Proof. Take $y \in (M)_1$ . Since the sequence $(\theta (p_n))_{n \in \mathbb {N}}$ converges strongly to $1_N$ , it suffices to show that for all $k \in \mathbb {N}$

$$\begin{align*}(\theta(y)-\theta_n(y)) \theta(p_k) \rightarrow 0 \text{ strongly as } n \rightarrow \infty.\end{align*}$$

Fix $k \in \mathbb {N}$ and $a \in N$ , such that $\tau _N(a^*a) < \infty $ . Given $\varepsilon>0$ , there exists $x \in X$ , such that

$$\begin{align*}\|\theta(x-y)\theta(p_k)\|_{\tau_N} < \frac{\varepsilon}{4\|a\|}.\end{align*}$$

Then there exists $n_0 \in \mathbb {N}$ , such that for all $n \geq n_0$

$$\begin{align*}\|(\theta(x p_k) - \theta_n ( x p_k))a\|_{\tau_N} < \frac{\varepsilon}{4} \text{ and } \|(\theta(p_k)-\theta_n(p_k))a\|_{\tau_N} < \frac{\varepsilon}{4}.\end{align*}$$

For all $n \geq n_0$ , we then get that

$$ \begin{align*} \|(\theta(y) - \theta_n(y))\theta(p_k)a\|_{\tau_N} &\leq \|\theta(x-y)\theta(p_k)a\|_{\tau_N} + \|\theta_n(x-y)\theta_n(p_k)a\|_{\tau_N}\\ &\quad + \|(\theta(xp_k) - \theta_n(xp_k))a\|_{\tau_N} + \|\theta_n(y)(\theta(p_k) - \theta_n(p_k))a\|_{\tau_N}\\ &< 2\|a\| \|\theta(x-y)\theta(p_k)\|_{\tau_N} +\varepsilon/4 + \|(\theta(p_k)-\theta_n(p_k))a\|_{\tau_N} \\ &< \varepsilon. \end{align*} $$

As k and a were arbitrary, this proves the claim.

Proof. We note right away that the first two conditions above can always be tested on self-adjoint elements, hence, one can equivalently state them with the strong- $*$ topology. Denote

$$\begin{align*}\mathfrak{m} := \{x \in M \mid \tau_M(x^*x) < \infty\} \subset M.\end{align*}$$

We let $L^2(M,\tau _M)$ denote the GNS-Hilbert space of M with respect to $\tau _M$ . Similarly, we use the notation $L^2(N, \tau _N)$ . Choose a countable subset $X = \{x_n\}_{n \in \mathbb {N}}$ in $(M)_1$ , such that $X \cap \mathfrak {m}$ is $\|\cdot \|_{\tau _M}$ -dense in $(\mathfrak {m})_1$ . Take a strongly dense sequence $\{y_n\}_{n \in \mathbb {N}}$ in $(N)_1$ . Choose an increasing sequence of compact subsets $K_n \subseteq G$ , such that the union is all of G.

We are going to create a map $\theta : M\to N$ via an inductive procedure. For the first step, we choose $x_{1,1} \in (M)_1$ and $z_1 \in \mathcal {U}(N)$ , such that

• • $ \|z_1 \rho (x_1) z_1^* - \rho (x_1)\|^\sharp _\phi \leq 1/2$ ;

• • $\|y_1 - z_1\rho (x_{1,1})z_1^*\|_\phi \leq 1/2$ ;

• • $\max _{g \in K_1} \|\beta _g(z_1^*) - z_1^*\|_\phi \leq 1/2$ .

Now assume that after the n-th step of the induction, we have found and ${\{x_{l,j}\}_{j \leq l \leq n} \subset (M)_1}$ , such that

1. (1) for ;

2. (2) for and ;

3. (3) for ;

4. (4) and .

Then, by our assumptions, we can find $z_{n+1} \in \mathcal {U}(N)$ and $\{x_{n+1,j}\}_{j \leq n+1}\subset (M)_1$ , such that

• • for ;

• • for and ;

• • for ;

• • and.

