1. Introduction
The Kirchberg 
$\mathcal {O}_{2}$-absorption theorem [Reference KirchbergKir, Reference Kirchberg and PhillipsKP00] states that the Cuntz algebra 
$\mathcal {O}_{2}$ tensorially absorbs all unital simple separable nuclear 
${\rm C}^{\ast }$-algebras 
$A$; 
$A \otimes \mathcal {O}_{2} \cong \mathcal {O}_{2}$. This is one of the fundamental ingredients of Phillips's proof of the classification theorem of Kirchberg algebras [Reference PhillipsPhi00] (see [Reference KirchbergKir] for Kirchberg's original approach). Here recall that a 
${\rm C}^{\ast }$-algebra is said to be a Kirchberg algebra if it is purely infinite simple, separable, nuclear. We refer the reader to the book [Reference RørdamRor02] for basic facts and background on the Kirchberg algebras.
Considering the special roles of this beautiful, deep, surprising theorem of Kirchberg, and also observing the recent developments on the classification of 
${\rm C}^{\ast }$-dynamical systems (see e.g. [Reference Izumi and MatuiIM18, Reference Izumi and MatuiIM19, Reference SzabóSza18c]), it is a natural and important attempt to establish an equivariant analogue of the 
$\mathcal {O}_{2}$-absorption theorem. (For history and background of the study of group actions on operator algebras, we refer the reader to [Reference IzumiIzu10] or the introduction of [Reference SzabóSza18c] and references therein.) Indeed, among other things, recently Szabó [Reference SzabóSza18c] established the theorem for countable amenable groups. Here we extend the theorem to countable exact groups. The precise statement is as follows.
Main Theorem Let 
$G$ be a countable exact group. Let 
$\delta : G \curvearrowright \mathcal {O}_{2}$ be an equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer action with the quasi-central approximation property (QAP) [Reference Buss, Echterhoff and WillettBEW20]. Then for any action 
$\alpha : G \curvearrowright A$ on a unital simple separable nuclear 
${\rm C}^{\ast }$-algebra, the diagonal action 
$\alpha \otimes \delta$ is strongly cocycle conjugate to 
$\delta$.
For the definition of the QAP, see §§ 2.1 and 2.2. Note that such a 
$\delta$ always exists (see Example B below or [Reference SuzukiSuz19a]), and (therefore) all the assumptions cannot be removed (cf. Remark 3.10). Thus one can regard this theorem as the equivariant 
$\mathcal {O}_{2}$-absorption theorem.
As a rephrasing of the Main Theorem, we obtain the following uniqueness theorem.
Theorem A Let 
$G$ be a countable exact group. Then, up to strong cocycle conjugacy, there is a unique equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer, QAP action 
$G\curvearrowright \mathcal {O}_{2}$.
It is notable that, as all the three conditions are stable under cocycle conjugacy, the above theorem gives the first abstract characterization of a (strong) cocycle conjugacy class of pointwise outer actions of a non-amenable group on a simple operator algebra.
We emphasize that all the previously known classification results of simple equivariant operator algebras (in both 
${\rm C}^{\ast }$- and 
$W^\ast$-cases) use amenability of the acting group in substantial ways. Theorem A is the first positive evidence that a fruitful classification theory of 
${\rm C}^{\ast }$-dynamical systems could be developed even beyond amenable groups.
Theorem B Here we give motivating examples of actions which satisfy the assumptions of the Main Theorem; cf. [Reference SuzukiSuz19a]. Let 
$G$ be a countable exact group. For each 
$n\in \mathbb {N}$, choose a (topologically) amenable action 
$\eta _n : G \curvearrowright X_n$ on a compact metrizable space 
$X_n$ [Reference Brown and OzawaBO88, Theorem 5.1.7]. For each 
$n\in \mathbb {N}$, choose a unital embedding 
$\iota _n : C(X_n) \mathop {\rtimes _{{\mathrm {r}}, \eta _n}} G \rightarrow \mathcal {O}_{2}$ [Reference Kirchberg and PhillipsKP00, Reference KirchbergKir]. Each 
$\iota _n$ defines a unitary representation 
$u_n : G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ by the restriction. Then the diagonal action
\[ \bigotimes\limits_{n \in \mathbb{N}} {\rm Ad}(u_n) : G \curvearrowright \bigotimes\limits_{n \in \mathbb{N}} \mathcal{O}_{2} \]defines an action
\[ \delta : G \curvearrowright \mathcal{O}_{2} \]
via an isomorphism 
${\bigotimes} _{n \in \mathbb {N}} \mathcal {O}_{2} \cong \mathcal {O}_{2}$ [Reference RørdamRor02]. It is clear from the construction that 
$\delta$ satisfies the assumptions of the Main Theorem. Thus, as a particular consequence of the Main Theorem, the strong cocycle conjugacy class of 
$\delta$ is independent on the choices of 
$\eta _n$ and 
$\iota _n$. This already implies the following non-trivial consequence: for any amenable action 
$\eta : G \curvearrowright X$ on a compact metrizable space and for any choices of 
$\eta _n$ and 
$\iota _n$, there is a unital 
$G$-equivariant embedding
\[ (C(X), \eta) \rightarrow ((\mathcal{O}_{2})_\omega, \delta_\omega). \]Our proof of the Main Theorem mostly follows that of Szabó's theorem [Reference SzabóSza18c, Theorem C]. However at a few crucial steps, amenability of the acting group is used in essential ways. In particular, this includes the most important step Lemma 2.3 of [Reference SzabóSza18c]. This lemma is used to carry out Connes's two-by-two matrix trick [Reference ConnesCon77] (see [Reference SzabóSza18c, Lemmas 2.4 and 5.1]). Unfortunately the statement of the lemma no longer holds true for non-amenable groups; see Remark 3.9. However, we will see in Lemma 3.1 that, under appropriate assumptions, the statement is still valid. This is the main contribution of the present article toward the Main Theorem.
As an intermediate result in the proof of the Main Theorem, we also obtain the following useful characterizations of the QAP in an important case. We use this result together with Lemma 3.1 to deduce the key result Corollary 3.7.
Theorem C Let 
$\alpha : G \curvearrowright A$ be an action of a countable exact group on a unital simple separable nuclear 
${\rm C}^{\ast }$-algebra. Let 
$\omega$ be a free ultrafilter on 
$\mathbb {N}$. Then the following conditions are pairwise equivalent.
(1) The
$\alpha$ has the QAP.(2) There is a sequence
$(\xi _n)_{n=1}^\infty$ in $\ell ^2(G, A_\omega )$ satisfying $\lim _{n \rightarrow \infty } \langle \xi _n, \tilde {\alpha }_g(\xi _n)\rangle =1$ for all $g\in G$.(3) There is a sequence
$(k_n : G \rightarrow A_\omega )_{n=1}^\infty$ of finitely supported positive definite functions satisfying $\lim _{n\rightarrow \infty } k_n(g) =1$ for all $g\in G$.(4) The induced action
$\alpha ^\omega : G \curvearrowright A^\omega$ on the ultrapower $A^\omega$ has the QAP.(5) The induced action
$\alpha _\omega : G \curvearrowright A_\omega$ on the central sequence algebra $A_\omega$ has the QAP.(6) For any separable
$G$-${\rm C}^{\ast }$-subalgebra $B\subset A^\omega$, the restriction of $\alpha ^\omega$ to $A^\omega \cap B'$ has the QAP.
We stress that even for some actions in Example B (see also [Reference SuzukiSuz19a, Reference SuzukiSuz20a]), validity of conditions (2) to (6) are not obvious. The crucial implication used in the present article is (1) 
$\Rightarrow$ (4); cf. Lemma 3.1 and Corollary 3.7.
