Strongly outer actions of certain torsion-free amenable groups on the Razak–Jacelon algebra

Abstract

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 $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this article, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of Γ on the Razak–Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó’s result for strongly self-absorbing C$^*$-algebras.

1. Introduction

Let $\mathcal{W}$ be the Razak–Jacelon algebra studied in [15] (see also [33]). By classification results in [2] and [6] (see also [30]), $\mathcal{W}$ 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 [40] for group actions on strongly self-absorbing C$^*$-algebras (see also [12–14, 22–24, 26, 37, 39] for pioneering works). We refer the reader to [11] 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 [7]. 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 [40, 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 [40, 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 [31] of strongly outer $\mathcal{W}$-absorbing actions of countable discrete amenable groups. Actually, we use the following theorem that is a slight variant of [31, 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:

1. (i) there exists a unital $*$-homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,

2. (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 $F(\mathcal{W})^{\alpha}$,

3. (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 [19] and [20], Herman-Ocneanu’s argument in [9], and homotopy type arguments in [28] 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 [25–27, 34–36, 42].

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 α_g 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 τ_A. Let $(\pi_{\tau_{A}}, H_{\tau_{A}})$ be the Gelfand–Naimark–Segal representation of τ_A. 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 [8] and [26] 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 [17] 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 M_ω is the von Neumann algebraic central sequence algebra (or the asymptotic centralizer) of M.

For a tracial state τ_A 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 [28, proposition 2.1]. Put $J_{\tau_{A}}:=\{x\in F(A)\; |\; \tau_{A, \omega}(x^*x)=0\}$. If A is separable and τ_A 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 [18, theorem 3.3]. In this article, the reindexing argument and the diagonal argument (or Kirchberg’s ɛ-test [17, lemma A.1]) are frequently used. We refer the reader to [1, Section 1.3] and [32, 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 [31, 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 [31, theorem 3.6], [31, proposition 3.11], and [31, proposition 3.12]. Note that these propositions are based on results in [25–27, 34–36, 42].

Proposition 2.1. Let α be an outer action of a countable discrete amenable group on $\mathcal{W}$.

1. (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. (2) The fixed point algebra $F(\mathcal{W})^{\alpha}$ is monotracial.

3. (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:

1. (i) there exists a unital $*$-homomorphism from $M_{2}(\mathbb{C})$ to $F(\mathcal{W})^{\alpha}$,

2. (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 [17, corollary 1.13]. Hence [38, corollary 3.8] (see also [41]) implies this cocycle conjugacy. Using this observation, proposition 2.1 and definition 2.2 instead of $M_{2^{\infty}}$-stability of $\mathcal{W}$, [28, proposition 4.1], [28, theorem 5.3], and [28, theorem 5.8], we obtain the following theorem by essentially the same arguments as in the proofs of [28, proposition 4.2], [28, theorem 5.7], and [28, corollary 5.11] (or [29, 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. (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. (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. (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 [31, theorem 8.1]. Note that one of the main techniques in the proof of [31, theorem 8.1] is Szabó’s approximate cocycle intertwining argument [43] (see also [5]).

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. [31, proposition 4.2], [31, theorem 4.5], and [31, theorem 8.1] imply the only if part. The if part is an immediate consequence of [31, 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, [16, 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 [32, 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 [32, 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 [4, 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, [26, 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 [28, lemma 6.2]. See also [24, 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 [3, 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 [28, lemma 6.2]. (See also [24, 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 [28, proposition 4.1], [28, proposition 4.2], [28, theorem 5.8], and [28, lemma 6.2], we obtain the following Rohlin type theorem by essentially the same arguments in the proofs of [28, lemma 6.3] and [28, theorem 6.4]. Note that these arguments are based on [19] and [20].

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 [9, theorem 1] (see also remarks after [9, lemma 1] and [10, 21]) 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 [28] (or [30, 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 $\mathbb{Z}$, [30, 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.