We carry on inductively and obtain a sequence of unitaries $(z_n)_{n \in \mathbb {N}}$ in $\mathcal {U}(N)$ and a family $\{ x_{n,j} \}_{n\in \mathbb {N}, j\leq n}\subset (M)_1$ . For each $n \in \mathbb {N}$ , we define and the normal $*$ -homomorphism ${\theta _n\colon M \rightarrow N}$ by $\theta _n = \mathrm {Ad}(u_n) \circ \rho $ . For $n> m$ and , we get

We see that for all $j\in \mathbb {N}$ , the sequence $(\theta _n(x_j))_{n \in \mathbb {N}}$ is norm-bounded and Cauchy with respect to $\|\cdot \|_\phi ^\sharp $ . This means that it converges to some element in N in the strong- $*$ -operator topology. A similar calculation using (2) shows that for $n> m \geq l \geq j$

(2.1) $$ \begin{align}\|\theta_n(x_{l,j}) - \theta_m(x_{l,j})\|^\sharp_{\phi} < \sum_{k=m}^{n-1} 2^{-k-1}, \end{align} $$

so the sequence $(\theta _n(x_{l,j}))_{n \in \mathbb {N}}$ also converges in the strong- $*$ -operator topology for all $j \leq l$ . Since $\theta _n$ is a $*$ -homomorphism for all $n \in \mathbb {N}$ , we conclude that, restricted to the C $^*$ -algebra $A\subset M$ generated by $\{x_n\}_{n \in \mathbb {N}} \cup \{x_{l,j}\}_{j \leq l}$ , the sequence $(\theta _n)_{n \in \mathbb {N}}$ converges point- $*$ -strongly to a $*$ -homomorphism $\theta '\colon A \rightarrow N$ . Since A contains a $\|\cdot \|_{\tau _M}$ -dense subset of $\mathfrak {m}$ , and clearly $\tau _N\circ \theta '=\tau _M|_A$ , there is a unique isometry $T\colon L^2(M, \tau _M) \rightarrow L^2(N, \tau _N)$ induced from the formula $T[a]=[\theta '(a)]$ for all $a\in A\cap \mathfrak {m}$ . Then the normal $*$ -homomorphism

$$\begin{align*}\theta \colon M \rightarrow N \colon x \mapsto T x T^*\end{align*}$$

extends $\theta '$ and $\left (\theta _n \big \lvert _{\mathfrak {m}}\right )_{n \in \mathbb {N}}$ converges point-strongly to $\theta \lvert _{\mathfrak {m}}$ .

We claim that $\theta $ is an isomorphism. Clearly, $\tau _N\circ \theta =\tau _M$ , and so $\theta $ is injective. By applying (3), we find for all $m \geq j$ that

Combining this with (2.1) for $l=m$ and $n \rightarrow \infty $ , we find that

$$\begin{align*}\|\theta(x_{m,j}) - y_j\|_\phi \leq \|\theta'(x_{m,j}) - \theta_m(x_{m,j})\|_\phi + \|\theta_m(x_{m,j}) - y_j\|_\phi \leq 2^{-m} + 2^{-m} = 2^{-m+1}.\end{align*}$$

Since the $y_j$ are strongly dense in the unit ball of N and $\theta $ is normal, this implies surjectivity of $\theta $ . By Lemma 2.1, it then follows that $\theta _n\to \theta $ point-strongly as $n\to \infty $ . Since $\theta _n$ is a unitary perturbation of $\rho $ for each n, this implies $\rho |_{\mathcal {Z}(M)}=\theta _n|_{\mathcal {Z}(M)}\to \theta |_{\mathcal {Z}(M)}$ and, in particular, $\rho (\mathcal {Z}(M))=\theta (\mathcal {Z}(M))=\mathcal {Z}(N)$ .

For $n> m$ and $g \in K_{m+1}$ , we have

From this calculation, we see that for every $g \in G$ , the sequences are Cauchy with respect to $\|\cdot \|^\sharp _\phi $ , with uniformity on compact sets. It follows that for every $g \in G$ , the strong- $*$ limit exists in $\mathcal {U}(N)$ and that this convergence is uniform (w.r.t. $\|\cdot \|^\sharp _\phi $ ) on compact sets. Since $\beta $ is point-strong continuous, this implies the continuity of the assignment . Moreover, for each $g \in G$ and $x \in M$ , we have the equalities of limits with respect to the strong operator topology:

It follows that is a cocycle conjugacy.