We also discuss an equivariant analogue of the Kirchberg 
$\mathcal {O}_{\infty }$-absorption theorem [Reference KirchbergKir, Reference Kirchberg and PhillipsKP00]. Recall that the theorem states that the Cuntz algebra 
$\mathcal {O}_{\infty }$ is tensorially absorbed by any Kirchberg algebra 
$A$; 
$A\otimes \mathcal {O}_{\infty } \cong A$. This is another main ingredient of the proof of the Kirchberg–Phillips classification theorem in [Reference PhillipsPhi00]. Therefore, similar to the 
$\mathcal {O}_{2}$-absorption theorem, it is desirable to obtain an equivariant analogue of the 
$\mathcal {O}_{\infty }$-absorption theorem. Indeed Szabó obtains such results for countable amenable groups (see Theorems B and 3.5 in [Reference SzabóSza18c]). In this article, we prove the following result for QAP actions on unital Kirchberg algebras.
We first give model (absorbed) actions on 
$\mathcal {O}_{\infty }$.
Definition D Let 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{\infty })$ be a unitary representation of a group 
$G$. We say that 
$u$ factors through 
$\mathcal {O}_{2}$ if there is a unitary representation 
$v : G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ and a (non-unital) embedding 
$\iota : \mathcal {O}_{2}\rightarrow \mathcal {O}_{\infty }$ satisfying
\[ u_g= \iota(v_g)+ (1_{\mathcal{O}_{\infty}}-\iota(1_{\mathcal{O}_{2}})) \quad{\rm for\ all}\ g\in G. \]
Note that by the Kirchberg 
$\mathcal {O}_{2}$-embedding theorem (applied to the reduced group 
${\rm C}^{\ast }$-algebra ), every countable exact group admits such a faithful representation. Note that unlike Example 3.6 in [Reference SzabóSza18c], here we do not assume the injectivity of the induced 
$\ast$-homomorphism on 
${\mathrm {C}}^{\ast }(G)$.
Example E (cf. Example 3.6 in [Reference SzabóSza18c])
Let 
$G$ be a countable group. Let 
$(u_n)_{n=1}^\infty$ be a sequence of unitary representations of 
$G$ on 
$\mathcal {O}_{\infty }$ which factor through 
$\mathcal {O}_{2}$. Then the diagonal action
\[ \bigotimes\limits_{n\in \mathbb{N}} {\rm Ad}(u_n): G \curvearrowright \bigotimes\limits_{\mathbb{N}} \mathcal{O}_{\infty} \]defines an action
\[ \gamma : G \curvearrowright \mathcal{O}_{\infty} \]
via an isomorphism 
${\bigotimes} _{\mathbb {N}} \mathcal {O}_{\infty } \cong \mathcal {O}_{\infty }$ [Reference RørdamRor02]. We remark that 
$\gamma$ admits an invariant state. Thus, when 
$G$ is non-amenable, 
$\gamma$ cannot have the QAP.
Theorem F. Let 
$G$ be a countable exact group. Let 
$\alpha : G \curvearrowright A$ be a pointwise outer, QAP action on a unital Kirchberg algebra. Then for any action 
$\gamma$ in Example E, 
$\alpha$ is strongly cocycle conjugate to 
$\alpha \otimes \gamma$. In particular 
$\alpha$ is equivariantly 
$\mathcal {O}_{\infty }$-absorbing.
2. Preliminaries
Here we recall some definitions and facts, and fix the notation.
2.1 Hilbert ${\rm C}^{\ast }$-bimodule $\ell ^2(G, A)$
Let 
$G$ be a discrete group. Let 
$A$ be a 
${\rm C}^{\ast }$-algebra. Define the linear space
\[ \ell^2(G, A) := \bigg\{ (\xi_g)_{g\in G} \in \prod_G A: \sum_{g\in G} \xi_g^\ast \xi _g\ {\rm converges\ in\ norm}\bigg\}. \]
For 
$a\in A$ and 
$\xi \in \ell ^2(G, A)$, define 
$a\xi , \xi a \in \ell ^2(G, A)$ to be
\[ (a\xi)_g := a \xi_g,\quad (\xi a)_g:=\xi_g a\quad{\rm for}\ g\in G. \]
For 
$\xi , \eta \in \ell ^2(G, A)$, set
\[ \langle \xi, \eta\rangle := \sum_{g\in G} \xi_g^\ast \eta_g \in A. \]
Note that these operations make 
$\ell ^2(G, A)$ a Hilbert 
${\rm C}^{\ast }$-bimodule over 
$A$. For 
$\xi \in \ell ^2(G, A)$, set
\[ \|\xi \|:= \| \langle \xi, \xi\rangle\|^{1/2}. \]
This defines a complete norm on 
$\ell ^2(G, A)$.
Let 
$\alpha : G \curvearrowright A$ be an action. Then define the norm-preserving action 
$\tilde {\alpha } : G \curvearrowright \ell ^2(G, A)$ to be
\[ \tilde{\alpha}_g(\xi)_h := \alpha_g(\xi_{g^{{-}1} h}) \quad{\rm for}\ g, h\in G, \xi \in \ell^2(G, A). \]
This action is used to formulate amenability-type conditions for 
${\rm C}^{\ast }$-dynamical systems; see [Reference Anantharaman-DelarocheAna87, Reference Anantharaman-DelarocheAna02, Reference Brown and OzawaBO88, Reference Buss, Echterhoff and WillettBEW20] for instance.
2.2 Amenable actions on ${\rm C}^{\ast }$-algebras
Recently, in [Reference SuzukiSuz19a], we discovered actions of non-amenable groups on simple 
${\rm C}^{\ast }$-algebras with properties which should be regarded as amenability of 
${\rm C}^{\ast }$-dynamical systems. A few applications of such actions are already found in 
${\rm C}^{\ast }$-algebra theory; see [Reference SuzukiSuz20a, Reference SuzukiSuz20b, Reference SuzukiSuz19b]. We believe that this novel phenomenon provides a new rich field in 
${\rm C}^{\ast }$-algebra theory. One of the motivations behind the present article is to examine a new evidence supporting this my expectation from a different angle.
While the definitive definition of amenability is still not known for (non-commutative) 
${\rm C}^{\ast }$-dynamical systems, in [Reference Buss, Echterhoff and WillettBEW20], motivated by examples constructed in [Reference SuzukiSuz19a], a new amenability-type condition, called the quasi-central approximation property (QAP), is introduced for 
${\rm C}^{\ast }$-dynamical systems. (We remark that the situation is totally different in the von Neumann algebra setting. In the realm of von Neumann algebras, the definitive definition is already known, and non-amenable discrete groups cannot act amenably on a factor [Reference Anantharaman-DelarocheAna79].) For simplicity, here we recall the definition of the QAP only in the unital and discrete case. See [Reference Buss, Echterhoff and WillettBEW20] for the general case. Recall that a discrete group action 
$\alpha : G \curvearrowright A$ on a unital 
${\rm C}^{\ast }$-algebra is said to have the quasi-central approximation property (QAP) if there is a net 
$(\xi _i)_{i\in I}$ in 
$\ell ^2(G, A)$ satisfying:
(1)
$\lim _{i\in I} \langle \xi _i, \tilde {\alpha }_g (\xi _i) \rangle = 1_A$ for all $g\in G$;(2)
$\lim _{i\in I} \|a \xi _i - \xi _i a\|=0$ for all $a\in A$.
For actions on commutative 
${\rm C}^{\ast }$-algebras, it is not hard to see that the QAP is equivalent to the (topological) amenability (cf. [Reference Brown and OzawaBO88, Lemma 4.3.7]). It is also clear that the QAP is stable under taking equivariant inductive limits. In particular, the actions constructed in Example B have the QAP.
