# Minimal compact group actions on C∗-algebras with simple fixed point
algebras

Masaki Izumi
Graduate School of Science
Kyoto University
Sakyo-ku, Kyoto 606-8502, Japan Supported in part by JSPS KAKENHI Grant Number
JP20H01805

###### Abstract

The notion of qausi-product actions of a compact group on a C∗-algebra was
introduced by Bratteli et al. in their attempt to seek an equivariant analogue
of Glimm’s characterization of non-type I C∗-algebras. We show that a faithful
minimal action of a second countable compact group on a separable C∗-algebra
is quasi-product whenever its fixed point algebra is simple. This was
previously known only for compact abelian groups and for profinite groups. Our
proof relies on a subfactor technique applied to finite index inclusions of
simple C∗-algebras in the purely infinite case, and also uses ergodic actions
of compact groups in the general case. As an application, we show that if
moreover the fixed point algebra is a Kirchberg algebra, such an action is
always isometrically shift-absorbing, and hence is classifiable by the
equivariant KK-theory due to a recent result of Gabe-Szabó.

In memory of the late Professor Huzihiro Araki

##  1 Introduction

Compact group actions on C∗-algebras have been extensively studied, especially
in relation to mathematical physics in the early history of operator algebras,
as internal symmetries in models in physics are often described by compact
group actions (see, for example, [13], [14]). Araki-Haag-Kastler-Takesaki’s
work [1] on chemical potential is one of the highlights in this subject in the
70’s, which inspired a lot of subsequent work both in mathematics and physics.

In the pure mathematics side, the most notable class of compact group actions
on C∗-algebras is probably quasi-product actions introduced by Bratteli et al.
[5], [2], [3]. Roughly speaking, while the classical Glimm theorem
characterizes non-type I C∗-algebras as those having sub-quotients isomorphic
to the CAR algebra (see [40, Theorem 6.7.3] for the precise statement), quasi-
product actions are, by definition, those having invariant-quotients
equivariantly isomorphic to the UHF algebras with product type actions. [3,
Theorem 1] shows that ten conditions on a compact group action are mutually
equivalent, which can be taken as the definition (and characterization) of a
quasi-product action. The existence of an invariant pure state is one of them,
which is very useful for practical applications because it assures that every
invariant state can be approximated by invariant pure states (see [6]).

A Galois correspondence for minimal actions of compact groups on factors was
established by Izumi-Longo-Popa [25]. Its C∗-counterpart was recently obtained
by Mukohara [37] for quasi-product actions with simple fixed point algebras.
She also showed that the fixed point inclusions for such actions are
C∗-irreducible in the sense of Rørdam [47]. This also proves the importance of
quasi-product actions.

One of the purposes of this paper is to prove the following result, which
shows that the class of quasi-product actions is broad and relatively easy to
recognize:

######  Theorem 1.1.

Let $\alpha$ be a faithful action of a second countable compact group $G$ on a
separable C∗-algebra $A$. We assume that $\alpha$ is minimal in the sense that
the relative commutant ${A^{\alpha}}^{\prime}\cap M(A)$ is trivial, where
$A^{\alpha}$ is the fixed point algebra of $\alpha$ and $M(A)$ is the
multiplier algebra of $A$. Assume further that $A^{\alpha}$ is simple. Then
$\alpha$ is quasi-product. (Note that $A$ is necessarily simple thanks to [50,
Lemma 24], [3, Theorem 1,(3)].)

When $G$ is abelian, the statement of the theorem is reduced to the proper
outerness of the dual action (see [2, Theorem 1]), which in turn follows from
Kishimoto’s result [33, Lemma 1.1] for automorphisms. In the general case, the
dual action consists of endomorphisms whose proper outerness is technically
much subtler as was observed in [3] and [23]. In fact, the author challenged
the problem of generalizing Kisimoto’s result to irreducible endomorphisms of
finite index, and ended up with a result under the assumption of finite depth
condition [23, Theorem 7.5], which implies the statement of Theorem 1.1 for
profinite $G$. This work is to challenge the problem again after more than 20
years.

Classification of group actions on Kirchberg algebras is a growing subject and
a recent ground breaking result of Gabe-Szabó [17] established an equivariant
version of the Kirchberg-Phillips classification theorem [42] (see also [16],
[24], [27], [28], [36], [39], [49] for related results). They showed that
amenable isometrically shift-absorbing actions of second countable locally
compact groups on Kirchberg algebras are completely classified by equivariant
$KK$-equivalence. An action of a locally compact group $G$ on a C∗-algebra $A$
is said to be isometrically shift-absorbing if the quasi-free action on the
Cuntz algebra $\mathcal{O}_{\infty}$ arising from the infinite direct sum of
the regular representation of $G$ equivariantly embeds in the central sequence
algebra of $A$. For a countable discrete group, this condition is equivalent
to the outerness of the action (see [26]). However, for other classes of
groups, it is a non-trivial task to characterize when this property holds. In
fact, for the real numbers $G=\mathbb{R}$, it is equivalent to the Rokhlin
property (see [17, Corollary 6.15], [49]).

When $G$ is compact, every isometrically shift-absorbing action is quasi-
product by [3, Theorem 1,(8)], and Mukohara [37] showed that it is a minimal
action having the fixed point algebra purely infinite and simple. We show the
converse when the fixed point algebra is a Kirchberg algebra, as an
application of Theorem 1.1.

######  Theorem 1.2.

Let $G$ be a second countable compact group, and let $\alpha$ be a faithful
minimal action of $G$ on a separable C∗-algebra $A$ whose fixed point algebra
is a Kirchberg algebra. Then $\alpha$ is isometrically shift-absorbing. (Note
that $A$ is necessarily a Kirchbeg algebra thanks to [10, Proposition 3].)

This paper is organized as follows. In Section 2, we summarize the basics of
properly outer endomorphisms, inclusions of simple C∗-algebras of finite
indices, and quasi-product actions of compact groups. A common key notion in
the above three subjects is, what we call, the property (BEK) for an inclusion
of prime C∗-algebras introduced in Bratteli-Elliott-Kishimoto [3].

In Section 3, we show that every irreducible inclusion of purely infinite
simple C∗-algebras of finite index has the property (BEK). For the proof we
crucially use an ultraproduct technique and the fact that every irreducible
inclusion of simple C∗-algebras of finite index is C∗-irreducible in the sense
of Rørdam. As a consequence, we prove the proper outerness of every
irreducible proper endomorphism of finite index in the case of purely infinite
simple C∗-algebras. This immediately implies Theorem 1.1 in the case of purely
infinite $A^{\alpha}$. The general case is reduced to this case by a tensor
product trick in Section 4, but the reduction argument is rather complicated.
In fact, we use ergodic actions of $G$, which appear for a tensor categorical
reason (see Remark 4.6).

In Section 5 we show an equivariant version of the completely positive
approximation property (CPAP) of a nuclear C∗-algebra with a compact group
action, as preparation for the proof of Theorem 1.2. Theorem 5.4 is of
interest in its own right as a Fejér type approximation of a general compact
group. We prove Theorem 1.2 in Section 6 by showing an equivariant version of
Kirchberg’s dilation theorem for unital completely positive (ucp) maps of
Kirchberg algebras.

In Section 7, we treat quasi-free actions of compact groups on the Cuntz
algebras as applications of our main results.

##  2 Preliminaries

###  2.1 Notation

We use the following notation throughout the paper. For a C∗-algebra $A$, we
denote by $U(A)$ the unitary group of $A$. For $u\in U(A)$, we denote by
$\operatorname{Ad}u$ the automorphism of $A$ defined by
$\operatorname{Ad}u(x)=uxu^{*}$ for $x\in A$. An automorphism of $A$ is called
inner if it is of the form $\operatorname{Ad}u$, and otherwise it is called
outer. We denote by $A_{1}$ the unit ball of $A$, and by $A_{+}$ the set of
positive elements in $A$. We denote by $M(A)$ the multiplier algebra of $A$.
For two C∗-algebras $A$, $B$, we denote by $A\otimes B$ the minimal tensor
product. An action $\alpha$ of a topological group $G$ on a C∗-algebra $A$ is
a continuous homomorphism from $G$ into the automorphism group
$\operatorname{Aut}(A)$ of $A$ equipped with the point norm topology. We
denote the fixed point algebra of $\alpha$ by $A^{\alpha}$, or $A^{G}$ when
there is no possibility of confusion.

An inclusion of C∗-algebras $A\supset B$ is called irreducible if the relative
commutant $M(A)\cap B^{\prime}$ is trivial. We denote by
$\operatorname{End}(A)$ the set of endomorphisms of $A$, and call
$\rho\in\operatorname{End}(A)$ irreducible if the inclusion $A\supset\rho(A)$
is irreducible.

For a C∗-algebra $A$ and a free ultra-filter
$\omega\in\beta\mathbb{N}\setminus\mathbb{N}$, we define

$A^{\infty}=\ell^{\infty}(\mathbb{N},A)/c_{0}(\mathbb{N},A),$
$A^{\omega}=\ell^{\infty}(\mathbb{N},A)/c_{\omega}(\mathbb{N},A),$
$c_{\omega}(A)=\\{(x_{n})\in\ell^{\infty}(\mathbb{N},A);\;\lim_{n\to\omega}\|x_{n}\|=0\\}.$

We treat $A$ as a subalgebra of $A^{\infty}$ and $A^{\omega}$ as usual.

We denote by $\mathbb{M}_{n}(\mathbb{C})$ the $n$ by $n$ matrix algebra. We
denote by $\mathbb{K}(H)$ the set of compact operators on a Hilbert space $H$,
and denote $\mathbb{K}=\mathbb{K}(\ell^{2})$. We denote
$\mathbb{T}=\\{z\in\mathbb{C};\;|z|=1\\}$. We use the capital Greek letters
$\Pi$, $\Sigma$, etc. for representations of C∗-algebras, and reserve $\pi$,
$\sigma$, etc. for representations of compact groups.

###  2.2 Properly outer endomorphisms

######  Definition 2.1.

Let $A$ be a C∗-algebra. We say that $\rho\in\operatorname{End}(A)$ is
properly outer if the following holds: for any $a\in A$ and non-zero
hereditary C∗-subalgebra $C$ of $A$,

$\inf\\{\|ca\rho(c)\|;\;c\in C_{+},\;\|c\|=1\\}=0.$

Kishimoto [33, Lemma 1.1] showed that if $A$ is simple, any outer automorphism
of $A$ is properly outer. It was observed in [3, p.322] that the same argument
shows

######  Lemma 2.2.

Let $A$ be a separable simple C∗-algebra, and assume that
$\rho\in\operatorname{End}(A)$ is not properly outer. Then for any irreducible
representation $\Pi$ of $A$, the composition $\Pi\circ\rho$ contains $\Pi$ as
a subrepresentation.

The following theorem of Bratteli-Elliott-Kishimoto is one of the main
technical results in their analysis of quasi-product actions of compact
groups.

######  Theorem 2.3 ([3, Theorem 3.1]).

Let $A\supset B$ be an inclusion of separable C∗-algebras. Then the following
conditions are equivalent.

  * $(1)$

For any $x,y\in A$,

$\sup_{b\in B_{1}}\|xby\|=\|x\|\|y\|.$

  * $(2)$

There exists $\delta>0$ such that for any $x,y\in A$,

$\sup_{b\in B_{1}}\|xby\|\geq\delta\|x\|\|y\|.$

  * $(3)$

There exists an faithful irreducible representation $\Pi$ of $A$ whose
restriction to $B$ is irreducible.

######  Definition 2.4.

We call the condition of $A\supset B$ in Theorem 2.3 the property (BEK).

######  Remark 2.5.

The property (BEK) forces the inclusion $A\supset B$ to be irreducible.

The following lemma is shown in [3, p.322] when there exists $n\geq 2$ such
that $\rho^{n}$ is reducible. For the sake of completeness, we give a proof in
full generality.

######  Lemma 2.6.

Let $A$ be a separable simple C∗-algebra and let
$\rho\in\operatorname{End}(A)$. We assume that the image of $\rho$ is a proper
irreducible subalgebra of $A$ and the inclusion $A\supset\rho(A)$ has the
property (BEK). Then $\rho$ is properly outer.

###### Proof.

Assume on the contrary that $\rho$ is not properly outer. Since
$A\supset\rho(A)$ has the property (BEK), there exists an irreducible
representation $(\Pi,H)$ of $A$ whose restriction to $\rho(A)$ is irreducible,
which means that $\Pi\circ\rho$ is irreducible. Since $\rho$ is not properly
outer, the composition $\Pi\circ\rho$ contains $\Pi$, and so $\Pi\circ\rho$ is
unitarily equivalent to $\Pi$, and there exits a unitary $u\in B(H)$, unique
up to scalar multiple, satisfying $\Pi\circ\rho=\operatorname{Ad}u\circ\Pi$.
If there exists $n\geq 2$ such that $\rho^{n}$ is reducible, on one hand
$\Pi\circ\rho^{n}$ is unitarily equivalent to $\Pi$ and on the other hand it
is reducible, which is a contradiction. Thus $\rho^{n}$ is irreducible for all
$n\in\mathbb{N}$.

Proof of [33, Lemma 1.1] shows that there exist $a\in A\setminus\\{0\\}$,
$\delta>0$, and a non-zero hereditary C∗-subalgebra $C\subset A$ such that for
any $c\in C$,

$\Pi(c^{*})(\Pi(a)u+u^{*}\Pi(a^{*}))\Pi(c)\geq\delta\Pi(c^{*}c).$ (2.1)

For non-negative integers $n$, we set $A_{n}=u^{*n}\Pi(A)u^{n}$. Then
$\\{A_{n}\\}_{n=0}^{\infty}$ is a strictly increasing sequence of simple
C∗-algebras. Let $B$ be the norm closure of its union, which is simple. We
introduce $\theta\in\operatorname{Aut}(B)$ by the restriction of
$\operatorname{Ad}u$ to $B$. By construction, we have $\theta(A_{n})=A_{n-1}$
and $\theta(\Pi(x))=\Pi(\rho(x))$ for all $x\in A$.

We claim that $\theta^{n}$ is outer for all $n\in\mathbb{N}$. Assume on the
contrary that there exists $n\in\mathbb{N}$ with $\theta^{n}$ inner. Then
there exists $v\in U(B)$ satisfying $\theta^{n}=\operatorname{Ad}v$. Since
$\theta^{n}(\Pi(x))=\operatorname{Ad}u^{n}(\Pi(x))$, we see that
$v^{*}u^{n}\in\Pi(A)^{\prime}=\mathbb{C}$, and there exists $c\in\mathbb{T}$
satisfying $v=cu^{n}$. For any $\varepsilon>0$, there exist $m\in\mathbb{N}$
and $w\in U(A_{m})$ satisfying $\|v-w\|<\varepsilon$. Then
$\operatorname{Ad}u^{m}(w)\in\Pi(A)$ and

$\|cu^{n}-\operatorname{Ad}u^{m}(w)\|=\|\operatorname{Ad}u^{m}(cu^{n}-w)\|<\varepsilon.$

Since $\varepsilon>0$ is arbitrary, this means $u^{n}\in\Pi(A)$, and
$\rho^{n}$ is an inner automorphism of $A$, which contradicts
$A\supsetneqq\rho(A)$. Thus the claim is shown.

By the claim, the crossed product $B\rtimes_{\theta}\mathbb{Z}$ is simple, and
it is canonically isomorphic to the C∗-algebra $C^{*}(B\cup\\{u\\})$ generated
by $B$ and $u$. Thus there exists a conditional expectation
$E:C^{*}(B\cup\\{u\\})\to B$ satisfying $E(u)=0$. This contradicts Eq.(2.1),
and we get the statement. ∎

###  2.3 Finite index inclusions of simple C∗-algebras

Our basic reference for inclusions of simple C∗-algebras is [23].

Let $A\supset B$ be an inclusion of C∗-algebras with a conditional expectation
$E:A\to B$. Pimsner-Popa [43] defined the index of $E$, denoted by
$\operatorname{Ind}E$, by the best constant $\lambda\geq 1$ such that the map
$E-\lambda^{-1}\operatorname{id}:A\to A$ is positive (note that [23,
Definition 2.1] contains typographic errors). When there is no such a constant
$\lambda$, we set $\operatorname{Ind}E=\infty$. When $B$ has no non-zero
finite dimensional representation, we can replace positivity by complete
positivity in the definition (see [23, Lemma 2.2]).

Watatani [52] introduced an alternative definition of the index of $E$ in
terms of a quasi-basis, a generalization of the Pimsner-Popa basis. His
definition is particularly well-behaved when we discuss the C∗-basic
construction, which we will do now. The two definitions coincide when $A$ and
$B$ are infinite dimensional simple C∗-algebras [23, Theorem 3.2, Corollary
3.7], which we always assume in what follows.