For the last part of the statement, assume that $\tau _N$ is finite. Then our previous calculations show that in the above situation, $\rho $ is approximately unitarily equivalent to $\theta $ . Conversely, suppose $\rho $ is approximately unitarily equivalent to a cocycle conjugacy . In particular, there exists a sequence $(u_n)_{n \in \mathbb {N}} \in \mathcal {U}(N)$ , such that $\|u_n\rho (x)u_n^* - \theta (x)\|_{\tau _N} \rightarrow 0$ for all $x \in M$ and uniformly over compact subsets of G. Choose a sequence $\{y_n\}_{n \in \mathbb {N}} \subset (N)_1$ that is strongly dense in $(N)_1$ . For all $k,n \in \mathbb {N}$ , define $x_{n,k} = \theta ^{-1}(u_ny_ku_n^*)$ . Then choose an increasing sequence $(m(n))_{n \in \mathbb {N}} \subset \mathbb {N}$ , such that

$$ \begin{align*} \lim_{n \rightarrow \infty} \| \theta(x_{n,k}) - u_{m(n)} \rho (x_{n,k}) u_{m(n)}^*\|_{\tau_N} = 0 \quad \text{for } k \in \mathbb{N}.\end{align*} $$

Define $w_n := u_n^*u_{m(n)}$ . One can check that these satisfy the assumptions in the lemma.

Proof. Fix a faithful normal state $\phi $ on M. Let $\pi \colon (\mathcal {R},\delta ) \rightarrow (M_{\omega ,\alpha },\alpha _{\omega })$ be a unital equivariant $*$ -homomorphism. We obtain an induced a map on the algebraic tensor product

$$\begin{align*}\mathcal{R} \odot M \rightarrow M^\omega_\alpha \text{ via } x \otimes m \mapsto \pi(x) m. \end{align*}$$

Since for each $m \in M_+$ the map $x \mapsto \phi ^\omega (\pi (x)m)$ defines a positive tracial functional on $\mathcal {R}$ , we see that it must be equal to some multiple of the unique tracial state $\tau $ on $\mathcal {R}$ , and hence, we get for each $x \in \mathcal {R}$ and $m \in M$ that

$$\begin{align*}\phi^\omega(\pi(x) m) = \tau(x) \phi(m) = (\tau \otimes \phi)(x \otimes m).\end{align*}$$

So we see that the map sends the faithful normal state $\tau \otimes \phi $ to $\phi ^\omega $ , and hence, it extends to a unital normal $*$ -homomorphism $\mathcal {R} \bar {\otimes } M \rightarrow M^\omega _\alpha $ , which, moreover, is $(\delta \otimes \alpha )$ -to- $\alpha ^\omega $ equivariant. In this way, we get a unital equivariant normal $*$ -homomorphism

$$\begin{align*}(\mathcal{R} \bar{\otimes} \mathcal{R} \bar{\otimes}M, \delta \otimes \delta \otimes \alpha) \rightarrow (\mathcal{R} \bar{\otimes} M^\omega_\alpha, \delta \otimes \alpha^\omega),\end{align*}$$

given by $x_1 \otimes x_2 \otimes m \mapsto x_1 \otimes (\phi (x_2) m)$ . Composing with the canonical inclusion map $\iota \colon \mathcal {R} \bar {\otimes } M^\omega _\alpha \mapsto (\mathcal {R} \bar {\otimes } M)^\omega _{\delta \otimes \alpha }$ , we get a unital and equivariant normal $*$ -homomorphism

$$\begin{align*}\Phi\colon (\mathcal{R} \bar{\otimes} \mathcal{R} \bar{\otimes} M, \delta \otimes \delta \otimes \alpha) \rightarrow ((\mathcal{R} \bar{\otimes} M)^\omega_{\delta \otimes \alpha}, (\delta \otimes \alpha)^\omega),\end{align*}$$

such that

$$\begin{align*}\Phi(x \otimes 1_{\mathcal{R}} \otimes m) = x \otimes m \text{ for all } x \in \mathcal{R}, m \in M, \end{align*}$$

and

$$\begin{align*}\Phi(1_{\mathcal{R}} \otimes \mathcal{R} \otimes M) \subset \iota(1_{\mathcal{R}} \otimes M^\omega_\alpha). \end{align*}$$