In Theorem C, for some important cases, we obtain useful characterizations of the QAP in terms of the (relative) central sequence algebra. This is another important step to prove the Main Theorem; cf. Corollary 3.7.
Here we record the following basic observation on the QAP. The proof is easy and straightforward and we leave details to the reader.
Proposition 2.1 Let 
$\alpha$ and 
$\beta$ be an action of a discrete group 
$G$ on a unital 
${\rm C}^{\ast }$-algebra 
$A$ such that 
$\alpha _g \circ \beta _g^{-1}$ is inner for all 
$g\in G$. Then 
$\alpha$ has the QAP if and only if so does 
$\beta$. In particular, the 
$QAP$ is stable under cocycle conjugacy.
Another object used to formulate amenability-type conditions of actions is positive definite functions; see e.g. [Reference Anantharaman-DelarocheAna87, Reference Anantharaman-DelarocheAna02, Reference Brown and OzawaBO88, Reference Buss, Echterhoff and WillettBEW20]. Recall that for a 
${\rm C}^{\ast }$-dynamical system 
$\alpha : G \curvearrowright A$, a function 
$f : G \rightarrow A$ is said to be positive definite [Reference Anantharaman-DelarocheAna87] if it satisfies the following condition: for any finite sequence 
$g_1, \ldots , g_n \in G$, the matrix 
$(\alpha _{g_i}(f(g_i^{-1} g_j)))_{1\leq i, j \leq n} \in \mathbb {M}_n(A)$ is positive.
2.3 Ultrapowers and central sequence algebras
Throughout this article, we fix a free ultrafilter 
$\omega$ on 
$\mathbb {N}$. For a 
${\rm C}^{\ast }$-algebra 
$A$, the ultrapower 
$A^\omega$ of 
$A$ with respect to 
$\omega$ is defined to be
\[ A^\omega := \ell^\infty(\mathbb{N}, A) / \Big\{(x_n)_{n=1}^\infty \in \ell^\infty(\mathbb{N}, A) : \lim_{n\rightarrow \omega} \| x_n\|=0 \Big\}. \]
(We alert the reader to the fact that some authors (including [Reference SzabóSza18c]) use the symbol 
$A_\omega$ for the 
${\rm C}^{\ast }$-algebraic ultrapower. In this article, following e.g. [Reference Izumi and MatuiIM18, Reference Izumi and MatuiIM19], we employ the notation 
$A^\omega$ which is standard at least in von Neumann algebra theory.) For 
$(x_n)_{n=1}^\infty \in \ell ^\infty (\mathbb {N}, A)$, denote by 
$[x_n]_n^\omega$ its image in 
$A^\omega$.
We regard 
$A$ as a 
${\rm C}^{\ast }$-subalgebra of 
$A^\omega$ via the embedding 
$a \mapsto [(a, a, a, \ldots )]_n^\omega$. The relative commutant 
${\rm C}^{\ast }$-algebra
\[ A_\omega:=A^\omega \cap A' \]
is said to be the central sequence algebra of 
$A$ (with respect to 
$\omega$).
For any discrete group action 
$\alpha : G \curvearrowright A$, we have the actions 
$\alpha ^\omega : G \curvearrowright A^\omega$, 
$\alpha _\omega : G \curvearrowright A_\omega$ inherited from the pointwise application of 
$\alpha$ to 
$\ell ^\infty (\mathbb {N}, A)$. When the underlying 
${\rm C}^{\ast }$-algebra is obvious from the context, we simply denote 
$\alpha ^\omega$, 
$\alpha _\omega$, and their restrictions to 
$G$-
${\rm C}^{\ast }$-subalgebras (typically relative commutant 
${\rm C}^{\ast }$-subalgebras) by 
$\alpha$ for short.
2.4 Exact groups
Recall that a discrete group 
$G$ is said to be exact [Reference Kirchberg and WassermannKW99] if the reduced group 
${\rm C}^{\ast }$-algebra 
${\mathrm {C}}_{\mathrm {r}}^{\ast }(G)$ is exact. We refer the reader to [Reference Brown and OzawaBO88, Reference OzawaOza06] for introduction and applications of this rich property. Here we recall only a few things. First, it should be stressed that, comparing to amenable groups, the class of exact groups is fairly huge. For instance the class includes all hyperbolic groups and linear groups. Moreover the class is stable under taking extensions, subgroups, increasing unions, amalgamated free products. All the currently known non-exact groups are constructed by a probabilistic method [Reference GromovGro03, Reference OsajdaOsa20]. By Ozawa's theorem [Reference OzawaOza00], exactness of a countable group is equivalent to the existence of a (topologically) amenable action on a compact metrizable space (see also [Reference Brown and OzawaBO88, Theorem 5.1.7]).
2.5 (Strong) cocycle conjugacy
Let 
$\alpha : G \curvearrowright A$ be a discrete group action on a unital 
${\rm C}^{\ast }$-algebra. A map 
$u : G \rightarrow \mathcal {U}(A)$ is said to be an 
$\alpha$-cocycle if it satisfies 
$u_{gh}=u_g \alpha _g(u_h)$ for all 
$g, h\in G$. Each 
$\alpha$-cocycle 
$u$ defines a new action 
$\alpha ^u : G \curvearrowright A$, called the cocycle perturbation of 
$\alpha$ by 
$u$, by the formula 
$\alpha ^u_g:= {\rm Ad}(u_g) \circ \alpha _g$ for 
$g\in G$. Two actions 
$\alpha : G \curvearrowright A$ and 
$\beta : G \curvearrowright B$ are said to be cocycle conjugate if 
$\alpha$ is conjugate to a cocycle perturbation of 
$\beta$, that is, if there is a 
$\beta$-cocycle 
$w$ and an isomorphism 
$\varphi : B \rightarrow A$ satisfying 
$\alpha _g = \varphi \circ \beta _g^w \circ \varphi ^{-1}$ for all 
$g\in G$. Clearly, if two actions 
$\alpha$ and 
$\beta$ are cocycle conjugate, then 
$\alpha _\omega$ and 
$\beta _\omega$ are conjugate.
Two actions 
$\alpha$ and 
$\beta$ are said to be strongly cocycle conjugate if there is an isomorphism 
$\varphi : B \rightarrow A$, a 
$\beta$-cocycle 
$w$, and a sequence 
$(u_n)_{n=1}^\infty$ in 
$\mathcal {U}(B)$ satisfying
\[ \alpha_g = \varphi \circ \beta_g^w \circ \varphi^{{-}1} \quad {\rm and} \quad w_g = \lim_{n\rightarrow \infty} u_n \beta_g(u_n)^\ast \quad {\rm for\ all}\ g\in G. \]For an advantage of strong cocycle conjugacy rather than just cocycle conjugacy, see Lemma 4.2 of [Reference SzabóSza18a] for instance. Note that these two relations are reflexive, symmetric, and transitive.
2.6 Saturated ${\rm C}^{\ast }$-dynamical systems
The definition of saturated 
${\rm C}^{\ast }$-dynamical systems is introduced by Szabó [Reference SzabóSza18c] as an equivariant analogue of [Reference Farah and HartFH13]. Roughly speaking, this is an abstract formulation of some important properties shared by ultraproduct actions and their restrictions to (relative) central sequence algebras. These actions are thus typical examples of saturated actions. This concept allows us to considerably reduce the number of times to repeat standard reindexation arguments. We thus employ this concept in the article. We refer the reader to § 1 of [Reference SzabóSza18c] for the definition and basic facts of saturated actions.