For a given conditional expectation $E:A\to B$ of finite index, we can
introduce a Hilbert $B$-module $\mathcal{E}_{E}$ as follows. We let
$\mathcal{E}_{E}=A$ as a right-$B$ module, and introduce $B$-valued inner
product by $\langle{x,y}\rangle_{E}=E(x^{*}y)$. The Pimsner-Popa inequality
assures that $\mathcal{E}_{E}$ is already complete. When $x\in A$ is regarded
as an element in $\mathcal{E}_{E}$, we often denote $\eta_{E}(x)$ to avoid
possible confusion. We denote by $\mathbb{L}_{B}(\mathcal{E}_{E})$ the
C∗-algebra of adjointable $B$-module maps on $\mathcal{E}_{E}$. We regard $A$
as a C∗-subalgebra of $\mathbb{L}_{B}(\mathcal{E}_{E})$ through the left
multiplication. The Jones projection $e_{E}\in\mathbb{L}_{B}(\mathcal{E}_{E})$
is defined by $e_{E}\eta_{E}(x)=\eta_{E}(E(x))$ for $x\in A$, which belongs to
the commutant of $B$. Then we have the relation $e_{E}xe_{E}=E(x)e_{E}$ for
all $x\in A$. The C∗-basic construction of $A\supset B$ is the norm closure of
the linear span of $Ae_{E}A$, which coincides with the set of “compact
operators” $\mathbb{K}_{B}(\mathcal{E}_{E})$. Under the assumption of
$\operatorname{Ind}E<\infty$ and the simplicity of $B$, we always have
$\mathbb{K}_{B}(\mathcal{E}_{E})\supset A$ and there exists a conditional
expectation $E_{1}:\mathbb{K}_{B}(\mathcal{E}_{E})\to A$ given by
$E_{1}(e_{E})=1/\operatorname{Ind}E$, which is called the dual conditional
expectation of $E$ (see[23, Corollary 3.4]). We have
$\operatorname{Ind}E_{1}=\operatorname{Ind}E$.

When there exists a conditional expectation $E:A\to B$ of finite index, there
exists a unique conditional expectation $E_{0}:A\to B$ satisfying
$\operatorname{Ind}E_{0}\leq\operatorname{Ind}E$ for all faithful $E$ (see
[20], [52]). We call $E_{0}$ the minimal conditional expectation of $A\supset
B$. We denote $[A:B]_{0}=\operatorname{Ind}E_{0}$ and call it the minimum
index of $A\supset B$. If moreover $A\supset B$ is irreducible, there is only
one conditional expectation.

The following result is a consequence of the second dual approach developed in
[23], which will be used in one of our main technical results (see [23,
Theorem 3.3]).

######  Lemma 2.7.

Let $A\supset B$ be an irreducible inclusion of simple C∗-algebras with a
conditional expectation $E:A\to B$ of finite index. Then every intermediate
C∗-subalgebra between $A$ and $B$ is simple, that is, the inclusion $A\supset
B$ is C∗-irreducible in the sense of Rørdam [47, Definition 3.1].

Next we give a brief account of sector theory for C∗-algebras developed in
[23, Section 4]. For two simple $C^{*}$-algebras $A$ and $B$, we denote by
$\operatorname{Mor}(B,A)_{0}$ the set of homomorphisms from $B$ into $A$ whose
image has a finite index. When $A$ and $B$ are non-unital, every
$\rho\in\operatorname{Mor}(B,A)_{0}$ uniquely extends to a strictly continuous
homomorphism from $M(B)$ to $M(A)$, which we denote by the same symbol $\rho$.
For $\rho\in\operatorname{Mor}(B,A)_{0}$, we denote by $E_{\rho}$ the minimal
conditional expectation $E_{\rho}:A\to\rho(B)$, and denote
$d(\rho)=[A:\rho(B)]_{0}^{1/2}$, which is called the statistical dimension of
$\rho$. We say that two homomorphisms $\rho,\sigma\in\operatorname{Mor}(B,A)$
are equivalent if there exists a unitary $u\in M(A)$ satisfying
$\rho=\operatorname{Ad}u\circ\sigma$.

For $\rho,\sigma\in\operatorname{Mor}(B,A)_{0}$, we denote by $(\rho,\sigma)$
the intertwiner space

$(\rho,\sigma)=\\{v\in M(A);\;\forall x\in B,\;v\rho(x)=\sigma(x)v\\},$

which is always finite dimensional. When $\rho$ is irreducible, it is a
Hilbert space with $\langle{v,w}\rangle=w^{*}v$.

We mainly work on the following two classes of C∗-algebras:

  * •

$\mathcal{C}_{1}$: the class of simple stable $\sigma$-unital C∗-algebras.

  * •

$\mathcal{C}_{2}$: the class of unital purely infinite simple C∗-algebras in
the Cuntz standard form, that is $[1_{A}]_{0}=0$ in $K_{0}(A)$.

Assume that $A$ and $B$ are C∗-algebras belonging to either $\mathcal{C}_{1}$
or $\mathcal{C}_{2}$. Then every $\rho\in\operatorname{Mor}(B,A)_{0}$ has its
conjugate $\overline{\rho}\in\operatorname{Mor}(A,B)_{0}$, uniquely determined
up to equivalence and characterized by the following property: there exist
isometries $R_{\rho}\in(\operatorname{id}_{B},\overline{\rho}\rho)$ and
$\overline{R}_{\rho}\in(\operatorname{id}_{A},\rho\overline{\rho})$ satisfying

${\overline{R}_{\rho}}^{*}\rho(R_{\rho})=\frac{1}{d(\rho)},\quad{R_{\rho}}^{*}\overline{\rho}(\overline{R}_{\rho})=\frac{1}{d(\rho)}.$

The inclusion $B\supset\overline{\rho}(A)$ is isomorphic to the dual inclusion
$\mathbb{K}_{\rho(B)}(\mathcal{E}_{E_{\rho}})\supset A$ of $A\supset\rho(B)$
(see [23, Lemma 4.4]).

For $\rho,\sigma\in\operatorname{Mor}(B,A)_{0}$, their direct sum
$\rho\oplus\sigma\in\operatorname{Mor}(B,A)_{0}$, which is uniquely determined
up to unitary equivalence, is defined as follows: we choose two isometries
$s_{1},s_{2}\in M(A)$ satisfying the Cuntz algebra $\mathcal{O}_{2}$ relation
$s_{1}s_{1}^{*}+s_{2}s_{2}^{*}=1$ and let

$(\rho\oplus\sigma)(x)=s_{1}\rho(x)s_{1}^{*}+s_{2}\sigma(x)s_{2}^{*}.$

With these operations together with composition as monoidal product, the two
sets $\operatorname{End}(A)_{0}$ and $\operatorname{End}(B)_{0}$ are rigid
C∗-tensor categories and $\operatorname{Mor}(A,B)_{0}$ and
$\operatorname{Mor}(B,A)_{0}$ are their module categories. In particular, the
Frobenius-reciprocity holds (see [22]).

One of advantages to work in the classes $\mathcal{C}_{i}$, $i=1,2$, is that
we have the following crossed product type decomposition. Let $A\supset B$ be
an irreducible inclusion of simple C∗-algebras belonging to either
$\mathcal{C}_{1}$ or $\mathcal{C}_{2}$ with a conditional expectation $E:A\to
B$ of finite index. Let $\iota:B\hookrightarrow A$ be the inclusion map and
let

$[\overline{\iota}\iota]=\bigoplus_{\xi\in\Xi}n_{\xi}[\rho_{\xi}]$

be the irreducible decomposition, where $n_{\xi}$ is the multiplicity of
$\rho_{\xi}\in\operatorname{End}(B)_{0}$. We may and do assume $0\in\Xi$ and
$\rho_{0}=\operatorname{id}_{B}$. Since $\iota$ is irreducible, we have
$n_{0}=1$. By the Frobenius reciprocity,

$\dim(\iota,\iota\rho_{\xi})=\dim(\overline{\iota}\iota,\rho_{\xi})=n_{\xi}.$

Let $\\{V(\xi)\\}_{i=1}^{n_{\xi}}$ be an orthonormal basis of
$(\iota,\iota\rho_{\xi})$. Then every $x\in A$ is uniquely decomposed as

$x=\sum_{\xi\in\Xi}\sum_{i=1}^{n_{\xi}}x(\xi)_{i}V(\xi)_{i},$
$x(\xi)_{i}=d(\rho_{\xi})E(xV(\xi)_{i}^{*})\in B,$

(see [23, p.124]). Note that $x(0)_{1}=E(x)$, and

$b(x-E(x))b=\sum_{\xi\in\Xi\setminus\\{0\\}}\sum_{i=1}^{n_{\xi}}bx(\xi)_{i}\rho_{\xi}(b)V(\xi)_{i},\quad
b\in B.$

For two representations $(\Pi_{1},H_{1})$, $(\Pi_{2},H_{2})$ of a C∗-algebra
$A$, we denote by $\operatorname{Hom}_{A}(\Pi_{1},\Pi_{2})$ the set of
intertwiners

$\\{T\in\mathbb{B}(H_{1},H_{2});\;\forall x\in
A,\;T\Pi_{1}(x)=\Pi_{2}(x)T\\}.$

######  Lemma 2.8.

  * $(1)$

Let $A\supset B$ be an inclusions of simple C∗-algebras of finite index, and
let $\Pi$ be an irreducible representation of $A$. Then the restriction of
$\Pi$ to $B$ is a finite direct sum of irreducible representations.

  * $(2)$

Let $A$ and $B$ be C∗-algebras belonging to $\mathcal{C}_{1}$ or
$\mathcal{C}_{2}$, and let $\rho\in\operatorname{Mor}(B,A)_{0}$. Let $\Phi$
and $\Psi$ be irreducible representations of $A$ and $B$ respectively. Then

$\dim\operatorname{Hom}_{A}(\Phi\circ\rho,\Psi)=\dim\operatorname{Hom}_{A}(\Phi,\Psi\circ\overline{\rho}).$

###### Proof.

The statements follow from [23, Lemma 5.1, Lemma 5.2]. ∎

###  2.4 Quasi-product actions

For a compact group $G$, we denote by $\mathcal{U}(G)$ the category of finite
dimensional unitary representations of $G$. We choose and fix a transversal
$\widehat{G}$ of the set of equivalence classes of irreducible unitary
representations of $G$. For $\pi\in\widehat{G}$, we denote
$\chi_{\pi}(g)=\operatorname{Tr}\pi(g)$ and $d(\pi)=\dim\pi$. We choose and
fix an orthonormal basis $\\{\xi(\pi)\\}_{i=1}^{d(\pi)}$ of the representation
space $H_{\pi}$ of $\pi$, and identify $\pi(g)$ with its matrix representation
$(\pi_{ij}(g))_{ij}$. For $\pi,\sigma,\mu\in\widehat{G}$, we let
$N_{\pi,\sigma}^{\mu}=\dim\operatorname{Hom}_{G}(\mu,\pi\otimes\sigma)$.

Let $\alpha$ be an action of $G$ on a C∗-algebra $A$. For $\pi\in\widehat{G}$,
we define $P_{\pi}:A\to A$ by

$P_{\pi}(x)=d(\pi)\int_{G}\overline{\chi_{\pi}(g)}\alpha_{g}(x)dg,$

where $dg$ is the normalized Haar measure. Then we have $P_{\pi}\circ
P_{\sigma}=\delta_{\pi,\sigma}P_{\pi}$ for $\pi,\sigma\in\widehat{G}$. The
image of $P_{\pi}$ is called the spectral subspace corresponding to
$\pi\in\widehat{G}$, and we denote it by $A^{\alpha}(\pi)$. The Peter-Weyl
theorem together with the Hahn-Banach theorem implies that the linear span of
$\cup_{\pi\in\widehat{G}}A^{\alpha}(\pi)$ is dense in $A$. Or alternatively,
we can see it more directly from a Fejér type approximation (see Theorem 5.4).
Let

$A^{\alpha}_{1}(\pi)=\\{x=(x_{1},x_{2},\ldots,x_{d(\pi)})\in
A^{d(\pi)};\;\alpha_{g}(x)=x\pi(g)\\}.$

Then $A^{\alpha}(\pi)\neq\\{0\\}$ if and only if
$A^{\alpha}_{1}(\pi)\neq\\{0\\}$.

Let $E_{\alpha}=P_{1}$, which is a conditional expectation from $A$ to the
fixed point algebra $A^{\alpha}$. From the compactness of $G$, we get the
following well-known lemma:

######  Lemma 2.9.

If $\\{u_{\lambda}\\}_{\lambda\in\Lambda}$ is an approximate units of $A$, so
is $\\{E_{\alpha}(u_{\lambda})\\}_{\lambda\in\Lambda}$.

The lemma implies $\overline{A^{\alpha}A}^{\|\cdot\|}=A$, and we have the
inclusion relation $M(A^{\alpha})\subset M(A)$. We denote the extension of
$\alpha_{g}$ to $M(A)$ by the same symbol $\alpha_{g}$. Then the map
$g\mapsto\alpha_{g}(T)$ is is continuous in the strict topology for all $T\in
M(A)$. We have $M(A^{\alpha})=M(A)^{\alpha}$ (see, for example, [41, Lemma
2]).

Bratteli-Elliott-Kishimoto [3, Theorem 1] showed that 10 conditions on a
compact group action are mutually equivalent, which are the defining
conditions of a quasi-product action. We name some of them under the
assumption of Theorem 1.1 now. Recall that $\alpha$ is said to be minimal if
the inclusion $A\supset A^{\alpha}$ is irreducible, which is a necessary
condition for $\alpha$ to be quasi-product because of the condition (1) below.
Note that the crossed product $A\rtimes_{\alpha}G$ is automatically simple
under the assumption of Theorem 1.1 (see [37, Proposition A]).

######  Theorem 2.10 (Quasi-product action).

Let $G$ be a second countable compact group, and let $\alpha$ be a faithful
minimal action of $G$ on a separable C∗-algebra $A$ whose fixed point algebra
$A^{\alpha}$ is simple. Then the following conditions are equivalent:

  * $(1)$

The inclusion $A\supset A^{\alpha}$ has the property (BEK).

  * $(2)$

There exists an $\alpha$-invariant pure state of $A$.

  * $(3)$

For each $\pi\in\widehat{G}$, there exists a sequence
$\\{y_{n}\\}_{n=1}^{\infty}$ in $A^{\alpha}_{1}(\pi)$ such that

    * $(\mathrm{i})$

$\|y_{n1}\|=1$ for all $n\in\mathbb{N}$,

    * $(\mathrm{ii})$

$\\{y_{n1}\\}_{n=1}^{\infty}$ is a central sequence in $A$,

    * $(\mathrm{iii})$

$\limsup_{n}\|ay_{n1}\|\geq\|a\|/d(\pi)$ for all $a\in A$.

  * $(4)$

The dual endomorphisms $\hat{\alpha}_{\pi}$,
$\pi\in\widehat{G}\setminus\\{1\\}$, of the stabilized crossed product
$(A\rtimes_{\alpha}G)\otimes\mathbb{K}$ are properly outer.

######  Remark 2.11.

When the above equivalent conditions hold, the C∗-algebra $A$ is simple too,
which follows from Landstad’s observation [50, Lemma 24] because (1) implies
that $A$ is prime. We can also use (4) to show that $A\otimes\mathbb{K}$ is
simple as in the case of a crossed product by a discrete group (see [33,
Theorem 3.1]).

The condition (4) is the most tractable for our purpose. On the other hand,
the definition of $\hat{\alpha}_{\pi}$ in [3] is rather tedious, and we
reformulate it, following the original formulation by Roberts [45], in the
special case where $A^{\alpha}$ belongs to either $\mathcal{C}_{1}$ or
$\mathcal{C}_{2}$.

By a Hilbert space $\mathcal{H}$ in $M(A)$, we mean a finite dimensional
subspace of $M(A)$ such that it is a Hilbert space with respect to the inner
product given by $\langle{V,W}\rangle=W^{*}V$ for $V,W\in\mathcal{H}$. For a
Hilbert space $\mathcal{H}$ in $M(A)$, we choose an orthonormal basis
$\\{\psi(\mathcal{H})_{i}\\}_{i}$ of $\mathcal{H}$ and define the support
$s(\mathcal{H})$ of $\mathcal{H}$ by

$\sum_{i}\psi(\mathcal{H})_{i}\psi(\mathcal{H})_{i}^{*},$

which does not depend on the choice of $\\{\psi(\mathcal{H})_{i}\\}_{i}$. For
two Hilbert spaces $\mathcal{H}_{1},\mathcal{H}_{2}$ in $M(A)$, we denote by
$\mathcal{H}_{1}\mathcal{H}_{2}$ and $\mathcal{H}_{1}\mathcal{H}_{2}^{*}$ the
linear spans of $\\{VW;\;V\in\mathcal{H}_{1},\;W\in\mathcal{H}_{2}\\}$ and
$\\{VW^{*};\;V\in\mathcal{H}_{1},\;W\in\mathcal{H}_{2}\\}$ respectively. Then
$\mathcal{H}_{1}\mathcal{H}_{2}$ is identified with the tensor product
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, and
$\mathcal{H}_{1}\mathcal{H}_{2}^{*}$ is identified with
$\mathbb{B}(\mathcal{H}_{2},\mathcal{H}_{1})$ through left multiplication.
Occasionally, we consider infinite dimensional Hilbert spaces in $M(A)$, and
in that case we take the norm closure to define
$\mathcal{H}_{1}\mathcal{H}_{2}$ and $\mathcal{H}_{1}\mathcal{H}_{2}^{*}$.
Then $\mathcal{H}_{1}\mathcal{H}_{2}$ is a Hilbert space again and
$\mathcal{H}_{1}\mathcal{H}_{2}^{*}$ is isometric to the space of compact
operators $\mathbb{K}(\mathcal{H}_{2},\mathcal{H}_{1})$.

If a Hilbert space $\mathcal{H}$ in $M(A)$ is globally invariant under
$\alpha$, it carries a unitary representation of $G$, and hence $\mathcal{H}$
is considered as an object in the category $\mathcal{U}(G)$. We denote by
$\mathcal{U}(G,\alpha)$ the set of globally $\alpha$-invariant Hilbert spaces
in $M(A)$ with support 1. For
$\mathcal{H}_{1},\mathcal{H}_{2}\in\mathcal{U}(G,\alpha)$, we define the space
of morphisms $(\mathcal{H}_{1},\mathcal{H}_{2})_{G}$ from $\mathcal{H}_{1}$ to
$\mathcal{H}_{2}$ in the category $\mathcal{U}(G,\alpha)$ by

$(\mathcal{H}_{1},\mathcal{H}_{2})_{G}:=\mathcal{H}_{2}\mathcal{H}_{1}^{*}\cap
M(A^{\alpha}).$

In this way, we get a tensor subcategory $\mathcal{U}(G,\alpha)$ of
$\mathcal{U}(G)$.

