{\nu})\right\|_{\varphi}^{2}=\varphi\circ\alpha({x^{\nu}}^{*}x^{\nu})=\left\|x^{\nu}\right\|_{\varphi\circ\alpha}^{2}$
$\left\|[\varphi,\alpha(x^{\nu})\right\|=\left\|[\varphi\circ\alpha,x^{\nu}]\right\|\,.$

The notation for $C^{*}$-algebras is different. Let $A$ be a $C^{*}$-algebra.
We write $A_{\infty}=\ell^{\infty}({\mathbb{N}},A)/C_{0}({\mathbb{N}},A)$. One
could replace $\infty$ with a free ultrafilter, but we don’t need to do so
here. Denote the quotient map by
$\pi_{\infty}:\ell^{\infty}({\mathbb{N}},A)\to A_{\infty}$. Now, $A$ embeds in
$A_{\infty}$ as the $\pi_{\infty}$-image of the constant sequences in
$\ell^{\infty}({\mathbb{N}},A)$, and the central sequence algebra, denoted by
$A_{\infty}\cap A^{\prime}$ is the $\pi_{\infty}$-image of central sequences
in $\ell^{\infty}({\mathbb{N}},A)$. The (two sided) annihilator
$\text{Ann}(A,A_{\infty})$ is the closed ideal of $A_{\infty}\cap A^{\prime}$
defined as the $\pi_{\infty}$-image of sequences
$(a_{n})_{n}\in\ell^{\infty}({\mathbb{N}},A)$ such that
$\|a_{n}a\|+\|aa_{n}\|\underset{{n\to\infty}}{\longrightarrow}0$, for all
$a\in A$. Denote $F(A):=(A_{\infty}\cap A^{\prime})/\text{Ann}(A,A_{\infty})$,
then any automorphism $\alpha$ of $A$ induces an automorphism of $F(A)$, which
we also denote by slight abuse of notation by $\alpha$.

We now recall the notion of $C^{*}$-to-$W^{*}$-ultraproduct
$(A,\rho)_{\omega}$, for a $C^{*}$-algebra $A$ and state $\rho$, see
[ando2016non] for more details. Let $L_{\rho}(A)$ be the closed left ideal of
$\ell^{\infty}({\mathbb{N}},A)$ consisting of those
$(a_{v})\in\ell^{\infty}({\mathbb{N}},A)$ such that

$\lim_{v\to\omega}\rho(a_{v}^{*}a_{v})=0\,.$

Define $\mathcal{I}_{\rho}(A):=L_{\rho}(A)\cap L_{\rho}^{*}(A)$. Denote by
$N(\mathcal{I}_{\rho}(A))$ the normalizer of $\mathcal{I}_{\rho}(A)$ in
$\ell^{\infty}({\mathbb{N}},A)$. The $C^{*}$-to-$W^{*}$-ultraproduct
$(A,\rho)_{\omega}$ is defined to be the quotient $C^{*}$-algebra
$N(\mathcal{I}_{\rho}(A))/\mathcal{I}_{\rho}(A)$.

Next, we consider inclusions $N\subset M$ of unital, separable von Neumann
algebras with $B\subseteq M$ a weakly dense C*-subalgebra. Let $\varphi$ be a
faithful normal state on $M$. Note that we do not necessarily have
$N^{\omega}\subset M^{\omega}$. We may, nonetheless, define

$B^{\prime}\cap N^{\omega}:=\\{(x^{v})\in
N_{\omega}(N):\,\,\|x^{v}b-bx^{v}\|_{\varphi}^{\\#}\underset{v\to\omega}{\longrightarrow}0,\,\,\forall
b\in B\\}/\mathcal{I}_{\omega}(N)$

We recall the following.

######  Proposition 2.7.

[ando2016non, Proposition 3.4, Proposition 4.15] Let $\varphi$ be a faithful
normal state on a $W^{*}$-algebra $M$, and let $A\subset N$ be a weakly dense
$C^{*}$-subalgebra. Set $\rho:=\varphi|_{A}$. Then there exists a
$*$-isomorphism from the $C^{*}$-to-$W^{*}$-ultraproduct $(A,\rho)_{\omega}$
onto $N^{\omega}$ which maps $A^{\prime}\cap(A,\rho)_{\omega}$ onto
$N^{\prime}\cap N^{\omega}$.

The following simple lemma lets us approximate projections in $M_{\omega}$
with ones from $M$, as done in [connes1975outer, Lemma 1.1.5].

######  Lemma 2.8.

lem:connes Let $\varphi$ be a faithful normal state on $M$. Let
$\\{E_{j}\\}_{j=1}^{N}$ be a partition of unity in $M_{\omega}$. Then, there
exists a sequence of partition of unity
$\\{e_{1}^{\nu},e_{2}^{\nu},\ldots,e_{N}^{\nu}\\}_{\nu}$ in $M$ such that for
$j=1,2,\ldots,N$, $\\{e_{j}^{\nu}\\}_{\nu}$ is the representative sequence for
$E_{j}$.

###### Proof.

Assume first $N=2$. Let $E_{1},E_{2}\in M_{\omega}$ be a partition of unity.
Using [connes1975outer, Lemma 1.1.5], take $(e_{1}^{\nu})_{\nu}$ to be the
representative sequence of $E_{1}$ consisting of projections $e_{1}^{\nu}\in
M$. Note that $(1-e_{1}^{\nu})_{\nu}$ is a representative sequence for
$E_{2}$, so $\\{e_{1}^{\nu},1-e_{1}^{\nu}\\}_{\nu}$ forms a sequence of
partition of unity in $M$. Now, Let $\\{E_{1},E_{2},\ldots,E_{N}\\}$ be a
partition of unity in $M_{\omega}$. Since $E_{1}+E_{2}$ is a projection in
$M_{\omega}$, we can choose a representative sequence $(e_{1}^{\nu})_{\nu}$
for $E_{1}$ consisting of projections, then take $(x_{2}^{\nu})_{\nu}$ to be
the representative sequence for $E_{1}+E_{2}$ consisting of projections
$x_{2}^{\nu}$, which we may assume satisfies $e_{1}^{\nu}\leq x_{2}^{\nu}$
(otherwise change $x_{2}^{\nu}$ outside a neighborhood of $\omega$ to
$e_{1}^{\nu}$). Then, $(x_{2}^{\nu}-e_{1}^{\nu})_{\nu}$ is a representative
sequence for $E_{2}$ consisting of projections in $M$. Repeating this argument
for $E_{1},E_{2},E_{3}$, we choose $(x_{3}^{\nu})_{\nu}$ to be representative
sequence for $E_{1}+E_{2}+E_{3}$ such that $x_{2}^{\nu}\leq x_{3}^{\nu}$. Then
$(x_{3}^{\nu}-x_{2}^{\nu})_{\nu}$ is a representative sequence for $E_{3}$
consisting of projections in $M$. Continuing in this manner, for
$E_{1},E_{2},\ldots,E_{N-1}$ we get a representative sequence
$(x_{N-1}^{\nu})_{\nu}-x_{N-2}^{\nu}$ consisting of projections in $M$ for
$E_{N-1}$. Finally, $(1-x_{N-1}^{\nu})_{\nu}$ forms a representative sequence
for $E_{N}$ consisting of projections in $M$. This gives us partitions of
unity as required.

∎

###  2.3. Free and centrally free actions

Let $M$ be a separable von Neumann algebra. An automorphism
$\theta\in\mathrm{Aut}(M)$ is called properly outer if $\theta|_{PMP}$ is not
inner for any nonzero invariant central projection $P$. We say that
$\theta\in\mathrm{Aut}(M)$ is strongly outer if the restriction of $\theta$ to
the relative commutant of any countable $\theta$-invariant subset of $M$ is
properly outer. Let us denote by $\mathcal{CT}(M)$ the collection of all
centrally trivial automorphisms on $M$:

$\mathcal{CT}(M)=\left\\{\theta\in\mathrm{Aut}(M):\theta_{\omega}=\text{id}\in\mathrm{Aut}\left(M_{\omega}\right)\right\\}\,.$

An automorphism $\theta\in\mathrm{Aut}(M)$ is called properly centrally non-
trivial if for any invariant central projection $P\in M$, its restriction to
$PMP$ is not centrally trivial.

Let $\Gamma$ be a discrete group acting on a von Neumann algebra $M$ by
automorphisms. The action $\alpha:\Gamma\to\mathrm{Aut}(M)$ is said to be free
if $\alpha_{g}$ is properly outer for any $g\neq e$. The action $\alpha$ is
said to be strongly free if $\alpha_{g}$ is strongly outer for any $g\neq e$.
We say that $\alpha$ is centrally free if $\alpha_{g}$ is properly centrally
non-trivial for any $g\neq e$. We recall the following.

######  Lemma 2.9.

[ocneanu2006actions, Lemma 5.6] lem:ocneanu_stronglyouter Let $M$ be a
separable von Neumann algebra. Let $\alpha$ be a properly centrally nontrivial
automorphism of $M$, then $\alpha_{\omega}$ is a strongly outer automorphism
of $M_{\omega}$.

###  2.4. Crossed product $C^{*}$-algebras

Let $\Gamma$ be a discrete group, $A$ be an unital $C^{*}$-algebra, and
$\alpha\colon\Gamma\to\mathrm{Aut}(A)$ be an action. We denote the canonical
conditional expectation by $\mathbb{E}\colon A\rtimes_{\alpha,r}\Gamma\to A$.
(See [BroOza08, Proposition 4.1.9].) We shall denote the canonical unitaries
in $A\rtimes_{\alpha,r}\Gamma$ by $\\{\lambda_{s}\\}_{s\in\Gamma}$. They
satisfy $\lambda_{s}a\lambda_{s}^{*}=\alpha_{s}(a)$ for any $s\in\Gamma$ and
any $a$ in the canonical copy of $A$ in the reduced crossed product.

###  2.5. Rokhlin dimension

We recall the definition of the Rokhlin dimension for actions of finite groups
and of ${\mathbb{Z}}$ from [hirshberg2015rokhlin, hirshberg_phillips]. For
actions of ${\mathbb{Z}}$, we use the single tower version.

######  Definition 2.10.

Let $A$ be a separable $C^{*}$-algebra.

  1. (1)

Suppose $\alpha\in\mathrm{Aut}(A)$ (thought of as an action of
${\mathbb{Z}}$). We say that $\alpha$ has _Rokhlin dimension_ $d$ if $d$ is
the least non-negative integer such that for any $\varepsilon>0$, for any
finite subset $F\subseteq A$ and for any $N\in{\mathbb{N}}$ there exist
positive contractions $\\{f_{k}^{(l)}\mid k=1,2,\ldots,N\,;\,l=0,1,2,\ldots
d\\}$ in $A$ such that for all $x\in F$ we have

    1. (a)

$\|(\sum_{l=0}^{d}\sum_{k=1}^{N}f_{k}^{(l)})x-x\|<\varepsilon$.

    2. (b)

For all $j\neq k$ and for all $l$ we have
$\|(f_{k}^{(l)}f_{j}^{(l)})x\|<\varepsilon$.

    3. (c)

For all $l$ and for all $k$ we have
$\|(\alpha(f_{k}^{(l)})-f_{k+1}^{(l)})x\|<\varepsilon$ with the convention
that $f_{N+1}^{(l)}=f_{1}^{(l)}$.

    4. (d)

For any indices $k,l$ we have $\|[x,f_{k}^{(l)}]\|<\varepsilon$.

  2. (2)

Suppose $G$ is a finite group and suppose $\alpha\colon G\to\mathrm{Aut}(A)$.
We say that $\alpha$ has _Rokhlin dimension_ $d$ if $d$ is the least non-
negative integer such that for any $\varepsilon>0$ and for any finite subset
$F\subseteq A$ there exist positive contractions $\\{f_{g}^{(l)}\mid g\in
G\,;\,l=0,1,2,\ldots d\\}$ in $A$ such that for all $x\in F$ we have

    1. (a)

$\|(\sum_{l=0}^{d}\sum_{g\in G}f_{g}^{(l)})x-x\|<\varepsilon$.

    2. (b)

For all $g\neq h$ in $G$ and for all $l$ we have
$\|(f_{g}^{(l)}f_{h}^{(l)})x\|<\varepsilon$.

    3. (c)

For all $l$ and for all $g,h\in G$ we have
$\|(\alpha_{g}(f_{h}^{(l)})-f_{gh}^{(l)})x\|<\varepsilon$.

    4. (d)

For any index $l$ and for any $g\in G$ we have
$\|[x,f_{g}^{(l)}]\|<\varepsilon$.

For this paper, it would be enough to consider the restriction to any singly
generated subgroup, so we make the following definition.