Since $\delta $ has approximately inner half-flip, we can choose a sequence of unitaries $(v_n)_{n \in \mathbb {N}}$ in $\mathcal {R} \bar {\otimes } \mathcal {R}$ , such that $\max _{g \in K}\|v_n - (\delta \otimes \delta )_g(v_n)\|_{\tau \otimes \tau } \rightarrow 0$ for all compact subsets $K \subseteq G$ and ${\|x \otimes 1_{\mathcal {R}} - v_n (1_{\mathcal {R}} \otimes x)v_n^*\|_{\tau \otimes \tau } \rightarrow 0}$ for all $x \in \mathcal {R}$ .

Define $u_n := \Phi (v_n \otimes 1_M) \subset (\mathcal {R} \bar {\otimes } M)^\omega _{\delta \otimes \alpha }$ . This sequence of unitaries satisfies

• • $[u_n, 1_{\mathcal {R}} \otimes m] = \Phi ([v_n \otimes 1_M, 1_{\mathcal {R} \bar {\otimes } \mathcal {R}} \otimes m]) = 0$ for all $m \in M$ ;

• • $\Phi (1_{\mathcal {R}} \otimes x \otimes m) \in \iota (1_{\mathcal {R}} \otimes M^\omega _\alpha )$ and

  $$ \begin{align*} \lim_{n \rightarrow \infty} u_n \Phi(1_{\mathcal{R}} \otimes x \otimes m) u_n^* &=\lim_{n \rightarrow \infty} \Phi((v_n \otimes 1_M)(1_{\mathcal{R}} \otimes x \otimes m)(v_n^* \otimes 1_M))\\ &=\Phi(x \otimes 1_{\mathcal{R}} \otimes m) \\&= x \otimes m, \end{align*} $$
  where the limit is taken with respect to the strong operator topology;

• • $\displaystyle \max _{g \in K} \|u_n^* - (\delta \otimes \alpha )^\omega _g(u_n^*)\|_{(\tau \otimes \phi )^\omega } = \max _{g \in K} \|(v_n^* - (\delta \otimes \delta )_g(v_n^*)) \otimes 1_M\|_{\tau \otimes \tau \otimes \phi } \to 0$ for all compact $K \subseteq G$ .

Each $u_n$ can be lifted to a sequence of unitaries $(z_n^{(k)})_{k\in \mathbb {N}}$ in $\mathcal {E}_{\delta \otimes \alpha }^\omega \cap \mathcal {N}_\omega (\mathcal {R} \bar {\otimes } M)$ . Applying a diagonal sequence argument to the $(z_n^{(k)})_{k\in \mathbb {N}}$ and using Lemma 1.12, we can obtain a sequence of unitaries $(w_n)_{n \in \mathbb {N}}$ in $\mathcal {R} \bar {\otimes } M$ , such that

• • $\mathrm {Ad}(w_n)(1_{\mathcal {R}} \otimes m) - 1_{\mathcal {R}} \otimes m \rightarrow 0$ strongly for all $m \in M$ .

• • $\inf _{m \in (M)_1}\|x - w_n(1_{\mathcal {R}} \otimes m)w_n^*\|_{\tau \otimes \phi } \rightarrow 0$ for $x \in (\mathcal {R} \bar {\otimes }M)_1$ .

• • $\max _{g \in K} \|w_n^* - (\delta \otimes \alpha )_g(w_n^*)\|_{\tau \otimes \phi } \rightarrow 0$ for every compact subset $K \subseteq G$ .

We conclude that the map $1_{\mathcal {R}} \otimes \mathrm {id}_M\colon (M, \alpha ) \rightarrow (\mathcal {R} \bar {\otimes } M, \delta \otimes \alpha )$ satisfies all the necessary conditions to apply Lemma 2.2. This completes the proof.