For $\mathcal{H}\in\mathcal{U}(G,\alpha)$, we define
$\rho_{\mathcal{H}}\in\operatorname{End}(A)_{0}$ by

$\rho_{\mathcal{H}}(x)=\sum_{i}\psi(\mathcal{H})_{i}x\psi(\mathcal{H})_{i}^{*},$

which does not depend on the choice of the orthonormal basis, and hence
$\rho_{\mathcal{H}}$ commutes with $\alpha_{g}$. By construction, we have
$[\rho_{\mathcal{H}}]=\dim\mathcal{H}[\operatorname{id}_{A}]$, and
$d(\rho_{\mathcal{H}})=\dim\mathcal{H}$. We define
$\check{\alpha}_{\mathcal{H}}\in\operatorname{End}(A^{\alpha})_{0}$ to be the
restriction of $\rho_{\mathcal{H}}$ to $A^{\alpha}$. Note that
$Vx=\check{\alpha}_{\mathcal{H}}(x)V$ holds for all $x\in A^{\alpha}$ and
$V\in\mathcal{H}$. Since
$(\rho_{\mathcal{H}_{1}},\rho_{\mathcal{H}_{2}})=\mathcal{H}_{2}\mathcal{H}_{1}^{*}$,
we have
$(\mathcal{H}_{1},\mathcal{H}_{2})_{G}\subset(\check{\alpha}_{\mathcal{H}_{1}},\check{\alpha}_{\mathcal{H}_{2}})$,
and the irreducibility of $A\supset A^{\alpha}$ implies the equality of the
two sets. Since
$\rho_{\mathcal{H}_{1}}\circ\rho_{\mathcal{H}_{2}}=\rho_{\mathcal{H}_{1}\mathcal{H}_{2}}$,
we have
$\check{\alpha}_{\mathcal{H}_{1}\mathcal{H}_{2}}=\check{\alpha}_{\mathcal{H}_{1}}\circ\check{\alpha}_{\mathcal{H}_{2}}$.
Setting $\check{\alpha}_{T}=T$ for
$T\in(\mathcal{H}_{1},\mathcal{H}_{2})_{G}$, we get a tensor functor
$\check{\alpha}:\mathcal{U}(G,\alpha)\to\operatorname{End}(A^{\alpha})_{0}$,
which is in fact an embedding of $\mathcal{U}(G,\alpha)$ into
$\operatorname{End}(A^{\alpha})_{0}$.

Using the assumption that $A^{\alpha}$ belongs to either $\mathcal{C}_{1}$ or
$\mathcal{C}_{2}$, we can show that $\mathcal{U}(G,\alpha)$ has a direct sum,
up to equivalence, and that every object in $\mathcal{U}(G,\alpha)$ is a
direct sum of simple objects. Indeed, for
$\mathcal{H}_{1},\mathcal{H}_{2}\in\mathcal{U}(G,\alpha)$, we choose
isometries $s_{1},s_{2}\in M(A^{\alpha})$ satisfying the $\mathcal{O}_{2}$
relation $s_{1}s_{1}^{*}+s_{2}s_{2}^{*}=1$, and set
$\mathcal{H}_{1}\oplus\mathcal{H}_{2}=s_{1}\mathcal{H}_{1}+s_{2}\mathcal{H}_{2}\in\mathcal{U}(G,\alpha)$.
Then we have

$\check{\alpha}_{\mathcal{H}_{1}\oplus\mathcal{H}_{2}}(x)=s_{1}\check{\alpha}_{\mathcal{H}_{1}}(x)s_{1}^{*}+s_{2}\check{\alpha}_{\mathcal{H}_{2}}(x)s_{2}^{*},$

which is compatible with the definition of a direct sum in
$\operatorname{End}(A^{\alpha})_{0}$. Let
$\mathcal{H}\in\mathcal{U}(G,\alpha)$ and let $\\{p_{i}\\}_{i=1}^{n}$ be a set
of minimal projections in
$(\mathcal{H},\mathcal{H})_{G}=(\check{\alpha}_{\mathcal{H}},\check{\alpha}_{\mathcal{H}})$
whose summation is 1. Then [23, Lemma 4.1] shows that $p_{i}$ is equivalent to
1 in $M(A^{\alpha})$, and there exist isometries $V_{i}\in M(A^{\alpha})$
satisfying $V_{i}V_{i}^{*}=p_{i}$. Now
$\mathcal{H}_{i}:=V_{i}^{*}\mathcal{H}\in\mathcal{U}(G,\alpha)$, and
$\mathcal{H}$ is equivalent to the direct sum of $\mathcal{H}_{i}$,
$i=1,2,\ldots,n$.

######  Lemma 2.12.

Under the assumption of Theorem 1.1, assume that $A^{\alpha}$ belongs to
either $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$. Then for every representation
$\sigma\in\mathcal{U}(G)$, there exists
$\mathcal{H}_{\sigma}\in\mathcal{U}(G,\alpha)$ equivalent to $\sigma$.

###### Proof.

This essentially follows from [1, Lemma III 3.4]. ∎

######  Remark 2.13.

If we only assume that $A^{\alpha}$ is unital simple purely infinite in the
above lemma, we can still have a globally $\alpha$-invariant Hilbert space
equivalent to $\sigma$, not necessarily with support 1. Indeed, choosing a
non-zero projection $e\in A^{\alpha}$ with $[e]_{0}=0$ in $K_{0}(A^{\alpha})$
and applying the lemma to $eAe$, we get
$\mathcal{H}_{\sigma}\in\mathcal{U}(G,\alpha|_{eAe})$ equivalent to $\sigma$.
We choose an isometry $V\in A^{\alpha}$ with $VV^{*}\leq e$. Then
$\mathcal{H}_{\sigma}V$ is the desired Hilbert space.

We call the functor $\check{\alpha}$ the pre-dual action of $\alpha$. For each
$\pi\in\widehat{G}$, we fix $\mathcal{H}_{\pi}\in\mathcal{U}(G,\alpha)$
equivalent to $\pi$, and denote
$\check{\alpha}_{\pi}=\check{\alpha}_{\mathcal{H}_{\pi}}$ for simplicity. We
arrange the orthonormal basis $\\{\psi(\mathcal{H}_{\pi})_{i}\\}_{i}$ so that
it is consistent with the orthonormal basis $\\{\xi(\pi)_{i}\\}_{i}$ of
$H_{\pi}$ we have already chosen, and denote
$\psi(\pi)_{i}=\psi(\mathcal{H}_{\pi})_{i}$ for simplicity.

Let $\overline{\pi}$ be the complex conjugate representation of
$\pi\in\widehat{G}$. We choose an orthonormal basis
$\\{\xi(\overline{\pi})_{i}\\}_{i}$ of the representation space of
$\overline{\pi}$ so that $\overline{\pi}_{ij}(g)=\overline{\pi_{ij}(g)}$
holds. When $\pi$ and $\overline{\pi}$ are inequivalent, we can arrange
$\overline{\pi}$ to be a member of $\widehat{G}$. When $\pi$ and
$\overline{\pi}$ are equivalent, we have two specially chosen orthonormal
bases $\\{\psi(\pi)_{i}\\}_{i}$ and $\\{\psi(\overline{\pi})_{i}\\}_{i}$ of
$\mathcal{H}_{\pi}$ corresponding to $\\{\xi(\pi)_{i}\\}_{i}$ and
$\\{\xi(\overline{\pi})_{i}\\}_{i}$ respectively .

For $\pi\in\widehat{G}$, we set

$R_{\pi}=\frac{1}{\sqrt{d(\pi)}}\sum_{i=1}^{d(\pi)}\psi(\overline{\pi})_{i}\psi(\pi)_{i}\in(\operatorname{id},\check{\alpha}_{\overline{\pi}}\check{\alpha}_{\pi}),$
$\overline{R}_{\pi}=\frac{1}{\sqrt{d(\pi)}}\sum_{i=1}^{d(\pi)}\psi(\pi)_{i}\psi(\overline{\pi})_{i}\in(\operatorname{id},\check{\alpha}_{\pi}\check{\alpha}_{\overline{\pi}}),$

which are isometries satisfying

$\overline{R}_{\pi}^{*}\check{\alpha}_{\pi}(R_{\pi})=R_{\pi}^{*}\check{\alpha}_{\overline{\pi}}(\overline{R}_{\pi})=\frac{1}{d(\pi)}.$

This in particular shows that $d(\check{\alpha}_{\pi})=d(\pi)$ as
$\check{\alpha}_{\pi}$ is irreducible, and more generally shows
$d(\check{\alpha}_{\mathcal{H}})=\dim\mathcal{H}$ by the additivity of the
statistical dimension.

The standard left inverse $\phi_{\pi}:A^{\alpha}\to A^{\alpha}$ of
$\check{\alpha}_{\pi}$ is given by

$\phi_{\pi}(x)=\frac{1}{d(\pi)}\sum_{i=1}^{d(\pi)}\psi(\pi)_{i}^{*}x\psi(\pi)_{i},\quad
x\in A^{\alpha},$

which is also expressed as
$\phi_{\pi}(x)=R_{\pi}^{*}\check{\alpha}_{\overline{\pi}}(x)R_{\pi}$. Then the
minimal conditional expectation for the inclusion
$A^{\alpha}\supset\check{\alpha}_{\pi}(A^{\alpha})$ can be expressed as
$\check{\alpha}_{\pi}\circ\phi_{\pi}$.

Now we can reformulate the (4) part of Theorem 2.10 as follows:

######  Lemma 2.14.

Under the assumption of Theorem 1.1, assume that $A^{\alpha}$ belongs to
either $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$. Then $\alpha$ is quasi-product
if and only if the endomorphisms $\check{\alpha}_{\pi}$,
$\pi\in\widehat{G}\setminus\\{1\\}$, are properly outer.

We denote by $A_{0}$ the linear span of
$\cup_{\pi\in\widehat{G}}A^{\alpha}(\pi)$, which is a $G$-invariant dense
$*$-subalgebra of $A$. A direct computation shows the following:

######  Lemma 2.15.

Under the assumption of Theorem 1.1, assume that $A^{\alpha}$ belongs to
either $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$. Then

  * $(1)$

For any $x\in A^{\alpha}$ and $\pi\in\widehat{G}$, we have

$P_{\pi}(x)=d(\pi)\sum_{i=1}^{d(\pi)}E_{\alpha}(x\psi(\pi)_{i}^{*})\psi(\pi)_{i}=d(\pi)\sum_{i=1}^{d(\pi)}\psi(\overline{\pi})_{i}^{*}E_{\alpha}(\psi(\overline{\pi})_{i}x).$

In particular, the linear span of
$\cup_{\pi\in\widehat{G}}A^{\alpha}\mathcal{H}_{\pi}$ is a dense
$*$-subalgebra of $A$.

  * $(2)$

Every $x\in A_{0}$ is uniquely expanded as

$x=\sum_{\pi\in\widehat{G}}\sum_{i=1}^{d(\pi)}x(\pi)_{i}\psi(\pi)_{i},$

where $x(\pi)_{i}=d(\pi)E_{\alpha}(x\psi(\pi)_{i}^{*})\in A^{\alpha}$.

Since we need the Doplicher-Roberts reconstruction theorem [12], we recall the
permutation symmetry $\theta$ of the category $\mathcal{U}(G,\alpha)$, which
is an assignment of a unitary
$\theta(\mathcal{H}_{1},\mathcal{H}_{2})\in(\mathcal{H}_{1}\mathcal{H}_{2},\mathcal{H}_{2}\mathcal{H}_{1})_{G}$
to each $\mathcal{H}_{1}.\mathcal{H}_{2}\in\mathcal{U}(G,\alpha)$ given by

$\theta(\mathcal{H}_{1},\mathcal{H}_{2})=\sum_{ij}\psi(\mathcal{H}_{2})_{i}\psi(\mathcal{H}_{1})_{i}\psi(\mathcal{H}_{2})_{j}^{*}\psi(\mathcal{H}_{1})_{i}^{*}.$

Then we have

$\theta(\mathcal{H}_{1},\mathcal{H}_{2})^{*}=\theta(\mathcal{H}_{2},\mathcal{H}_{1}),$
(2.2)
$\theta(\mathcal{H}_{1}\otimes\mathcal{H}_{2},\mathcal{H}_{3})=\theta(\mathcal{H}_{1},\mathcal{H}_{3})\rho_{\mathcal{H}_{1}}(\theta(\mathcal{H}_{2},\mathcal{H}_{3})),$
(2.3)
$\rho_{\mathcal{H}_{1}}(T)=\theta(\mathcal{H}_{3},\mathcal{H}_{1})T\theta(\mathcal{H}_{2},\mathcal{H}_{1})^{*},\quad\forall
T\in(\mathcal{H}_{2},\mathcal{H}_{3})_{G}.$ (2.4)

Before ending this section, we recall quasi-free actions on the Cuntz
algebras. The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal C∗-algebra
generated by a separable infinite dimensional Hilbert space $\mathcal{H}$ in a
C∗-algebra. By universality, every unitary representation of a group $G$ in
$\mathbb{B}(\mathcal{H})$ induces its action on $\mathcal{O}_{\infty}$, which
is called the quasi-free action arising from the unitary representation. We
define the Cunzt algebra $\mathcal{O}_{n}$, $n=2,3,\ldots$, and quasi-free
actions on them in a similar way with $\dim\mathcal{H}=n$ and an extra
condition that $\mathcal{H}$ has support 1.

##  3 Finite index inclusions with the property (BEK)

The following is our first main technical result.

######  Theorem 3.1.

Let $A\supset B$ be an irreducible inclusion of separable unital purely
infinite simple C∗-algebras with a conditional expectation $E:A\to B$ of
finite index. Then the inclusion $A\supset B$ has the property (BEK).

We need to prepare a few lemmata before proving the theorem. For two operators
$S,T$ acting on the same Hilbert space, we denote $[S,T]=ST-TS$.

######  Lemma 3.2.

Let $B$ be a C∗-algebra, let $(\Phi,K)$ be an irreducible representation of
$B$, and let $\delta>0$. Then if $T\in\mathbb{B}(K)_{1}$ satisfies

$\sup_{b\in B_{1}}\|[\Phi(b),T]\|\leq\delta,$

there exists $\lambda\in\mathbb{C}$ satisfying $\|T-\lambda 1_{K}\|\leq
3\delta$.

###### Proof.

By the Kaplansky density theorem, we have $\|[S,T]\|\leq\delta$ for all
$S\in\mathbb{B}(K)_{1}$. We assume $\dim K=\infty$ as the finite dimensional
case can be easily handled.

We first claim that $\|T\xi-\langle{T\xi,\xi}\rangle\xi\|\leq\delta$ holds for
every unit vector $\xi\in K$. The claim holds for an eigenvector of $T$.
Assume that $\xi$ and $T\xi$ are linearly independent, and let
$\lambda=\langle{T\xi,\xi}\rangle$ and
$\eta=\|T\xi-\lambda\xi\|^{-1}\left(T\xi-\lambda\xi\right)$. Then we can
choose a unitary $u_{\pm}\in\mathbb{B}(K)$ satisfying $u_{\pm}\xi=\eta$ and
$u_{\pm}\eta=\pm\xi$. We have

$[u_{\pm},T]\xi=u_{\pm}T\xi-T\eta=u_{\pm}(\lambda\xi+\|T\xi-\lambda\xi\|\eta)-T\eta=\lambda\eta-T\eta\pm\|T\xi-\lambda\xi\|\xi.$

Since $\|[u_{\pm},T]\|\leq\delta$, we get $\|T\xi-\lambda\xi\|\leq\delta$, and
the claim is shown.

Next we claim that if $\xi$ and $\eta$ are mutually orthogonal unit vectors in
$K$, we have
$|\langle{T\xi,\xi}\rangle-\langle{T\eta,\eta}\rangle|\leq\delta$. We choose a
unitary $u\in\mathbb{B}(K)$ satisfying $u\xi=\eta$ and $u\eta=\xi$. Note that
we have $u^{*}\eta=\xi$. Then the claim follows from

$\langle{[u,T]\xi,\eta}\rangle=\langle{T\xi,u^{*}\eta}\rangle-\langle{Tu\xi,\eta}\rangle=\langle{T\xi,\xi}\rangle-\langle{T\eta,\eta}\rangle.$

Finally, we fix a unit vector $\xi\in K$, and set
$\lambda=\langle{T\xi,\xi}\rangle$. Let $\eta$ be an arbitrary unit vector in
$K$. Choosing a unit vector $\zeta\in\\{\xi,\eta\\}^{\perp}$ and applying the
second claim, we get
$|\langle{T\xi,\xi}\rangle-\langle{T\eta,\eta}\rangle|\leq 2\delta$, which
together with the first claim implies

$\|T\eta-\lambda\eta\|\leq\|T\eta-\langle{T\eta,\eta}\rangle\eta\|+\|\langle{T\eta,\eta}\rangle\eta-\lambda\eta\|\leq
3\delta.$

Thus the statement is shown. ∎

######  Lemma 3.3.

Let $A\supset B$ be an inclusion of C∗-algebras with finite index, and let
$(\Pi,H)$ be an irreducible representation of $A$. Then there exists a
constant $C>0$ depending only on $\Pi$ satisfying the following property:
whenever $T\in\mathbb{B}(H)_{1}$ satisfies

$\sup_{b\in B_{1}}\|[\Pi(b),T]\|\leq\delta,$

there exists $T_{1}\in\Pi(B)^{\prime}$ satisfying $\|T_{1}-T\|\leq C\delta$.