######  Definition 2.11.

Let $A$ be a separable $C^{*}$-algebra. Let $\Gamma$ be a countable discrete
group, and suppose $\alpha\colon\Gamma\to\mathrm{Aut}(A)$ is an action. We say
that $\alpha$ has _pointwise finite relative Rokhlin dimension_ if for any
$s\in\Gamma\smallsetminus\\{e\\}$ the restriction of $\alpha$ to the subgroup
generated by $s$ has a finite Rokhlin dimension.

We now introduce a notion of Rokhlin dimension for an action on an inclusion
$A\subseteq B$. The key point here is that we want the Rokhlin towers to be in
$A$ but approximately central with respect to elements of $B$.

######  Definition 2.12.

Let $A\subseteq B$ be an inclusion of separable $C^{*}$-algebras.

  1. (1)

Suppose $\alpha\in\mathrm{Aut}(B)$ is an automorphism which leaves $A$
invariant (thought of as an action of ${\mathbb{Z}}$). We say that $\alpha$
has _relative Rokhlin dimension_ $d$ if $d$ is the least non-negative integer
such that for any $\varepsilon>0$, for any finite subset $F\subseteq B$ and
for any $N\in{\mathbb{N}}$ there exist positive contractions
$\\{f_{k}^{(l)}\mid k=1,2,\ldots,N\,;\,l=0,1,2,\ldots d\\}$ in $A$ such that
for all $x\in F$ we have

    1. (a)

$\|(\sum_{l=0}^{d}\sum_{k=1}^{N}f_{k}^{(l)})x-x\|<\varepsilon$.

    2. (b)

For all $j\neq k$ and for all $l$ we have
$\|(f_{k}^{(l)}f_{j}^{(l)})x\|<\varepsilon$.

    3. (c)

For all $l$ and for all $k$ we have
$\|(\alpha(f_{k}^{(l)})-f_{k+1}^{(l)})x\|<\varepsilon$ with the convention
that $f_{N+1}^{(l)}=f_{1}^{(l)}$.

    4. (d)

For any indices $k,l$ we have $\|[x,f_{k}^{(l)}]\|<\varepsilon$.

  2. (2)

Suppose $G$ is a finite group and suppose $\alpha\colon G\to\mathrm{Aut}(B)$
is an action which leaves $A$ invariant. We say that $\alpha$ has _relative
Rokhlin dimension_ $d$ if $d$ is the least non-negative integer such that for
any $\varepsilon>0$ and for any finite subset $F\subseteq B$ there exist
positive contractions $\\{f_{g}^{(l)}\mid g\in G\,;\,l=0,1,2,\ldots d\\}$ such
that for all $x\in F$ we have

    1. (a)

$\|(\sum_{l=0}^{d}\sum_{g\in G}f_{g}^{(l)})x-x\|<\varepsilon$.

    2. (b)

For all $g\neq h$ in $G$ and for all $l$ we have
$\|(f_{g}^{(l)}f_{h}^{(l)})x\|<\varepsilon$.

    3. (c)

For all $l$ and for all $g,h\in G$ we have
$\|(\alpha_{g}(f_{h}^{(l)})-f_{gh}^{(l)})x\|<\varepsilon$.

    4. (d)

For any index $l$ and for any $g\in G$ we have
$\|[x,f_{g}^{(l)}]\|<\varepsilon$.

  3. (3)

Let $\Gamma$ be a countable discrete group, and suppose
$\alpha\colon\Gamma\to\mathrm{Aut}(B)$ is an action which leaves $A$
invariant. We say that $\alpha$ has _pointwise finite relative Rokhlin
dimension_ if for any $s\in\Gamma\smallsetminus\\{e\\}$ the restriction of
$\alpha$ to the subgroup generated by $s$ has a finite Rokhlin dimension.

######  Remark 2.13.

Suppose $A\subseteq B$ is an inclusion of separable $C^{*}$-algebras such that
the natural inclusion from $A_{\infty}$ into $B_{\infty}$ descends to an
unital homomorphism $F(A)\subseteq F(B)$. It is immediate that an action on
$B$, which leaves $A$ invariant, has a finite Rokhlin dimension provided the
restriction to $A$ has a finite Rokhlin dimension. We will provide in Lemma
5.5 conditions which guarantee that we have such an unital homomorphism
$F(A)\subseteq F(B)$.

##  3\. von Neumann algebraic Rokhlin theorems

Connes’ noncommutative Rokhlin theorem [connes1975outer, Theorem 1.2.5] for
single automorphism is stated for automorphisms of finite von Neumann algebras
with an invariant trace. We need a more general version that does not assume
the trace is invariant. The proof of this stronger version goes through almost
verbatim as in [connes1975outer] (also see [TakesakiIII, Chapter XVII]), only
that the use of the Ergodic theoretic Rokhlin lemma for measure preserving
actions is replaced by the Rokhlin theorem for nonsingular actions
([ornstein1980ergodic]). We nonetheless provide a full proof for the reader’s
convenience. The Rokhlin theorem for essentially free measure class preserving
actions of amenable groups was announced in [ornstein1980ergodic]. There is a
short note in [ornstein1987entropy], where the authors outline how to
generalize the result from measure preserving systems to measure class
preserving ones. We could not find complete proof in published literature;
however, one can be found in a Ph.D. thesis by Jarrett [jarrett2018non,
Appendix B].

######  Proposition 3.1.

prop:connes Let $(N,\tau)$ be a finite von Neumann algebra, with $\tau$ a
faithful normal tracial state. Assume that
$\alpha:\mathbb{Z}\to\mathrm{Aut}(N)$ is free. Then, for any positive integer
$n$ and for any $\varepsilon>0$, there exists a partition of unity
$\\{E_{j}\\}_{j=1}^{n}$ such that

$\left\|\alpha_{1}(E_{j})-E_{j+1}\right\|_{\tau}<\varepsilon,\leavevmode\nobreak\
j=1,2,\ldots,n,$

where $E_{n+1}=E_{1}$

###### Proof.

Let $(N,\tau)$ be a finite von Neumann algebra. We can decompose $N$ into an
infinite direct sum $\overline{\bigoplus}_{k=0}^{\infty}N_{k}$ where each
summand is invariant, the action on the center $Z(N_{0})$ is free, the action
on the center $Z(N_{1})$ is trivial, and for each $k>0$, the action on
$Z(N_{k})$ is $k$-periodic and there is no non-zero invariant projection $P\in
Z(N_{k})$ such that the restriction to $PZ(N_{k})$ has a shorter period. It
suffices to prove the statements for each of those cases separately.

Let us first assume that $\alpha$ acts trivially on the center $Z(N)$. Let $n$
be a fixed positive integer and $\varepsilon>0$ be given. Denote by
$\mathcal{U}(N)$ the unitary group $N$. Recall that $\|x\|_{1}=\tau(|x|)$.
Consider the collection
$\mathcal{E}=\left\\{\left(E_{1},E_{2},\ldots,E_{n};U\right)\right\\}$, where

  1. (1)

$\left\\{E_{1},E_{2},\ldots,E_{n}\right\\}$ are mutually orthogonal equivalent
projections.

  2. (2)

$U\in\mathcal{U}(N)$ and
$\|U-1\|_{1}\leq\varepsilon\tau\left(\sum_{i=1}^{n}E_{i}\right)$.

  3. (3)

$U\alpha_{1}(E_{i})U^{*}=E_{i+1}$, $E_{n+1}=E_{1}$.

It follows from [connes1975outer, Lemma 1.2.7] that $\mathcal{E}$ is a non-
empty set. (It is assumed in [connes1975outer, Lemma 1.2.7] that the von
Neumann algebra is countably decomposable; however, this assumption is not
needed; see [TakesakiIII, Lemma 1.8, Chapter-XVII].) We define an order on
$\mathcal{E}$ by setting
$\left(E_{1},E_{2},\ldots,E_{n};U\right)\leq\left(E_{1}^{\prime},E_{2}^{\prime},\ldots,E_{n}^{\prime};U^{\prime}\right)$
if $E_{i}\leq E_{i}^{\prime}$ for each $1\leq i\leq n$ and
$\|U-U^{\prime}\|_{1}\leq\varepsilon\tau\left(\sum_{i=1}^{n}E_{i}^{\prime}-E_{i}\right)$.
Let $\mathcal{F}$ be a totally ordered subset of $\mathcal{E}$. The map
$\mathcal{O}:\mathcal{F}\to[0,1]$, defined by

$\mathcal{O}\left(E_{1},E_{2},\ldots,E_{n};U\right)=\tau\left(\sum_{i=1}^{n}E_{i}\right)$

is order-preserving. Therefore, $\mathcal{F}$ contains a cofinal sequence

$y_{m}=\left\\{\left(E_{1,m},E_{2,m},\ldots,E_{n,m};U_{m}\right)\right\\}.$

Using (2), we see that

$\displaystyle\left\|U_{m+1}-U_{m}\right\|_{1}\leq\varepsilon\tau\left(\sum_{i=1}^{n}E_{i,m+1}-E_{i,m}\right),$

and hence, $\sum_{m=1}^{\infty}\|U_{m+1}-U_{m}\|_{1}<\infty$. Since
$\mathcal{U}(N)$ is complete with respect to the $\|.\|_{1}$-norm, we see that
$\lim_{m\to\infty}U_{m}=U$ exists and is in $\mathcal{U}(N)$. Moreover,
$\\{E_{i,m}\\}_{m}$, an increasing sequence of projections, converges to a
projection $E_{i}$ for each $i=1,2,\ldots,n$. We now observe that
$\left(E_{1},E_{2},\ldots,E_{n};U\right)$ satisfies (1), (2) and (3).
Therefore, $\mathcal{F}$ admits a maximal element, which we denote by
$\left(E_{1},E_{2},\ldots,E_{n};U\right)$, by mild abuse of notation. We claim
that $\sum_{i=1}^{n}E_{i}=1$. Suppose not. Set $P=1-\sum_{i=1}^{n}E_{i}\neq
0$. From (3), it follows that $U\alpha_{1}(P)U^{*}=P$. Therefore,
$\tilde{\alpha}=\text{Ad}(U)\circ\alpha_{1}$ leaves $PNP$ invariant. Moreover,
$\tilde{\alpha}$ also acts freely on $PNP$. Denote
$\tau_{PNP}(\cdot)=\frac{1}{\tau(P)}\tau(\cdot)$ and we denote by
$\|.\|_{PNP}$ the $L^{1}$-norm associated with $\tau_{PNP}$. Using
[TakesakiIII, Lemma 1.8, Chapter-XVII], we can find mutually orthogonal
equivalent projections $\\{P_{1},P_{2},\ldots,P_{n}\\}$ in $NP$ and an unitary
$\tilde{U}\in\mathcal{U}(PNP)$ such that

$\left\|\tilde{U}-P\right\|_{1,PNP}<\varepsilon\tau_{P\mathcal{N}P}\left(\sum_{i=1}^{n}P_{i}\right)\text{
and }\tilde{U}\tilde{\alpha}(P_{j})\tilde{U}^{*}=P_{j+1},\leavevmode\nobreak\
j=1,2,\ldots,n.$

Consequently, we observe that
$\tilde{U}U\alpha_{1}(P_{i})U^{*}\tilde{U}^{*}=P_{i+1}$ for $i=1,2,\ldots,n$.
Letting $\tilde{U}U=V\in\mathcal{U}(PNP)$, we see that
$V\alpha_{1}(P_{i})V^{*}=P_{i+1}$ for each $i=1,2,\ldots,n$ with the
convention that $P_{n+1}=P_{1}$. Moreover, it also follows that

$\|V-E\|_{1,\tau_{NP}}\leq\varepsilon\tau_{{NP}}\left(\sum_{i=1}^{n}P_{i}\right).$

Let $E_{i}^{\prime}=E_{i}+P_{i}$ for each $i=1,2,\ldots,n$ and let
$U^{\prime}=(V+(1-P))U$. Then
$E_{1}^{\prime},E_{2}^{\prime},\ldots,E_{n}^{\prime}$ are mutually orthogonal
projections. Moreover,
$U^{\prime}\alpha_{1}(E_{i}^{\prime})U^{\prime*}=E^{\prime}_{i+1}$ for each
$i=1,2,\ldots,n$ with the convention that $E^{\prime}_{n+1}=E_{1}^{\prime}$.
Furthermore,

$\displaystyle\left\|V+(1-P)-1\right\|_{1}$
$\displaystyle=\tau(P)\|V-P\|_{1,\tau_{NP}}$
$\displaystyle\leq\varepsilon\tau(P)\tau_{{NP}}\left(\sum_{i=1}^{n}P_{i}\right)$
$\displaystyle=\varepsilon\tau\left(\sum_{i=1}^{n}P_{i}\right)$

Therefore, we have

$\displaystyle\left\|U^{\prime}-U\right\|_{1}$
$\displaystyle=\left\|\left(V+(1-P)-1\right)U\right\|_{1}$
$\displaystyle\leq\varepsilon\tau\left(\sum_{i=1}^{n}P_{i}\right)$

