1. Introduction
Let 
$\mathcal{W}$ be the Razak–Jacelon algebra studied in [Reference Jacelon15] (see also [Reference Razak33]). By classification results in [Reference Castillejos and Evington2] and [Reference Elliott, Gong, Lin and Niu6] (see also [Reference Nawata30]), 
$\mathcal{W}$ is the unique simple separable nuclear monotracial 
$\mathcal{Z}$-stable C
$^*$-algebra that is KK-equivalent to 
$\{0\}$. Also, 
$\mathcal{W}$ is regarded as a stably finite analog of the Cuntz algebra 
$\mathcal{O}_2$. More generally, we can consider that 
$\mathcal{W}$ is a non-unital analog of strongly self-absorbing C
$^*$-algebras. (Note that every strongly self-absorbing C
$^*$-algebra is unital by definition.) In this article, we study group actions on 
$\mathcal{W}$ and show an analogous result of Szabó’s result in [Reference Szabó40] for group actions on strongly self-absorbing C
$^*$-algebras (see also [Reference Izumi and Matui12–Reference Izumi and Matui14, Reference Matui22–Reference Matui and Sato24, Reference Matui and Sato26, Reference Szabó37, Reference Szabó39] for pioneering works). We refer the reader to [Reference Izumi11] for the importance and some difficulties of studying group actions on C
$^*$-algebras. Gabe and Szabó classified outer actions of countable discrete amenable groups on Kirchberg algebras up to cocycle conjugacy in [Reference Gabe and Szabó7]. In their classification, 
$\mathcal{O}_2$ and 
$\mathcal{O}_{\infty}$ play central roles. Hence it is natural to expect that 
$\mathcal{W}$ plays a central role in the classification theory of group actions on ‘classifiable’ stably finite (at least stably projectionless) C
$^*$-algebras.
Let 
$\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) 
$\mathfrak{C}$ contains the trivial group, (ii) 
$\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by 
$\mathbb{Z}$. (We say that Γ is an extension by 
$\mathbb{Z}$ if there exists an exact sequence 
$1\to H \to \Gamma \to \mathbb{Z}\to 1$.) Note that 
$\mathfrak{C}$ is the same class as in [Reference Szabó40, definition B]. It is easy to see that 
$\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-
$\mathbb{Z}$ groups, and 
$\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. Szabó showed that if 
$\Gamma\in \mathfrak{C}$ and 
$\mathcal{D}$ is a strongly self-absorbing C
$^*$-algebra, then there exists a unique strongly outer action of Γ on 
$\mathcal{D}$ up to cocycle conjugacy [Reference Szabó40, corollary 3.4]. In this article, we show an analogous result of this result. Indeed, the main theorem in this article is the following theorem.
Theorem A (theorem 4.3)
Let Γ be a countable discrete group in 
$\mathfrak{C}$, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. Then α is cocycle conjugate to 
$\mu^{\Gamma}\otimes \mathrm{id}_{\mathcal{W}}$ on 
$M_{2^{\infty}}\otimes\mathcal{W}$ where 
$\mu^{\Gamma}$ is the Bernoulli shift action of Γ on 
$\bigotimes_{g\in \Gamma}M_{2^{\infty}}\cong M_{2^{\infty}}$.
We say that an action α on 
$\mathcal{W}$ is 
$\mathcal{W}$-absorbing if there exists a simple separable nuclear monotracial C
$^*$-algebra A and an action β on A such that α is cocycle conjugate to 
$\beta\otimes\mathrm{id}_{\mathcal{W}}$ on 
$A\otimes\mathcal{W}$. The proof of the main theorem above is based on a characterization in [Reference Nawata31] of strongly outer 
$\mathcal{W}$-absorbing actions of countable discrete amenable groups. Actually, we use the following theorem that is a slight variant of [Reference Nawata31, theorem 8.1]. Note that 
$F(\mathcal{W})$ is Kirchberg’s central sequence C
$^*$-algebra of 
$\mathcal{W}$. Furthermore, 
$F(\mathcal{W})^{\alpha}$ is the fixed point algebra for the action on 
$F(\mathcal{W})$ induced by an action α on 
$\mathcal{W}$. Let 
$\mathrm{Sp}(x)$ denote the spectrum of x.