2.7 $\mathcal {D}$-absorbing actions
Let 
$\mathcal {D}$ be a strongly self-absorbing 
${\rm C}^{\ast }$-algebra [Reference Toms and WinterTW07]. (In this article we only consider the Cuntz algebra cases 
$\mathcal {D}=\mathcal {O}_{2}, \mathcal {O}_{\infty }$.) Let 
$\alpha$ be an action of a discrete group 
$G$ on a unital 
${\rm C}^{\ast }$-algebra 
$A$. Recall that 
$\alpha$ is said to be equivariantly 
$\mathcal {D}$-absorbing if 
$\alpha$ is cocycle conjugate to 
${\rm id}_{\mathcal {D}} \otimes \alpha : G \curvearrowright \mathcal {D} \otimes A$. An equivariant McDuff-type theorem (see [Reference Izumi and MatuiIM18, Theorem 4.11] or [Reference SzabóSza18a, Corollary 3.8]) shows that 
$\alpha$ is equivariantly 
$\mathcal {D}$-absorbing if and only if there is a unital embedding 
$\mathcal {D} \rightarrow (A_\omega )^\alpha$. Moreover, if this is the case, the theorem shows that 
$\alpha$ is in fact strongly cocycle conjugate to 
${\rm id}_{\mathcal {D}} \otimes \alpha$.
2.8 Notation
Here we fix the basic notation used in this article.
Throughout the article, all 
${\rm C}^{\ast }$-algebras are assumed to be non-zero. Let 
$A$, 
$B$ be unital 
${\rm C}^{\ast }$-algebras. Let 
$X$ be a normed space. Throughout the article, 
$G$ denotes a countable discrete exact group.
• For
$x, y \in X$ and $\epsilon >0$, let $x \approx _\epsilon y$ if $\|x-y\|<\epsilon$.• For
$x, y \in A$, set $[x, y]:= xy -yx$.• Denote by
$\mathcal {U}(A)$ the unitary group of $A$.• Denote by
$1_A$ the unit of $A$. When the considered ${\rm C}^{\ast }$-algebra $A$ is clear from the context, denote $1_A$ by $1$ for short.• For
$u\in \mathcal {U}(A)$, ${\rm Ad}(u)$ denotes the inner automorphism of $A$ defined by $u$; ${\rm Ad}(u)(x)=u x u^\ast$ for $x\in A$.• For a unitary representation
$u : G \rightarrow \mathcal {U}(A)$, ${\rm Ad}(u)$ denotes the inner action $G \curvearrowright A$ given by $g \mapsto {\rm Ad}(u_g)$.• The symbols ‘
$\otimes$’, ‘$\rtimes _{\mathrm {r}}$’ denote the minimal tensor product (of ${\rm C}^{\ast }$-algebras and completely positive maps) and the reduced crossed product respectively.• For an action
$\alpha : G \curvearrowright A$, $A^\alpha$ denotes the fixed-point algebra of $\alpha$;
\[ A^\alpha:=\{a\in A: \alpha_g(a)=a\ {\rm for\ all}\ g\in G\}. \]• For two actions
$\alpha : G \curvearrowright A$ and $\beta : G \curvearrowright B$, denote by $\alpha \otimes \beta : G \curvearrowright A\otimes B$ the diagonal action; $(\alpha \otimes \beta )_g := \alpha _g \otimes \beta _g$ for $g\in G$.• For a subset
$S \subset A$, set $A \cap S' := \{ a\in A: [a, s]=0\ {\rm for\ all}\ s\in S\}$.• For
$n\in \mathbb {N}$, denote by $\mathbb {M}_n(A)$ the ${\rm C}^{\ast }$-algebra of $n$-by-$n$ matrices over $A$. When $A=\mathbb {C}$, we denote it by $\mathbb {M}_n$ for short. We identify $\mathbb {M}_n(A)$ with $\mathbb {M}_n \otimes A$ via the canonical isomorphism.• For
$n\in \mathbb {N}$ and $1\leq i, j \leq n$, denote by $e_{i, j} \in \mathbb {M}_n(A)$ the matrix whose $(i, j)$-entry is $1_A$ and the other entries are $0$.• Let
$\alpha : G \curvearrowright A$. Let $B \subset A$ be a $G$-${\rm C}^{\ast }$-subalgebra. When there is no confusion, we denote the restriction action $G \curvearrowright B$ by the same symbol $\alpha$.• Denote by
$1_G$ the identity element of $G$.
3. Proof of the Main Theorem
The following key result is a generalization of [Reference SzabóSza18c, Lemma 2.3]. Because Følner sets no longer exist in our case, the statement involves technical assumptions (which may be regarded as a relative (weaker) version of the QAP). In fact, the full statement of Lemma 2.3 fails for non-amenable groups; see Remark 3.9. We will however see that the following statement is strong enough to obtain the Main Theorem.
Recall that 
$G$ denotes a countable exact group.
Lemma 3.1 Let 
$D$ be a unital 
${\rm C}^{\ast }$-algebra. Let 
$\alpha : G \curvearrowright D$ be a saturated action. Let 
$A \subset D$ be a separable 
$G$-
${\rm C}^{\ast }$-subalgebra satisfying the following conditions.
(1) The relative commutant
$D \cap A'$ is purely infinite simple.(2) The restriction action
$G \curvearrowright D \cap A'$ is pointwise outer.(3) For any finite subsets
$S\subset A$, $F\subset G$ and any positive number $\epsilon >0$, there is $\xi \in \ell ^2(G, D)$ satisfying:
•
$\langle \xi , \tilde {\alpha }_g(\xi ) \rangle \approx _\epsilon 1$ for all $g\in F$;•
$a\xi \approx _\epsilon \xi a$ for all $a\in S$.
Then the fixed-point algebra 
$(D \cap A') ^{\alpha }$ is purely infinite simple.
Proof. Let 
$a\in (D \cap A') ^{\alpha }$ be a positive element of norm one. We will find, following Lemma 2.3 of [Reference SzabóSza18c], a proper isometry 
$v\in (D \cap A') ^{\alpha }$ satisfying 
$v^\ast a v =1$.
By Proposition 2.2 of [Reference SzabóSza18c], we have a non-zero projection 
$p \in D \cap A'$ satisfying:
•
$\alpha _g(p) p=0$ for all $g\in G \setminus \{1_G\}$;•
$a\alpha _g(p)=\alpha _g(p)$ for all $g\in G$.
By replacing 
$p$ by its proper subprojection (which exists by (1)) if necessary, we may assume in addition that there is a non-zero projection 
$q \in D \cap A'$ with 
$q \alpha _g(p)=0$ for all 
$g\in G$. Take an isometry 
$t \in D \cap A'$ with 
$t^\ast p t=1$.
Let finite subsets 
$S \subset A$, 
$F \subset G$ and 
$\epsilon >0$ be given. Choose 
$\xi \in \ell ^2(G, D)$ as in (3) for the 
$S$, 
$F$, 
$\epsilon$. By standard perturbation arguments, we may assume in addition that 
$\xi$ is finitely supported and satisfies 
$\langle \xi , \xi \rangle =1$. Set
\[ v_{S, F}:= \sum_{g\in G} \alpha_g(pt)\xi_g \in D. \]Then direct computations imply
\[ v_{S, F}^\ast v_{S, F}= \sum_{g\in G} \xi_g ^\ast \xi_g=1. \]
Observe also that 
$qv_{S, F} = 0$. Hence 
$v_{S, F}$ is a proper isometry. It is clear from the choice of 
$p$ that 
$av_{S, F}=v_{S, F}$. Thus 
$v_{S, F}^\ast a v_{S, F}=1$.