Thus,

$\displaystyle\left\|U^{\prime}-1\right\|_{1}$
$\displaystyle\leq\left\|U^{\prime}-U\right\|_{1}+\left\|U-1\right\|_{1}$
$\displaystyle\leq\varepsilon\tau\left(\sum_{i=1}^{n}P_{i}\right)+\varepsilon\tau\left(\sum_{i=1}^{n}E_{i}\right)$
$\displaystyle=\varepsilon\tau\left(\sum_{i=1}^{n}E_{i}^{\prime}\right)\,.$

Therefore,
$\left(E_{1}^{\prime},E_{2}^{\prime},\ldots,E_{n}^{\prime};U^{\prime}\right)\in\mathcal{E}$
and in particular,

$\left(E_{1},E_{2},\ldots,E_{n};U\right)<\left(E_{1}^{\prime},E_{2}^{\prime},\ldots,E_{n}^{\prime};U^{\prime}\right).$

This contradicts the maximality of $\left(E_{1},E_{2},\ldots,E_{n};U\right)$.
Therefore, $\sum_{i=1}^{n}E_{i}=1$. Finally, we observe that

$\displaystyle\left\|\alpha_{1}(E_{j})-E_{j+1}\right\|_{2}^{2}$
$\displaystyle\leq\left\|\alpha_{1}(E_{j})-E_{j+1}\right\|\left\|\alpha_{1}(E_{j})-E_{j+1}\right\|_{1}$
$\displaystyle\leq 2\left\|\alpha_{1}(E_{j})-E_{j+1}\right\|_{1}$
$\displaystyle=2\left\|\alpha_{1}(E_{j})-U\alpha_{1}(E_{j})U^{*}\right\|_{1}$
$\displaystyle\leq 4\|U-1\|_{1}=4\varepsilon,\leavevmode\nobreak\
j=1,2,\ldots,n.$

This completes the proof for the case in which $\alpha_{1}$ acts trivially on
the center $Z(N)$.

If $\alpha_{1}|_{Z(N)}$ is essentially free, then we can use the Rokhlin
theorem for non-singular actions ([jarrett2018non, Theorem B.2.4]) to obtain
the required projections $\left\\{E_{1},E_{2},\ldots,E_{n}\right\\}$. It
remains to consider the case in which $\alpha_{1}$ is periodic with period
$p\geq 1$. Let $P$ be a projection in $Z(N)$ be such that
$\left\\{\alpha_{1}^{j}(P):0\leq j\leq p-1\right\\}$ is a partition of unity.
It then follows that $\alpha_{1}^{p}|_{NP}$ is free and leaves the center
$Z\left(NP\right)$ fixed. Therefore, using the arguments from before, we can
find a partition $\left\\{F_{1},F_{2},\ldots,F_{n}\right\\}$ of unity in $NP$
such that

$\left\|\alpha_{1}^{p}|_{NP}(F_{i})-F_{i+1}\right\|_{2}<\frac{\varepsilon}{p},\leavevmode\nobreak\
i=1,2,\ldots,n\text{ and }F_{n+1}=F_{1}.$

Let

$F_{0}=F_{n}\text{ and
}G_{ip+j}=\alpha_{1}^{j}\left(F_{i}\right),\leavevmode\nobreak\ 0\leq
j<p,0\leq i<n.$

Observe that $\left\\{G_{k}:0\leq k<np\right\\}$ is a partition of unity in
$N$. Moreover,
$\left\|\alpha_{1}(G_{k})-G_{k+1}\right\|_{2}<\frac{\varepsilon}{p}$. Let

$E_{i}=\sum_{k=0}^{p-1}G_{kn+i},\leavevmode\nobreak\ 1\leq i\leq n,\text{ with
the convention that }G_{pn}=G_{0}.$

It then follows that $\left\\{E_{i}:1\leq i\leq n\right\\}$ is a partition of
unity in $N$ and $\left\|\alpha_{1}(E_{i})-E_{i+1}\right\|_{2}\leq\varepsilon$
for each $i=1,2,\ldots,n$. As usual, $E_{n+1}=E_{1}$. This completes the
proof.

∎

prop:connes holds for finite von Neumann algebras, which is not the case here.
We bypass the finiteness assumption by working with the central sequence von
Neumann algebras $M_{\omega}$. This, in turn, gives us projections in
$M_{\omega}$ which are Rokhlin towers. To obtain the desired partition at the
level of the von Neumann algebra $M$ (which is not necessarily finite), we use
an index selection trick along with lem:connes.

######  Lemma 3.2.

relConnesrelative Let $N\subset M$ be an inclusion of von Neumann algebras
with separable preduals such that $1_{N}=1_{M}$. Let $B\subset M$ be a weakly
dense C*-subalgebra. Suppose that $N^{\prime}\cap N^{\omega}\subseteq
B^{\prime}\cap N^{\omega}$ and suppose $\varphi$ is a faithful normal state on
$M$. Let $\alpha:\mathbb{Z}\to\mathrm{Aut}(M)$ be an action such that
$\alpha(N)\subseteq N$ and such that
$\alpha_{\omega}:\mathbb{Z}\to\mathrm{Aut}(N_{\omega})$ is free. Then, given
$N\in\mathbb{N}$, there exists a sequence
$\\{e_{1}^{\nu},e_{2}^{\nu},\ldots,e_{N}^{\nu}\\}_{\nu}$ of partitions of
unity in $N$ such that

(1)
$\left\|\alpha_{1}(e_{j}^{\nu})-e_{j+1}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
j=1,2,\ldots,N.$ (2)
$\left\|e_{j}^{\nu}b-be_{j}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
b\in B\text{ and }j\in\\{1,2,\ldots,N\\}.$ (3)
$\left\|e_{j}^{\nu}be_{j}^{\nu}-be_{j}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
b\in B\text{ and }j\in\\{1,2,\ldots,N\\}.$

###### Proof.

Applying prop:connes to $N_{\omega}$, for $N\in{\mathbb{N}}$, for any
$n\in{\mathbb{N}}$, we obtain a partition of unity
$\\{E_{j}^{n}\\}_{j=1}^{N}\subseteq N_{\omega}\subseteq N^{\prime}\cap
N^{\omega}\subseteq B^{\prime}\cap N^{\omega}$ such that
$\left\|\alpha_{1}(E_{j}^{n})-E_{j+1}^{n}\right\|_{\varphi_{\omega}}<\frac{1}{n}$
for $j=1,2,\ldots,N$, with $E_{N+1}^{n}=E_{1}^{n}$. Applying an index
selection trick (e.g. [ocneanu2006actions, Section 5.5]) to the elements
$\\{E_{1}^{n},E_{2}^{n},\ldots,E_{N}^{n}\\}$ gives us a partition of unity
$\\{E_{j}\\}_{j=1}^{N}\subseteq N_{\omega}\subseteq N^{\prime}\cap N^{\omega}$
such that

$\alpha_{1}(E_{j})=E_{j+1}\text{ for }j={1,2,\ldots,N},\text{
with}\,E_{N+1}=E_{1}\,.$

Finally, as a consequence of lem:connes, and the fact that
$\left\|.\right\|_{\varphi}^{\\#}$ defines the *-strong operator topology on
bounded subsets of $M$, we obtain a sequence of partition of unity
$e_{1}^{\nu},e_{2}^{\nu},\ldots,e_{N}^{\nu}$ in $N$, where
$\\{e_{j}^{\nu}\\}_{\nu}$ is a representative sequence for $E_{j}$ in
$N^{\prime}\cap N^{\omega}$, such that for all $j\in\\{1,2,\ldots,N\\}$ we
have

$\alpha_{1}(e_{j}^{\nu})-e_{j+1}^{\nu}\underset{{\nu}\rightarrow\omega}{\longrightarrow}0\text{
*-strongly}$

with $e_{N+1}^{\nu}=e_{1}^{\nu}$ for all ${\nu}$. Equivalently,

$\left\|\alpha_{1}(e_{j}^{\nu})-e_{j+1}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\rightarrow\omega}{\longrightarrow}0\,.$

This establishes equation (1). Equation (2) follows from the condition
$N^{\prime}\cap N^{\omega}\subseteq B^{\prime}\cap N^{\omega}$. Consequently,
using equation (2), we see that

$\displaystyle\left\|e_{j}^{\nu}be_{j}^{\nu}-e_{j}^{\nu}b\right\|_{\varphi}$
$\displaystyle\leq\left\|be_{j}^{\nu}-e_{j}^{\nu}b\right\|_{\varphi}\underset{{\nu}\rightarrow\omega}{\longrightarrow}0,$

and thus

$\displaystyle\left\|e_{j}^{\nu}be_{j}^{\nu}-be_{j}^{\nu}\right\|_{\varphi}$
$\displaystyle\leq\left\|e_{j}^{\nu}be_{j}^{\nu}-e_{j}^{\nu}b\right\|_{\varphi}+\left\|e_{j}^{\nu}b-be_{j}^{\nu}\right\|_{\varphi}$
$\displaystyle\leq
2\left\|be_{j}^{\nu}-e_{j}^{\nu}b\right\|_{\varphi}\underset{{\nu}\rightarrow\omega}{\longrightarrow}0.$

This establishes equation (3), thereby completing the proof. ∎

######  Theorem 3.3.

thm:relativizedconnes Let $A\subseteq B$ be separable $C^{*}$-algebras such
that $A$ contains an approximate identity for $B$. Let $\omega$ be a free
ultrafilter on $\mathbb{N}$. Let $\alpha:{\mathbb{Z}}\to\mathrm{Aut}(B)$ be an
action which leaves $A$ invariant. Suppose for every central
$\alpha^{**}$-invariant projection $P$ in $B^{**}$ such that $PB^{**}P$ is
separable, we have

$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$

and $\alpha^{**}_{\omega}:{\mathbb{Z}}\to\mathrm{Aut}((PA^{**}P)_{\omega})$ is
free. Then, for any $N\in{\mathbb{N}}$ and for any finite set of elements
$S\subseteq B$, there exists a net of mutually orthogonal, contractive,
positive elements
$\\{p_{1}^{\beta},p_{2}^{\beta},\ldots,p_{N}^{\beta}\\}_{\beta}$ in $A$ with
$p_{N+1}^{\beta}=p_{1}^{\beta}$, such that for all $j\in\\{1,2,\ldots,N\\}$
and for all $b\in S$ the following hold:

(4) $\displaystyle\alpha_{1}(p_{j}^{\beta})-p_{j+1}^{\beta}$ $\displaystyle\to
0\text{ *-strongly }.$ (5) $\displaystyle bp_{j}^{\beta}-p_{j}^{\beta}b$
$\displaystyle\to 0\text{ *-strongly}.$ (6) $\displaystyle
p_{j}^{\beta}bp_{j}^{\beta}-bp_{j}^{\beta}$ $\displaystyle\to 0\text{
*-strongly}.$ (7) $\displaystyle\sum_{l=1}^{N}p_{j}^{\beta}-1$
$\displaystyle\to 0\text{ *-strongly }.$ (8)
$\displaystyle(p_{j}^{\beta})^{2}-p_{j}^{\beta}$ $\displaystyle\to 0\text{
*-strongly }.$

###### Proof.

We prove this in two steps.
Step 1: We claim that for any $N\in{\mathbb{N}}$ and any finite set
$S\subseteq B$ and for any $*$-strong neighborhood $V$ of $0$ in $B^{**}$
there exist mutually orthogonal projections $e_{1},e_{2},\ldots e_{N}$ in
$A^{**}$ such that for any $j\in\\{1,2,\ldots,N\\}$ and for any $b\in S$ the
following hold:

  1. (A)

$\alpha_{1}(e_{j})-e_{j+1}\in V\text{ with }e_{N+1}=e_{1}.$

  2. (B)

$e_{j}b-be_{j}\in V$

  3. (C)

$e_{j}be_{j}-be_{j}\in V$

  4. (D)

$\sum_{j=1}^{N}e_{j}-1\in V$

relConnesrelative does not directly apply, as $A^{**}$ and $B^{**}$ typically
does not have separable preduals. This is corrected as follows. Pick a normal
state $\varphi$ and $\varepsilon>0$ be such that $\\{b\in
B^{**}\mid\|b\|_{\varphi}^{\\#}<\varepsilon\\}\subseteq V$.