Theorem B (theorem 2.4)
Let α be a strongly outer action of a countable discrete amenable group Γ on 
$\mathcal{W}$. Then α is cocycle conjugate to 
$\mu^{\Gamma}\otimes\mathrm{id}_{\mathcal{W}}$ on 
$M_{2^{\infty}}\otimes \mathcal{W}$ if and only if α satisfies the following properties:
(i) there exists a unital
$*$-homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,(ii) if x and y are normal elements in
$F(\mathcal{W})^{\alpha}$ such that $\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and $0 \lt \tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any $f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$, then x and y are unitary equivalent in $F(\mathcal{W})^{\alpha}$,(iii) there exists an injective
$*$-homomorphism from $\mathcal{W}\rtimes_{\alpha}\Gamma$ to $\mathcal{W}$.
We use a first cohomology vanishing type theorem (corollary 3.4) for showing that if 
$\Gamma\in \mathfrak{C}$ and α is a strongly outer action of Γ on 
$\mathcal{W}$, then 
$F(\mathcal{W})^{\alpha}$ satisfies the properties (i) and (ii) in the theorem above. Kishimoto’s techniques for Rohlin type theorems in [Reference Kishimoto19] and [Reference Kishimoto20], Herman-Ocneanu’s argument in [Reference Herman and Ocneanu9], and homotopy type arguments in [Reference Nawata28] enable us to show this first cohomology vanishing type theorem. Also, note that our arguments for 
$F(\mathcal{W})^{\alpha}$ are based on results that are shown by techniques around (equivariant) property (SI) in [Reference Matui and Sato25–Reference Matui and Sato27, Reference Sato34–Reference Sato36, Reference Szabó42].
2. Preliminaries
2.1. Notations and basic definitions
Let α and β be actions of a countable discrete group Γ on C
$^*$-algebras A and B, respectively. We say that α is conjugate to β if there exists a isomorphism φ from A onto B such that 
$\varphi\circ \alpha_g=\beta_g\circ \varphi$ for any 
$g\in \Gamma$. Note that α induces an action on the multiplier algebra M(A) of A. We denote it by the same symbol α. An α-cocycle on A is a map from Γ to the unitary group of M(A) such that 
$u_{gh}=u_{g}\alpha_g(u_h)$ for any 
$g,h\in \Gamma$. We say that α is cocycle conjugate to β if there exist an isomorphism φ from A onto B and a β-cocycle u such that 
$\varphi\circ \alpha_g=\mathrm{Ad}(u_g)\circ \beta_g \circ \varphi$ for any 
$g\in\Gamma$. An action α of Γ on A is said to be outer if αg is not an inner automorphism of A for any 
$g\in \Gamma\setminus \{\iota\}$ where ι is the identity of Γ. We denote by A α and 
$A\rtimes_{\alpha}\Gamma$ the fixed point algebra and the reduced crossed product C
$^*$-algebra, respectively.
Assume that A has a unique tracial state τA. Let 
$(\pi_{\tau_{A}}, H_{\tau_{A}})$ be the Gelfand–Naimark–Segal representation of τA. Then 
$\pi_{\tau_{A}}(A)^{\prime\prime}$ is a finite factor and α induces an action 
$\tilde{\alpha}$ on 
$\pi_{\tau_{A}}(A)^{\prime\prime}$. We say that α is strongly outer if 
$\tilde{\alpha}$ is an outer action on 
$\pi_{\tau_{A}}(A)^{\prime\prime}$. (We refer the reader to [Reference Gardella, Hirshberg and Vaccaro8] and [Reference Matui and Sato26] for the definition of strongly outerness for more general settings.)
We denote by 
$\mathcal{R}_0$ and 
$M_{2^{\infty}}$ the injective II1 factor and the canonical anticommutation relations (CAR) algebra, respectively.