###### Proof.

Thanks to Lemma 2.8, the restriction $\Pi|_{B}$ of $\Pi$ to $B$ is a finite
direct sum of irreducible representations. Let $(\Pi_{k},H_{k})$,
$k=1,2,\ldots,m$, be the mutually disjoint irreducible components of the
restriction $\Pi|_{B}$. Then we may assume

$H=\bigoplus_{k=1}^{m}H_{k}\otimes\mathbb{C}^{n_{k}},\quad\Pi(b)=\bigoplus_{k=1}^{m}\Pi_{k}(b)\otimes
1_{\mathbb{C}^{n_{k}}},\quad b\in B,$

where $n_{k}$ is the multiplicity of $\Pi_{i}$. Note that we have

$\Pi(B)^{\prime\prime}=\bigoplus_{k=1}^{m}\mathbb{B}(H_{k})\otimes
1_{\mathbb{C}^{n_{k}}}.$

Let $z_{k}=1_{H_{k}}\otimes 1_{\mathbb{C}^{n_{k}}}$. By the Kaplansky density
theorem, we have $\|[z_{k},T]\|\leq\delta$. Let

$\Delta(T)=\sum_{k=1}^{m}z_{k}Tz_{k}.$

Then $\|T-\Delta(T)\|\leq m\delta$.

We take a system of matrix units $\\{e^{(k)}_{ij}\\}_{1\leq i,j\leq n_{k}}$ in
$\mathbb{B}(\mathbb{C}^{n_{k}})$ and express $\Delta(T)$ as

$\Delta(T)=\bigoplus_{k=1}^{m}\sum_{1\leq i,j\leq n_{k}}T^{(k)}_{ij}\otimes
e^{(k)}_{ij},$

where $T^{(k)}_{ij}\in\mathbb{B}(H_{k})$. For $b\in B_{1}$, we have

$\|[T^{(k)}_{ij},\Pi_{k}(b)]\|\leq\|[\Delta(T),\Pi(b)]\|\leq(2m+1)\delta.$

By Lemma 3.2, there exist $\lambda^{(k)}_{ij}\in\mathbb{C}$ satisfying
$\|T^{(k)}_{ij}-\lambda^{(k)}_{ij}1_{H_{k}}\|\leq 3(2m+1)\delta$. Let

$T_{1}=\bigoplus_{k=1}^{m}\left(1_{H_{k}}\otimes\sum_{1\leq i,j\leq
n_{k}}\lambda^{(k)}_{ij}e^{(k)}_{ij}\right).$

Then $T_{1}\in\Pi(B)^{\prime}$, and

$\|T_{1}-T\|\leq\|T_{1}-\Delta(T)\|+\|\Delta(T)-T\|\leq
3(2m+1)\delta\max_{1\leq k\leq m}n_{k}^{2}+m\delta,$

which shows the statement. ∎

###### Proof of Theorem 3.1.

When $B$ is not unital, it is stable because it is purely infinite and simple.
In this case, we can choose a projection $p\in B$ satisfying

$(A,B,E)\cong(pAp\otimes\mathbb{K},pBp\otimes\mathbb{K},E|_{pAp}\otimes\operatorname{id}_{\mathbb{K}}).$

Thus we may and do assume that $A$ and $B$ are unital to prove the statement.

Assume on the contrary that the inclusion $A\supset B$ does not have the
property (BEK). Then for each natural number $n\in\mathbb{N}$, there exist
$x_{n},y_{n}\in A$ with $\|x_{n}\|=\|y_{n}\|=1$ satisfying

$\sup_{b\in B_{1}}\|x_{n}by_{n}\|\leq\frac{1}{n}.$

We choose a free ultra-filter $\omega\in\beta\mathbb{N}\setminus\mathbb{N}$.
Note that $A^{\omega}$ is purely infinite and simple (see [46, Proposition
6.2.6]). Then $A^{\omega}\supset B^{\omega}$ is an inclusion of purely
infinite simple C∗-algebras with a conditional expectation
$E^{\omega}:A^{\omega}\to B^{\omega}$ given by

$E^{\omega}([(x_{n})])=[(E(x_{n}))].$

Note that we have $\operatorname{Ind}E^{\omega}=\operatorname{Ind}E$. We
define $x,y\in A^{\omega}$ by $x=[(x_{n})]$ and $y=[(y_{n})]$. Then
$\|x\|=\|y\|=1$ and $xB^{\omega}y=\\{0\\}$ by construction.

As was observed in [3, Section 3], the C∗-algebra generated by
$B^{\omega}xB^{\omega}$, $B^{\omega}yB^{\omega}$, and $B^{\omega}$ is not
prime. Thus Lemma 2.7 implies that $A^{\omega}\supset B^{\omega}$ is not
irreducible. Let $r=[(r_{n})]\in A^{\omega}\cap{B^{\omega}}^{\prime}$ with
$\|r\|=1$. Then we have

$\lim_{n\to\omega}\sup_{b\in B_{1}}\|br_{n}-r_{n}b\|=0.$

We choose an irreducible representation $(\Pi,H)$ of $A$. Then Lemma 3.3 shows
that there exist $Q_{n}\in\Pi(B)^{\prime}$ satisfying

$\lim_{n\to\omega}\|\Pi(r_{n})-Q_{n}\|=0.$

We may assume that $\\{Q_{n}\\}$ is bounded. Since $\Pi(B)^{\prime}$ is finite
dimensional, the norm limit $\lim_{n\to\omega}Q_{n}$ exists. Since $B$ is
simple, the restriction of $\Pi$ to $B$ is faithful, and the norm limit
$\lim_{n\to\omega}r_{n}$ exists too. This means that $r\in A\cap
B^{\prime}=\mathbb{C}$ and $A^{\omega}\supset B^{\omega}$ is irreducible,
which is a contradiction. ∎

######  Corollary 3.4.

Let $A$ be a separable purely infinite simple C∗-algebra, and assume that
$\rho\in\operatorname{End}(A)_{0}$ is irreducible and $d(\rho)>1$. Then $\rho$
is properly outer.

Although the following result is not used later, it is of interest in its own
right.

######  Theorem 3.5.

Let $A\supset B$ be an irreducible inclusion of separable simple C∗-algebras
with a conditional expectation $E:A\to B$ of finite index. Let
$\mathbb{K}_{B}(\mathcal{E}_{E})\supset A$ be its basic construction, and let
$E_{1}:\mathbb{K}_{B}(\mathcal{E}_{E})\to A$ be the dual conditional
expectation. Let $\iota:B\hookrightarrow A$ be the inclusion map. Then the
following conditions are equivalent:

  * $(1)$

The inclusion $A\supset B$ has the property (BEK).

  * $(2)$

The inclusion $\mathbb{K}_{B}(\mathcal{E}_{E})\supset A$ has the property
(BEK).

  * $(3)$

For any $x\in A$ and any non-zero hereditary C∗-subalgebra $C$ of $B$,

$\inf\\{\|c(x-E(x))c\|;\;c\in C_{+},\;\|c\|=1\\}=0.$

  * $(4)$

For any $y\in\mathbb{K}_{B}(\mathcal{E}_{E})$ and any non-zero hereditary
C∗-subalgebra $D$ of $A$,

$\inf\\{\|d(y-E_{1}(y))d\|;\;d\in D_{+},\;\|d\|=1\\}=0.$

If moreover $B$ belongs to either the class $\mathcal{C}_{1}$ or
$\mathcal{C}_{2}$, the above conditions are further equivalent to the
following two conditions:

  * $(5)$

Every irreducible component of $\iota\overline{\iota}$ not equivalent to
$\operatorname{id}_{A}$ is properly outer.

  * $(6)$

Every irreducible component of $\overline{\iota}\iota$ not equivalent to
$\operatorname{id}_{B}$ is properly outer.

###### Proof.

Since the conditions (1)-(4) persist after taking tensor product with
$\mathbb{K}$ and passing to the corners by a non-zero projection in $M(B)$, we
may and do assume that $A,B\in\mathcal{C}_{1}$ to prove the theorem.

(1)$\Rightarrow$(5). We choose an irreducible representation $\Pi$ of $A$
whose restriction to $B$ is irreducible. Then

$1=\dim\operatorname{Hom}_{B}(\Pi\circ\iota,\Pi\circ\iota)=\dim\operatorname{Hom}_{A}(\Pi,\Pi\circ\iota\overline{\iota}).$

Thus if $\rho\in\operatorname{End}(A)_{0}$ is an irreducible component not
equivalent to $\operatorname{id}_{A}$, we have
$\dim\operatorname{Hom}_{A}(\Pi,\Pi\circ\rho)=0$, which shows that $\rho$ is
properly outer by Lemma 2.2.

(5)$\Rightarrow$(4). Note that the dual inclusion
$\mathbb{K}_{B}(\mathcal{E}_{E})\supset A$ is isomorphic to
$B\supset\overline{\iota}(A)$. Thus the statement follows from the crossed
product type decomposition of the inclusion
$\mathbb{K}_{B}(\mathcal{E}_{E})\supset A$ (see Section 2.3).

(4)$\Rightarrow$(2). We denote $\lambda=(\operatorname{Ind}E)^{-1}$. Let
$x_{1},x_{2}\in\mathbb{K}_{B}(\mathcal{E}_{E})$ with $\|x_{1}\|=\|x_{2}\|=1$.
Our task is to show $\sup_{a\in A_{1}}\|x_{1}ax_{2}\|\geq\lambda^{2}$. Since
$\|x_{1}ax_{2}\|=\||x_{1}|a|x_{2}|\|$, we may and do assume that $x_{1},x_{2}$
are positive. By the Pimsner-Popa inequality, we have
$\|E_{1}(x_{i})\|\geq\lambda$ for $i=1,2$. Let $0<\varepsilon$ be a
sufficiently small constant. Then by assumption we can choose $a_{1},a_{2}\in
A_{+}$ of norm 1 satisfying

$\|a_{i}(x_{i}-E_{1}(x_{i}))a_{i}\|\ <\varepsilon,$
$\|a_{i}E_{1}(x_{i})a_{i}\|>\|E_{1}(x_{i})\|-\varepsilon\geq\lambda-\varepsilon,$

for $i=1,2$. Since $A$ is simple, there exists $a\in A_{1}$ satisfying

$\|a_{1}E_{1}(x_{1})a_{1}aa_{2}E_{1}(x_{2})a_{2}\|\geq(\lambda-\varepsilon)^{2},$

and so

$\displaystyle\|x_{1}a_{1}aa_{2}x_{2}\|$
$\displaystyle\geq\|a_{1}x_{1}a_{1}aa_{2}x_{2}a_{2}\|$
$\displaystyle\geq\|a_{1}E_{1}(x_{1})a_{1}aa_{2}E_{1}(x_{2})a_{2}\|-2\varepsilon$
$\displaystyle\geq(\lambda-\varepsilon)^{2}-2\varepsilon.$

Since $\varepsilon>0$ is arbitrary, we obtain (2).

The implications (2)$\Rightarrow$(6)$\Rightarrow$(3)$\Rightarrow$(1) follow
from the same arguments. ∎

######  Remark 3.6.

It was shown in [23, Theorem 7.5, Corollary 7.6] that the above condition
holds for every finite depth inclusion. The condition (3) is adapted as the
definition of the outerness of $E$ in [38], which is essentially the same as
the pinching property in [47, Definition 3.13].

##  4 Proof of Theorem 1.1

As a corollary of Corollary of 3.4, we get the following.

######  Corollary 4.1.

Theorem 1.1 is true if $A^{\alpha}$ is purely infinite.

###### Proof.

We choose a non-zero projection $e\in A^{\alpha}$ with $[e]_{0}=0$ in
$K_{0}(A^{\alpha})$. Then the inclusion $A\supset A^{\alpha}$ is isomorphic to
$eAe\otimes\mathbb{K}\supset eA^{\alpha}e\otimes\mathbb{K}$ if $A^{\alpha}$ is
not unital, and to a corner inclusion of it if $A^{\alpha}$ is unital. Thus
$A\supset A^{\alpha}$ has the property (BEK) if and only if $eAe\supset
eA^{\alpha}e$ has the same property, and we may and do assume that
$A^{\alpha}\in\mathcal{C}_{2}$ to prove the statement.

Theorem 3.4 implies that the endomorphisms $\check{\alpha}_{\pi}$,
$\pi\in\widehat{G}\setminus\\{1\\}$, $d(\pi)>1$, are properly outer. Assume
that $\pi\in\widehat{G}\setminus\\{1\\}$ has $d(\pi)=1$. Note that
$\check{\alpha}_{\pi}$ is an automorphism and it is outer thanks to the
irreducibility of $A\supset A^{\alpha}$. Thus it is properly outer by
Kishimoto’s theorem [33, Lemma 1.1]. ∎

######  Remark 4.2.

When $A^{\alpha}$ belongs to $\mathcal{C}_{2}$, we can apply Doplicher-
Roberts’ construction [12, Theorem 5.1] to $\mathcal{A}=A^{\alpha}$,
$\Delta=\\{\check{\alpha}_{\mathcal{H}}\\}_{\mathcal{H}\in\mathcal{U}(G,\alpha)}$,
and
$\varepsilon(\check{\alpha}_{\mathcal{H}_{1}},\check{\alpha}_{\mathcal{H}_{2}})=\theta(\mathcal{H}_{1},\mathcal{H}_{2})$,
and we obtain the universal C∗-algebra $\mathcal{B}$ containing $\mathcal{A}$
with a $G$-action $\widetilde{\alpha}$ satisfying the following properties
(1)-(4):

  * (1)

$\mathcal{A}=\mathcal{B}^{\widetilde{\alpha}}$.

  * (2)

For each $\mathcal{H}\in\mathcal{U}(G,\alpha)$, there exists a copy
$\widetilde{\mathcal{H}}\in\mathcal{U}(G,\widetilde{\alpha})$, as a $G$-space,
of $\mathcal{H}$ such that $\mathcal{B}$ is generated by $\mathcal{A}$ and
$\cup_{\mathcal{H}\in\mathcal{U}(G,\alpha)}\widetilde{\mathcal{H}}$.

  * (3)

The Hilbert space $\widetilde{\mathcal{H}}$ implements
$\check{\alpha}_{\mathcal{H}}$ in the sense that
$Vx=\check{\alpha}_{\mathcal{H}}(x)V$ holds for all
$V\in\widetilde{\mathcal{H}}$ and $x\in\mathcal{A}$.

  * (4)

$\varepsilon(\check{\alpha}_{\mathcal{H}_{1}},\check{\alpha}_{\mathcal{H}_{2}})=\theta(\widetilde{\mathcal{H}}_{1},\widetilde{\mathcal{H}}_{2})$.

Moreover the inclusion $\mathcal{B}\supset\mathcal{A}$ is irreducible.

By the universality of $\mathcal{B}$, there exists a $G$-equivariant
surjection from $\mathcal{B}$ onto $A$ that is the identity on $A^{\alpha}$.
On the other hand, we can apply Corollary 4.1 to $\mathcal{B}$ and see that
$\mathcal{B}$ is simple. Therefore the map from $\mathcal{B}$ to $A$ is an
isomorphism. This means that $A$ has universality with respect to (1)-(4),
which we will use in the proof of Theorem 1.1 in the general case.

As a consequence of Corollary 4.1, we get the following lemma. Recall that if
$\beta$ is an ergodic action of $G$ on a C∗-algebra $B$, i.e.
$B^{\beta}=\mathbb{C}$, the C∗-algebra $B$ is necessarily nuclear (see [50,
Lemma 22], [10, Proposition 3]).

######  Lemma 4.3.

Let the notation be as in the assumption of Theorem 1.1. Let $\beta$ be an
ergodic action of $G$ on a C∗-algebra $B$, and let $\gamma$ be the diagonal
action $\gamma_{g}=\alpha_{g}\otimes\beta_{g}$ of $G$ on $A\otimes B$. Then
the fixed point algebra $(A\otimes B)^{\gamma}$ is simple. More over there
exists a conditional expectation $F:(A\otimes B)^{\gamma}\to
A^{\alpha}\otimes\mathbb{C}$.

###### Proof.

Note that the restriction $F$ of $E_{\alpha}\otimes E_{\beta}$ to $(A\otimes
B)^{\gamma}$ is a faithful conditional expectation from $(A\otimes
B)^{\gamma}$ onto $A^{\alpha}\otimes\mathbb{C}$.

Replacing $A$ and $\alpha$ with $A\otimes\mathcal{O}_{\infty}$ and
$\alpha\otimes\operatorname{id}_{\mathcal{O}_{\infty}}$ respectively, we may
assume that $A^{\alpha}$ is purely infinite. Moreover, passing to the corner
by a projection in $A^{\alpha}$, we may and do assume that
$A^{\alpha}\in\mathcal{C}_{2}$.