Let $P$ be the smallest central $\alpha^{**}$-invariant projection which
dominates the central support of $\varphi$, that is, $P$ can be obtained by
taking the join of all iterates of central support of $\varphi$ under
$\alpha$. We may assume without loss of generality that $\varphi|_{PA^{**}P}$
is a faithful normal state (otherwise replace it by
$\frac{1}{2}\varphi+\sum_{n\in{\mathbb{Z}}\smallsetminus\\{0\\}}\frac{1}{5^{|n|}}\varphi\circ\alpha^{n}$
and replace $\varepsilon$ by $\varepsilon/2$). Notice that $PA^{**}P$ and
$PB^{**}P$ have separable preduals, since $A$ and $B$ both are separable
$C^{*}$-algebras. Using the fact that
$\alpha^{**}_{\omega}:{\mathbb{Z}}\to\mathrm{Aut}((PA^{**}P)_{\omega})$ is
free and $PBP$ is weakly dense in $PB^{**}P$, by relConnesrelative, for
$N\in{\mathbb{N}}$, we get a partition of unity $e_{1},e_{2},\ldots,e_{N}\in
PA^{**}P$ such that

(9)
$\left\|\alpha_{1}(e_{j})-e_{j+1}\right\|_{\varphi}^{\\#}<\varepsilon\,\,\forall
j=1,2,\ldots,N.$ (10)
$\left\|e_{j}b-be_{j}\right\|_{\varphi}^{\\#}<\varepsilon\,\,\forall b\in
S\text{ and }j\in\\{1,2,\ldots,N\\}.$ (11)
$\left\|e_{j}be_{j}-be_{j}\right\|_{\varphi}^{\\#}<\varepsilon\,\,\forall b\in
S\text{ and }j\in\\{1,2,\ldots,N\\}.$ (12)
$\left\|P-\sum_{j=1}^{N}e_{j}\right\|_{\varphi}^{\\#}=0.$

Therefore, those elements satisfy conditions (A),(B), (C) and (D) above.

Step 2: We now show that we can find a net of mutually orthogonal positive
elements in $A$ which satisfies the conclusion of thm:relativizedconnes.

Let $\varphi$ be a normal state on $B^{**}$ and let $\varepsilon>0$. We need
to show that we can find mutually orthogonal positive elements
$p_{1},p_{2},\ldots,p_{N}$ in $A$, with $p_{N+1}=p_{1}$ such that for
$j=1,2,\ldots,N$ and any $b\in S$ we have

(13) $\displaystyle\|\alpha_{1}(p_{j})-p_{j+1}\|_{\varphi}^{\\#}<\varepsilon,$
(14) $\displaystyle\|bp_{j}-p_{j}b\|_{\varphi}^{\\#}<\varepsilon,$ (15)
$\displaystyle\|p_{j}bp_{j}-bp_{j}\|_{\varphi}^{\\#}<\varepsilon,$ (16)
$\displaystyle\|1-\sum_{l=1}^{N}p_{j}\|_{\varphi}^{\\#}<\varepsilon,$ (17)
$\displaystyle\|(p_{j})^{2}-p_{j}\|_{\varphi}^{\\#}<\varepsilon.$

Denote by $\sigma:{\mathbb{Z}}\to C({\mathbb{Z}}_{N})$ the translation action,
that is, $\sigma_{l}(f)(s)=f(l-s)$ for $s,l\in{\mathbb{Z}}$. For
$j\in\\{1,2,\ldots,N\\}$ let $f_{j}\in C({\mathbb{Z}}_{N})$ be the projection
in $C({\mathbb{Z}}_{N})$ such that $f_{j}(j)=1$ and zero otherwise, with
$f_{N+1}=f_{1}$. Using Step 1, we can find a net of equivariant order zero
maps

$\Theta^{\nu}:(C({\mathbb{Z}}_{N}),\sigma,{\mathbb{Z}})\to(A^{**},\alpha^{**},{\mathbb{Z}})$

which satisfy

$\displaystyle\alpha^{**}_{1}(\Theta^{\nu}(f_{j}))$
$\displaystyle-\Theta^{\nu}(\sigma_{1}(f_{j}))\to 0\text{ *-strongly }$
$\displaystyle\Theta^{\nu}(f_{i})b$ $\displaystyle-b\Theta^{\nu}(f_{i})\to
0\text{ *-strongly, }\forall b\in S$

Fix $\nu$ such that for all $b\in S$ and for $j=1,2,\ldots,N$ we have

(18)
$\displaystyle\|\alpha_{1}(\Theta^{\nu}(f_{j}))-\Theta^{\nu}(f_{j+1})\|_{\varphi}^{\\#}<\varepsilon,$
(19)
$\displaystyle\|b\Theta^{\nu}(f_{j})-\Theta^{\nu}(f_{j})b\|_{\varphi}^{\\#}<\varepsilon,$
(20)
$\displaystyle\|\Theta^{\nu}(f_{j})b\Theta^{\nu}(f_{j})-b\Theta^{\nu}(f_{j})\|_{\varphi}^{\\#}<\varepsilon,$
(21)
$\displaystyle\|1-\sum_{l=1}^{N}\Theta^{\nu}(f_{j})\|_{\varphi}^{\\#}<\varepsilon.$

We can now use the Kaplansky density-type lemma from
[hirshberg2012decomposable, Lemma 1.1], to get a net
$(\Psi^{\mu}_{\nu})_{\mu}$ of order zero maps $C({\mathbb{Z}}_{N})\to A$ such
that $\Psi^{\mu}_{\nu}(f)\to\Theta^{\nu}(f)$ *-strongly for all $f\in
C({\mathbb{Z}}_{N})$. Define

$\Psi^{\mu}_{\nu}(f_{j})=p_{j,\nu}^{\mu}\,\text{ for }\,j\in{1,2,\ldots,N}$

Then $p_{j,\nu}^{\mu}$ are positive elements in $A$ and
$p_{i,{\nu}}^{\mu}p_{j,\nu}^{\mu}=0$ whenever $i\neq j$. Thus, we get a net
$(p_{1}^{\eta},p_{2}^{\eta},\ldots p_{N}^{\eta})_{\eta}$ of mutually
orthogonal positive elements in $A$. It now readily follows that there exists
$\eta_{0}$ such that for any $\eta\geq\eta_{0}$, the elements
$p_{1}^{\eta},p_{2}^{\eta},\ldots p_{N}^{\eta}$ satisfy the required
conditions above. ∎

Ocneanu [ocneanu2006actions] proved a generalized Rokhlin-type theorem for
centrally free actions of amenable groups. We show that if the group in
question admits large monotiles, one can somewhat strengthen the conclusions
of [ocneanu2006actions, Theorem 6.1]. In the paper, we only need the statement
for finite cyclic groups, but the proof of the general statement is similar.
The statement appears below in thm:relativizedocneanu.

######  Proposition 3.4.

prtunityinvna Let $\Gamma$ be a countable amenable group. Let $N\subset M$ be
von Neumann algebras with separable preduals such that $1_{N}=1_{M}$. Let
$B\subset M$ be weakly dense C*-subalgebra. Let
$\alpha:\Gamma\to\mathrm{Aut}(M)$ be an action such that $\alpha(N)\subseteq
N$ and $\alpha$ restricted on $N$ is centrally free. Let $N^{\prime}\cap
N^{\omega}\subseteq B^{\prime}\cap N^{\omega}$ and $\varphi$ be a faithful
normal state on $M$ such that $\alpha|_{Z(N)}$ leaves $\varphi|_{Z(N)}$
invariant. Let $T$ be a monotile for $\Gamma$. Then there exists a sequence
$\\{e_{t}^{\nu}\\}_{t\in T,\nu\in{\mathbb{N}}}$ of partitions of unity in $N$
such that

(22)
$\left\|\alpha_{ts^{-1}}(e_{s}^{\nu})-e_{t}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
s,t\in T,$ (23)
$\left\|e_{t}^{\nu}b-be_{t}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
b\in B\;\forall t\in T,$ (24)
$\left\|e_{t}^{\nu}be_{t}^{\nu}-be_{t}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,\,\forall
b\in B\;\forall t\in T.$

###### Proof.

By prop:monotileableepsilon, $T$ is an $\varepsilon$-paving set for any
$\varepsilon$. Using [ocneanu2006actions, Chapter 6] along with
lem:ocneanu_stronglyouter with $\varepsilon=\frac{1}{n}$, we get a sequence of
partitions of unity $\\{E_{t}^{(n)}\\}_{t\in T}$ in $N_{\omega}\subset
N^{\prime}\cap N^{\omega}$ such that

$\lvert\alpha_{ts^{-1}}(E_{s}^{(n)})-E_{t}^{(n)}\rvert_{\varphi}<\frac{5|T|}{\sqrt{n}}\,\,\forall
s,t\in T$

and

$[E_{t}^{(n)},\alpha_{g}(E_{s}^{(n)})]=0,\,\,\forall g\in\Gamma\text{ and
}s,t\in T.$

Apply the index selection trick (see [ocneanu2006actions, Section 5.5]) to the
elements $\\{E_{t}^{(n)}\\}_{t\in T}$ obtained above to get a partition of
unity $\\{E_{t}\\}_{t\in T}$ such that

$\alpha_{ts^{-1}}(E_{s})=E_{t}\,\,\forall s,t\in T$

and

$[E_{t}^{n},\alpha_{g}(E_{s}^{n})]=0,\,\,\forall g\in\Gamma\text{ and }s,t\in
T.$

Using lem:connes, we obtain a sequence of partition of unity
$\\{e_{t}^{\nu}\\}_{t\in T}$ in $N$ such that $\\{e_{t}^{\nu}\\}_{\nu}$ is a
representative sequence for $E_{t}$ in $N_{\omega}\subset N^{\prime}\cap
N^{\omega}$. Thus, for any $s,t\in T$, we have

$\alpha_{ts^{-1}}(e_{s}^{\nu})-e_{t}^{\nu}\underset{{\nu}\to\omega}{\longrightarrow}0\text{
*-strongly}.$

Equivalently,

$\left\|\alpha_{ts^{-l}}(e_{s}^{\nu})-e_{t}^{\nu}\right\|_{\varphi}^{\\#}\underset{{\nu}\to\omega}{\longrightarrow}0\,.$

This establishes equation (22). Equation (23) follows from the fact that
$N^{\prime}\cap N^{\omega}\subseteq B^{\prime}\cap N^{\omega}$. Consequently,
using equation (23), we see that

$\displaystyle\left\|e_{t}^{\nu}be_{t}^{\nu}-e_{t}^{\nu}b\right\|_{\varphi}$
$\displaystyle\leq\left\|be_{t}^{\nu}-e_{t}^{\nu}b\right\|_{\varphi}\underset{{\nu}\to\omega}{\longrightarrow}0,$

so

$\displaystyle\left\|e_{t}^{\nu}be_{t}^{\nu}-be_{t}^{\nu}\right\|_{\varphi}$
$\displaystyle\leq\left\|e_{t}^{\nu}be_{t}^{\nu}-e_{t}^{\nu}b\right\|_{\varphi}+\left\|e_{t}^{\nu}b-be_{t}^{\nu}\right\|_{\varphi}$
$\displaystyle\leq
2\left\|be_{t}^{\nu}-e_{t}^{\nu}b\right\|_{\varphi}\underset{{\nu}\to\omega}{\longrightarrow}0.$

This establishes equation (24), thereby completing the proof. ∎

######  Theorem 3.5.

thm:relativizedocneanu Let $A\subseteq B$ be separable $C^{*}$-algebras such
that $A$ contains an approximate identity for $B$. Let $\omega$ be a free
ultrafilter on ${\mathbb{N}}$. Let $\alpha:{\mathbb{Z}}_{N}\to\mathrm{Aut}(B)$
be an action which leaves $A$ invariant. Suppose for every central
$\alpha^{**}$-invariant projection $P$ in $B^{**}$ such that $PB^{**}P$ is
separable, we have

$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$

and
$\alpha^{**}_{\omega}:{\mathbb{Z}}_{N}\to\mathrm{Aut}((PA^{**}P)_{\omega})$ is
strongly free. Then, for any finite set of elements $S\subseteq B$, there
exists a net of mutually orthogonal, positive elements
$\\{p_{1}^{\beta},p_{2}^{\beta},\ldots,p_{N}^{\beta}\\}_{\beta}$ in $A$, with
$p_{N+1}^{\beta}=p_{1}^{\beta}$, such that the following hold for all
$j\in\\{1,2,\ldots,N\\}$ and for all $b\in S$:

(25) $\displaystyle\alpha_{1}(p_{j}^{\beta})-p_{j+1}^{\beta}$
$\displaystyle\to 0\text{ *-strongly },$ (26) $\displaystyle
bp_{j}^{\beta}-p_{j}^{\beta}b$ $\displaystyle\to 0\text{ *-strongly},$ (27)
$\displaystyle p_{j}^{\beta}bp_{j}^{\beta}-bp_{j}^{\beta}$ $\displaystyle\to
0\text{ *-strongly},$ (28) $\displaystyle\sum_{l=1}^{N}p_{j}^{\beta}-1$
$\displaystyle\to 0\text{ *-strongly },$ (29)
$\displaystyle(p_{j}^{\beta})^{2}-p_{j}^{\beta}$ $\displaystyle\to 0\text{
*-strongly }.$

###### Proof.

As in the proof of thm:relativizedconnes, given a normal state $\varphi$ on
$B^{**}$ and $\varepsilon>0$, one needs to find mutually orthogonal positive
elements $p_{1},p_{2},\ldots,p_{N}\in A$ such that for $j=1,2,\ldots,N$ and
all $b\in S$ we have

(30) $\displaystyle\|\alpha_{1}(p_{j})-p_{j+1}\|^{\\#}_{\varphi}<\varepsilon,$
(31) $\displaystyle\|bp_{j}-p_{j}b\|^{\\#}_{\varphi}<\varepsilon,$ (32)
$\displaystyle\|p_{j}bp_{j}-bp_{j}\|^{\\#}_{\varphi}<\varepsilon,$ (33)
$\displaystyle\|\sum_{l=1}^{N}p_{j}-1\|^{\\#}_{\varphi}<\varepsilon,$ (34)
$\displaystyle\|(p_{j})^{2}-p_{j}\|^{\\#}_{\varphi}<\varepsilon.$

By replacing $\varphi$ by its average over its iterates, we may assume without
loss of generality that it is invariant. The rest of the proof proceeds just
as the proof of thm:relativizedconnes, using prtunityinvna and the fact that
the entire group is a monotile in the case of a finite group. ∎

##  4\. Structure of Intermediate Algebras

The goal of this section is to prove mainintmresult. We begin with the
following preparatory lemma.