2.2. Fixed point algebras of Kirchberg’s central sequence C$^*$-algebras
Let ω be a free ultrafilter on 
$\mathbb{N}$, and put
\begin{equation*}
A^{\omega}:= \ell^{\infty}(\mathbb{N}, A)/\{\{x_n\}_{n\in\mathbb{N}}\in \ell^{\infty}(\mathbb{N}, A)\; |
\; \lim_{n\to\omega} \|x_n \|=0 \}.
\end{equation*} We denote by 
$(x_n)_n$ a representative of an element in A ω. We identify A with the C
$^*$-subalgebra of A ω consisting of equivalence classes of constant sequences. Set
\begin{equation*}\mathrm{Ann}(A,A^\omega):=\{(x_n)_n\in A^\omega\cap A^{\prime}\;\vert\;\lim_{n\rightarrow\omega}\Arrowvert x_na\Arrowvert=0\,\text{for any }a\in A\}.\end{equation*} Then 
$\mathrm{Ann}(A, A^{\omega})$ is an closed ideal of 
$A^{\omega}\cap A^{\prime}$, and define
\begin{equation*}
F(A):= A^{\omega}\cap A^{\prime}/\mathrm{Ann}(A, A^{\omega}).
\end{equation*}See [Reference Kirchberg17] for basic properties of F(A). For a finite von Neumann algebra M, put
\begin{equation*}
M^{\omega}:=\ell^{\infty}(\mathbb{N}, M)/\{\{x_n\}_{n\in\mathbb{N}}\in \ell^{\infty}(\mathbb{N}, M)
\; |\; \lim_{n\to\omega}\| x_n\|_2=0\}
\end{equation*}and
\begin{equation*}
M_{\omega}:=M^{\omega}\cap M^{\prime}.
\end{equation*}Note that we identify M with the subalgebra of M ω consisting of equivalence classes of constant sequences and Mω is the von Neumann algebraic central sequence algebra (or the asymptotic centralizer) of M.
For a tracial state τA on A, define a map 
$\tau_{A, \omega}$ from F(A) to 
$\mathbb{C}$ by 
$\tau_{A, \omega}([(x_n)_n])=\lim_{n\to\omega}\tau_A(x_n)$ for any 
$[(x_n)_n]\in F(A)$. Then 
$\tau_{A, \omega}$ is a well-defined tracial state on F(A) by [Reference Nawata28, proposition 2.1]. Put 
$J_{\tau_{A}}:=\{x\in F(A)\; |\; \tau_{A, \omega}(x^*x)=0\}$. If A is separable and τA is faithful, then 
$\pi_{\tau_{A}}$ induces an isomorphism from 
$F(A)/J_{\tau_{A}}$ onto 
$\pi_{\tau_{A}}(A)^{\prime\prime}_{\omega}$ by essentially the same argument as in the proof of [Reference Kirchberg and Rørdam18, theorem 3.3]. In this article, the reindexing argument and the diagonal argument (or Kirchberg’s ɛ-test [Reference Kirchberg17, lemma A.1]) are frequently used. We refer the reader to [Reference Bosa, Brown, Sato, Tikuisis, White and Winter1, Section 1.3] and [Reference Ocneanu32, Chapter 5] for details of these arguments. Every action α of a countable discrete group on A induces an action on F(A). We denote it by the same symbol α for simplicity. Note that if α on A are cocycle conjugate to β on B, then α on F(A) are conjugate to β on F(B). If A is simple, separable, and monotracial, then 
$\tilde{\alpha}$ also induces an action on 
$\pi_{\tau_{A}}(A)^{\prime\prime}_{\omega}$. We also denote it by the same symbol 
$\tilde{\alpha}$. By [Reference Nawata31, proposition 3.9], we see that 
$\pi_{\tau_{A}}$ induces an isomorphism from 
$F(A)^{\alpha}/J_{\tau_{A}}^{\alpha}$ onto 
$(\pi_{\tau_{A}}(A)^{\prime\prime})_{\omega}^{\tilde{\alpha}}$.
The following proposition is an immediate consequence of [Reference Nawata31, theorem 3.6], [Reference Nawata31, proposition 3.11], and [Reference Nawata31, proposition 3.12]. Note that these propositions are based on results in [Reference Matui and Sato25–Reference Matui and Sato27, Reference Sato34–Reference Sato36, Reference Szabó42].