Proof. The ‘if’ direction follows immediately from the previous proposition. To prove the other direction, we assume that $\delta $ is strongly self-absorbing and reproduce an argument analogous to [Reference Toms and Winter52, Proposition 1.4] and [Reference Szabó48, Proposition 5.5]. Denote the unique tracial state on $\mathcal {R}$ by $\tau $ . Let be a cocycle conjugacy and $u_n \in \mathcal {U}(\mathcal {R} \bar {\otimes } \mathcal {R})$ a sequence of unitaries, such that

(3.1)

and

$$\begin{align*}\lim_{n \rightarrow \infty} \|\phi(x) - u_n(x \otimes 1) u_n^*\|_{\tau \otimes \tau} = 0 \text{ for all } x \in \mathcal{R}. \end{align*}$$

Note that

As a consequence of (3.1), then for every compact $K \subseteq G$ , one has

(3.2) $$ \begin{align}\lim_{n \rightarrow \infty} \max_{g \in K} \sup_{x \in (\mathcal{R})_1} \|(\mathrm{Ad}(u_n^*) \circ \phi \circ \delta_g)(x) - ((\delta \otimes \delta)_g \circ \mathrm{Ad}(u_n^*) \circ \phi)(x)\|_{\tau \otimes \tau} = 0.\end{align} $$

In particular, applying this to $x = \phi ^{-1}(u_n)$ and using that $(\tau \otimes \tau ) = \tau \circ \phi ^{-1}$ yields

$$\begin{align*}\lim_{n \rightarrow \infty} \max_{g \in K}\|(\mathrm{Ad}(\phi^{-1}(u_n^*)) \circ \delta_g) (\phi^{-1}(u_n)) -(\phi^{-1}\circ (\delta \otimes \delta)_g)(u_n)\|_\tau = 0.\end{align*}$$

Combining this with (3.1) again, one gets

(3.3)

First, we prove that $\delta $ has approximately inner half-flip. Define the cocyle morphism . Note that

Applying the equivariant flip automorphism to both sides of this equivalence, we get that

(3.4)

We also get

By transitivity, we get that ${1_{\mathcal {R}} \otimes \mathrm {id}_{\mathcal {R}}\approx _{\mathrm {u}} \mathrm {id}_{\mathcal {R}} \otimes 1_{\mathcal {R}}}$ .

Next, we prove the existence of a unital equivariant $*$ -homomorphism $(\mathcal {R}, \delta ) \rightarrow (\mathcal {R}_{\omega ,\delta }, \delta _\omega )$ . Define the sequence of trace-preserving $*$ -homomorphisms

$$\begin{align*}\chi_n = \phi^{-1} \circ \mathrm{Ad}(u_n) \circ (1_{\mathcal{R}} \otimes \mathrm{id}_{\mathcal{R}}).\end{align*}$$

We conclude from (3.2) that for all $x \in \mathcal {R}$

$$\begin{align*}\lim_{n \rightarrow \infty} \max_{g \in K} \|\delta_g(\chi_n(x)) - \chi_n(\delta_g(x))\|_{\tau} =0.\end{align*}$$

From this and the fact that all $\chi _n$ are trace-preserving, it also follows that $(\chi _n(x))_{n \in \mathbb {N}}$ belongs to $\mathcal {E}^\omega _\alpha $ . Moreover, for any $x,y \in \mathcal {R}$

$$ \begin{align*} \lim_{n \rightarrow \infty} \|[x, \chi_n(y)] \|_\tau &= \lim_{n \rightarrow \infty}\|[\phi(x), u_n(1 \otimes y) u_n^*]\|_{\tau \otimes \tau}\\ &= \lim_{n \rightarrow \infty}\|u_n[x \otimes 1, 1 \otimes y] u_n^*\|_{\tau \otimes \tau}\\ &=0. \end{align*} $$

So, the maps $\chi _n$ induce a unital equivariant $*$ -homomorphism $(\mathcal {R}, \delta ) \rightarrow (\mathcal {R}_{\omega ,\delta }, \delta _\omega )$ .

Proof. The implication (1) $\Rightarrow $ (2) is tautological. Since strong self-absorption implies approximately inner half-flip by Proposition 3.8, the implication (3) $\Rightarrow $ (1) follows from Proposition 3.7.