######  Lemma 4.1.

correctiveepsilon Let $A\subseteq B$ be unital, separable $C^{*}$-algebras
such that $A$ contains an approximate identity for $B$. Let $\omega$ be a free
ultrafilter on ${\mathbb{N}}$. Let $\Gamma$ be a discrete countable group, and
let and $\alpha:\Gamma\to\mathrm{Aut}(B)$ be an action which leaves $A$
invariant. Suppose for every central $\alpha^{**}$-invariant projection $P$ in
$B^{**}$ such that $PB^{**}P$ is separable, we have

$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$

and $\alpha^{**}:\Gamma\to\mathrm{Aut}(PA^{**}P)$ is centrally free. For any
$s\in\Gamma\setminus\\{e\\}$, there exists $N\in{\mathbb{N}}$ (depending only
on $s$), such that for every finite set $F\subseteq B$, for any finite set of
normal states
$\varphi_{1},\varphi_{2},\ldots,\varphi_{M}\in(B^{**}\overline{\rtimes}_{\alpha^{**}}\Gamma)_{*}$
and for any $\delta>0$, there exist mutually orthogonal, positive contractions
$p_{1},p_{2},\ldots,p_{N}\in A$ such that the following hold

  1. (a)

$\left\|\sum_{l=1}^{N}p_{l}x\lambda_{s}p_{l}\right\|_{\varphi_{j}}<\delta,\leavevmode\nobreak\
x\in F\,\,\,j=1,2,\ldots,M$

  2. (b)

$\left\|\sum_{l=1}^{N}p_{l}xp_{l}-x_{i}\right\|_{\varphi_{j}}<\delta,\leavevmode\nobreak\
x\in F,\,\,\,j=1,2,\ldots,M$

###### Proof.

Let $\Gamma_{s}$ be the subgroup of $\Gamma$ generated by $s\neq e\in\Gamma$.
Then, $\Gamma_{s}\cong{\mathbb{Z}}$ or ${\mathbb{Z}}_{N}$ (if order of $s$ is
$N$). If $s$ has infinite order, fix any $N\geq 2$ (for example, $N=2$ would
do). By thm:relativizedocneanu (if $\Gamma_{s}\cong{\mathbb{Z}}_{N}$) or by
thm:relativizedconnes (if $\Gamma_{s}\cong{\mathbb{Z}}$) there exists a net of
mutually orthogonal, positive contractions
$\\{p_{1}^{\beta},p_{2}^{\beta},\ldots,p_{N}^{\beta}\\}_{\beta}$ in $A$, such
that for all $l\in\\{1,2,\ldots,N\\}$ and for all $x\in F$, we have

$\displaystyle\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}$ $\displaystyle\to
0\text{ *-strongly, with }p_{N+1}^{\beta}=p_{1}^{\beta},$ $\displaystyle
xp_{l}^{\beta}-p_{l}^{\beta}x$ $\displaystyle\to 0\text{ *-strongly},$
$\displaystyle p_{l}^{\beta}xp_{l}^{\beta}-xp_{l}^{\beta}$ $\displaystyle\to
0\text{ *-strongly},$ $\displaystyle\sum_{l=1}^{N}p_{l}^{\beta}-1$
$\displaystyle\to 0\text{ *-strongly },$
$\displaystyle(p_{l}^{\beta})^{2}-p_{l}^{\beta}$ $\displaystyle\to 0\text{
*-strongly }.$

For any $b\in B$ and any $y\in B$, define positive linear functionals
$\varphi_{j}^{s}$ and $\varphi_{j}^{y}$ by
$\varphi_{j}^{s}(b):=\varphi_{j}({\lambda_{s}}^{*}b\lambda_{s})$,
$\varphi_{j}^{y}(b)=\varphi_{j}(y^{*}by)$. Let $L=\text{max}\\{\|x\|\\}_{x\in
F}$. Given $\delta>0$, choose $\beta_{0}$ such that for all
$\beta\geq\beta_{0}$ for $l=1,2,\ldots,N$, for $j=1,2,\ldots M$ and for all
$x\in F$ we have

$\displaystyle\left\|\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right\|_{\varphi_{j}^{s}}<\frac{\delta}{2NL},$
$\displaystyle\left\|xp_{l+1}^{\beta}-p_{l+1}^{\beta}x\right\|_{\varphi_{j}^{s}}<\frac{\delta}{2N},$
$\displaystyle\left\|xp_{l}^{\beta}-p_{l}^{\beta}x\right\|_{\varphi_{j}}<\frac{\delta}{3N},$
$\displaystyle\left\|(p_{l}^{\beta})^{2}-p_{l}^{\beta}\right\|_{\varphi_{j}^{x}}<\frac{\delta}{3N},$
$\displaystyle\left\|\sum_{l=1}^{N}p_{l}^{\beta}-1\right\|_{\varphi_{j}^{x}}<\frac{\delta}{3N}.$

Let $\beta>\beta_{0}$. For any $x\in F$ and for any $j=1,2,\ldots M$, we have

$\displaystyle\left\|\sum_{l=1}^{N}p_{l}^{\beta}x\lambda_{s}p_{l}^{\beta}\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\left\|p_{l}^{\beta}x\lambda_{s}p_{l}^{\beta}\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\left\|p_{l}^{\beta}x\alpha_{s}(p_{l}^{\beta})\lambda_{s}-p_{l}^{\beta}xp_{l+1}^{\beta}\lambda_{s}\right\|_{\varphi_{j}}$
$\displaystyle+\sum_{l=1}^{N}\left\|p_{l}^{\beta}xp_{l+1}^{\beta}\lambda_{s}-p_{l}^{\beta}p_{l+1}^{\beta}x\lambda_{s}\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\left\|p_{l}^{\beta}\right\|\left\|x\right\|\left\|\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)\lambda_{s}\right\|_{\varphi_{j}}$
$\displaystyle+\sum_{l=1}^{N}\left\|p_{l}^{\beta}\right\|\left\|\left(xp_{l+1}^{\beta}-p_{l+1}^{\beta}x\right)\lambda_{s}\right\|_{\varphi_{j}}.$

Note also that

$\displaystyle\left\|\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)\lambda_{s}\right\|_{\varphi_{j}}^{2}$
$\displaystyle=\varphi_{j}\left(\lambda_{s}^{*}\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)^{*}\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)\lambda_{s}\right)$
$\displaystyle=\varphi_{j}^{s}\left(\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)^{*}\left(\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right)\right)$
$\displaystyle=\left\|\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right\|_{\varphi_{j}^{s}}^{2}.$

Similarly, we obtain

$\left\|\left(xp_{l+1}^{\beta}-p_{l+1}^{\beta}x\right)\lambda_{s}\right\|_{\varphi_{j}}=\left\|xp_{l+1}^{\beta}-p_{l+1}^{\beta}x\right\|_{\varphi_{j}^{s}}.$

Therefore, it follows that

$\displaystyle\left\|\sum_{l=1}^{N}p_{l}^{\beta}x\lambda_{s}p_{l}^{\beta}\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\|x\|\left\|\alpha_{s}(p_{l}^{\beta})-p_{l+1}^{\beta}\right\|_{\varphi_{j}^{s}}+\sum_{l=1}^{N}\left\|xp_{l+1}^{\beta}-p_{l+1}x\right\|_{\varphi_{j}^{s}}$
$\displaystyle<\frac{\delta}{2}+\frac{\delta}{2}=\delta.$

Likewise, for $\beta>\beta_{0}$, for any $x\in F$ and for $j=1,2,\ldots,M$, we
have

$\displaystyle\left\|\sum_{l=1}^{N}p_{l}^{\beta}xp_{l}^{\beta}-x\right\|_{\varphi_{j}}$
$\displaystyle\leq\left\|\sum_{l=1}^{N}p_{l}^{\beta}xp_{l}^{\beta}-\sum_{l=1}^{N}(p_{l}^{\beta})^{2}x\right\|_{\varphi_{j}}+\left\|\sum_{l=1}^{N}\left((p_{l}^{\beta})^{2}-p_{l}^{\beta})\right)x\right\|_{\varphi_{j}}$
$\displaystyle+\left\|\left(\sum_{l=1}^{N}p_{l}^{\beta}-1\right)x\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\left\|p_{l}^{\beta}\right\|\left\|xp_{l}^{\beta}-p_{l}^{\beta}x\right\|_{\varphi_{j}}+\sum_{l=1}^{N}\left\|\left((p_{l}^{\beta})^{2}-p_{l}^{\beta})\right)x\right\|_{\varphi_{j}}$
$\displaystyle+\left\|\left(\sum_{l=1}^{N}p_{l}^{\beta}-1\right)x\right\|_{\varphi_{j}}$
$\displaystyle\leq\sum_{l=1}^{N}\|xp_{l}^{\beta}-p_{l}^{\beta}x\|_{\varphi_{j}}+\sum_{l=1}^{N}\left\|\left((p_{l}^{\beta})^{2}-p_{l}^{\beta}\right)\right\|_{\varphi_{j}^{x}}$
$\displaystyle+\left\|\left(\sum_{l=1}^{N}p_{l}-1\right)\right\|_{\varphi_{j}^{x}}$
$\displaystyle<\delta.$

Thus, we may take $\\{p_{1},p_{2},\ldots,p_{N}\\}$ to be
$\\{p_{1}^{\beta},p_{2}^{\beta},\ldots,p_{N}^{\beta}\\}$ for the above choice
of $\beta$. ∎

Given $s\in\Gamma\smallsetminus\\{s\\}$, and elements
$p_{1},p_{2}\ldots,p_{N}$ as above, we can define a completely positive
contraction $\Psi\colon B\rtimes_{\alpha,r}\Gamma\to
B\rtimes_{\alpha,r}\Gamma$ by $\Psi(x)=\sum_{l=1}^{N}p_{l}xp_{l}$. Using this,
under the assumptions of thm:relativizedocneanu and thm:relativizedconnes, we
can find a net of completely positive contractive contractions
$\Psi_{\beta}:B\rtimes_{\alpha,r}\Gamma\to B\rtimes_{\alpha,r}\Gamma$ such
that for any $x\in B$ we have

$\Psi_{\beta}(x\lambda_{s})\to 0\;*\text{-strongly},$

and

$\Psi_{\beta}(x)\to x\;*\text{-strongly}.$

######  Remark 4.2.

The following observation will be used in the proof of mainintmresult below:
for any $x\in B$ and for any $s\in\Gamma$ we have

$\displaystyle\|\alpha_{s}(x)\|_{\varphi_{s}}^{2}$
$\displaystyle=\varphi_{s}\left(\left(\lambda_{s}x\lambda_{s}^{*}\right)^{*}\left(\lambda_{s}x\lambda_{s}^{*}\right)\right)$
$\displaystyle=\varphi_{s}\left(\lambda_{s}x^{*}\lambda_{s}^{*}\lambda_{s}x\lambda_{s}^{*}\right)$
$\displaystyle=\varphi_{s}\left(\lambda_{s}x^{*}x\lambda_{s}^{*}\right)$
$\displaystyle=\varphi\left(\lambda_{s}^{*}\lambda_{s}x^{*}x\lambda_{s}^{*}\lambda_{s}\right)$
$\displaystyle=\varphi(x^{*}x)$ $\displaystyle=\|x\|^{2}_{\varphi}$

###### Proof of mainintmresult.

Let $C$ be an intermediate $C^{*}$-algebra of the form

$A\rtimes_{\alpha,r}\Gamma\subseteq C\subseteq B\rtimes_{\alpha,r}\Gamma.$

We want to show that $\mathbb{E}(C)\subset C$.