We show that 
$v_{S, F}$ almost commutes with 
$S$ and is almost invariant under 
$F$ (the precise meanings are self-explaining below). For 
$x\in S$, as 
$pt\in D \cap A'$, we have
\[ [x, v_{S, F}] =\sum_{g\in G} \alpha_g(pt) (x\xi_g - \xi_g x). \]Hence
\[ \|[x, v_{S, F}]\|^2=\|[x, v_{S, F}]^\ast [x, v_{S, F}]\|= \bigg\|\sum_{g\in G} (x\xi_g - \xi_g x)^\ast (x\xi_g - \xi_g x)\bigg\| =\| x\xi - \xi x\|^2 < \epsilon^2. \]
For 
$s\in F$, we have
\[ \alpha_s(v_{S, F})= \sum_{g\in G} \alpha_{sg}(pt) \alpha_s(\xi_g) = \sum_{g\in G} \alpha_g(pt) \alpha_s(\xi_{s^{{-}1}g}) = \sum_{g\in G} \alpha_g(pt) \tilde{\alpha}_s(\xi)_g. \]Therefore computations similar to the previous one yield
\[ \|v_{S, F}- \alpha_s(v_{S, F})\| = \| \xi - \tilde{\alpha}_s(\xi)\|<\sqrt{2\epsilon}. \] Now, since 
$\alpha$ is saturated, 
$G$ is countable, and 
$A$ is separable, this guarantees the existence of a proper isometry 
$v \in (D\cap A')^\alpha$ with 
$v^\ast a v=1$.
Remark 3.2 We remark that the proof of Lemma 2.1 in [Reference SzabóSza18c], which Proposition 2.2 of [Reference SzabóSza18c] (thus Lemma 3.1) depends on, needs a slight correction. To explain this, let us use the same notation as in [Reference SzabóSza18c, Lemma 2.1]. The problem is that, in the proof, the condition 
$qa=q$ is not confirmed. (This does not follow from 
$q\in \overline {aAa}$.) However this is easily fixed as follows. Since 
$D$ is saturated, by standard applications of functional calculus, one can find a positive element 
$a_0$ in 
$D$ with 
$a_0 a= a_0$, 
$\|a_0 \| =1$. Applying the proof of [Reference SzabóSza18c, Lemma 2.1] to 
$a_0$ instead of 
$a$, we obtain the desired 
$q$ (for the original 
$a$).
We next show a variant of Lemmas 2.4 and 2.9 in [Reference SzabóSza18c]. Because of the restrictions in Lemma 3.1, we need to choose all objects more carefully.
Lemma 3.3 Let 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ be a unitary representation. Let 
$\alpha : G \curvearrowright \mathcal {O}_{2}$ be an equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer, QAP action. Then there exist a unital 
$\ast$-homomorphism 
$\nu : \mathcal {O}_{2} \rightarrow ((\mathcal {O}_{2})_\omega )^\alpha$ and 
$w\in \mathcal {U}( (\mathcal {O}_{2})_\omega )$ satisfying 
$\nu (u_g)= w \alpha _g(w^\ast )$ for all 
$g\in G$.
Proof. Since a cocycle conjugation induces the conjugation of the actions on the central sequence algebras, we may assume that 
$\alpha$ is of the form
\[ {\rm id}_{\mathcal{O}_{2}} \otimes \alpha_0 : G \curvearrowright \mathcal{O}_{2} \otimes \mathcal{O}_{2} \]
for some QAP action 
$\alpha _0$. Put 
$A:=\mathcal {O}_{2} \otimes 1_{\mathcal {O}_{2}}$, 
$B := 1_{\mathcal {O}_{2}}\otimes \mathcal {O}_{2}$, 
$C:=\mathcal {O}_{2} \otimes \mathcal {O}_{2}$. Take two unital 
$\ast$-homomorphisms 
$\nu , \mu : \mathcal {O}_{2} \rightarrow A_\omega$ satisfying 
$[\nu (x), \mu (y)]=0$ for all 
$x, y\in \mathcal {O}_{2}$ [Reference RørdamRor02]. Regarding 
$\nu$ as a map into 
$(C_\omega )^\alpha$, we will show that this 
$\nu$ satisfies the desired condition.
Define 
$\beta : G \curvearrowright \mathbb {M}_2(C^\omega )$ to be
\[ \beta_g := {\rm Ad} \bigg(\begin{array}{cc} 1 & 0 \\ 0 & \nu(u_g) \end{array} \bigg) \circ ({\rm id}_{\mathbb{M}_2} \otimes \alpha_g) \quad{\rm for}\ g\in G. \]
Note that 
$\beta$ is saturated by [Reference SzabóSza18c, Corollary 1.10 and Proposition 1.12]. Set
\[ D:=\mathbb{M}_2(C_\omega) \subset \mathbb{M}_2(C^\omega). \]
Since 
$\mu (\mathcal {O}_{2}) \subset (C_\omega )^\alpha \cap \nu (\mathcal {O}_{2})'$, we have a unital inclusion
\[ 1_{\mathbb{M}_2} \otimes \mu(\mathcal{O}_{2}) \subset D^\beta \cap \{e_{1, 1}, e_{2, 2}\}'. \]This shows
\[ [e_{1, 1}]_0 = [e_{2, 2}]_0 =0 \quad{\rm in}\ K_0(D^\beta). \] We next show that, with respect to 
$\beta$, the inclusion 
$1_{\mathbb {M}_2} \otimes C \subset \mathbb {M}_2(C^\omega )$ fulfills the assumptions of Lemma 3.1. Since 
$D= \mathbb {M}_2(C^\omega ) \cap (1_{\mathbb {M}_2} \otimes C)'$, this claim together with Lemma 3.1 shows that 
$D^\beta$ is purely infinite simple. Condition (1) follows from [Reference Kirchberg and PhillipsKP00, Proposition 3.4]. Condition (2) follows from [Reference SzabóSza18c, Theorem 3.1] (cf. [Reference NakamuraNak00]). To show condition (3), fix finite subsets 
$T \subset 1_{\mathbb {M}_2}\otimes B$, 
$F \subset G$ and a positive number 
$\epsilon >0$. Then, as 
$\alpha _0$ has the QAP, one can take 
$\xi \in \ell ^2(G, 1_{\mathbb {M}_2} \otimes B) \subset \ell ^2(G, \mathbb {M}_2(C^\omega ))$ satisfying:
•
$\langle \xi , (\widetilde {{\rm id}_{\mathbb {M}_2} \otimes \alpha })_g(\xi ) \rangle \approx _\epsilon 1$ for all $g\in F$;•
$b\xi \approx _\epsilon \xi b$ for all $b\in T$.
Observe that, for any 
$g\in G$, as the values of 
$\xi$ commute with 
$\scriptsize{\begin{pmatrix} 1 & 0 \\ 0 & \nu (u_g) \end{pmatrix}\in \mathbb {M}_2(A_\omega)}$, we have 
$(\widetilde {{\rm id}_{\mathbb {M}_2} \otimes \alpha })_g(\xi ) =\tilde {\beta }_g(\xi )$. Hence 
$\langle \xi , \tilde {\beta }_g(\xi ) \rangle \approx _\epsilon 1$ for all 
$g\in F$. We also note that 
$a \xi = \xi a$ for all 
$a\in 1_{\mathbb {M}_2} \otimes A$. As 
$(1_{\mathbb {M}_2} \otimes A) \cup ( 1_{\mathbb {M}_2} \otimes B)$ generates 
$1_{\mathbb {M}_2} \otimes C$ as a 
${\rm C}^{\ast }$-algebra, this proves condition (3) of the inclusion.
Hence, by Lemma 3.1, 
$D^\beta$ is purely infinite simple. Therefore, by [Reference CuntzCun81], one can take a partial isometry 
$v\in D^\beta$ satisfying 
$vv^\ast =e_{1, 1}, v^\ast v =e_{2, 2}$. Certainly this element is of the form 
$v= e_{1, 2} \otimes w$ for some 
$w\in \mathcal {U}(C_\omega )$. Since 
$v$ is 
$\beta$-invariant, direct computations (cf. [Reference SzabóSza18c, Lemma 2.4]) show that 
$\nu (u_g)= w \alpha _g(w)^\ast$ for all 
$g\in G$. We thus obtain the desired element 
$w$.