We first recall basic facts about ergodic actions (see [19, Proposition
2.1,Theorem 4.1], [51, Theorem 1]). The conditional expectation
$E_{\beta}:B\to B^{\beta}=\mathbb{C}$ is a trace, and we denote it by $\tau$
for simplicity. There exist non-negative integers $m_{\pi}\leq d(\pi)$ with
$\dim B^{\beta}(\pi)=m_{\pi}d(\pi)$ for $\pi\in\widehat{G}$. We can choose
$X(\pi)_{a}\in B^{\beta}_{1}(\pi)$, $a=1,2,\cdots,m_{\pi}$ such that
$\\{X(\pi)_{ai}\\}_{a,i}$ form an orthonormal basis of $B^{\beta}(\pi)$ with
respect to the inner product given by $\tau$. Then we have

$\sum_{i=1}^{d(\pi)}X(\pi)_{ai}X(\pi)_{bi}^{*}=\delta_{a,b}d(\pi).$

Let

$W(\pi)_{a}=\frac{1}{\sqrt{d(\pi)}}\sum_{i=1}^{d(\pi)}\psi(\pi)_{i}\otimes
X(\pi)_{ai}^{*}.$

Then $W(\pi)_{a}$, $a=1,2,\ldots,m_{\pi}$, are isometries in $(A\otimes
B)^{\gamma}$ with mutually orthogonal ranges. Since the linear span of
$\cup_{\pi_{1},\pi_{2}}A^{\alpha}(\pi_{1})\otimes B^{\beta}(\pi_{2})$ is dense
in $A\otimes B$, the linear span of
$\cup_{\pi_{1},\pi_{2}}E_{\gamma}(A^{\alpha}(\pi_{1})\otimes
B^{\beta}(\pi_{2}))$ is dense in $(A\otimes B)^{\gamma}$. On the other hand,
we have $A^{\alpha}(\pi)=A^{\alpha}\mathcal{H}_{\pi}$, and
$E_{\gamma}(A^{\alpha}(\pi_{1})\otimes B^{\beta}(\pi_{2}))$ survives only when
$\pi_{2}$ is equivalent to $\overline{\pi_{1}}$. Thus we can see that the
linear span of $\cup_{\pi,a}(A^{\alpha}\otimes 1)W(\pi)_{a}$ is dense in
$(A\otimes B)^{\gamma}$. Note that

$W(\pi)_{a}(x\otimes 1)=(\check{\alpha}_{\pi}(x)\otimes 1)W(\pi)_{a},$

holds for all $x\in A^{\alpha}$. Since $F$ is faithful and
$\check{\alpha}_{\pi}$, $\pi\in\widehat{G}\setminus\\{1\\}$, are all properly
outer, we can conclude that $(A\otimes B)^{\gamma}$ is simple as in the case
of a crossed product by a discrete group. ∎

We proceed to the proof of Theorem 1.1 in the general case, which requires
several reduction steps. Since the inclusion $A\supset A^{\alpha}$ has the
property (BEK) if and only if $A\otimes\mathbb{K}\supset
A^{\alpha}\otimes\mathbb{K}$ has the same property, we may assume
$A^{\alpha}\in\mathcal{C}_{1}$ to prove Theorem 1.1. Our task is to show that
$\check{\alpha}_{\pi}$ is properly outer for $\pi\in\widehat{G}$ with
$d(\pi)>1$.

We fix $\pi_{0}\in\widehat{G}$ with $d(\pi_{0})>1$ and assume on the contrary
that $\check{\alpha}_{\pi_{0}}$ is not properly outer. Then the proof of [23,
Theorem 7.5], which is an adaptation of Kishimoto’s argument in [33] to the
endomorphism case, shows that there would exist $b\in M(A^{\alpha})$, a non-
zero hereditary C∗-subalgebra $C$ of $A^{\alpha}$, and $\delta>0$ such that
for any irreducible representation $(\Pi,H)$ of $A^{\alpha}$, there exists a
finite rank self-adjoint operator $T$ on $H$ and isometry
$V\in\operatorname{Hom}_{A}(\Pi,\Pi\circ\check{\alpha}_{\pi_{0}})$ satisfying

$\Pi(c)^{*}(\Pi(b)V+V^{*}\Pi(b)^{*}+T)\Pi(c)\geq\delta\Pi(c^{*}c),$

for all $c\in C$. Let
$\mathcal{L}_{\pi}=\operatorname{Hom}_{A^{\alpha}}(\Pi,\Pi\circ\check{\alpha}_{\pi})$,
which is a finite dimensional Hilbert space in $\mathbb{B}(H)$. Let $D$ be the
C∗-algebra generated by
$\cup_{\pi\in\widehat{G}}\Pi(A^{\alpha})\mathcal{L}_{\pi}$. As was observed in
the proof of [23, Theorem 7.5], if $D$ is simple and there exists a
conditional expectation $F$ from $D$ onto $\Pi(A^{\alpha})$, we can get a
contradiction. Therefore Theorem 1.1 is reduced to the following lemma:

######  Lemma 4.4.

Under the assumption in Theorem 1.1, we assume $A^{\alpha}\in\mathcal{C}_{1}$.
Let $(\Pi,H)$ be an irreducible representation of $A^{\alpha}$, let
$\mathcal{L}_{\pi}=\operatorname{Hom}_{A^{\alpha}}(\Pi,\Pi\circ\check{\alpha}_{\pi})$
for $\pi\in\widehat{G}$, and let $D$ be the C∗-algebra generated by
$\cup_{\pi\in\widehat{G}}\Pi(A^{\alpha})\mathcal{L}_{\pi}$. Then $D$ is simple
and there exists a conditional expectation from $D$ onto $\Pi(A^{\alpha})$.

For simplicity, we suppress the symbol $\Pi$ in $\Pi(A^{\alpha})$, and treat
$A^{\alpha}$ as a subalgebra of $\mathbb{B}(H)$ irreducibly acting on $H$. For
each $\pi\in\widehat{G}$, we choose an orthonormal basis
$\\{V(\pi)_{p}\\}_{p=1}^{m_{\pi}}$ of $\mathcal{L}_{\pi}$. Note that we have
$m_{1}=1$, and we can choose $V(1)_{1}=1$. Then as in [23, Lemma 7.1], we can
show that there exist
$C_{(\pi,p),(\sigma,q)}^{(\mu,r)}\in(\check{\alpha}_{\mu},\check{\alpha}_{\pi}\check{\alpha}_{\sigma})$
and $c_{(\pi,p),(\overline{\pi},q)}\in\mathbb{C}$ satisfying

$V(\pi)_{p}V(\sigma)_{q}=\sum_{\mu,r}C_{(\pi,p),(\sigma,q)}^{(\mu,r)}V(\mu)_{r},$
(4.1)
$V(\pi)_{p}^{*}=\sum_{q}c_{(\pi,p),(\overline{\pi},q)}R_{\pi}^{*}V(\overline{\pi})_{q}.$
(4.2)

Let $D_{0}$ be the linear span of
$\cup_{\pi\in\widehat{G}}A^{\alpha}\mathcal{L}_{\pi}$. Then the above
relations show that $D_{0}$ is a dense $*$-subalgebra of $D$, and every
approximate unit of $A^{\alpha}$ is an approximate unit of $D$. In particular,
we have $\overline{A^{\alpha}D}^{\|\cdot\|}=D$.

We prove Lemma 4.4 by using Doplicher-Roberts’ endomorphism crossed product in
[12]. As it is formulated only for unital C∗-algebras, we show that Lemma 4.4
is reduced to the same statement with $A^{\alpha}\in\mathcal{C}_{2}$.

We first show that Lemma 4.4 is reduced to the case where $A^{\alpha}$ is
purely infinite. To see it, we fix an irreducible representation
$(\Pi_{1},H_{1})$ of $\mathcal{O}_{\infty}$, and replace $A$ and $(\Pi,H)$
with $A\otimes\mathcal{O}_{\infty}$ and $(\Pi\otimes\Pi_{1},H\otimes H_{1})$
respectively. If Lemma 4.4 holds for purely infinite
$A^{\alpha}\in\mathcal{C}_{1}$, we see that $D\otimes\mathcal{O}_{\infty}$ is
simple and there exists a conditional expectation from
$D\otimes\mathcal{O}_{\infty}$ onto $A^{\alpha}\otimes\mathcal{O}_{\infty}$.
This implies that $D$ is simple and there exists a conditional expectation
from $D$ onto $A^{\alpha}$.

Next we show that Lemma 4.4 for purely infinite $A^{\alpha}\in\mathcal{C}_{1}$
is reduced to the same statement for $A^{\alpha}\in\mathcal{C}_{2}$. We choose
a system of matrix units $\\{e_{i,j}\\}_{i,j\geq 1}$ in $A^{\alpha}$ such that
$[e_{11}]_{0}=0$ in $K_{0}(A^{\alpha})$ and
$\\{\sum_{i=1}^{n}e_{ii}\\}_{n=1}^{\infty}$ is an approximate unit of
$A^{\alpha}$, and hence converges to 1 in the strict topology. We denote
$e=e_{11}$ and $p_{n}=\sum_{i=1}^{n}e_{ii}$ for simplicity. Note that $e$ is a
full projection of $D$ because $\\{p_{n}\\}_{n=1}^{\infty}$ is an approximate
unit of $D$ too. Note that $\\{\check{\alpha}_{\pi}(p_{n})\\}_{n=1}^{\infty}$
is an approximate unit of $A^{\alpha}$ as
$\check{\alpha}_{\pi}\in\operatorname{End}(A^{\alpha})_{0}$.

For each $\pi\in\widehat{G}$, we choose a partial isometry $v_{\pi}\in
A^{\alpha}$ satisfying $v_{\pi}v_{\pi}^{*}=e$ and
$v_{\pi}^{*}v_{\pi}=\check{\alpha}_{\pi}(e)$. Then

$U_{\pi}=\sum_{i=1}^{\infty}e_{i1}v_{\pi}\check{\alpha}_{\pi}(e_{1i})$

converges to a unitary in $M(A^{\alpha})$ in the strict topology, and
$\operatorname{Ad}U_{\pi}\cdot\check{\alpha}_{\pi}(e_{ij})=e_{ij}$. Replacing
$\mathcal{H}_{\pi}$ with $U_{\pi}\mathcal{H}_{\pi}$ implies replacing
$\mathcal{L}_{\pi}$ with $U_{\pi}\mathcal{L}_{\pi}$, which does not change
$D$. Thus we may do assume $\check{\alpha}_{\pi}(e_{ij})=e_{ij}$, and

$(D\supset A^{\alpha},\check{\alpha}_{\pi})\cong(eDe\otimes\mathbb{K}\supset
eA^{\alpha}e\otimes\mathbb{K},\check{\alpha}_{\pi}|_{eA^{\alpha}e}\otimes\operatorname{id}_{\mathbb{K}}).$

Therefore if the same statement as in Lemma 4.4 for
$A^{\alpha}\in\mathcal{C}_{2}$ holds, we can see that $eDe$ is simple and
there exists a conditional expectation from $eDe$ onto $eA^{\alpha}e$, which
shows that Lemma 4.4 holds.

Now Theorem 1.1 is reduced to the following lemma:

######  Lemma 4.5.

Under the assumption in 1.1, we assume $A^{\alpha}\in\mathcal{C}_{2}$. Let
$(\Pi,H)$ be an irreducible representation of $A^{\alpha}$, let
$\mathcal{L}_{\pi}=\operatorname{Hom}_{A^{\alpha}}(\Pi,\Pi\circ\check{\alpha}_{\pi})$
for $\pi\in\widehat{G}$, and let $D$ be the C∗-algebra generated by
$\Pi(A^{\alpha})$ and $\cup_{\pi\in\widehat{G}}\mathcal{L}_{\pi}$. Then $D$ is
simple and there exists a conditional expectation from $D$ onto
$\Pi(A^{\alpha})$.

######  Remark 4.6.

Let $\operatorname{Rep}(A^{\alpha})_{0}$ be the category representations of
$A^{\alpha}$ unitarily equivalent to finite direct sums of irreducible
representations of $A^{\alpha}$. Then Lemma 2.8 shows that
$\operatorname{Rep}(A^{\alpha})_{0}$ is a module category of
$\mathcal{U}(G,\alpha)$ via
$\Pi\otimes\mathcal{H}:=\Pi\circ\check{\alpha}_{\mathcal{H}}$. It is known
that there is a correspondence between module categories of $\mathcal{U}(G)$
and ergodic $G$-actions (see [9, Theorem 6.4]). In fact, we prove Lemma 4.5 by
describing the structure of $D$ in terms of an ergodic $G$-action.

From now on, we assume $A^{\alpha}\in\mathcal{C}_{2}$. We choose an
orthonormal basis $\\{V(\pi)_{p}\\}_{p}$ of $\mathcal{L}_{\pi}$ for each
$\pi\in\widehat{G}$. Then there exists
$C_{(\pi,p),(\sigma,q)}^{(\mu,r)}\in(\check{\alpha}_{\mu},\check{\alpha}_{\pi}\check{\alpha}_{\sigma})$
and $c_{(\pi,p),(\overline{\pi},q)}\in\mathbb{C}$ satisfying Eq.(4.1) and
(4.2).

Instead of working on $D$ directly, we first consider its universal
counterpart. Let $\mathcal{D}$ be the universal C∗-algebra generated by
$A^{\alpha}$ and isometries $V^{\prime}(\pi)_{p}$, $\pi\in\widehat{G}$,
$p=1,2,\cdots,m_{\pi}$, satisfying the following relations:

$V^{\prime}(\pi)_{p}^{*}V^{\prime}(\pi)_{q}=\delta_{p,q},$ (4.3)
$V^{\prime}(\pi)_{p}x=\check{\alpha}_{\pi}(x)V^{\prime}(\pi)_{p},\quad x\in
A^{\alpha},$ (4.4)
$V^{\prime}(\pi)_{p}V^{\prime}(\sigma)_{q}=\sum_{\mu,r}C_{(\pi,p),(\sigma,q)}^{(\mu,r)}V^{\prime}(\mu)_{r},$
(4.5)
$V^{\prime}(\pi)_{p}^{*}=\sum_{q}c_{(\pi,p),(\overline{\pi},q)}R_{\pi}^{*}V^{\prime}(\overline{\pi})_{q}.$
(4.6)

######  Lemma 4.7.

Let the notation be as above.

  * $(1)$

For each $\mathcal{H}\in\mathcal{U}(G,\alpha)$, there exists
$\check{\alpha}^{\prime}_{\mathcal{H}}\in\operatorname{End}(\mathcal{D})$
satisfying
$\check{\alpha}^{\prime}_{\mathcal{H}}|_{A^{\alpha}}=\check{\alpha}_{\mathcal{H}}$,
and
$\check{\alpha}^{\prime}_{\mathcal{H}}(V^{\prime}(\pi)_{p})=\theta(\pi,\mathcal{H})V^{\prime}(\pi)_{p}$.

  * $(2)$

We set $\check{\alpha}^{\prime}_{T}=T$ for
$T\in(\mathcal{H}_{1},\mathcal{H}_{2})_{G}$. Then $\check{\alpha}^{\prime}$ is
a tensor functor from $\mathcal{U}(G,\alpha)$ to
$\operatorname{End}(\mathcal{D})$.

###### Proof.

(1) By the universality of $\mathcal{D}$, it suffices to show that
$\check{\alpha}^{\prime}_{\mathcal{H}}$ preserves the relations (4.3)-(4.6).
Since $\theta(\pi,\mathcal{H})$ is unitary, the relation (4.3) is preserved.
We can see that (4.4) is preserved from

$\theta(\pi,\mathcal{H})V^{\prime}(\pi)_{p}\check{\alpha}_{\pi}(x)=\theta(\pi,\mathcal{H})\check{\alpha}_{\pi}\check{\alpha}_{\mathcal{H}}(x)V^{\prime}(\pi)_{p}=\check{\alpha}_{\mathcal{H}}\check{\alpha}_{\pi}(x)\theta(\pi,\mathcal{H})V^{\prime}(\pi)_{p}.$

To see that (4.5) is preserved, it suffices to show

$\theta(\pi,\mathcal{H})V^{\prime}(\pi)_{p}\theta(\sigma,\mathcal{H})V^{\prime}(\sigma)_{q}=\sum_{\mu,r}\check{\alpha}_{\mathcal{H}}(C_{(\pi,p),(\sigma,q)}^{(\mu,r)})\theta(\mu,\mathcal{H})V^{\prime}(\mu)_{r}.$

The left-hand side is equal to

$\displaystyle\theta(\pi,\mathcal{H})\check{\alpha}_{\pi}(\theta(\sigma,\mathcal{H}))V^{\prime}(\pi)_{p}V^{\prime}(\sigma)_{q}=\theta(\pi\otimes\sigma,\mathcal{H})\sum_{\mu,r}C_{(\pi,p),(\sigma,q)}^{(\mu,r)}V^{\prime}(\mu)_{r}$
$\displaystyle=\sum_{\mu,r}\left(\theta(\pi\otimes\sigma,\mathcal{H})C_{(\pi,p),(\sigma,q)}^{(\mu,r)}\theta(\mathcal{H},\mu)\right)\theta(\mu,\mathcal{H})V^{\prime}(\mu)_{r},$

which coincides with the right-hand side thanks to Eq.(2.4). Finally to see
that the relation (4.6) is preserved, it suffices to show

$V^{\prime}(\pi)_{p}^{*}\theta(\mathcal{H},\pi)=\sum_{q}c_{(\pi,p),(\overline{\pi},q)}\check{\alpha}_{\mathcal{H}}(R_{\pi}^{*})\theta(\overline{\pi},\mathcal{H})V^{\prime}(\overline{\pi})_{q}.$

The left-hand side is

$\displaystyle\sum_{q}c_{(\pi,p),(\overline{\pi},q)}R_{\pi}^{*}V^{\prime}(\overline{\pi})_{q}\theta(\mathcal{H},\pi)=\sum_{q}c_{(\pi,p),(\overline{\pi},q)}R_{\pi}^{*}\check{\alpha}_{\overline{\pi}}(\theta(\mathcal{H},\pi))V^{\prime}(\overline{\pi})_{q}$
$\displaystyle=\sum_{q}c_{(\pi,p),(\overline{\pi},q)}\left(R_{\pi}^{*}\check{\alpha}_{\overline{\pi}}(\theta(\mathcal{H},\pi))\theta(\mathcal{H},\overline{\pi})\right)\theta(\overline{\pi},\mathcal{H})V^{\prime}(\overline{\pi})_{q}.$

Since Eq.(2.3) and(2.4) imply

$\theta(\overline{\pi},\mathcal{H})\check{\alpha}_{\overline{\pi}}(\theta(\pi,\mathcal{H}))R_{\pi}=\theta(\overline{\pi}\otimes\pi,\mathcal{H})R_{\pi}=\check{\alpha}_{\mathcal{H}}(R_{\pi}),$

the relation (4.6) is preserved, and $\check{\alpha}^{\prime}_{\mathcal{H}}$
is well-defined.