We first prove part (1) of the theorem, that is, we assume for every central
$\alpha^{**}$-invariant projection $P\in B^{**}$ such that $PB^{**}P$ is
separable, we have

$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$

and $\alpha^{**}:\Gamma\to\mathrm{Aut}(PA^{**}P)$ is centrally free.

We do so in two steps. First, for an arbitrary element $x\in C$, we claim that
there exists bounded net $\\{x^{\gamma}\\}_{\gamma}\in C$ such that
$x^{\gamma}\to\mathbb{E}(x)$ $\sigma$-strongly in
$B^{**}\overline{\rtimes}_{\alpha^{**}}\Gamma$. To that end, fix $x\in C$ and
$\delta>0$. Pick $b_{1},b_{2},\ldots,b_{m}\in B$ and
$s_{1},s_{2},\ldots,s_{m}\in\Gamma\setminus\\{e\\}$ such that

(35)
$\left\|x-\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}-\mathbb{E}(x)\right\|<\frac{\delta}{3}.$

For $i=1,2,\ldots,m$, let $\Gamma_{s_{i}}$ be the subgroup of $\Gamma$
generated by $s_{i}$. Let $N_{i}$ be the order of $s_{i}$ if it is finite or
some arbitrary integer greater than $1$ if it is infinite. Now let $\varphi$
be a normal state on $B^{**}\overline{\rtimes}_{\alpha^{**}}\Gamma$. We show
for any $\varepsilon>0$ and for $j\in 1,2,\ldots,m$ there exist contractions
$q_{1},q_{2},\ldots,q_{Q}\in A$ such that:

(36)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{j}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}\right\|_{\varphi}$
$\displaystyle\leq\sum_{i=1}^{j}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}\right\|_{\varphi}<\varepsilon$

and

(37)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle<\varepsilon\,.$

We prove this by induction on $j$. For $j=1$, this is correctiveepsilon. Let
us now assume that the claim holds for $j-1$. Choose contractions
$q_{1},q_{2},\ldots,q_{Q}\in A$ such that equations (36) and (37) hold for
$j-1$ in place of $j$, with the following parameters:

(38)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{j-1}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}\right\|_{\varphi}$
$\displaystyle\leq\sum_{i=1}^{j-1}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}\right\|_{\varphi}<\frac{\varepsilon}{12N_{j}}$
(39)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle<\frac{\varepsilon}{6}.$

For $i=1,2,\ldots,j$, define
$x_{i}:=\sum_{l=1}^{Q}q_{l}b_{i}\alpha_{s_{i}}(q_{l}^{*})$. Define normal
states $\varphi_{i}$ on $B^{**}\overline{\rtimes}_{\alpha^{**}}\Gamma$ by
$\varphi_{i}(y):=\varphi(\lambda_{s_{i}}^{*}y\lambda_{s_{i}})$. Apply
correctiveepsilon, for $s_{j}$, $x_{j}$ and $\\{\alpha_{s_{i}}^{-1}(x_{i})\mid
i=1,2,\ldots,j-1\\}$, to choose mutually orthogonal positive contractions
$\\{p_{1},p_{2},\ldots,p_{N_{j}}\\}$ such that:

(40)
$\left\|\sum_{k=1}^{N_{j}}p_{k}x_{j}\lambda_{s_{j}}p_{k}\right\|_{\varphi}<\frac{\varepsilon}{6},$
(41)
$\left\|\sum_{k=1}^{N_{j}}p_{k}\left(\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}\right)p_{k}-\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}\right\|_{\varphi}<\frac{\varepsilon}{6},$

and for $i=1,2,\ldots,j-1$ and for $k=1,2,\ldots,N_{j}$ we have

(42)
$\left\|\alpha_{s_{i}}^{-1}(x_{i})p_{k}-p_{k}\alpha_{s_{i}}^{-1}(x_{i})\right\|_{\varphi}<\frac{\varepsilon}{12N_{j}(j-1)}.$

By Remark 4.2, equation 42 implies that for $i=1,2,\ldots,j-1$ and for
$k=1,2,\ldots,N_{j}$ we have

(43)
$\left\|x_{i}\alpha_{s_{i}}(p_{k})-\alpha_{s_{i}}(p_{k})x_{i}\right\|_{\varphi_{i}}<\frac{\varepsilon}{12N_{j}(j-1)}.$

For $L:=l+(k-1)Q$ with $l\in\\{1,2,\ldots,Q\\}$ and
$k\in\\{1,2,\ldots,N_{j}\\}$, we set $r_{L}:=p_{k}q_{l}\in A$. Observe that

$\displaystyle\left\|\sum_{L=1}^{N_{j}Q}r_{L}\left(\sum_{i=1}^{j-1}b_{i}\lambda_{s_{i}}\right){r_{L}}^{*}\right\|_{\varphi}=$
$\displaystyle\left\|\sum_{k=1}^{N_{j}}\sum_{l=1}^{Q}p_{k}q_{l}\left(\sum_{i=1}^{j-1}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}p_{k}\right\|_{\varphi}$
$\displaystyle\leq\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}p_{k}\right\|_{\varphi}$
$\displaystyle=\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\alpha_{s_{i}}(q_{l}^{*})\alpha_{s_{i}}(p_{k})\right\|_{\varphi_{i}}$
$\displaystyle=\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|x_{i}\alpha_{s_{i}}(p_{k})\right\|_{\varphi_{i}}$
$\displaystyle\leq\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|x_{i}\alpha_{s_{i}}(p_{k})-\alpha_{s_{i}}(p_{k})x_{i}\right\|_{\varphi_{i}}+\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|\alpha_{s_{i}}(p_{k})x_{i}\right\|_{\varphi_{i}}$
$\displaystyle\leq\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|x_{i}\alpha_{s_{i}}(p_{k})-\alpha_{s_{i}}(p_{k})x_{i}\right\|_{\varphi_{i}}+\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\|\alpha_{s_{i}}(p_{k})\|\|x_{i}\|_{\varphi_{i}}$
$\displaystyle\leq\sum_{i=1}^{j-1}\sum_{k=1}^{N_{j}}\left\|x_{i}\alpha_{s_{i}}(p_{k})-\alpha_{s_{i}}(p_{k})x_{i}\right\|_{\varphi_{i}}+N_{j}\sum_{i=1}^{j-1}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}\right\|_{\varphi}$
$\displaystyle\stackrel{{\scriptstyle(\ref{equ:firstmainthm2ind}),(\ref{equ:firstmainthmind})}}{{<}}\frac{\varepsilon}{12}+\frac{\varepsilon}{12}=\frac{\varepsilon}{6}.$

At the same time, let us also observe that

$\displaystyle\left\|\sum_{L=1}^{N_{j}Q}r_{L}b_{j}\lambda_{s_{j}}{r_{L}}^{*}\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{k=1}^{N_{j}}\sum_{l=1}^{Q}p_{k}q_{l}b_{j}\lambda_{s_{j}}q_{l}^{*}p_{k}\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{k=1}^{N_{j}}p_{k}\left(\sum_{l=1}^{Q}q_{l}b_{j}\alpha_{s_{j}}q_{l}^{*}\right)\lambda_{s_{j}}p_{k}\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{k=1}^{N_{j}}p_{k}x_{j}\lambda_{s_{j}}p_{k}\right\|_{\varphi}\stackrel{{\scriptstyle(\ref{equ:secmainthmind})}}{{<}}\frac{\varepsilon}{6}.$

Hence, using the triangle inequality, we obtain that

(44)
$\displaystyle\left\|\sum_{L=1}^{N_{j}Q}r_{L}\left(\sum_{i=1}^{j}b_{i}\lambda_{s_{i}}\right){r_{L}}^{*}\right\|_{\varphi}=\left\|\sum_{k=1}^{N_{j}}\sum_{l=1}^{Q}p_{k}q_{l}\left(\sum_{i=1}^{j}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}p_{k}\right\|_{\varphi}<\frac{\varepsilon}{3}.$

Furthermore,

$\displaystyle\left\|\sum_{L=1}^{N_{j}Q}r_{L}\mathbb{E}(x){r_{L}}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{k=1}^{N_{j}}\sum_{l=1}^{Q}p_{k}q_{l}\mathbb{E}(x)q_{l}^{*}p_{k}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle\leq\left\|\sum_{k=1}^{N_{j}}p_{k}\left(\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}\right)p_{k}-\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}\right\|_{\varphi}$
$\displaystyle+\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
(45)
$\displaystyle\stackrel{{\scriptstyle(\ref{equ:secmainthm2ind}),(\ref{equ:firstmainthmparttwoind})}}{{<}}\frac{\varepsilon}{6}+\frac{\varepsilon}{6}=\frac{\varepsilon}{3}.$

This concludes proof by induction. For $\delta>0$, a state $\varphi$ as above
and for $j=m$, choose $q_{1},q_{2},\ldots,q_{Q}$ as above, and set
$x_{\delta,\varphi}:=\sum_{l=1}^{Q}q_{l}xq_{l}^{*}$. Then

$\displaystyle\left\|x_{\delta,\varphi}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{l=1}^{Q}q_{l}xq_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle=\left\|\sum_{l=1}^{Q}q_{l}\left(x-\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}-\mathbb{E}(x)\right)q_{l}^{*}+\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}+\sum_{l=1}^{Q}q_{l}\left(\mathbb{E}(x)\right)q_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle\leq\left\|x-\left(\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}\right)-\mathbb{E}(x)\right\|+\left\|\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}\right\|_{\varphi}+\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|_{\varphi}$
$\displaystyle<\delta.$

Since $\delta>0$ is arbitrary and $x_{\delta,\varphi}\in C$, we see that we
can find a net of elements $\\{x^{\gamma}\\}_{\gamma}\subset C$ such that
$x^{\gamma}\to\mathbb{E}(x)$ $\sigma$-strongly and, and in particular,
$\sigma$-weakly.

Let $\mathcal{K}=\text{conv}(\\{x^{\gamma}\\}_{\gamma})$. Using the Hahn-
Banach separation theorem, we conclude that $\mathbb{E}(c)$ lies in the norm
closure of $\mathcal{K}$. Therefore, $\mathbb{E}(C)\subset C$. It now follows
from [Suz17, Proposition 3.4] that $C=\mathbb{E}(C)\rtimes_{\alpha,r}\Gamma$.

We now prove part (2) of the theorem; that is, we assume that the action
$\alpha$ has a pointwise finite relative Rokhlin dimension. The idea of the
proof is a somewhat simpler version of the proof of the first case, as our
approximations are done in the norm.

As before, fix $x\in C$ and $\delta>0$. Pick $b_{1},b_{2},\ldots,b_{m}\in B$
and $s_{1},s_{2},\ldots,s_{m}\in\Gamma\setminus\\{e\\}$ such that

(46)
$\left\|x-\sum_{i=1}^{m}b_{i}\lambda_{s_{i}}-\mathbb{E}(x)\right\|<\frac{\delta}{3}.$

We now show for any $\varepsilon>0$ and for $j\in 1,2,\ldots,m$ there exist
contractions $q_{1},q_{2},\ldots,q_{Q}\in A$ such that:

(47)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{j}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}\right\|$
$\displaystyle\leq\sum_{i=1}^{j}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}\right\|<\varepsilon$

and

(48)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|$
$\displaystyle<\varepsilon\,.$

This is sufficient to prove the claim.

We prove this by induction on $j$. For $j=1$, we let $N$ be the order of
$s_{1}$ if it is finite or some arbitrary integer greater than $2$ if $s_{1}$
has infinite order. Let $d_{1}$ be the relative Rokhlin dimension of the
restriction of $\alpha$ to the group generated by $s_{1}$. Set
$F=\\{\mathbb{E}(x),\mathbb{E}(x)^{*},b_{1},b_{1}^{*}\\}$. Set
$\eta=\varepsilon/3(d_{1}+1)N$. Pick positive contractions $\\{f_{k}^{(l)}\mid
k=1,2,\ldots,N\,,\,l=0,1,\ldots,d_{1}\\}$ such that for all $x\in F$

  1. (1)

$\|(\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}f_{k}^{(l)})x-x\|<\eta$.

  2. (2)

For all $j\neq k$ and for all $l$ we have
$\|(f_{k}^{(l)}f_{j}^{(l)})x\|<\eta$.

  3. (3)

For all $l$ and for all $k$ we have
$\|(\alpha_{s_{1}}(f_{k}^{(l)})-f_{k+1}^{(l)})x\|<\eta$ with the convention
that $f_{N+1}^{(l)}=f_{1}^{(l)}$.