Corollary 3.4 Let 
$\alpha : G \curvearrowright \mathcal {O}_{2}$ be an equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer, QAP action. Then for any unitary representation 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{2})$, there is a unital 
$G$-equivariant embedding 
$(\mathcal {O}_{2}, {\rm Ad}(u)) \rightarrow ((\mathcal {O}_{2})_\omega , \alpha _\omega )$.
Proof. By Lemma 3.3, we have a unital 
$\ast$-homomorphism 
$\nu : \mathcal {O}_{2} \rightarrow ((\mathcal {O}_{2})_\omega )^\alpha$ and 
$w\in \mathcal {U}((\mathcal {O}_{2})_\omega )$ satisfying 
$w \alpha _g(w)^\ast = \nu (u_g)$ for all 
$g\in G$. Set 
$\mu := {\rm Ad}(w^\ast )\circ \nu$. Then direct computations (cf. [Reference SzabóSza18c, Lemma 2.9]) show that 
$\mu$ gives the desired embedding.
We now prove Theorem C. Before the proof, we record the following universality-like property of QAP actions, which is of independent interest.
Proposition 3.5 Let 
$\alpha : G \curvearrowright A$ be a QAP action on a unital simple separable nuclear 
${\rm C}^{\ast }$-algebra. Then for any action 
$\beta : G \curvearrowright B$ on a unital separable exact 
${\rm C}^{\ast }$-algebra, there is a 
$G$-equivariant unital completely positive map from 
$B$ into 
$A_\omega$. Moreover, when we additionally assume that 
$\alpha$ is pointwise outer and equivariantly 
$\mathcal {O}_{2}$-absorbing, the stated maps can be taken to be an embedding.
Proof. By the observation in the first paragraph of the proof of [Reference SzabóSza18c, Proposition 5.3], we only need to consider the case 
$(B, \beta )= (\mathcal {O}_{2}, {\rm Ad}(u))$ for unitary representations 
$u: G \rightarrow \mathcal {U}(\mathcal {O}_{2})$. (In [Reference SzabóSza18c] the statement is restricted to amenable groups, but the same proof works for exact groups after employing the reduced 
${\rm C}^{\ast }$-completion instead of the full one; see [Reference Brown and OzawaBO88, Theorem 10.2.9].)
The last statement follows from this observation and Corollary 3.4.
Let 
$u: G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ be a unitary representation. Take an equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer action 
$\sigma : G \curvearrowright \mathcal {O}_{2}$ which admits an invariant state 
$\varphi$. (For instance, take the diagonal action of the trivial action on 
$\mathcal {O}_{2}$ and the Bernoulli shift action 
$G \curvearrowright {\bigotimes} _G \mathcal {O}_{2}$.) Then, since 
$A \otimes \mathcal {O}_{2} \cong \mathcal {O}_{2}$, the diagonal action 
$\alpha \otimes \sigma$ satisfies the assumptions of Corollary 3.4. Therefore we have a unital 
$G$-equivariant embedding
\[ \iota : (\mathcal{O}_{2}, {\rm Ad}(u)) \rightarrow ((A \otimes \mathcal{O}_{2})_\omega, \alpha \otimes \sigma). \]
As 
$\varphi$ is 
$\sigma$-invariant, the map 
${\rm id}_{A} \otimes \varphi : A \otimes \mathcal {O}_{2} \rightarrow A$ induces a 
$G$-equivariant unital completely positive map
\[ \Phi : ((A \otimes \mathcal{O}_{2})_\omega, \alpha \otimes \sigma) \rightarrow (A_\omega, \alpha);\quad [x_n]_n^\omega \mapsto [({\rm id}_A \otimes \varphi)(x_n)]_n^\omega. \]
The composite 
$\Phi \circ \iota$ gives the desired map.
Proof of Theorem C Clearly (6) implies (4) and (5). By [Reference Anantharaman-DelarocheAna87, Proposition 2.5], (3) implies (2). We show that (1) implies (3), (2) implies (6), (4) implies (1), and (5) implies (1), which complete the proof.
$(1)\Rightarrow (3)$. Assume that 
$\alpha$ has the QAP. Take an amenable action 
$\eta : G \curvearrowright X$ on a compact metrizable space ([Reference OzawaOza00], [Reference Brown and OzawaBO88, Theorem 5.1.7]). By Proposition 3.5, we have a 
$G$-equivariant unital completely positive map 
$\Phi : C(X) \rightarrow A_\omega$. Since 
$\eta$ is amenable, by [Reference Brown and OzawaBO88, Theorem 4.4.3], there is a sequence 
$(f_n : G \rightarrow C(X))$ of finitely supported positive definite functions with 
$\lim _{n\rightarrow \infty } f_n(g) =1$ for all 
$g\in G$. Since 
$\Phi$ is completely positive and 
$G$-equivariant, the functions 
$\Phi \circ f_n : G \rightarrow A_\omega$ are positive definite. As 
$\Phi$ is unital and continuous, we have
\[ \lim_{n \rightarrow \infty}(\Phi \circ f_n)(g)=1 \quad{\rm for\ all}\ g\in G. \]
This confirms condition (3) of 
$\alpha$.
$(2)\Rightarrow (6)$. Assume that 
$\alpha$ satisfies condition (2). Let 
$B \subset A^\omega$ be a separable 
$G$-
${\rm C}^{\ast }$-subalgebra. Take any finite subset 
$F\subset G$ and finite sequence 
$a(1), \ldots , a(k) \in A^\omega$. Fix a positive number 
$\epsilon >0$. Choose a dense sequence 
$(b(n))_{n=1}^\infty$ of 
$B$.
For each 
$i=1, \ldots , k$, take a representing sequence 
$(a(i)_n)_{n\in \mathbb {N}}$ of 
$a(i)$. We also fix a representing sequence 
$(b(i)_n)_{n\in \mathbb {N}}$ of 
$b(i)$ for each 
$i\in \mathbb {N}$. Choose 
$\xi \in \ell ^2(G, A_\omega )$ satisfying 
$\langle \xi , \tilde {\alpha }_g(\xi )\rangle \approx _{\epsilon } 1$ for all 
$g\in F$. By a standard perturbation argument, we may assume in addition that 
$\xi$ is finitely supported. Choose a bounded sequence 
$(\xi _{n})_{n=1}^\infty$ in 
$\ell ^2(G, A)$ satisfying:
• for any
$n \in \mathbb {N}$, ${\rm supp}(\xi _{n}) \subset {\rm supp}(\xi )$;• for each
$g\in G$, $[\xi _{n, g}]_n^\omega = \xi _{g}$.
Then 
$\lim _{n\rightarrow \omega } \| \langle \xi _{n}, \tilde {\alpha }_g(\xi _{n}) \rangle -1\|< \epsilon$ for all 
$g\in F$, 
$\lim _{n \rightarrow \omega } \| a\xi _{n} - \xi _{n} a\| =0$ for all 
$a\in A$. By these two conditions, for each 
$n\in \mathbb {N}$, one can choose 
$k(n)\in \mathbb {N}$ satisfying 
$\langle \xi _{k(n)}, \tilde {\alpha }_g(\xi _{k(n)}) \rangle \approx _{\epsilon } 1$ for all 
$g\in F$ and 
$\| x \xi _{k(n)} - \xi _{k(n)} x\| < 1/n$ for all 
$x= a(1)_n, \ldots , a(k)_n, b(1)_n, \ldots , b(n)_n$. Define 
$\eta \in \ell ^2(G, A^\omega )$ to be 
$\eta _g:= [\xi _{k(n), g}]_n^\omega$ for 
$g\in G$. Then by the choice of the 
$k(n)$, we obtain:
•
$\langle \eta , \tilde {\alpha }_g(\eta )\rangle \approx _{\epsilon } 1$ for all $g\in F$;•
$a_i \eta = \eta a_i$ for all $i=1, \ldots , k$;•
$\eta \in \ell ^2(G, A^\omega \cap B')$.