(2) It suffices to show that $T\in(\mathcal{H}_{1},\mathcal{H}_{2})_{G}$
satisfies

$T\theta(\pi,\mathcal{H}_{1})V^{\prime}(\pi)_{p}=\theta(\pi,\mathcal{H}_{2})V^{\prime}(\pi)_{p}T.$

The right-hand side is

$\theta(\pi,\mathcal{H}_{2})\check{\alpha}_{\pi}(T)V^{\prime}(\pi)_{p}=T\theta(\pi,\mathcal{H}_{1})V^{\prime}(\pi)_{p},$

by Eq.(2.4), and the equality holds. ∎

###### Proof of Lemma 4.5.

We apply [12, Theorem 5.1] to $\mathcal{A}=\mathcal{D}$,
$\Delta=\\{\check{\alpha}^{\prime}_{\mathcal{H}}\\}_{\mathcal{H}\in\mathcal{U}(G,\alpha)}$,
and
$\varepsilon(\check{\alpha}^{\prime}_{\mathcal{H}_{1}},\check{\alpha}^{\prime}_{cH_{2}})=\theta(\mathcal{H}_{1},\mathcal{H}_{2})$.
Although [12, Theorem 5.1] is stated under the condition that $\mathcal{A}$
has trivial center, their construction itself works without this condition.
Thanks to Remark 4.2, the resulting algebra $\mathcal{B}$ satisfies the
following properties:

  * (1)

The C∗-algebra $\mathcal{B}$ is generated by $\mathcal{D}$ and $A$ with a
common C∗-subalgebra $A^{\alpha}$.

  * (2)

For every $\mathcal{H}\in\mathcal{U}(G,\alpha)$, the Hilbert space
$\mathcal{H}$ implements $\check{\alpha}^{\prime}_{\mathcal{H}}$.

  * (3)

There exists a $G$-action $\alpha^{\prime}$ on $\mathcal{B}$ extending
$\alpha$ such that $\mathcal{B}^{\alpha^{\prime}}=\mathcal{D}$.

For $\pi\in\widehat{G}$, let
$Y(\pi)_{pi}=V^{\prime}(\pi)_{p}^{*}\psi(\pi)_{i}$. Then
$Y(\pi)_{a}\in\mathcal{B}^{\alpha^{\prime}}(\pi)$. Since $\mathcal{D}$ is
generated by $A^{\alpha}$ and $\cup_{\pi\in\widehat{G}}\mathcal{L}_{\pi}$, and
$A$ is generated by $A^{\alpha}$ and
$\cup_{\pi\in\widehat{G}}\mathcal{H}_{\pi}$, the C∗-algebra $\mathcal{B}$ is
generated by $A$ and $\\{Y(\pi)_{pi}\\}_{p,i}$. We claim that $Y(\pi)_{pi}$
commutes with $A$. It commutes with $\mathcal{A}^{\alpha}$ by construction.
Since the linear span of
$\cup_{\sigma\in\widehat{G}}A^{\alpha}\mathcal{H}_{\sigma}$ is dense in $A$,
it suffices to show that $Y(\pi)_{pi}$ commutes with $\psi(\sigma)_{j}$.
Indeed,

$\displaystyle\psi(\sigma)_{j}Y(\pi)_{pi}=\check{\alpha}^{\prime}_{\sigma}(V^{\prime}(\pi)_{p}^{*})\psi(\sigma)_{j}\psi(\pi)_{i}$
$\displaystyle=V^{\prime}(\pi)_{p}^{*}\theta(\sigma,\pi)\psi(\sigma)_{j}\psi(\pi)_{i}=V^{\prime}(\pi)_{p}^{*}\psi(\pi)_{i}\psi(\sigma)_{j}=Y(\pi)_{pi}\psi(\sigma)_{j},$

which shows the claim.

We next claim that the linear span $B_{0}$ of $\\{Y(\pi)_{ai}\\}_{\pi,a,i}$.is
a $*$-algebra. Indeed,

$\displaystyle Y(\pi)_{pi}^{*}$
$\displaystyle=\psi(\pi)_{i}^{*}V^{\prime}(\pi)_{p}=\sqrt{d(\pi)}R_{\pi}^{*}\psi(\overline{\pi})_{i}V^{\prime}(\pi)_{p}=\sqrt{d(\pi)}R_{\pi}^{*}\check{\alpha}^{\prime}_{\overline{\pi}}(V^{\prime}(\pi)_{p})\psi(\overline{\pi})_{i}$
$\displaystyle=\sqrt{d(\pi)}R_{\pi}^{*}\theta(\pi,\overline{\pi})V^{\prime}(\pi)_{p}\psi(\overline{\pi})_{i}=\sqrt{d(\pi)}\overline{R}_{\pi}^{*}V^{\prime}(\pi)_{p}\psi(\overline{\pi})_{i}$
$\displaystyle=\sqrt{d(\pi)}\overline{R}_{\pi}^{*}\sum_{q}\overline{c((\pi,p),(\overline{\pi},q))}V^{\prime}(\overline{\pi})_{q}^{*}R_{\pi}\psi(\overline{\pi})_{i}$
$\displaystyle=\sqrt{d(\pi)}\sum_{q}\overline{c((\pi,p),(\overline{\pi},q))}V^{\prime}(\overline{\pi})_{q}^{*}\check{\alpha}_{\overline{\pi}}(\overline{R}_{\pi}^{*})R_{\pi}\psi(\overline{\pi})_{i}$
$\displaystyle=\frac{1}{\sqrt{d(\pi)}}\sum_{q}\overline{c((\pi,p),(\overline{\pi},q))}Y(\overline{\pi})_{qi},$

shows that $B_{0}$ is closed under $*$. For the product, we have

$\displaystyle
Y(\sigma)_{qj}Y(\pi)_{pi}=V^{\prime}(\sigma)_{q}^{*}\psi(\sigma)_{j}V^{\prime}(\pi)_{p}^{*}\psi(\pi)_{i}=V^{\prime}(\sigma)_{q}^{*}\check{\alpha}^{\prime}_{\sigma}(V^{\prime}(\pi)_{p}^{*})\psi(\sigma)_{j}\psi(\pi)_{i}$
$\displaystyle=V^{\prime}(\sigma)_{q}^{*}V^{\prime}(\pi)_{p}^{*}\theta(\sigma,\pi)\psi(\sigma)_{j}\psi(\pi)_{i}=\sum_{\mu,r}V^{\prime}(\mu)_{r}^{*}{C_{(\pi,p),(\sigma,q)}^{(\mu,r)}}^{*}\psi(\pi)_{i}\psi(\sigma)_{j}$
$\displaystyle=\sum_{\mu,r}\sum_{\nu,k}d(\nu)V^{\prime}(\mu)_{r}^{*}{C_{(\pi,p),(\sigma,q)}^{(\mu,r)}}^{*}E_{\alpha}(\psi(\pi)_{i}\psi(\sigma)_{j}\psi(\nu)_{k}^{*})\psi(\nu)_{k}.$

Since
$C_{(\pi,p),(\sigma,q)}^{(\mu,r)}\in(\check{\alpha}_{\mu},\check{\alpha}_{\pi}\check{\alpha}_{\sigma})$
and
$E_{\alpha}(\psi(\pi)_{i}\psi(\sigma)_{j}\psi(\nu)_{k}^{*})\in(\check{\alpha}_{\nu},\check{\alpha}_{\pi}\check{\alpha}_{\sigma})$,
the product
${C_{(\pi,p),(\sigma,q)}^{(\mu,r)}}^{*}E_{\alpha}(\psi(\pi)_{i}\psi(\sigma)_{j}\psi(\nu)_{k}^{*})$
survives only if $\mu=\nu$, and

$\displaystyle Y(\sigma)_{qj}Y(\pi)_{pi}$
$\displaystyle=\sum_{\mu,r,s}d(\mu)\langle{E_{\alpha}(\psi(\pi)_{i}\psi(\sigma)_{j}\psi(\mu)_{k}^{*}),C_{(\pi,p),(\sigma,q)}^{(\mu,r)}}\rangle
V^{\prime}(\mu)_{r}^{*}\psi(\mu)_{k}$
$\displaystyle=\sum_{\mu,r,s}d(\mu)\langle{E_{\alpha}(\psi(\pi)_{i}\psi(\sigma)_{j}\psi(\mu)_{k}^{*}),C_{(\pi,p),(\sigma,q)}^{(\mu,r)}}\rangle
Y(\mu)_{rk}.$

Therefore the claim is shown.

Let $B$ be the closure of $B_{0}$ and let $\beta$ be the restriction of
$\alpha^{\prime}$ to $B$. Then $B^{\beta}(\pi)$ is the linear span of
$\\{Y(\pi)_{ai}\\}_{a,i}$, and in particular we have $B^{\beta}=\mathbb{C}$,
that is, the action $\beta$ is ergodic. Thus $B$ is nuclear. Since $A$ is
simple and $\mathcal{B}$ is generated by two mutually commuting C∗-subalgebras
$A$ and $B$, we can identify $\mathcal{B}$ with $A\otimes B$. Thus Lemma 4.3
implies that $\mathcal{D}=\mathcal{B}^{\alpha^{\prime}}$ is simple and there
exists a conditional expectation from $\mathcal{D}$ onto $A^{\alpha}$. By the
universality of $\mathcal{D}$, there exists an isomorphism from $\mathcal{D}$
onto $D$ that is the identity on $A^{\alpha}$. Thus Lemma 4.5 is shown. ∎

##  5 Fejér type approximation for compact groups

In this section, we assume that $G$ is a second countable compact group, and
we use the notation in Section 2.4. We always use the normalized Haar measure
of $G$. We equip $C(G)$ with a $G$-action by the left translation
$L_{g}(f)(h)=f(g^{-1}h)$ for $f\in C(G)$. We denote by $\lambda$ the left
regular representation of $G$. For $f\in C(G)$, we denote by
$M_{f}\in\mathbb{B}(L^{2}(G))$ the multiplication operator by $f$.

We first establish a $G$-equivariant CPAP of $C(G)$, which may be interpreted
as a Fejér type approximation. The orthogonality relation shows that we have a
completely orthonormal system
$\\{\sqrt{d(\pi)}\pi_{ij}\\}_{\pi\in\widehat{G},\;1\leq i,j\leq d(\pi)}$ of
$L^{2}(G)$. We denote by

$\\{E_{\sqrt{d(\pi)}\pi_{ij},\sqrt{d(\sigma)}\sigma_{kl}}\\}_{\pi,\sigma\in\widehat{G},\;1\leq
i,j\leq d(\pi),\;1\leq k,l\leq d(\sigma)}$

the corresponding system of matrix units in $\mathbb{B}(L^{2}(G))$. For a
finite subset $F\subset\widehat{G}$, we define a finite rank projection
$P_{F}\in\mathbb{B}(L^{2}(G))$ commuting with $\lambda_{g}$ by

$P_{F}=\sum_{\pi\in F}\sum_{1\leq i,j\leq
d(\pi)}E_{\sqrt{d(\pi)}\pi_{ij},\sqrt{d(\pi)}\pi_{ij}}.$

Mimicking the construction in the proof of [7, Theorem 2.6.8] for the reduced
group C∗-algebra of a discrete group, we define ucp maps
$\varphi_{F}:C(G)\to\mathbb{B}(P_{F}L^{2}(G))$ and
$\varphi^{\prime}_{F}:\mathbb{B}(P_{F}L^{2}(G))\to C(G)$ by
$\varphi_{F}(f)=P_{F}M_{f}P_{F}$, and

$\varphi^{\prime}_{F}(E_{\sqrt{d(\pi)}\pi_{ij},\sqrt{d(\sigma)}\sigma_{kl}})=\frac{1}{w(F)}\sqrt{d(\pi)d(\sigma)}\pi_{ij}\overline{\sigma_{kl}},$
$w(F)=\sum_{\pi\in F}d(\pi)^{2}.$

Then $\varphi_{F}$ and $\varphi^{\prime}_{F}$ are $G$-equivariant in the sense
that $\varphi_{F}\circ
L_{g}=\operatorname{Ad}(P_{F}\lambda_{g})\circ\varphi_{F}$ and
$\varphi^{\prime}_{F}\circ\operatorname{Ad}(P_{F}\lambda_{g})=L_{g}\circ\varphi^{\prime}_{F}$
hold.

We define a kernel function $K_{F}\in C(G)$ associated with $F$ by

$K_{F}(g)=\frac{1}{w(F)}\left|\sum_{\pi\in F}d(\pi)\chi_{\pi}(g)\right|^{2}.$

Recall that the convolution $f_{1}*f_{2}$ of $f_{1},f_{2}\in C(G)$ is defined
by

$f_{1}*f_{2}(g)=\int_{G}f_{1}(h)f_{2}(h^{-1}g)dh=\int_{G}f_{1}(gh^{-1})f_{2}(h)dh.$

######  Lemma 5.1.

For $f\in C(G)$, we have

$\varphi^{\prime}_{F}\circ\varphi_{F}(f)=f*K_{F}.$

###### Proof.

We have

$\displaystyle\varphi_{F}(f)=\sum_{\pi,\sigma\in
F}\sum_{i,j,k,l}\langle{M_{f}\sqrt{d(\sigma)}\sigma_{kj},\sqrt{d(\pi)}\pi_{ij}}\rangle
E_{\sqrt{d(\pi)}\pi_{ij},\sqrt{d(\sigma)}\sigma_{kl}}$
$\displaystyle=\sum_{\pi,\sigma\in
F}\sum_{i,j,k,l}\sqrt{d(\pi)d(\sigma)}\int_{G}f(h)\sigma_{kl}(h)\overline{\pi_{ij}(h)}dhE_{\sqrt{d(\pi)}\pi_{ij},\sqrt{d(\sigma)}\sigma_{kl}},$
$\displaystyle\varphi^{\prime}_{F}\circ\varphi_{F}(f)(g)=\frac{1}{w(F)}\sum_{\pi,\sigma\in
F}\sum_{i,j,k,l}d(\pi)d(\sigma)\int_{G}f(h)\sigma_{kl}(h)\overline{\sigma_{kl}(g)}\overline{\pi_{ij}(h)}\pi_{ij}(g)dh$
$\displaystyle=\frac{1}{w(F)}\sum_{\pi,\sigma\in
F}d(\pi)d(\sigma)\int_{G}f(h)\overline{\chi_{\sigma}(h^{-1}g)}\chi_{\pi}(h^{-1}g)dh=\int_{G}f(h)K_{F}(h^{-1}g)dh,$

which shows the statement. ∎

Popa [44] introduced the notion of amenability for rigid C∗-tensor categories
in the context of subfactors. As in the case of discrete groups, he formulated
a Følner sequence and showed that amenability is equivalent to the existence
of a Følner sequence. It was observed in [21] that the amenability is a
property depending only on the fusion algebra structure of the category. The
following version of the definition of a Følner sequence was formulated in
[21, Definition 4.5], and it is equivalent to Popa’s original definition.

######  Definition 5.2.

A sequence $\\{F_{n}\\}_{n=1}^{\infty}$ of finite subsets of $\widehat{G}$ is
said to be a Følner sequence if for every $\pi\in\widehat{G}$, the following
holds:

$\lim_{n\to\infty}\frac{1}{w(F_{n})}\left(\sum_{\sigma\in F_{n}}\sum_{\mu\in
F_{n}^{c}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}+\sum_{\sigma\in
F_{n}^{c}}\sum_{\mu\in
F_{n}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}\right)=0,$

where $F_{n}^{c}$ is the complement of $F_{n}$.

It is well-known that the representation category of $G$ is amenable, and
$\widehat{G}$ always has a Følner sequence (see, for example, [48, Theorem
4.5], [35, subsection 2.3]). One easy way to see it is as follows. Since every
finitely generated subcategory of $\mathcal{U}(G)$ is equivalent to the
representation category of a closed subgroup of $SU(n)$, it has polynomial
growth, and hence has a Følner sequence (see [21, Example 7.6]). Therefore the
whole category has a Følner sequence too.

######  Example 5.3.

When $G=\mathbb{T}$, we have $\widehat{\mathbb{T}}=\mathbb{Z}$, and
$F_{n}=\\{0,1,\ldots,n\\}$, $n\in\mathbb{N}$, give a Følner sequence. The
function $K_{F_{n}}$ in this case is

$K_{F_{n}}(e^{it})=\frac{1}{n+1}\left|\sum_{k=0}^{n}e^{ikt}\right|^{2}=\frac{1}{n+1}\frac{\sin^{2}\frac{(n+1)t}{2}}{\sin^{2}\frac{t}{2}},$

which is nothing but the Fejér kernel.