Proposition 2.1. Let α be an outer action of a countable discrete amenable group on 
$\mathcal{W}$.
(1) The Razak–Jacelon algebra
$\mathcal{W}$ has property (SI) relative to α, that is, if a and b are positive contractions in $F(\mathcal{W})^{\alpha}$ satisfying $\tau_{\mathcal{W}, \omega}(a)=0$ and $\inf_{m\in\mathbb{N}}\tau_{\mathcal{W}, \omega}(b^m) \gt 0$, then there exists an element s in $F(\mathcal{W})^{\alpha}$ such that bs = s and $s^*s=a$.(2) The fixed point algebra
$F(\mathcal{W})^{\alpha}$ is monotracial.(3) If a and b are positive elements in
$F(\mathcal{W})^{\alpha}$ satisfying $d_{\tau_{\mathcal{W}, \omega}}(a) \lt d_{\tau_{\mathcal{W}, \omega}}(b)$, then there exists an element r in $F(\mathcal{W})^{\alpha}$ such that $r^*br=a$.
Definition 2.2. Let α be an action of a countable discrete group Γ on 
$\mathcal{W}$. We say that α has property W if α satisfies the following properties:
(i) there exists a unital
$*$-homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,(ii) if x and y are normal elements in
$F(\mathcal{W})^{\alpha}$ such that $\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and $0 \lt \tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any $f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$, then x and y are unitary equivalent in $F(\mathcal{W})^{\alpha}$.
Note that if there exists a unital 
$*$-homomorphism from 
$M_{2}(\mathbb{C})$ to 
$F(\mathcal{W})^{\alpha}$, then α on 
$\mathcal{W}$ is cocycle conjugate to 
$\alpha\otimes\mathrm{id}_{M_{2^{\infty}}}$ on 
$\mathcal{W}\otimes M_{2^{\infty}}$. Indeed, there exists a unital 
$*$-homomorphism from 
$M_{2^{\infty}}$ to 
$F(\mathcal{W})^{\alpha}$ by a similar argument as [Reference Kirchberg17, corollary 1.13]. Hence [Reference Szabó38, corollary 3.8] (see also [Reference Szabó41]) implies this cocycle conjugacy. Using this observation, proposition 2.1 and definition 2.2 instead of 
$M_{2^{\infty}}$-stability of 
$\mathcal{W}$, [Reference Nawata28, proposition 4.1], [Reference Nawata28, theorem 5.3], and [Reference Nawata28, theorem 5.8], we obtain the following theorem by essentially the same arguments as in the proofs of [Reference Nawata28, proposition 4.2], [Reference Nawata28, theorem 5.7], and [Reference Nawata28, corollary 5.11] (or [Reference Nawata29, corollary 5.5]).
Theorem 2.3 Let α be an outer action of a countable discrete amenable group on 
$\mathcal{W}$. Assume that α has property W.
(1) For any
$\theta\in [0,1]$, there exists a projection p in $F(\mathcal{W})^{\alpha}$ such that $\tau_{\mathcal{W}, \omega}(p)=\theta$.(2) For any unitary element u in
$F(\mathcal{W})^{\alpha}$, there exists a continuous path of unitaries $U: [0,1]\to F(\mathcal{W})^{\alpha}$ such that
\begin{equation*}
U(0)=1,\quad U(1)=u \quad \text{and} \quad \mathrm{Lip}(U)\leq 2\pi
\end{equation*}where
$\mathrm{Lip}(U)$ is the Lipschitz constant of U, that is, the smallest positive number satisfying $\| U(t)- U(s)\| \leq \mathrm{Lip}(U)|t-s|$ for any $t, s\in [0,1]$.(3) If p and q are projections in
$F(\mathcal{W})^{\alpha}$ such that $0 \lt \tau_{\mathcal{W}, \omega}(p)=\tau_{\mathcal{W}, \omega}(q)$, then p and q are Murray–von Neumann equivalent.