This proves condition (6).
$(4)\Rightarrow (1)$. Assume that 
$\alpha ^\omega$ has the QAP. Then for any finite subsets 
$F \subset G$, 
$S\subset A$, and any 
$\epsilon >0$, one can find an element 
$\xi \in \ell ^2(G, A^\omega )$ with:
•
$\langle \xi , \tilde {\alpha }_g(\xi )\rangle \approx _{\epsilon } 1$ for all $g\in F$;•
$\xi$ is finitely supported;•
$\|x\xi -\xi x\| < \epsilon$ for all $x\in S$.
Choose a bounded sequence 
$(\xi _{n})_{n=1}^\infty$ in 
$\ell ^2(G, A)$ with:
•
${\rm supp}(\xi _n) \subset {\rm supp}(\xi )$ for all $n\in \mathbb {N}$;•
$[\xi _{n, g}]_n^\omega = \xi _g$ for all $g\in G$.
Then by the choice of 
$\xi$, one can find 
$n \in \mathbb {N}$ satisfying:
•
$\langle \xi _n, \tilde {\alpha }_g(\xi _n)\rangle \approx _{\epsilon } 1$ for all $g\in F$;•
$\|x\xi _n-\xi _n x\| < \epsilon$ for all $x\in S$.
This shows that 
$\alpha$ has the QAP.
$(5)\Rightarrow (1)$. The proof of this implication is similar to that of 
$(4) \Rightarrow (1)$. We therefore omit details.
Remark 3.6 By the proof of Theorem C and obvious implications, for any countable group action on a unital separable 
${\rm C}^{\ast }$-algebra, conditions (2) to (6) are pairwise equivalent, and these conditions imply condition (1).
Now by Lemma 3.1 and Theorem C, we conclude the following useful result.
Corollary 3.7 Let 
$\alpha : G \curvearrowright A$ be a pointwise outer, QAP action on a unital Kirchberg algebra. Then for any unital simple separable nuclear 
$G$-
${\rm C}^{\ast }$-subalgebra 
$B \subset A^\omega$, the fixed-point algebra 
$(A^\omega \cap B')^\alpha$ is purely infinite simple.
Proof. By Lemma 3.1, it suffices to show that the inclusion 
$B \subset A^\omega$ satisfies the assumptions of Lemma 3.1.
Condition (1) follows from Theorem 1.17 of [Reference SzabóSza18c] (cf. Proposition 3.4 of [Reference Kirchberg and PhillipsKP00]). Condition (2) follows from Theorem 3.1 of [Reference SzabóSza18c] (cf. [Reference NakamuraNak00]) and standard reindexation arguments (cf. [Reference SzabóSza18c, Lemmas 5.1 and 5.2]). By Theorem C, 
$\alpha ^\omega$ has the QAP. In particular, the inclusion satisfies condition (3).
The rest of the proof of the Main Theorem is mostly identical to that of Theorem C in [Reference SzabóSza18c] modulo using Lemma 3.1, Theorem C, and Corollary 3.7 in place of [Reference SzabóSza18c, Lemma 2.3].
Lemma 3.8 (cf. [Reference SzabóSza18c, Lemma 5.1])
Let 
$D$ be a unital purely infinite simple 
${\rm C}^{\ast }$-algebra. Let 
$\alpha : G \curvearrowright D$ be a saturated, QAP action. Assume that any separable 
$G$-
${\rm C}^{\ast }$-subalgebra 
$A\subset D$ satisfies the following conditions.
• There is a unital embedding
$\mathcal {O}_{2} \rightarrow (D \cap A')^\alpha$.• The restriction action
$G \curvearrowright D \cap A'$ is faithful.
Let 
$\beta : G \curvearrowright B$ be an action on a unital simple separable nuclear 
${\rm C}^{\ast }$-algebra 
$B$. Then any two unital 
$G$-equivariant 
$\ast$-homomorphisms 
$(B, \beta ) \rightarrow (D, \alpha )$ are 
$G$-unitarily equivalent.
Proof. The proof is the same as the original proof of [Reference SzabóSza18c, Lemma 5.1], after using Lemma 3.1 in place of [Reference SzabóSza18c, Lemma 2.3].
Proof of the Main Theorem We first observe that the statement in the first paragraph of [Reference SzabóSza18c, Lemma 5.2] holds true without amenability of the acting group. (This is an easy consequence of [Reference SzabóSza18c, Theorem 3.1] (cf. [Reference NakamuraNak00]) and standard reindexation arguments.) Now the Main Theorem follows by the same proof as that of [Reference SzabóSza18c, Theorem 5.5] (cf. [Reference SzabóSza18a, Lemma 2.1]), after using Lemma 3.8 and the second statement of Proposition 3.5 instead of [Reference SzabóSza18c, Lemma 5.1] and [Reference SzabóSza18c, Corollary 5.4] respectively.
Remark 3.9 Here we remark that, for countable non-amenable exact groups 
$G$, there is a pointwise outer, saturated action of 
$G$ on a unital purely infinite simple 
${\rm C}^{\ast }$-algebra whose fixed-point algebra fails to be simple.
To see this, choose an equivariantly 
$\mathcal {O}_{2}$-absorbing, pointwise outer action 
$\alpha : G \curvearrowright \mathcal {O}_{2}$ which admits an invariant state. (For instance, take the diagonal action of the trivial action on 
$\mathcal {O}_{2}$ and the Bernoulli shift action 
$G \curvearrowright {\bigotimes} _{G} \mathcal {O}_{2}$.) Then notice that 
$\alpha _\omega$ also admits an invariant state. Take a unitary representation 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ whose adjoint action 
${\rm Ad}(u)$ does not have an invariant state; for instance choose one of any 
$u_n$ from Example B.
Then the statement of Corollary 3.4, hence that of Lemma 3.3, fails for these 
$\alpha$ and 
$u$. Reviewing the proof of Lemma 3.3, notice that the QAP of the considered action is used only to show the simplicity of the fixed-point algebra 
$D^\beta$ of a pointwise outer, saturated action 
$\beta : G \curvearrowright D$ on a unital purely infinite simple 
${\rm C}^{\ast }$-algebra (cf. Lemma 3.1). Indeed, starting the argument in the proof to these 
$\alpha$ and 
$u$, (using the notation in the proof of Lemma 3.3,) one can still show that the (non-zero) projections 
$e_{1, 1}$ and 
$e_{2, 2}$ are properly infinite in 
$D^\beta$ and satisfy 
$[e_{1, 1}]_0=[e_{2, 2}]_0$ in 
$K_0(D^\beta )$. Therefore, by [Reference CuntzCun81], just from the simplicity of 
$D^\beta$, one can conclude that these two projections are Murray–von Neumann equivalent in 
$D^\beta$. As we have seen in the proof of Lemma 3.3, this proves the statement of Lemma 3.3 for 
$\alpha$ and 
$u$. This is a contradiction. Thus 
$D^\beta$ is not simple.