In order to prove (2) $\Rightarrow $ (3), it is enough to show that there exists a unital equivariant $*$ -homomorphism $(\mathcal {R}, \delta ) \rightarrow ((M \bar {\otimes } \mathcal {R})_{\omega ,\alpha \otimes \delta }, (\alpha \otimes \delta )_\omega )$ . We know there exists a unital equivariant $*$ -homomorphism $(\mathcal {R}, \delta ) \rightarrow (\mathcal {R}_{\omega ,\delta }, \delta _\omega )$ by Proposition 3.8. Since the latter is unitally and equivariantly contained in $((M \bar {\otimes } \mathcal {R})_{\omega ,\alpha \otimes \delta }, (\alpha \otimes \delta )_\omega )$ , this finishes the proof.

Proof. Consider the induced faithful action $\gamma : G_0\curvearrowright M_\omega $ via $\gamma _{gN}=\alpha _{\omega ,g}$ . Then clearly the claim is equivalent to finding a $G_0$ -equivariant unital $*$ -homomorphism $(\mathcal R,\delta )\to (M_\omega ,\gamma )$ . Let us introduce some notation. Let $(x_n)_{n \in \mathbb {N}} \in \ell ^\infty (M)$ be a sequence representing an element $X \in M_\omega $ . Then we set $\tau _\omega (X) = \lim _{n \rightarrow \omega } x_n$ , where the limit is taken in the $\sigma $ -weak topology. Since M is a factor and $\tau _\omega (X)$ is central, this limit belongs to $\mathbb {C}$ . For any $\phi \in M_*$ we have

$$\begin{align*}\phi^\omega(X) = \lim_{n \rightarrow \omega}\phi(x_n) = \phi(\tau_\omega(X)) = \tau_\omega(X).\end{align*}$$

In particular, $\tau _\omega $ defines a normal faithful tracial state on $M_\omega $ , and we denote $\|X\|_1 = \tau _\omega (|X|)$ .

Since M is McDuff, we can find a unital $*$ -homomorphism $\Phi : \mathcal {R} \rightarrow M_\omega $ . Fix $\varepsilon>0$ and a symmetric finite subset containing the neutral element. By [Reference Ocneanu31, Lemmas 5.6 and 5.7], we are allowed to apply [Reference Masuda25, Theorem 4.8] to the action $\gamma \colon G_0\curvearrowright M_\omega $ . So if is a finite $(S,\varepsilon )$ -invariant subset, then there exists a partition of unity of projections $\{E_g\}_{g \in F} \subset M_\omega $ , such that

(3.5) $$ \begin{align} \sum_{g \in s^{-1}F \cap F} \|\gamma_{s}(E_g) - E_{sg}\|_1 &< 4\varepsilon^{1/2} \text{ for all } s \in S; \end{align} $$

(3.6) $$ \begin{align} \sum_{g \in F\setminus s^{-1}F} \|E_g\|_1 &< 3\varepsilon^{1/2} \text{ for all } s \in S; \end{align} $$

(3.7) $$ \begin{align} [E_g, \gamma_h(X)] &= 0 \text{ for all } g \in F,\ h\in G_0,\ X \in \Phi(\mathcal{R}). \end{align} $$

Define

$$\begin{align*}\Psi: \mathcal{R} \rightarrow M_\omega \text{ via } \Psi(x) = \sum_{g\in F} \gamma_{g}(\Phi(\delta_{g}^{-1}(x))) E_g. \end{align*}$$

This is a unital trace-preserving $*$ -homomorphism because the projections $E_g$ form a partition of unity and condition (3.7). For $s \in S$ and $x \in \mathcal {R}$ , we use conditions (3.5) and (3.6) to observe

$$ \begin{align*} \|\gamma_{s}(\Psi(x)) - \Psi(\delta_s(x))\|_1 &= \Big\| \sum_{g \in F} \gamma_{sg}(\Phi(\delta_g^{-1}(x))) \gamma_{s}(E_g) - \sum_{g \in s^{-1}F} \gamma_{sg}(\Phi(\delta_{g}^{-1}(x))) E_{sg} \Big\|_1 \\&\leq \sum_{g \in F \cap s^{-1}F} \left\| \gamma_{sg}(\Phi(\delta_g^{-1}(x))) ( \gamma_{s}(E_g) - E_{sg}) \right\|_1\\& \quad + \sum_{g \in F\setminus s^{-1}F} \|\gamma_g(\Phi(\delta_g^{-1}(x))) E_g\|_1 + \sum_{g \in F \setminus sF} \|\gamma_{g}(\Phi(\delta_{s^{-1}g}^{-1}(x)))E_g\|_1 \\&< 10\varepsilon^{1/2}\|x\|. \end{align*} $$

Since we can do this for arbitrary $\varepsilon>0$ and , the claim follows via a standard reindexing trick.