For any countable discrete group Γ, let 
$\mu^{\Gamma}$ be the Bernoulli shift action of Γ on 
$\bigotimes_{g\in \Gamma}M_{2^{\infty}}\cong M_{2^{\infty}}$. The following theorem is a slight variant of [Reference Nawata31, theorem 8.1]. Note that one of the main techniques in the proof of [Reference Nawata31, theorem 8.1] is Szabó’s approximate cocycle intertwining argument [Reference Szabó43] (see also [Reference Elliott5]).
Theorem 2.4 Let α be a strongly outer action of a countable discrete amenable group Γ on 
$\mathcal{W}$. Then α is cocycle conjugate to 
$\mu^{\Gamma}\otimes\mathrm{id}_{\mathcal{W}}$ on 
$M_{2^{\infty}}\otimes \mathcal{W}$ if and only if α has property W and there exists an injective 
$*$-homomorphism from 
$\mathcal{W}\rtimes_{\alpha}\Gamma$ to 
$\mathcal{W}$.
Proof. [Reference Nawata31, proposition 4.2], [Reference Nawata31, theorem 4.5], and [Reference Nawata31, theorem 8.1] imply the only if part. The if part is an immediate consequence of [Reference Nawata31, theorem 8.1] and theorem 2.3.
3. First cohomology vanishing type theorem
In this section, we shall show a first cohomology vanishing type theorem (corollary 3.4). This is a corollary of a Rohlin type theorem (theorem 3.3).
The following lemma is well-known among experts. See, for example, [Reference Katayama16, theorem 4.8] for a similar (but not the same) result. For the reader’s convenience, we shall give a proof based on Ocneanu’s classification theorem [Reference Ocneanu32, corollary 1.4].
Lemma 3.1. Let Γ be a countable discrete amenable group, and let N be a normal subgroup of Γ. If γ is an outer action of Γ on the injective II1 factor 
$\mathcal{R}_0$ and 
$g_0\notin N$, then 
$\gamma_{g_0}$ induces a properly outer automorphism of 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$.
Proof. Since N is a normal subgroup, it is clear that 
$\gamma_{g_0}$ induces an automorphism of 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. First, we shall show that 
$\gamma_{g_0}$ is not trivial as an automorphism of 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. Let π be the quotient map from Γ to 
$\Gamma/ N$, and let β be the Bernoulli shift action of 
$\Gamma /N$ on 
$\mathcal{R}_0\cong \bigotimes_{\pi (g)\in \Gamma /N}\mathcal{R}_0$. Define an action δ of Γ on 
$\mathcal{R}_0\cong \mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0}$ by 
$\delta_g:= \gamma_g\otimes \beta_{\pi (g)}$ for any 
$g\in\Gamma$. By Ocneanu’s classification theorem [Reference Ocneanu32, corollary 1.4], γ on 
$\mathcal{R}_0$ and δ on 
$\mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0}$ are cocycle conjugate. Hence there exists an isomorphism Φ from 
$(\mathcal{R}_0)_{\omega}$ onto 
$(\mathcal{R}_0 \bar{\otimes}\mathcal{R}_{0})_{\omega}$ such that 
$\Phi\circ \gamma_g=\delta_g\circ \Phi$ for any 
$g\in \Gamma$. Since 
$\beta_{\pi (g_0)}$ is an outer automorphism of 
$\mathcal{R}_0$, there exists an element 
$(x_n)_n$ in 
$(\mathcal{R}_0)_{\omega}$ such that 
$(\beta_{\pi(g_0)}(x_n))_n\neq (x_n)_n$ by [Reference Connes4, theorem 3.2]. Put 
$(y_n)_n:=\Phi^{-1}((1\otimes x_n)_n)\in (\mathcal{R}_0)_{\omega}$. Then it is easy to see that we have 
$(y_n)_n\in (\mathcal{R}_0)_{\omega}^{\gamma|_N}$ and 
$(\gamma_{g_0}(y_n))_n\neq (y_n)_n$. Finally, we shall show that 
$\gamma_{g_0}$ is properly outer as an automorphism of 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. Since 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$ is a factor (see, for example, [Reference Matui and Sato26, lemma 4.1]), it is enough to show that 
$\gamma_{g_0}$ is outer as an automorphism of 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$. In particular, we shall show that for any element 
$(u_n)_n$ in 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$, there exists an element 
$(z_n)_n$ in 
$(\mathcal{R}_0)_{\omega}^{\gamma|_N}$ such that 
$(u_nz_n)_n=(z_nu_n)_n$ and 
$(\gamma_{g_0}(z_n))_n\neq (z_n)_n$. Taking a suitable subsequence of 
$(y_n)_n$ (or the reindexing argument), we obtain the desired element 
$(z_n)_n$. Consequently, the proof is complete.