Remark 3.10 On the one hand, similar to amenable groups ([Reference IzumiIzu04], [Reference SzabóSza18c, Remark 5.8]), a pointwise outer, QAP action of a non-amenable group on 
$\mathcal {O}_{2}$ is not necessary equivariantly 
$\mathcal {O}_{2}$-absorbing. To see this, recall from the proof of [Reference SuzukiSuz20a, Theorem 5.1] that the free group 
$\mathbb {F}_\infty$ admits a pointwise outer, QAP action 
$\alpha$ on a unital Kirchberg algebra 
$A$ whose reduced crossed product is stably isomorphic to 
$\mathcal {O}_{\infty }$. As observed in [Reference IzumiIzu04], there is a pointwise outer action 
$\beta : \mathbb {Z}_2 \curvearrowright \mathcal {O}_{2}$ whose reduced crossed product has non-zero K-theory. Then the product action 
$\zeta : \mathbb {F}_\infty \times \mathbb {Z}_2 \curvearrowright A \otimes \mathcal {O}_{2} \cong \mathcal {O}_{2}$ is pointwise outer, QAP, but its reduced crossed product has non-zero K-theory. Thus 
$\zeta$ is not equivariantly 
$\mathcal {O}_{2}$-absorbing. On the other hand, this phenomenon is expected to be related to torsions of the acting groups. Works on amenable group actions (see e.g. [Reference Izumi and MatuiIM19, Reference MatuiMat08, Reference SzabóSza19]) suggest that the equivariant 
$\mathcal {O}_{2}$-absorbing assumption could be removed from the Main Theorem for certain torsion free groups.
4. Some equivariant $\mathcal {O}_{\infty }$-absorption results for exact groups
In this section we prove Theorem F, an equivariant variant of the Kirchberg 
$\mathcal {O}_{\infty }$-absorption theorem for QAP actions. This is an extension of another main result in [Reference SzabóSza18c, Theorems B and 3.4, Corollary 3.7] obtained for amenable groups. However we remark that Theorem F is not as satisfactory as the Main Theorem for non-amenable groups because the absorbed model actions themselves do not have the QAP. That said, we also expect that this result gives a new insight of amenable actions on Kirchberg algebras.
Proof of Theorem F We first observe that it suffices to show the following statement. For any unitary representation 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{\infty })$ which factors through 
$\mathcal {O}_{2}$ (see Definition D), there is a unital 
$G$-equivariant 
$\ast$-homomorphism 
$(\mathcal {O}_{\infty }, {\rm Ad}(u)) \rightarrow (A_\omega , \alpha )$. Indeed once we have confirmed the claim, standard reindexation arguments guarantee the existence of a unital 
$G$-equivariant 
$\ast$-homomorphism 
${\bigotimes} _{\mathbb {N}}(\mathcal {O}_{\infty }, \gamma ) \rightarrow (A_\omega , \alpha )$. By Proposition 5.3 of [Reference SzabóSza18b], 
${\bigotimes} _{\mathbb {N}}\gamma$ is strongly self-absorbing. Therefore Theorem 3.7 of [Reference SzabóSza18a] or Theorem 4.11 of [Reference Izumi and MatuiIM18] completes the proof.
Let 
$u : G \rightarrow \mathcal {U}(\mathcal {O}_{\infty })$ be a unitary representation factoring through 
$\mathcal {O}_{2}$. Take a unitary representation 
$v: G \rightarrow \mathcal {U}(\mathcal {O}_{2})$ and an embedding 
$\iota : \mathcal {O}_{2} \rightarrow \mathcal {O}_{\infty }$ satisfying 
$u_g = \iota (v_g)+(1_{\mathcal {O}_{\infty }}-\iota (1_{\mathcal {O}_{2}}))$ for all 
$g\in G$.
By Corollary 3.7, 
$(A_\omega )^\alpha$ is purely infinite simple. In particular one can find a unital 
$\ast$-homomorphism 
$\mu _0 : \mathcal {O}_{\infty } \rightarrow (A_\omega )^\alpha$. Set 
$p:= \mu _0(\iota (1_{\mathcal {O}_{2}})) \in (A_\omega )^\alpha$. Set 
$D:= p A_\omega p$. Note that 
$A_\omega$, thus 
$D$, is purely infinite simple [Reference Kirchberg and PhillipsKP00, Proposition 3.4]. Denote by 
$\zeta : G \curvearrowright D$ the restriction action of 
$\alpha _\omega$. Then, as 
$D^\zeta = p(A_\omega )^\alpha p$, 
$D^\zeta$ is purely infinite simple. Note that 
$\mu _0( \iota (\mathcal {O}_{2})) \subset D^\zeta$. By Theorem C, 
$\alpha _\omega$, therefore 
$\zeta$, has the QAP.
Define 
$\beta : G \curvearrowright \mathbb {M}_2(D)$ to be
\[ \beta_g:={\rm Ad} \bigg(\begin{array}{cc} p & 0 \\ 0 & \mu_0(u_g)p \end{array} \bigg) \circ ({\rm id}_{\mathbb{M}_2} \otimes \zeta_g) \quad {\rm for}\ g\in G. \]
Observe that, as 
$\zeta$ has the QAP, so does 
$\beta$. By [Reference SzabóSza18c, Corollary 1.10 and Proposition 1.12], 
$\beta$ is saturated. Since 
$\mu _0(u_g)p=\mu _0(\iota (v_g))$, one can find a sequence 
$(\nu _n : \mathcal {O}_{2} \rightarrow \mu _0(\iota (\mathcal {O}_{2})))_{n=1}^\infty$ of unital 
$\ast$-homomorphisms with 
$\lim _{n\rightarrow \infty } [\nu _n(x), \mu _0(u_g)p ] =0$ for all 
$g\in G$. Since 
$\beta$ is saturated, one can take a unital embedding
\[ \mathcal{O}_{2} \rightarrow \mathbb{M}_2(D)^\beta \cap \{e_{1, 1}, e_{2, 2}\}'. \]Hence
\[ [e_{1, 1}]_0 =[e_{2, 2}]_0=0 \quad {\rm in}\ K_0(\mathbb{M}_2(D)^\beta). \]
Since 
$\beta$ is saturated and satisfies the QAP, Lemma 3.1 yields that 
$\mathbb {M}_2(D)^\beta$ is purely infinite simple. Thus, by [Reference CuntzCun81], one can find a unitary 
$w_0\in \mathcal {U}(D)$ satisfying 
$e_{1, 2} \otimes w_0\in \mathbb {M}_2(D)^\beta$. Direct computations (cf. [Reference SzabóSza18c, Lemma 2.9]) show that 
$\mu _0(\iota (v_g))=w_0\alpha _g(w_0^\ast )$ for all 
$g\in G$. Set 
$w:= w_0 + (1-p) \in \mathcal {U}(A_\omega )$. Then 
$\mu _0(u_g) = w \alpha _g(w^\ast )$.
Now define 
$\mu := {\rm Ad}(w^\ast ) \circ \mu _0 : \mathcal {O}_{\infty } \rightarrow A_\omega$. Then direct computations show that
\[ \mu \circ {\rm Ad}(u_g) = \alpha_g \circ \mu \quad {\rm for\ all}\ g\in G. \]As explained in the first paragraph of the proof, this completes the proof.
5. Concluding remarks
The Main Theorem concludes the existence of the largest action on Kirchberg algebras for exact groups (with respect to the tensor products, up to strong cocycle conjugacy). For 
$\mathcal {O}_{\infty }$, certainly it is more desirable to obtain an equivariant 
$\mathcal {O}_{\infty }$-absorption theorem for some model action with the QAP instead of our present 
$\gamma$. More precisely, to understand (pointwise outer) QAP actions (on Kirchberg algebras) more deeply, it is desirable to find the smallest QAP actions on Kirchberg algebras in the same sense. However we remark that even the existence of a QAP action on 
$\mathcal {O}_{\infty }$ is not clear. At the moment we know such actions only for free groups (and therefore for some groups constructed from free groups); see [Reference SuzukiSuz20a, proof of Theorem 5.1].
Acknowledgements
The author is grateful to the referee for his/her careful reading and helpful suggestions. This work was supported by JSPS KAKENHI Early-Career Scientists (No. 19K14550).