######  Theorem 5.4.

Let $\\{F_{n}\\}_{n=1}^{\infty}$ be a Følner sequence for $\widehat{G}$. Then
for all $f\in C(G)$,

$\lim_{n\to\infty}\|\varphi^{\prime}_{F_{n}}\circ\varphi_{F_{n}}(f)-f\|_{\infty}=0.$

###### Proof.

Since the linear span of the matrix coefficients of the irreducible
representations is uniformly dense in $C(G)$, it suffice to show the statement
for $f=\pi_{ij}$ with $\pi\in\widehat{G}$. For a finite set
$F\subset\widehat{G}$, we have

$\displaystyle\pi_{ij}*K_{F}(g)=\frac{1}{w(F)}\sum_{k=1}^{d(\sigma)}\pi_{ik}(g)\sum_{\sigma,\mu\in
F}d(\sigma)d(\mu)\int_{G}\overline{\pi_{jk}(h)}\chi_{\sigma}(h)\overline{\chi_{\mu}(h)}dh$
$\displaystyle=\frac{1}{w(F)}\sum_{k=1}^{d(\sigma)}\pi_{ik}(g)\sum_{\sigma,\mu\in
F}d(\sigma)d(\mu)\sum_{\nu\in\widehat{G}}N_{\sigma,\overline{\mu}}^{\nu}\int_{G}\overline{\pi_{jk}(h)}\chi_{\nu}(h)dh$
$\displaystyle=\frac{\pi_{ij}(g)}{w(F)}\sum_{\sigma,\mu\in
F}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\sigma,\overline{\mu}}^{\pi}.$

Since $N_{\sigma,\overline{\mu}}^{\pi}=N_{\pi,\mu}^{\sigma}$, and
$\sum_{\sigma\in\widehat{G}}d(\sigma)N_{\pi,\mu}^{\sigma}=d(\pi)d(\mu)$, we
get

$\displaystyle\frac{1}{w(F)}\sum_{\sigma,\mu\in
F}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\sigma,\overline{\mu}}^{\pi}=\frac{1}{w(F)}\sum_{\sigma,\mu\in
F}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}$
$\displaystyle=\frac{1}{w(F)}\sum_{\mu\in
F}\left(\sum_{\sigma\in\widehat{G}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}-\sum_{\sigma\in
F^{c}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}\right)$
$\displaystyle=1-\frac{1}{w(F)}\sum_{\mu\in F}\sum_{\sigma\in
F^{c}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}.$

This shows

$\|\pi_{ij}-\varphi^{\prime}_{F_{n}}\circ\varphi_{F_{n}}(\pi_{ij})\|_{\infty}\leq\frac{\|\pi_{ij}\|_{\infty}}{w(F_{n})}\sum_{\mu\in
F_{n}}\sum_{\sigma\in
F_{n}^{c}}\frac{d(\sigma)d(\mu)}{d(\pi)}N_{\pi,\mu}^{\sigma}\to
0,\quad(n\to\infty).$

∎

As an application, we show that every nuclear C∗-algebra with a compact group
action has the following equivariant CPAP.

######  Lemma 5.5.

Let $B$ be a unital nuclear C∗-algebra with a $G$-action $\beta$. Then for any
finite subset $F\subset B$ and $\varepsilon>0$, there exists $n\in\mathbb{N}$,
ucp maps $\varphi:A\to\mathbb{M}_{n}(\mathbb{C})$,
$\varphi^{\prime}:\mathbb{M}_{n}(\mathbb{C})\to B$, and a unitary
representation $u$ of $G$ in $\mathbb{M}_{n}(\mathbb{C})$ such that
$\varphi\circ\beta_{g}=\operatorname{Ad}u(g)\circ\varphi$,
$\varphi^{\prime}\circ\operatorname{Ad}u(g)=\beta_{g}\circ\varphi^{\prime}$,
and $\|\varphi^{\prime}\circ\varphi(x)-x\|<\varepsilon$ for all $x\in F$.

###### Proof.

Since $B$ is nuclear and $\beta_{G}(F)$ is a compact set, there exists $n_{0}$
and ucp maps $\varphi_{0}:B\to\mathbb{M}_{n_{0}}(\mathbb{C})$ and
$\varphi^{\prime}_{0}:\mathbb{M}_{n_{0}}(\mathbb{C})\to B$ satisfying

$\|\varphi^{\prime}_{0}\circ\varphi_{0}(\beta_{g}(x))-\beta_{g}(x)\|<\varepsilon/2,\quad\forall
g\in G,\;\forall x\in F.$

We define ucp maps $\Phi:B\to C(G)\otimes\mathbb{M}_{n_{0}}(\mathbb{C})$ and
$\Phi^{\prime}:C(G)\otimes\mathbb{M}_{n_{0}}(\mathbb{C})\to B$ by
$\Phi(x)(g)=\varphi_{0}(\beta_{g^{-1}}(x))$ and
$\Phi^{\prime}(f)=\int_{G}\beta_{g}(\varphi^{\prime}_{0}(f(g)))dg$. Then
$\Phi$ and $\Phi^{\prime}$ are equivariant in the sense that
$\Phi\circ\beta_{g}=(L_{g}\otimes\operatorname{id})\circ\Phi$ and
$\Phi^{\prime}\circ(L_{g}\otimes\operatorname{id})=\beta_{g}\circ\Phi^{\prime}$
hold. For $x\in F$, we have

$\|\Phi^{\prime}\circ\Phi(x)-x\|=\|\int_{G}\beta_{g}(\varphi^{\prime}_{0}\circ\varphi_{0}(\beta_{g^{-1}}(x)))dg-x\|<\varepsilon/2.$

Now the statement follows from Theorem 5.4. ∎

##  6 Proof of Theorem 1.2

We first establish an equivariant version of Kirchberg’s dilation theorem
[30], [31, Proposition 1.7], as an application of Theorem 1.1 and Lemma 5.5.

######  Theorem 6.1.

Under the assumption of Theorem 1.2, we assume that $A$ is unital. Let
$\nu:A\to A$ be a $G$-equivariant ucp map. Then for every finite set $F\subset
A$ and $\varepsilon>0$, there exists $V$ in $A^{\alpha}$ such that
$\|\nu(x)-V^{*}xV\|<\varepsilon$ for all $x\in F$.

###### Proof.

Since $A$ is nuclear, Lemma 5.5 shows that there exist $n\in\mathbb{N}$, a
unitary representation $u$ of $G$ in $\mathbb{M}_{n}(\mathbb{C})$, and
$G$-equivariant ucp maps $\varphi:A\to\mathbb{M}_{n}(\mathbb{C})$ and
$\varphi^{\prime}:\mathbb{M}_{n}(\mathbb{C})\to A$ satisfying
$\|\varphi^{\prime}\circ\varphi(x)-\nu(x)\|<\varepsilon/2$ for all $x\in F$,
where $\mathbb{M}_{n}(\mathbb{C})$ is equipped with a $G$-action given by
$\operatorname{Ad}u_{g}$.

Thanks to Remark 2.13, there exist globally $\alpha$-invariant Hilbert spaces
$\mathcal{H}$ and $\overline{\mathcal{H}}$ with orthonormal bases
$\\{\psi_{i}\\}_{i=1}^{n}$ and $\\{\overline{\psi}_{i}\\}_{i=1}^{n}$
respectively satisfying $\alpha_{g}(\psi_{i})=\sum_{j}u_{ji}(g)\psi_{j}$, and
$\alpha_{g}(\overline{\psi}_{i})=\sum_{j}\overline{u_{ji}(g)}\overline{\psi}_{j}$.
We let $F_{1}=\cup_{i,j}\overline{\psi}_{i}F\overline{\psi}_{j}^{*}$.

Let $\\{\xi_{i}\\}_{i=1}^{n}$ be the standard basis of $\mathbb{C}^{n}$ and
let $\\{e_{ij}\\}_{i,j}$ be the corresponding system of matrix units. We
define a state $\omega$ of $A$ by

$\omega(x)=\frac{1}{n}\sum_{1\leq i,j\leq
n}\langle{\varphi(\overline{\psi}_{i}^{*}x\overline{\psi}_{j})\xi_{j},\xi_{i}}\rangle.$

Then a direct computation shows that $\omega$ is $\alpha$-invariant. We have

$\varphi^{\prime}\circ\varphi(x)=n\sum_{i,j}\omega(\overline{\psi}_{i}x\overline{\psi}_{j}^{*})\varphi^{\prime}(e_{ij}).$

We claim that there exists an isometry $v\in A^{\alpha}$ satisfying

$\|\omega(x)-v^{*}yv\|<\delta,\quad\forall y\in F_{1},$

where $\delta=\varepsilon/(2n^{3})$ for our purpose. Thanks to Theorem 1.1,
the action $\alpha$ is quasi-product, and $\omega$ can be approximated by
$\alpha$-invariant pure states in the weak$*$ topology (see [6, Theorem 3.1]).
Thus there exists an $\alpha$-invariant pure state $\omega_{0}$ of $A$
satisfying $|\omega_{0}(y)-\omega(y)|<\delta/2$ for all $y\in F_{1}$. Let
$L=\\{y\in A;\;\omega_{0}(y^{*}y)=0\\}$. Then $L\cap L^{*}$ is a globally
$\alpha$-invariant hereditary C∗-subalgebra of $A$, and there exists an
approximate units $\\{u_{n}\\}_{n}$ of $L\cap L^{*}$ in $A^{\alpha}$ by Lemma
2.9. Let $a_{n}=1-u_{n}$. Then $\\{a_{n}\\}_{n}$ is a decreasing sequence in
$A^{\alpha}_{+}$ of norm 1 such that
$\\{\|a_{n}(x-\omega_{0}(x))a_{n}\|\\}_{n}$ converges to 0 for all $x\in A$
because $x-\omega_{0}(x)\in\ker\omega_{0}=L+L^{*}$. Since $A^{\alpha}$ is
purely infinite, we can see from this that there exist isometries
$\\{v_{n}\\}_{n}$ in $A^{\alpha}$ such that
$\\{v_{n}^{*}xv_{n}-\omega_{0}(x)\\}_{n}$ converges to 0 for all $x\in A$.
Thus the claim is shown.

Next we claim that there exists an isometry $w\in A^{\alpha}$ satisfying

$\varphi^{\prime}(e_{ij})=w^{*}\psi_{i}\psi_{j}^{*}w,\quad 1\leq i,j\leq n.$

Since $\varphi^{\prime}$ is completely positive, we have
$a:=(\varphi^{\prime}(e_{ij}))\in\mathbb{M}(A)_{n}$ is positive, and a direct
computation shows that it is fixed by
$\alpha_{g}\otimes\operatorname{Ad}\overline{u}_{g}$. Since $b=a^{1/2}$ is
fixed by $\alpha_{g}\otimes\operatorname{Ad}\overline{u}_{g}$, we get

$w:=\sum_{j,k}\psi_{j}\overline{\psi}_{k}b_{kj}\in A^{\alpha}.$

Now we have

$w^{*}\psi_{i}\psi_{j}^{*}w=\sum_{k}b_{ki}^{*}b_{kj}=\varphi(e_{ij}),$

and the claim is shown.

Let

$V=\sqrt{n}\sum_{j=1}^{n}\overline{\psi}_{j}^{*}v\psi_{j}^{*}w,$

which is an element in $A^{\alpha}$. Then for $x\in F$,

$\|\varphi^{\prime}\circ\varphi(x)-V^{*}xV\|=\|n\sum_{i,j}w^{*}\psi_{i}(\omega(\overline{\psi}_{i}x\overline{\psi}_{j}^{*})-v^{*}\overline{\psi}_{i}x\overline{\psi}_{j}^{*}v)\psi_{j}^{*}w\|<\varepsilon/2,$

and the statement is shown. ∎

###### Proof of Theorem 1.2.

Let $e\in A^{\alpha}$ be a non-zero projection with $[e]_{0}=0$ in $K_{0}(A)$.
To show that $\alpha$ is isometrically shift-absorbing, it suffices to show
that the restriction of $\alpha$ to $eAe$ has the same property. Thus we may
and do assume $A^{\alpha}\in\mathcal{C}_{2}$.

We denote by $\alpha^{\infty}_{g}$ the automorphism of $A^{\infty}$ induce by
$\alpha_{g}$. Since the infinite direct sum of the regular representation is
equivalent to $\bigoplus_{\pi\in\widehat{G}}\pi^{\oplus\infty}$, it suffices
to show that there exists a Hilbert space $\mathcal{K}_{\pi}$ in
$A^{\infty}\cap A^{\prime}$ globally $\alpha^{\infty}$-invariant and
equivalent to $\pi$ for each $\pi\in\widehat{G}$ (not necessarily with support
1), and there exists a unital embedding of $\mathcal{O}_{\infty}$ into
$(A^{\alpha})^{\infty}\cap A^{\prime}$. Indeed, we choose an orthonormal basis
$\\{\eta(\pi)\\}_{i=1}^{d(\pi)}$ of $\mathcal{K}_{\pi}$ for each
$\pi\in\widehat{G}$, and isometries
$\\{S_{\pi,n}\\}_{(\pi,n)\in\widehat{G}\times\mathbb{N}}$ in
$(A^{\alpha})^{\infty}\cap A^{\prime}$ with mutually orthogonal ranges. Then

$\\{S_{\pi,n}\eta(\pi)_{i};\;\pi\in\widehat{G},\;1\leq i\leq
d(\pi),\;n\in\mathbb{N}\\}$

give a desired set of generators of $\mathcal{O}_{\infty}$ in $A^{\infty}\cap
A^{\prime}$.

For each $\mathcal{H}\in\mathcal{U}(G,\alpha)$, we apply Theorem 6.1 to
$\rho_{\mathcal{H}}$, and obtain an isometry
$V_{\mathcal{H}}\in(A^{\alpha})^{\infty}$ satisfying
$\rho_{\mathcal{H}}(x)=V_{\mathcal{H}}^{*}xV_{\mathcal{H}}$ for all $x\in A$.
We set $\mathcal{K}_{\mathcal{H}}=V_{\mathcal{H}}\mathcal{H}$, which is a
Hilbert space in $A^{\infty}$, and the restriction of $\alpha^{\infty}$ to
$\mathcal{K}_{\mathcal{H}}$ is equivalent to $\mathcal{H}$. For $x\in A$, we
have

$\psi(\mathcal{H})_{i}^{*}V_{\mathcal{H}}^{*}xV_{\mathcal{H}}\psi(\mathcal{H})_{i}=\psi(\mathcal{H})_{i}^{*}\rho_{\mathcal{H}}(x)\psi(\mathcal{H})_{i}=x,$

and

$\displaystyle[x,V_{\mathcal{H}}\psi(\mathcal{H})_{i}]^{*}[x,V_{\mathcal{H}}\psi(\mathcal{H})_{i}]$
$\displaystyle=\psi(\mathcal{H})_{i}^{*}V_{\mathcal{H}}^{*}x^{*}xV_{\mathcal{H}}\psi(\mathcal{H})_{i}-\psi(\mathcal{H})_{i}^{*}V_{\mathcal{H}}^{*}x^{*}V_{\mathcal{H}}\psi(\mathcal{H})_{i}x-x^{*}\psi(\mathcal{H})_{i}^{*}V_{\mathcal{H}}^{*}xV_{\mathcal{H}}\psi(\mathcal{H})_{i}+x^{*}x$
$\displaystyle=0.$

Thus the C∗-condition implies $\mathcal{K}_{\mathcal{H}}\subset A^{\infty}\cap
A^{\prime}$. We let $\mathcal{K}_{\pi}=\mathcal{K}_{\mathcal{H}_{\pi}}$.

When $\mathcal{H}$ is equivalent to $1\oplus 1$, we have
$\mathcal{K}_{\mathcal{H}}\subset(A^{\alpha})^{\infty}\cap A^{\prime}$. This
shows that there exist two isometries with mutually orthogonal ranges in
$(A^{\alpha})^{\infty}\cap A^{\prime}$, and there exists a unital embedding of
$\mathcal{O}_{\infty}$ in $(A^{\alpha})^{\infty}\cap A^{\prime}$. ∎

##  7 Quasi-free actions on the Cuntz algebras

Before ending this paper, we discuss quasi-free actions of a compact group $G$
on the Cuntz algebras $\mathcal{O}_{n}$ and $\mathcal{O}_{\infty}$ as
applications of our main results. We follow the convention in [15] for graph
C∗-algebras.

######  Example 7.1.

Let $\rho$ be a faithful unitary representation of a compact group $G$ in
$\mathbb{M}_{n}(\mathbb{C})$, and let $\alpha$ be the quasi-free action of $G$
arising from $\rho$ on $\mathcal{O}_{n}$. It is known that
$\mathcal{O}_{n}\cap({\mathcal{O}_{n}}^{U(n)})^{\prime}$ is trivial (see [4,
Theorem 3.2], [11, Corollary 3.3]), and hence $\alpha$ is minimal. Thus if
${\mathcal{O}_{n}}^{\alpha}$ is simple, Theorem 1.1 implies that $\alpha$ is
quasi-product. In fact, [11, Theorem 3.1] shows that
${\mathcal{O}_{n}}^{\alpha}$ is simple if $\rho(G)\subset SU(n)$. We can relax
this assumption to the following:

  * ($*$)

For every $\pi\in\widehat{G}$, there exists $n\geq 0$ such that $\pi$ is
contained in $\rho^{\otimes n}$.