Consider a semidirect product group 
$N\rtimes\mathbb{Z}$. For 
$g\in N$ and 
$m\in\mathbb{Z}$, let (g, m) denote an element in 
$N\rtimes \mathbb{Z}$. Note that we have 
$N\rtimes \mathbb{Z}=\{(g,m)\; |\; g\in N, m\in\mathbb{Z}\}$. The following lemma is an analogous lemma of [Reference Nawata28, lemma 6.2]. See also [Reference Matui and Sato24, theorem 3.4].
Lemma 3.2. Let Γ be a semidirect product 
$N\rtimes \mathbb{Z}$ where N is a countable discrete amenable group, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. Then for any 
$k\in\mathbb{N}$, there exists a positive contraction f in 
$F(\mathcal{W})^{\alpha|_{N}}$ such that
\begin{equation*}
\tau_{\mathcal{W}, \omega}(f)=\frac{1}{k}\quad \text{and} \quad f\alpha_{(\iota,j)}(f)=0
\end{equation*} for any 
$1\leq j \leq k-1$.
Proof. Since 
$\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime}$ is isomorphic to the injective II1 factor, lemma 3.1 implies that 
$\tilde{\alpha}_{(\iota, 1)}$ is an aperiodic automorphism of 
$(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime})_{\omega}^{\tilde{\alpha}|_N}$. Hence it follows from [Reference Connes3, theorem 1.2.5] that there exists a partition of unity 
$\{P_{j}\}_{j=1}^k$ consisting of projections in 
$(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime})_{\omega}^{\tilde{\alpha}|_N}$ such that 
$\tilde{\alpha}_{(\iota, 1)}(P_{j})=P_{j+1}$ for any 
$1\leq j \leq k-1$. Since 
$\pi_{\tau_{\mathcal{W}}}$ induces an isomorphism from 
$F(\mathcal{W})^{\alpha|_N}/J_{\tau_{\mathcal{W}}}^{\alpha|_N}$ onto 
$(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime})_{\omega}^{\tilde{\alpha}|_N}$ (see §2.2), there exists a positive contraction 
$[(e_n)_n]$ in 
$F(\mathcal{W})^{\alpha|_N}$ such that 
$(\pi_{\tau_{\mathcal{W}}}(e_n))_n=P_1$ in 
$(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime})_{\omega}^{\tilde{\alpha}|_N}$. Then we have
\begin{equation*}
\lim_{n\to\omega} \| \pi_{\tau_{\mathcal{W}}}(e_n \alpha_{(\iota ,j)}(e_n))\|_2=0
\quad
\text{and}
\quad
\tau_{\mathcal{W}, \omega}([(e_n)_n])=\tilde{\tau}_{\mathcal{W}, \omega}(P_1)=\dfrac{1}{k}
\end{equation*} for any 
$1\leq j \leq k-1$, where 
$\tilde{\tau}_{\mathcal{W}, \omega}$ is the induced tracial state on 
$(\pi_{\tau_{\mathcal{W}}}(\mathcal{W})^{\prime\prime})_{\omega}^{\tilde{\alpha}|_N}$ by 
$\tau_{\mathcal{W}}$. The rest of the proof is the same as [Reference Nawata28, lemma 6.2]. (See also [Reference Matui and Sato24, proposition 3.3].)