  4. (4)

For any indices $k,l$ we have $\|[x,f_{k}^{(l)}]\|<\eta$.

We may furthermore assume, by making $\eta$ smaller if needed and by using
polynomial approximations, that for all $x\in F$, we have

  1. (1)

For all $j\neq k$ and for all $l$ we have
$\left\|\left(\sqrt{f_{k}^{(l)}}\sqrt{f_{j}^{(l)}}\right)x\right\|<\eta$.

  2. (2)

For all $l$ and for all $k$ we have
$\left\|\left(\alpha_{s_{1}}\left(\sqrt{f_{k}^{(l)}}\right)-\sqrt{f_{k+1}^{(l)}}\right)x\right\|<\eta$.

  3. (3)

For any indices $k,l$ we have
$\left\|\left[x,\sqrt{f_{k}^{(l)}}\right]\right\|<\eta$.

So,

$\displaystyle\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}\sqrt{f_{k}^{(l)}}b_{1}\lambda_{s_{1}}\sqrt{f_{k}^{(l)}}\right\|$
$\displaystyle=\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}\sqrt{f_{k}^{(l)}}b_{1}\alpha_{s_{1}}(\sqrt{f_{k}^{(l)}})\lambda_{s_{1}}\right\|$
$\displaystyle<\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}\sqrt{f_{k}^{(l)}}b_{1}\sqrt{f_{k+1}^{(l)}}\lambda_{s_{1}}\right\|+(d_{1}+1)N\eta$
$\displaystyle<\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}\sqrt{f_{k}^{(l)}}\sqrt{f_{k+1}^{(l)}}b_{1}\lambda_{s_{1}}\right\|+2(d_{1}+1)N\eta$
$\displaystyle<3(d_{1}+1)N\eta=\varepsilon.$

and

$\displaystyle\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}\sqrt{f_{k}^{(l)}}\mathbb{E}(x)\sqrt{f_{k}^{(l)}}-\mathbb{E}(x)\right\|$
$\displaystyle<\left\|\sum_{l=0}^{d_{1}}\sum_{k=1}^{N}f_{k}^{(l)}\mathbb{E}(x)-\mathbb{E}(x)\right\|+(d_{1}+1)N\eta$
$\displaystyle<2(d_{1}+1)N\eta<\varepsilon.$

Thus, we found a finite collection of positive contractions as needed (which
we can relabel as $q_{1},q_{2},\ldots,q_{Q}$).

Let $N_{j}$ be the order of $s_{j}$ if it is finite, or some arbitrary integer
greater than $1$ otherwise. Using the induction hypothesis, choose
contractions $q_{1},q_{2},\ldots,q_{Q}\in A$ such that equations (47) and (48)
hold for $j-1$ in place of $j$, and with the following parameters:

(49)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\left(\sum_{i=1}^{j-1}b_{i}\lambda_{s_{i}}\right)q_{l}^{*}\right\|$
$\displaystyle\leq\sum_{i=1}^{j-1}\left\|\sum_{l=1}^{Q}q_{l}b_{i}\lambda_{s_{i}}q_{l}^{*}\right\|<\frac{\varepsilon}{12(d_{j}+1)N_{j}}$
(50)
$\displaystyle\left\|\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*}-\mathbb{E}(x)\right\|$
$\displaystyle<\frac{\varepsilon}{6}.$

For $i=1,2,\ldots,j$, define
$x_{i}:=\sum_{l=1}^{Q}q_{l}b_{i}\alpha_{s_{i}}(q_{l}^{*})$.

Where $d_{j}$ is the relative Rokhlin dimension of the restriction of $\alpha$
to the group generated by $s_{j}$.

Using the same argument as before, with $d_{j}$ being the Rokhlin dimension of
the restriction of the action to the subgroup generated by $s_{j}$, set
$\eta=\varepsilon/{36(d_{j}+1)(j-1)N_{j}}$, and pick positive contractions
$\\{f_{k}^{(l)}\mid k=1,2,\ldots,N_{j}\,,\,l=0,1,\ldots,d_{j}\\}$ such for any

$x\in\\{x_{j},x_{j}^{*},\sum_{l=1}^{Q}q_{l}\mathbb{E}(x)q_{l}^{*},\sum_{l=1}^{Q}q_{l}\mathbb{E}(x^{*})q_{l}^{*}\\}\cup\\{\alpha_{s_{i}}^{-1}(x_{i}),\alpha_{s_{i}}^{-1}(x_{i}^{*})\mid
i=1,2,\ldots,j\\}$

we have

  1. (1)

$\left\|(\sum_{l=0}^{d_{j}}\sum_{k=1}^{N_{j}}f_{k}^{(l)})x-x\right\|<\eta$.

  2. (2)

For all $j\neq k$ and for all $l$ we have
$\left\|(\sqrt{f_{k}^{(l)}}\sqrt{f_{j}^{(l)}})x\right\|<\eta$.

  3. (3)

For all $l$ and for all $k$ we have
$\left\|\left(\alpha_{s_{j}}\left(\sqrt{f_{k}^{(l)}}\right)-\sqrt{f_{k+1}^{(l)}}\right)x\right\|<\eta$.

  4. (4)

For any indices $k,l$ we have
$\left\|\left[x,\sqrt{f_{k}^{(l)}}\right]\right\|<\eta$.

We thus have

$\left\|\sum_{l=0}^{d_{j}}\sum_{k=1}^{N_{j}}\sqrt{f_{k}^{(l)}}x_{j}\lambda_{s_{j}}\sqrt{f_{k}^{(l)}}\right\|<3(d_{j}+1)N_{j}\eta$

and

$\left\|\sum_{l=0}^{d_{j}}\sum_{k=1}^{N_{j}}\sqrt{f_{k}^{(l)}}\sum_{r=1}^{Q}q_{r}\mathbb{E}(x)q_{r}^{*}\sqrt{f_{k}^{(l)}}-\sum_{r=1}^{Q}q_{r}\mathbb{E}(x)q_{r}^{*}\right\|<2(d_{j}+1)N_{j}\eta\,.$

Arguing as in the first case, we have

$\left\|\sum_{l=0}^{d_{j}}\sum_{k=1}^{N_{j}}\sum_{r=1}^{Q}\sqrt{f_{k}^{(l)}}q_{r}\left(\sum_{i=1}^{j}b_{i}\lambda_{s_{i}}\right)q_{r}^{*}\sqrt{f_{k}^{(l)}}\right\|<\frac{\varepsilon}{3}.$

and

$\left\|\sum_{l=0}^{d_{j}}\sum_{k=1}^{N_{j}}\sum_{r=1}^{Q}\sqrt{f_{k}^{(l)}}q_{r}\mathbb{E}(x)q_{r}^{*}\sqrt{f_{k}^{(l)}}-\mathbb{E}(x)\right\|<\frac{\varepsilon}{3}\,.$

The rest of the proof is similar to that of the first case. We give a short
explanation for readers’ convenience. For $\delta>0$ arbitrary and $j=m$,
choose contractions $q_{1},q_{2},\ldots,q_{Q}\in A$ as above. Set
$x_{\delta}:=\sum_{l=1}^{Q}{q_{l}}xq_{l}^{*}$. Then we obtain

$\left\|x_{\delta}-\mathbb{E}(x)\right\|<\delta$

This shows there exists a net of elements $\\{x_{\delta}\\}_{\delta}\subset C$
such that $x_{\delta}\rightarrow\mathbb{E}(x)$ in norm. Thus we conclude that
$\mathbb{E}(C)\subset C$ and hence $C=\mathbb{E}(C)\rtimes_{\alpha,r}\Gamma$
(by [Suz17, Proposition 3.4]). ∎

##  5\. Examples

We conclude the paper by giving a few natural classes of examples that satisfy
the hypotheses of Theorem LABEL:mainintmresult.

It is well known (see [gk, Theorem 6.2.11]) that if $A$ is a type I
$C^{*}$-algebra, then $A$ contains an essential ideal that has a continuous
trace. Moreover, $A$ has a composition series
$\\{I_{\theta}\,|\,0\leq\theta\leq\beta\\}$, such that
$I_{\theta+1}/I_{\theta}$ has Hausdorff spectrum for each $\theta<\beta$. It
can be further shown that if $\alpha\colon\Gamma\to\text{Aut}(A)$ is an action
on a type I $C^{*}$-algebra $A$, such that the induced action of $\Gamma$ on
$\hat{A}$ is free and a composition series as mentioned above is assumed to be
$\Gamma$-invariant, then the induced action of $\Gamma$ on $PA^{**}P$ is
centrally free. Here, $P$ is a central $\alpha^{**}$-invariant projection in
$B^{**}$ so that $PB^{**}P$ is separable.

We begin by studying free actions on a $C^{*}$-algebra with the Hausdorff
spectrum and show that the induced action on the cut down of its double dual
by a central invariant projection is centrally free.

######  Proposition 5.1.

prop_topologically_free Let $\Gamma$ be a countable discrete group. Let $A$ be
a separable $C^{*}$-algebra such that $\hat{A}$ is Hausdorff, and let
$\alpha:\Gamma\to\mathrm{Aut}(A)$ be an action that the induced action of
$\Gamma$ on $\hat{A}$ is free. Then for every central $\alpha^{**}$-invariant
projection $P\in A^{**}$ so that $PA^{**}P$ is separable, $\alpha^{**}$ acts
centrally freely on $PA^{**}P$.

###### Proof.

Let us suppose there is a central $\alpha^{**}$-invariant projection $P\in
A^{**}$ and $g\neq 0$ so that $PA^{**}P$ is separable and
$(\alpha^{**}_{g})_{\omega}$ acts trivially on $(PA^{**}P)_{\omega}$. Then,
$PC_{0}(\widehat{A})1_{A^{**}}P\subseteq
PZ(A^{**})P\subseteq(PA^{**}P)_{\omega}$. Because $C_{0}(\hat{A})1_{A^{**}}$
is weakly dense in $Z(A^{**})$, there exists a non-empty $\Gamma$-invariant
subset of $\hat{A}$ such that $\alpha_{g}$ acts trivially identity on it, and
in particular, the induced action of $\alpha$ on $\hat{A}$ is not free. ∎

######  Corollary 5.2.

Let $\alpha:\Gamma\to\mathrm{Aut}(B)$ be an action of countable discrete group
on a separable $C^{*}$-algebra $B$. Let $A\subseteq B$ be an inclusion of
$\Gamma$-invariant separable $C^{*}$-algebra $A$ which is type I
$C^{*}$-algebra. Furthermore, assume $A$ contains an approximate identity of
$B$ and has a $\Gamma$-invariant composition series
$\\{I_{\theta}\,|\,0\leq\theta\leq\beta\\}$ such that
$I_{\theta+1}/I_{\theta}$, for $\theta<\beta$ has Hausdorff spectrum and the
action of $\Gamma$ on $\hat{A}$ is free. Then for every central
$\alpha^{**}$-invariant projection $P\in B^{**}$ with $PB^{**}P$ is separable,
$\alpha^{**}$ acts centrally freely on $PA^{**}P$.

###### Proof.

Since $A$ contains an approximate identity of $B$, it suffices to show that
for every central $\alpha^{**}$-invariant projection $P\in A^{**}$ such that
$PA^{**}P$ is separable, the action $\alpha^{**}$ is centrally free on
$PA^{**}P$. The now proof follows from Proposition
LABEL:prop_topologically_free and the fact that
$A^{**}=J^{**}\oplus(A/J)^{**}$ for any ideal $J$ of $A$. ∎

######  Remark 5.3.

Note, when $\Gamma$ is finite and $A$ is a type I $C^{*}$-algebra, then one
can obtain a $\Gamma$-invariant composition series
$\\{I_{\theta}\,|\,0\leq\theta\leq\beta\\}$, such that
$I_{\theta+1}/I_{\theta}$, for $\theta<\beta$ has Hausdorff spectrum by a
small observation as follows. By [gk, Theorem 6.2.11], $A$ contains an
essential ideal $I$ such that $I$ has continuous trace and hence has Hausdorff
spectrum. Define $J:=\cap_{g\in\Gamma}\alpha_{g}(I)$. Then $J$ is a nonzero
closed two-sided ideal in $A$ having a Hausdorff spectrum. This gives the
start of our composition series. We continue the similar procedure for $A/J$
and proceed by transfinite induction. We do not know whether one can always
find $\Gamma$-invariant composition series of this form if $\Gamma$ is
infinite.

The following is a sufficient condition for pointwise finite Rokhlin
dimension.