[34, Theorem 7.1] shows that ${\mathcal{O}_{n}}^{\alpha}$ is isomorphic to a
corner of the graph algebra $C^{*}(\mathcal{G}_{\rho})$ of the following graph
$\mathcal{G}_{\rho}$: the vertex set $\mathcal{G}_{\rho}^{0}$ is
$\widehat{G}$, and the number of edges from $\sigma\in\widehat{G}$ to
$\pi\in\widehat{G}$ is $\dim\operatorname{Hom}_{G}(\pi,\rho\otimes\sigma)$.
Now [15, Theorem 3, 4] imply that ${\mathcal{O}_{n}}^{\alpha}$ is purely
infinite and simple under the condition ($*$), and our main results show that
$\alpha$ is isometrically shift-absorbing. It is very likely that ($*$) is
also necessary for ${\mathcal{O}_{n}}^{\alpha}$ to be purely infinite and
simple, but it does not seem to be known yet.

For the Cuntz algebra $\mathcal{O}_{\infty}$, we have the following statement,
previously known only for abelian $G$ (see [29, Proposition 7.4], [32, Theorem
5.1], [37, Corollary 4.17], and also [8] for a related result).

######  Proposition 7.2.

Let $(\rho,H)$ be a faithful unitary representation of a compact group $G$ on
a separable infinite dimensional Hilbert space $H$, and let $\alpha$ be the
corresponding quasi-free action of $G$ on $\mathcal{O}_{\infty}$. Then the
following conditions are equivalent:

  * $(1)$

$\alpha$ is isometrically shift-absorbing.

  * $(2)$

${\mathcal{O}_{\infty}}^{\alpha}$ is purely infinite and simple.

  * $(3)$

${\mathcal{O}_{\infty}}^{\alpha}$ is simple.

  * $(4)$

The Condition $(*)$ holds.

###### Proof.

Let

$(\mathcal{F}(H),\mathcal{F}(\rho))=(\bigoplus_{n=0}^{\infty}H^{\otimes
n},\bigoplus_{n=0}^{\infty}\rho^{\otimes n}).$

By convention $H^{\otimes 0}=\mathbb{C}\Omega$ where $\Omega$ is the vacuum
vector. We regard $\mathcal{O}_{\infty}$ as a concrete C∗-algebra acting on
the full Fock space $\mathcal{F}(H)$ generated by the left creation operators
$l(\xi)$, $\xi\in H$. Then the quasi-free action $\alpha$ is the restriction
of $\operatorname{Ad}\mathcal{F}(\rho)$ to $\mathcal{O}_{\infty}$. Since the
vacuum state is $\alpha$-invariant and pure on $\mathcal{O}_{\infty}$, if
${\mathcal{O}_{\infty}}^{\alpha}$ simple, the action $\alpha$ is quasi-
product, and in particular it is is minimal.

(1) $\Rightarrow$ (2). This is shown in [37, Example 4.5].

(2) $\Rightarrow$ (1). This follows from Theorem 1.2.

(2) $\Rightarrow$ (3). This is trivial.

(3) $\Rightarrow$ (4). Since ${\mathcal{O}_{\infty}}^{\alpha}$ is simple and
$\alpha$ is minimal, the crossed product
$\mathcal{O}_{\infty}\rtimes_{\alpha}G$ is simple (see [37, Proposition A]).
Thus the defining representation of $\mathcal{O}_{\infty}$ and
$\mathcal{F}(\rho)$ give a faithful representation of
$\mathcal{O}_{\infty}\rtimes_{\alpha}G$ on $\mathcal{F}(H)$, and hence
$\mathcal{F}(\rho)$ gives rise to a faithful representation of the group
C∗-algebra of $G$, which implies (4).

(4) $\Rightarrow$ (2). We show the implication by identifying
${\mathcal{O}_{\infty}}^{\alpha}$ with a corner of $C^{*}(\mathcal{G}_{\rho})$
as in the case of $\mathcal{O}_{n}$. Since the argument in [34] does not work
in this case, we give a more direct argument here. We let
$K=\bigoplus_{\pi\in\widehat{G}}H_{\pi}$, and equip $\mathbb{K}(K)$ with the
$G$-action given by $\operatorname{Ad}(\bigoplus_{\pi\in\widehat{G}}\pi)$. We
first embed $C^{*}(\mathcal{G}_{\rho})$ into
$(\mathcal{O}_{\infty}\otimes\mathbb{K}(K))^{G}$.

For each $\pi,\sigma\in\widehat{G}$, we fix an orthonormal basis $\\{e\\}$ of
$\operatorname{Hom}_{G}(H_{\pi},H\otimes H_{\sigma})$, where the inner product
of $e$ and $f$ is given by
$\langle{e,f}\rangle=f^{*}e\in\operatorname{End}_{G}(H_{\pi})=\mathbb{C}1_{\pi}$.
We identify $\\{e\\}$ with the set of edges from $\sigma$ to $\pi$, and
$\cup_{\pi,\sigma\in\widehat{G}}\\{e\\}$ with the edge set
$\mathcal{G}_{\rho}^{1}$. For $e$, we define $S_{e}$ to be the corresponding
element in

$\left(l(H)\otimes(H_{\sigma}\otimes
H_{\pi}^{*})\right)^{G}\subset(\mathcal{O}_{\infty}\otimes\mathbb{K}(K))^{G},$

where $H_{\sigma}\otimes H_{\pi}^{*}$ is identified with a subspace of
$\mathbb{K}(K)$ in a natural way. Then we have $S_{e}^{*}S_{e}=1\otimes
p_{\pi}$ and $S_{e}S_{e}^{*}\leq 1\otimes p_{\sigma}$, where $p_{\pi}$ is the
projection from $K$ onto $H_{\pi}$. If
$e_{1},e_{2},\ldots,e_{n}\in\mathcal{G}_{\rho}^{1}$ are distinct edges with a
common source $\sigma$, we have $S_{e_{i}}^{*}S_{e_{j}}=\delta_{i,j}1\otimes
p_{r(e_{i})}$, which shows that $\\{S_{e_{i}}S_{e_{i}}^{*}\\}_{i=1}^{n}$ are
mutually orthogonal subprojections of $1\otimes p_{\sigma}$. Thus
$\\{S_{e},1\otimes p_{\pi}\\}$ form a Cuntz-Krieger
$\mathcal{G}_{\rho}$-family, and there exists a surjection from the graph
algebra $C^{*}(\mathcal{G}_{\rho})$ onto the C∗-algebra $B$ generated by
$\\{S_{e}\\}_{e\in\mathcal{G}_{\rho}^{1}}$. [15, Theorem 3, 4] show that
$C^{*}(\mathcal{G}_{\rho})$ is purely infinite and simple thanks to the
condition ($*$), and so is $B$. Now our task is to show $(1\otimes
p_{1})B(1\otimes p_{1})={\mathcal{O}_{\infty}}^{\alpha}\otimes p_{1}$, which
implies (2).

Let $\mathcal{G}_{\rho}(1,\pi;n)$ be the set of paths $\xi$ in
$\mathcal{G}_{\rho}$ of length $n$ satisfying $s(\xi)=1$ and $r(\xi)=\pi$.
Then the linear span of $\\{S_{\xi}\\}_{\xi\in\mathcal{G}_{\rho}(1,\pi;n)}$ is
dense in $(l(H)^{n}\otimes(H_{1}\otimes H_{\pi}^{*}))^{G}$ by construction,
and so the linear span of

$\\{S_{\xi}S_{\eta}^{*};\;\xi\in\mathcal{G}_{\rho}(1,\pi;n),\;\eta\in\mathcal{G}_{\rho}(1,\pi;m),\;\pi\in\widehat{G}\\}$

is dense in $(l(H)^{n}{l(H)^{m}}^{*})^{G}\otimes p_{1}$. Thus $(1\otimes
p_{1})B(1\otimes p_{1})={\mathcal{O}_{\infty}}^{\alpha}\otimes p_{1}$. ∎

The following is a consequence of our main results and [17, Corollary 6.4],
and it is a generalization of [18, Theorem 5.1] in the finite group case.

######  Corollary 7.3.

For a compact group $G$, any two faithful quasi-free actions of $G$ on the
Cuntz algebras $\mathcal{O}_{\infty}$ satisfying the condition $(*)$ are
mutually conjugate. Such actions are absorbed through tensor product by every
faithful minimal action of $G$ on a Kirchberg algebra whose fixed point
algebra is a Kirchberg algebra. (Note that when $G$ is finite or $SU(n)$, the
condition $(*)$ is automatically satisfied.)

## Acknowledgments

The author would like to thank Miho Mukohara for stimulating discussions and
Narutaka Ozawa for useful discussions on the CPAP.

## References

  * [1] Araki, Huzihiro; Haag, Rudolf; Kastler, Daniel; Takesaki, Masamichi, Extension of KMS states and chemical potential, Comm. Math. Phys. 53 (1977), no.2, 97–134.
  * [2] Bratteli, Ola; Elliott, George A.; Evans, David E.; Kishimoto, Akitaka, Quasi-product actions of a compact abelian group on a C∗-algebra, Tohoku Math. J. (2) 41 (1989), no. 1, 133–161.
  * [3] Bratteli, Ola; Elliott, George A.; Kishimoto, Akitaka, Quasi-product actions of a compact group on a C∗-algebra, J. Funct. Anal. 115 (1993), no. 2, 313–343.
  * [4] Bratteli, O; Evans, David E., Derivations tangential to compact groups: the nonabelian case. Proc. London Math. Soc. (3) 52 (1986), no.2, 369–384.
  * [5] Bratteli, Ola; Kishimoto, Akitaka; Robinson, Derek W. Embedding product type actions into C∗-dynamical systems, J. Funct. Anal. 75 (1987), no. 1, 188–210.
  * [6] Bratteli, Ola; Kishimoto, Akitaka; Robinson, Derek W. Abundance of invariant and almost invariant pure states of C∗-dynamical systems, Comm. Math. Phys. 187 (1997), no. 3, 491–507.
  * [7] Brown, Nathanial P.; Ozawa, Narutaka, C∗-algebras and finite-dimensional approximations, Grad. Stud. Math., 88, American Mathematical Society, Providence, RI, 2008.
  * [8] Ceccherini, Tullio; Pinzari, Claudia, Canonical actions on $O_{\infty}$, J. Funct. Anal. 103 (1992), no.1, 26–39.
  * [9] De Commer, Kenny; Yamashita, Makoto, Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ. 28 (2013), No. 31, 1099–1138.
  * [10] Doplicher, Sergio; Longo, Roberto; Roberts, John. E.; Zsidó, Lázló, A remark on quantum group actions and nuclearity, Rev. Math. Phys. 14 (2002), no.7-8, 787–796.
  * [11] Doplicher, Sergio; Roberts, John E., Duals of compact Lie groups realized in the Cuntz algebras and their actions on C∗-algebras, J. Funct. Anal. 74 (1987), no.1, 96–120.
  * [12] Doplicher, Sergio; Roberts, John E. Endomorphisms of C∗-algebras, cross products and duality for compact groups, Ann. of Math. (2) 130 (1989), no.1, 75–119.
  * [13] Doplicher, Sergio; Roberts, John E. A new duality theory for compact groups. Invent. Math. 98 (1989), no.1, 157–218.
  * [14] Doplicher, Sergio; Roberts, John E. Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Comm. Math. Phys. 131 (1990), no.1, 51–107.
  * [15] Fowler, Neal J.; Laca, Marcelo; Raeburn, Iain, The C∗-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), no.8, 2319–2327.
  * [16] Gabe, James; Szabó, Gábor, The stable uniqueness theorem for equivariant Kasparov theory, preprint, arxiv:2202.09809.
  * [17] Gabe, James; Szabó, Gábor, The dynamical Kirchberg-Phillips theorem. to appear in Acta Mathematica, arXiv:2205.04933.
  * [18] Goldstein, Pavle ; Izumi, Masaki. Quasi-free actions of finite groups on the Cuntz algebra $\mathcal{O}_{\infty}$, Tohoku Math. J. (2), 63 (2011), 729–749.
  * [19] Høegh-Krohn, R.; Landstad, M. B.; Størmer, E. Compact ergodic groups of automorphisms. Ann. of Math. (2) 114 (1981), no.1, 75–86.
  * [20] Hiai, Fumio, Minimizing indices of conditional expectations onto a subfactor, Publ. Res. Inst. Math. Sci. 24 (1988), no.4, 673–678.
  * [21] Hiai, Fumio; Izumi, Masaki, Amenability and strong amenability for fusion algebras with applications to subfactor theory. Internat. J. Math. 9 (1998), no.6, 669–722.
  * [22] Izumi, Masaki, Subalgebras of infinite C∗-algebras with finite Watatani indices. II. Cuntz-Krieger algebras. Duke Math. J. 91 (1998), no.3, 409–461.
  * [23] Izumi, Masaki. Inclusions of simple C∗-algebras, J. Reine Angew. Math. 547 (2002), 97–138.
  * [24] Izumi, Masaki, Group actions on operator algebras, in Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, 1528–1548.
  * [25] Izumi, Masaki; Longo, Roberto; Popa, Sorin, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998), no.1, 25–63.
  * [26] Izumi, Masaki; Matui, Hiroki, $\mathbb{Z}^{2}$-actions on Kirchberg algebras, Adv. Math. 224 (2010), no. 2, 355–400.
  * [27] Izumi, Masaki; Matui, Hiroki, Poly-$\mathbb{Z}$ group actions on Kirchberg algebras I, Int. Math. Res. Not. IMRN. 2021, no. 16, 12077–12154.
  * [28] Izumi, Masaki; Matui, Hiroki, Poly-$\mathbb{Z}$ group actions on Kirchberg algebras II, Invent. Math. 224 (2021), no. 3, 699–766.
  * [29] Katsura, Takeshi, On crossed products of the Cuntz algebra $O_{\infty}$ by quasi-free actions of abelian groups, Operator algebras and mathematical physics (Constanţa, 2001), 209–233.
  * [30] Kirchberg, Eberhard, The classification of purely infinite $C^{*}$-algebras using Kasparov’s theorem, preprint, 1994.
  * [31] Kirchberg, Eberhard; Phillips, N. Christopher. Embedding of exact C∗-algebras in the Cuntz algebra $\mathcal{O}_{2}$, J. Reine Angew. Math. 525 (2000), 17–53.
  * [32] Kishimoto, Akitaka, Simple crossed products of C∗-algebras by locally compact abelian groups. Yokohama Math. J. 28 (1980), no.1-2, 69–85.
  * [33] Kishimoto, Akitaka, Outer automorphisms and reduced crossed products of simple C∗-algebras. Comm. Math. Phys. 81 (1981), no.3, 429–435.
  * [34] Kumjian, Alex; Pask, David; Raeburn, Iain; Renault, Jean, Graphs, groupoids, and Cuntz-Krieger algebras. J. Funct. Anal. 144 (1997), no.2, 505–541.
  * [35] Masuda, Toshihiko; Tomatsu, Reiji, Classification of minimal actions of a compact Kac algebra with amenable dual, Comm. Math. Phys. 274 (2007), no.2, 487–551.
  * [36] Meyer, Ralf, On the classification of group actions on C∗-algebras up to equivariant KK-equivalence, Ann. K-Theory, 6 (2021), no.2, 157–238.
  * [37] Mukohara, Miho, Inclusions of simple C∗-algebras arising from compact group actions, preprint, arXiv:2401.13989.
  * [38] Osaka, Hiroyuki, SP-property for a pair of C∗-algebras, J. Operator Theory 46 (2001), no.1, 159–171.
  * [39] Ozawa, Narutaka; Suzuki, Yuhei, On characterizations of amenable C∗-dynamical systems and new examples, Selecta Math. (N.S.) 27 (2021), no.5, Paper No. 92, 29.
  * [40] Pedersen, Gert K. C∗-algebras and their automorphism groups. Second edition. Edited and with a preface by Søren Eilers and Dorte Olesen, Pure Appl. Math. (Amst.) Academic Press, London, 2018.
  * [41] Peligrad, Costel, Properly outer and strictly outer actions of finitegroups on prime C∗-algebras, Internat. J. Math. (2024) 2450007, arXiv:2303.04232.
  * [42] Phillips, N. Christopher, A classification theorem for nuclear purely infinite simple $C^{*}$-algebras, Doc. Math. 5 (2000), 49–114.
  * [43] Pimsner, Mihai; Popa, Sorin, Entropy and index for subfactors, Ann. Sci.École Norm. Sup. (4) 19 (1986), no.1, 57–106.
  * [44] Popa, Sorin, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255.
  * [45] Roberts, John E. Cross products of von Neumann algebras by group duals, Symposia Mathematica, Vol. XX (Convegno sulle Algebre C∗ e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria K, INDAM, Rome, 1975), pp. 335–363 Academic Press, London-New York, 1976.
  * [46] Rørdam, Mikael, Classification of nuclear, simple C∗-algebras, Classification of nuclear C∗-algebras. Entropy in operator algebras, 1–145. Encyclopaedia Math. Sci., 126, Oper. Alg. Non-commut. Geom., 7.
  * [47] Rørdam, Mikael, Irreducible inclusions of simple C∗-algebras, Enseign. Math. 69 (2023), no. 3-4, 275–314.
  * [48] Ruan, Zhong-Jin, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no.2, 466–499.
  * [49] Szabó, Gábor, The classification of Rokhlin flows on C∗-algebras, Comm. Math. Phys. 382 (2021), no.3, 2015–2070.
  * [50] Wassermann, Antony, Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions. Canad. J. Math. 40 (1988), no. 6, 1482–1527.
  * [51] Wassermann, Antony, Ergodic actions of compact groups on operator algebras. I. General theory. Ann. of Math. (2) 130 (1989), no. 2, 273–319.
  * [52] Watatani, Yasuo, Index for C∗-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no.424.