Using proposition 2.1, theorem 2.3 (we need to assume that 
$\alpha|_N$ has property W), and lemma 3.2 instead of [Reference Nawata28, proposition 4.1], [Reference Nawata28, proposition 4.2], [Reference Nawata28, theorem 5.8], and [Reference Nawata28, lemma 6.2], we obtain the following Rohlin type theorem by essentially the same arguments in the proofs of [Reference Nawata28, lemma 6.3] and [Reference Nawata28, theorem 6.4]. Note that these arguments are based on [Reference Kishimoto19] and [Reference Kishimoto20].
Theorem 3.3 Let Γ be a semidirect product 
$N\rtimes \mathbb{Z}$ where N is a countable discrete amenable group, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. Assume that 
$\alpha|_{N}$ has property W. Then for any 
$k\in\mathbb{N}$, there exists a partition on unity 
$\{p_{1,i}\}_{i=0}^{k-1}\cup
\{p_{2,j}\}_{j=0}^{k}$ consisting of projections in 
$F(\mathcal{W})^{\alpha|_N}$ such that
\begin{equation*}
\alpha_{(\iota, 1)}(p_{1,i})=p_{1, i+1}\quad \text{and} \quad \alpha_{(\iota, 1)}(p_{2,j})=p_{2,j+1}
\end{equation*} for any 
$0\leq i\leq k-2$ and 
$0\leq j\leq k-1$.
Theorem 2.3, theorem 3.3, and Herman–Ocneanu’s argument [Reference Herman and Ocneanu9, theorem 1] (see also remarks after [Reference Herman and Ocneanu9, lemma 1] and [Reference Izumi10, Reference Kishimoto21]) imply the following corollary.
Corollary 3.4. Let Γ be a semidirect product 
$N\rtimes \mathbb{Z}$ where N is a countable discrete amenable group, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. Assume that 
$\alpha|_{N}$ has property W and S is a countable subset in 
$F(\mathcal{W})^{\alpha|_N}$. For any unitary element u in 
$F(\mathcal{W})^{\alpha|_N}\cap S^{\prime}$, there exists a unitary element v in 
$F(\mathcal{W})^{\alpha|_N}\cap S^{\prime}$ such that 
$u=v\alpha_{(\iota, 1)}(v)^{*}$.
4. Main theorem
In this section, we shall show the main theorem. Recall that 
$\mathfrak{C}$ is the smallest class of countable discrete groups with the following properties: (i) 
$\mathfrak{C}$ contains the trivial group, (ii) 
$\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by 
$\mathbb{Z}$. Note that if Γ is an extension of N by 
$\mathbb{Z}$, then Γ is isomorphic to a semidirect product 
$N\rtimes\mathbb{Z}$.
The following lemma is an easy consequence of the definition of property W and the diagonal argument.
Lemma 4.1. Let Γ be an increasing union 
$\bigcup_{m\in\mathbb{N}}\Gamma_m$ of discrete countable groups 
$\Gamma_m$, and let α be an action of Γ on 
$\mathcal{W}$. If 
$\alpha|_{\Gamma_m}$ has property W for any 
$m\in\mathbb{N}$, then α has property W.
The following lemma is an application of corollary 3.4.
Lemma 4.2. Let Γ be a semidirect product 
$N\rtimes \mathbb{Z}$ where N is a countable discrete amenable group, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. If 
$\alpha|_{N}$ has property W, then α has property W.