######  Lemma 5.4.

Let $A$ be a separable unital nuclear simple stably finite $Z$-stable
$C^{*}$-algebra such that the trace space $T(A)$ is a Bauer simplex with
finite-dimensional extreme boundary. Let $\Gamma$ be a discrete group and let
$\alpha\colon\Gamma\to\mathrm{Aut}(A)$ be an action such that for any
$s\in\Gamma\smallsetminus\\{e\\}$ and for any $\alpha_{s}$-invariant trace
$\tau$, the extension of $\alpha_{s}$ to the GNS closure
$\overline{A}^{\|\cdot\|_{\tau}}$ is outer. Then, the action $\alpha$ has a
pointwise finite Rokhlin dimension.

###### Proof.

This follows immediately from the combination of [GHV, Theorems A, B] and
[wouters, Theorem C]. ∎

We end this section by providing examples for which the hypotheses
$F(A)\subseteq F(B)$ or
$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$,
for any central projection in $P$ in $B^{**}$ such that $PB^{**}P$ is
separable, are automatic.

######  Lemma 5.5.

If $A\subseteq B$ is an inclusion of separable $C^{*}$-algebras such that $A$
contains an approximate identity of $B$, then the following conditions imply
that $F(A)\subseteq F(B)$. For any central projection $P$ in $B^{**}$ such
that $PB^{**}P$ is separable, we furthermore have
$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}=(PBP)^{\prime}\cap(PA^{**}P)^{\omega}$.

  1. (1)

$A$ is a subalgebra of the center of $B$ (so $A\cong C(X)$, where $X$ is a
compact Hausdorff space and $B$ is a unital $C(X)$-algebra).

  2. (2)

$B$ is any $C^{*}$-algebra completion of the algebraic tensor product of
$C\otimes_{\text{alg}}D$, where $D$ is unital, and $A=C\otimes 1_{D}$.

  3. (3)

Suppose $A$ is a separable $C^{*}$-algebra, let $H$ is a countable discrete
group, let $\beta\colon H\to A$ be a pointwise inner action, and assume
$B=A\rtimes_{\beta,r}H$.

(Notice that the third condition does not imply that the action is inner, so
$B$ need not be isomorphic to $A\otimes_{\min}C^{*}_{r}(H)$; for instance,
there is an example of an action of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
on $M_{2}$ such that the crossed product is isomorphic to $M_{4}$.)

As the part about the inclusion $F(A)\subseteq F(B)$ is straightforward, we
only provide details for the second part of the statement.

Proof of Part (1): This is immediate.

Proof of Part (2): Let $\varphi$ be the faithful normal state on $PB^{**}P$.
By Proposition 2.7, in order to show
$(PAP)^{\prime}\cap(PA^{**}P)^{\omega}\subseteq(P(A\otimes
D)P)^{\prime}\cap(PA^{**}P)^{\omega}$, it suffices to show
$(PAP)^{\prime}\cap(PAP,\varphi|_{PAP})_{\omega}\subseteq(P(A\otimes
D)P)^{\prime}\cap(PAP,\varphi|_{PAP})_{\omega}$. To this end, let
$(c_{n})_{n}\in\ell^{\infty}({\mathbb{N}},PAP)$ such that

$\lim_{n\to\omega}\|c_{n}a-ac_{n}\|^{\\#}_{\varphi|_{PAP}}=0,\,\,\,\forall
a\in A.$

Pick $(z_{1},z_{2},\ldots)\in\ell^{\infty}({\mathbb{N}},A)$ such that
$c_{n}=Pz_{n}P$ for $z_{n}\in A$. Let $y\in P(A\otimes D)P$, we show

$\lim_{n\to\omega}\|c_{n}y-yc_{n}\|^{\\#}_{\varphi|{P(A\otimes D)P}}=0.$

For this, set $M:=\text{sup}\\{\|z_{n}\|,n\in{\mathbb{N}}\\}$, which we may
assume is nonzero; otherwise there is nothing to prove. Take $\varepsilon>0$
and let $\sum_{j=1}^{k}(a_{j}\otimes d_{j})\in A\otimes D$ such that

$\left\|y-P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P\right\|<\frac{\varepsilon}{4M}.$

Now take $L=\text{max}\\{\|d_{j}\|,\,j=1,2,\ldots,k\\}$ and
$S=\text{max}\\{\|c_{n}a_{j}-a_{j}c_{n}\|^{1/2},\,n\in{\mathbb{N}},\,k=1,2,\ldots,k\\}$.
Now choose $W\in\omega$ such that for all $n\in W$ and for $j=1,2,\ldots,k$ we
have

$\|c_{n}a_{j}-a_{j}c_{n}\|^{1/2}_{\varphi|_{PAP}}<\frac{\varepsilon}{2kSL}.$

Note that for $x\otimes y\in A\otimes_{\text{min}}D$ we have

$\begin{split}\|P(x\otimes y)P\|_{\varphi|_{P(A\otimes
D)P}}^{2}&=|\varphi(P(x^{*}x\otimes y^{*}y)P)|\\\ &=|\varphi(P(1\otimes
y^{*}y)PP(x^{*}x\otimes 1)P)|\\\ &\leq\varphi(P(x^{*}xx^{*}x\otimes
1)P)^{1/2}\varphi(P(1\otimes y^{*}yy^{*}y)P)^{1/2}\\\ &\leq\|P(x\otimes
1)P\|\|y\|^{2}\|P(x\otimes 1)P\|_{\varphi|_{PAP}}.\end{split}$

For any $n\in W$ we have

$\begin{split}\|[c_{n},y]\|_{\varphi|_{P(A\otimes
D)P}}&\leq\|[c_{n},y-P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P]\|+\|[c_{n},P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P]\|_{\varphi|_{P(A\otimes D)P}}\\\ &\leq
2\|c_{n}\|\|y-P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P\|+\|[c_{n},P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P]\|_{\varphi|_{P(A\otimes D)P}}\\\
&\leq\frac{\varepsilon}{2}+\|[c_{n},P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P]\|_{\varphi|_{P(A\otimes D)P}},\end{split}$

and

$\begin{split}\|[c_{n},P\sum_{j=1}^{k}(a_{j}\otimes
d_{j})P]\|_{\varphi|_{P(A\otimes
D)P}}&\leq\|\sum_{j=1}^{k}P((z_{n}a_{j}-a_{j}z_{n})\otimes
d_{j})P\|_{\varphi|_{P(A\otimes D)P}}\\\
&\leq\sum_{j=1}^{k}\|P((z_{n}a_{j}-a_{j}z_{n})\otimes
d_{j})P\|_{\varphi|_{P(A\otimes D)P}}\\\
&\leq\sum_{j=1}^{k}\|c_{n}a_{j}-a_{j}c_{n}\|^{1/2}\|d_{j}\|\|c_{n}a_{j}-a_{j}c_{n}\|_{\varphi|_{PAP}}^{1/2}\\\
&\leq\frac{\varepsilon}{2}.\end{split}$

Thus, for all $n\in W$ we have $\|[c_{n},y]\|_{\varphi|_{P(A\otimes
D)P}}<\varepsilon$. Similarly, one gets
$\|[c_{n},y]^{*}\|_{\varphi|_{P(A\otimes
D)P}}\underset{n\to\omega}{\longrightarrow}0$, hence
$\|[c_{n},y]\|_{\varphi|_{P(A\otimes
D)P}}^{\\#}\underset{n\to\omega}{\longrightarrow}0$.

Proof of Part (3): Since $\beta$ acts by unitary conjugation, for each $s\in
H$, there exists a unitary element $u_{s}\in A$ such that
$\beta_{s}(a)=u_{s}au_{s}^{*}$ for $a\in A$. Following the same argument as in
the previous section, it suffices to show that for any faithful normal state
$\varphi$ on $P(A\rtimes_{\beta,r}H)^{**}P$ with $P$ a central projection, we
have

$(PAP)^{\prime}\cap(PAP,\varphi|_{PAP})_{\omega}\subset(P(A\rtimes_{\beta,r}H)P)^{\prime}\cap(PAP,\varphi|_{PAP})_{\omega}.$

Let $(c_{n})_{n}$ be a sequence in $\ell^{\infty}({\mathbb{N}},PAP)$ such that
$\|c_{n}a-ac_{n}\|_{\varphi|_{PAP}}^{\\#}\underset{n\to\omega}{\longrightarrow}0$
for all $a\in A$. Pick $(z_{1},z_{2},\ldots)\in\ell^{\infty}({\mathbb{N}},A)$
such that $c_{n}=Pz_{n}P$ for $z_{n}\in A$. Let $y\in
P(A\rtimes_{\beta,r}H)P$. We show that
$\|c_{n}y-yc_{n}\|_{\varphi|P(A\rtimes_{\beta,r}H)P}^{\\#}\underset{n\to\omega}{\longrightarrow}0$.
Since $(c_{n})$ is bounded, it suffices to show the above condition for a
dense set of $P(A\rtimes_{\beta,r}H)P$. To this end, fix
$e_{1},e_{2},\ldots,e_{k}\in A$, $s_{1},s_{2},\ldots,s_{k}\in H$ and
$\varepsilon>0$.

For $l=1,2,\ldots,k$, define states $\varphi_{l}$ on
$P(A\rtimes_{\beta,r}H)^{**}P$ by
$\varphi_{l}(z)=\varphi(P\lambda(s_{l})^{*}z\lambda(s_{l})P)$. Because
$\|\cdot\|_{\varphi}^{\\#}$ generates the *-strong topology on the unit ball
of $P(A\rtimes_{\beta,r}H)^{**}P$, there exists $W\in\omega$, such that for
all $n\in W$ and for all $l=1,2,\ldots k$ we have

$\|c_{n}e_{l}-e_{l}c_{n}\|_{\varphi_{l}|PAP}<\frac{\varepsilon}{2k}\,\,\text{and}\,\,\|u_{s_{l}}^{*}c_{n}-c_{n}u_{s_{l}}^{*}\|_{\varphi_{l}|PAP}<\frac{\varepsilon}{2\text{sup}\\{\|e_{l}\|\\}k}$

Then for all $n\geq N$, we have

$\begin{split}\|[c_{n},P\sum_{l=1}^{k}e_{l}\lambda(s_{l})P]\|_{\varphi|P(A\rtimes_{\beta,r}H)P}&=\|\sum_{l=1}^{k}P(z_{n}e_{l}\lambda(s_{l})-e_{l}\lambda(s_{l})z_{n})P\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\\\
&=\|\sum_{l=1}^{k}P(z_{n}e_{l}\lambda(s_{l})-e_{l}\beta_{s_{l}}(z_{n})\lambda(s_{l}))P\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\\\
&=\|\sum_{l=1}^{k}P(z_{n}e_{l}\lambda(s_{l})-e_{l}u_{s_{l}}z_{n}u_{s_{l}}^{*}\lambda(s_{l}))P\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\\\
&\leq\sum_{l=1}^{k}\|P(z_{n}e_{l}-e_{l}z_{n})PP\lambda(s_{l})P\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\\\
&+\sum_{l=1}^{k}\|e_{l}\|\|P(u_{s_{l}}^{*}z_{n}-z_{n}u_{s_{l}}^{*})PP\lambda(s_{l})P\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\\\
&=\sum_{l=1}^{k}\|c_{n}e_{l}-e_{l}c_{n}\|_{\varphi_{l}|PAP}+\sum_{l=1}^{k}\|e_{l}\|\|u_{s_{l}}^{*}c_{n}-c_{n}u_{s_{l}}^{*}\|_{\varphi_{l}|PAP}\\\
&<\varepsilon.\end{split}$

Similar calculations show that
$\|[c_{n},P\sum_{l=1}^{k}e_{l}\lambda(s_{l})P]^{*}\|_{\varphi|P(A\rtimes_{\beta,r}H)P}\underset{n\to\omega}{\longrightarrow}0$.
∎

We conclude by stating two special cases of the conditions above to provide a
sample of self-contained statements.

######  Corollary 5.6.

Let $D$ be a separable unital $C^{*}$-algebra. Let $A$ be a separable nuclear
$C^{*}$-algebra, and let $B=A\otimes D$. Let $\Gamma$ be a countable discrete
group, and let $\alpha\colon\Gamma\to\mathrm{Aut}(B)$ be an action that leaves
$A$ invariant. In the following two cases, any intermediate subalgebra
$A\rtimes_{\alpha,r}\Gamma\subseteq C\subseteq B\rtimes_{\alpha,r}\Gamma$ is a
crossed product:

  1. (1)

$A$ is type I, $\Gamma$ is finite, and the action of $\Gamma$ on $\hat{A}$ is
free.

  2. (2)

$A$ be a separable unital nuclear simple stably finite $Z$-stable
$C^{*}$-algebra such that the trace space $T(A)$ is a Bauer simplex with
finite-dimensional extreme boundary and for any
$s\in\Gamma\smallsetminus\\{e\\}$ and for any $\alpha_{s}$-invariant trace
$\tau$, the extension of $\alpha_{s}$ to the GNS closure
$\overline{A}^{\|\cdot\|_{\tau}}$ is outer.

## References