Proof. (i): Let $\tau $ be the unique tracial state on $\mathcal R$ , which we may use to define the 1-norm $\|\cdot \|_1=\tau (|\cdot |)$ on $\mathcal R$ . Set $\delta ^{(2)}=\delta \otimes \delta : G_1\curvearrowright \mathcal R\bar {\otimes }\mathcal R=:\mathcal R^{(2)}$ , which is also an outer action. Since the flip automorphism $\sigma $ on $\mathcal R^{(2)}$ is known to be approximately inner, we may pick a unitary $U\in \mathcal {U}(\mathcal R^{(2)\omega })$ with $UxU^*=\sigma (x)$ for all $x\in \mathcal R^{(2)}$ . By [Reference Connes11, Theorem 3.2], the induced action $\delta ^{(2)\omega }: G_1\curvearrowright \mathcal R^{(2)}_\omega $ is faithful. We may hence argue exactly as in the proof of Lemma 3.10 and apply Masuda’s noncommutative Rokhlin lemma. So let be a symmetric finite set and $\varepsilon>0$ . If is a finite $(S,\varepsilon )$ -invariant subset, then there exists a partition of unity of projections $\{E_g\}_{g \in F} \subset \mathcal R^{(2)}_\omega $ , such that

(3.8) $$ \begin{align} \sum_{g \in s^{-1}F \cap F} \|\delta^{(2)\omega}_{s}(E_g) - E_{sg}\|_1 &< 4\varepsilon^{1/2} \text{ for all } s \in S; \end{align} $$

(3.9) $$ \begin{align} \sum_{g \in F\setminus s^{-1}F} \|E_g\|_1 &< 3\varepsilon^{1/2} \text{ for all } s \in S; \end{align} $$

(3.10) $$ \begin{align} [E_g, x] &= 0 \text{ for all } g \in F,\ x\in\{ \delta^{(2)\omega}_h(U)\}_{h\in G_1}. \end{align} $$

Define $W = \sum _{g \in F} \delta ^{(2)\omega }_{g}(U) E_g$ . This is also a unitary in $\mathcal R^{(2)\omega }$ implementing the flip $\sigma $ because the projections $E_g$ form a partition of unity and condition (3.10). For $s \in S$ , we use conditions (3.8) and (3.9) to observe

$$ \begin{align*} \|\delta^{(2)\omega}_{s}(W) - W\|_1 &= \Big\| \sum_{g \in F} \delta^{(2)\omega}_{sg}(U) \delta^{(2)\omega}_{s}(E_g) - \sum_{g \in s^{-1}F} \delta^{(2)\omega}_{sg}(U) E_{sg} \Big\|_1 \\ &\leq \sum_{g \in F \cap s^{-1}F} \left\| \delta^{(2)\omega}_{sg}(U) ( \delta^{(2)\omega}_{s}(E_g) - E_{sg}) \right\|_1\\ & \qquad + \sum_{g \in F\setminus s^{-1}F} \| \delta^{(2)\omega}_{g}(U) E_g\|_1 + \sum_{g \in F \setminus sF} \|\delta^{(2)\omega}_{g}(U) E_{g}\|_1 \\ &< 10\varepsilon^{1/2}. \end{align*} $$

Since we can do this for arbitrary $\varepsilon>0$ and , we can use a reindexing trick to obtain a unitary $W\in \mathcal {U}((\mathcal R^{(2)\omega })^{\delta ^{(2)\omega }})$ with $WxW^*=\sigma (x)$ for all $x\in \mathcal R^{(2)}$ . In particular, $\delta $ has approximately inner half-flip. If we apply Lemma 3.10 for $G=G_1$ , $N=\{1\}$ and $\delta $ in place of $\alpha $ , it follows with Theorem 3.8 that $\delta $ is strongly self-absorbing. If $\gamma \colon G_1\curvearrowright \mathcal R$ is another outer action, then the same follows for $\gamma $ . By applying Lemma 3.10 and Corollary 3.9 twice, we obtain that $\gamma $ and $\delta $ absorb each other, hence, they are cocycle conjugate.