Proof. (i) There exists a unital 
$*$-homomorphism φ from 
$M_{2}(\mathbb{C})$ to 
$F(\mathcal{W})^{\alpha|_N}$ by the assumption. Let 
$\{e_{ij}\}_{i,j=1}^{2}$ be the standard matrix units of 
$M_{2}(\mathbb{C})$. Since we have 
$0 \lt \tau_{\mathcal{W}, \omega}(\varphi(e_{11}))
=\tau_{\mathcal{W}, \omega}(\alpha_{(\iota,1)}(\varphi(e_{11})))$, there exists an element w in 
$F(\mathcal{W})^{\alpha|_N}$ such that 
$w^*w
=\alpha_{(\iota, 1)}(\varphi(e_{11}))$ and 
$ww^*=\varphi(e_{11})$ by theorem 2.3. Put 
$u:= \sum_{i=1}^{2}\varphi(e_{i1})w\alpha_{(\iota ,1)}(\varphi(e_{1i}))$. Then u is a unitary element in 
$F(\mathcal{W})^{\alpha|_N}$ such that 
$\alpha_{(\iota, 1)}(\varphi(x))=u^*\varphi(x)u$ for any 
$x\in M_{2}(\mathbb{C})$. By corollary 3.4, there exists a unitary element v in 
$F(\mathcal{W})^{\alpha|_N}$ such that 
$u=v\alpha_{(\iota ,1)}(v)^*$. We have 
$\alpha_{(\iota, 1)}(v^*\varphi(x)v)= v^*\varphi(x)v$ for any 
$x\in M_{2}(\mathbb{C})$. Hence the map ψ defined by 
$\psi(x):=v^*\varphi(x)v$ for any 
$x\in M_{2}(\mathbb{C})$ is a unital 
$*$-homomorphism from 
$M_{2}(\mathbb{C})$ to 
$F(\mathcal{W})^{\alpha}$. (ii) Let x and y be normal elements in 
$F(\mathcal{W})^{\alpha}$ such that 
$\mathrm{Sp}(x)=\mathrm{Sp}(y)$ and 
$0 \lt \tau_{\mathcal{W}, \omega} (f(x))=\tau_{\mathcal{W}, \omega}(f(y))$ for any 
$f\in C(\mathrm{Sp}(x))_{+}\setminus\{0\}$. Since x and y are also elements in 
$F(\mathcal{W})^{\alpha|_N}$, there exists a unitary element u in 
$F(\mathcal{W})^{\alpha|_N}$ such that 
$uxu^*=y$ by the assumption. Note that 
$u\alpha_{(\iota, 1)}(u)^*$ is a unitary element in 
$F(\mathcal{W})^{\alpha|_N}\cap \{y\}^{\prime}$. Hence corollary 3.4 implies that there exists a unitary element v in 
$F(\mathcal{W})^{\alpha|_N}\cap \{y\}^{\prime}$ such that 
$u\alpha_{(\iota, 1)}(u)^*=v\alpha_{(\iota, 1)}(v)^*$. We have 
$\alpha_{(\iota ,1)}(v^*u)=v^*u$ and 
$v^*uxu^*v= v^*yv=y$. Therefore, x and y are unitary equivalent in 
$F(\mathcal{W})^{\alpha}$. By (i) and (ii), α has property W.
The following theorem is the main theorem in this article.
Theorem 4.3 Let Γ be a countable discrete group in 
$\mathfrak{C}$, and let α be a strongly outer action of Γ on 
$\mathcal{W}$. Then α is cocycle conjugate to 
$\mu^{\Gamma}\otimes \mathrm{id}_{\mathcal{W}}$ on 
$M_{2^{\infty}}\otimes\mathcal{W}$.
Proof. Every action of the trivial group on 
$\mathcal{W}$ has property W by results in [Reference Nawata28] (or [Reference Nawata30, theorem 3.8]). By lemmas 4.1 and 4.2, we see that α has property W. Note that this implies that 
$\mathcal{W}\rtimes_{\alpha}\Gamma$ is 
$M_{2^{\infty}}$-stable because α is cocycle conjugate to 
$\alpha\otimes\mathrm{id}_{M_{2^{\infty}}}$ on 
$\mathcal{W}\otimes M_{2^{\infty}}$. Since the class of separable nuclear C
$^*$-algebras that are KK-equivalent to 
$\{0\}$ is closed under countable inductive limits and crossed products by 
$\mathbb{Z}$, [Reference Nawata30, theorem 6.1] implies that 
$\mathcal{W}\rtimes_{\alpha}\Gamma$ is isomorphic to 
$\mathcal{W}$. Therefore, we obtain the conclusion by theorem 2.4.
The following corollary is an immediate consequence of the theorem above.
Corollary 4.4. Let Γ be a countable discrete group in 
$\mathfrak{C}$. Then there exists a unique strongly outer action of Γ on 
$\mathcal{W}$ up to cocycle conjugacy.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 20K03630.
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