(ii): Define N to be the subgroup of all elements $g\in G$ , such that $\alpha _{\omega ,g}$ is trivial, and set $G_0=G/N$ with quotient map $\pi ^0: G\to G_0$ . By assumption, we have $N\subseteq H$ , hence, $G_1$ can be viewed as a quotient of $G_0$ via a map $\pi ^{0\to 1}: G_0\to G_1$ . Then, $\pi =\pi ^{0\to 1}\circ \pi ^0$ and the action $\delta _{\pi ^{0\to 1}}:=\delta \circ \pi ^{0\to 1}$ is a $G_0$ -action with $(\delta _{\pi ^{0\to 1}})_{\pi ^0}=\delta _\pi $ . By Lemma 3.10, it follows that there exists an equivariant unital $*$ -homomorphism $(\mathcal R,\delta _\pi ) \to (M_\omega ,\alpha _\omega )$ . Since $\delta $ was strongly self-absorbing, so is $\delta _\pi $ as a G-action, and the claim follows by Corollary 3.9.

Proof. By assumption, there is a sequence of unitaries $z_n\in \mathcal {U}(\mathcal {R}\bar {\otimes }\mathcal {R})$ , such that

(4.1) $$ \begin{align} \|\Phi(x) - z_n(x\otimes 1_{\mathcal R})z_n^*\|_2\to 0\quad\text{for all } x\in\mathcal R \end{align} $$

and

(4.2) $$ \begin{align} \max_{h\in K} \|\mathbb{U}_h - z_n(\delta^{(2)}\otimes\delta^{(2)})_h(z_n)^*\|_2\to 0 \quad\text{for every compact } K\subseteq G_2. \end{align} $$

By definition, we have

$$\begin{align*}\Psi = (\theta^{-1}\otimes\Delta)\circ (\operatorname{id}_{M_2}\otimes\Phi)\circ\theta \end{align*}$$

and

If we consider the sequence of unitaries

$$\begin{align*}y_n=(\theta^{-1}\otimes\Delta)(1_{M_2}\otimes z_n), \end{align*}$$

then we can observe with (4.1) that

$$\begin{align*}\operatorname{Ad}(y_n^*)\circ\Psi \to (\theta^{-1}\otimes\Delta)\circ (\operatorname{id}_{M_2}\otimes\operatorname{id}_{\mathcal R}\otimes 1_{\mathcal R})\circ\theta = \operatorname{id}_{M_1}\otimes 1_{\mathcal R} \end{align*}$$

as well as

$$\begin{align*}\operatorname{Ad}(y_n)\circ(\operatorname{id}_{M_1}\otimes 1_{\mathcal R}) = \operatorname{Ad}(y_n)\circ(\theta^{-1}\otimes\Delta)\circ (\operatorname{id}_{M_2}\otimes\operatorname{id}_{\mathcal R}\otimes 1_{\mathcal R})\circ\theta \to \Psi \end{align*}$$

point-strongly. Moreover, given $g\in G_1$ , the fact that is the inverse of leads to the equation . If we combine this with (4.1) and (4.2), we can see that

in the strong- $*$ operator topology, uniformly over compact subsets. Analogously, we observe the convergence

uniformly over compact sets. This finishes the proof.

Proof. We note that by following the argument outlined in Example 4.6 and by applying Corollary 3.9, we see that $\alpha \simeq _{\mathrm {cc}} \alpha \otimes \delta $ implies that one can find a cocycle conjugacy between $\alpha $ and $\alpha \otimes \delta $ that induces the identity map on X. Hence, (1) implies (2).

In order to prove the other implication, assume (2) holds. To verify (1), we will show the existence of a measurable field of $*$ -isomorphisms ${X \ni x \mapsto \theta _x \colon B_x \rightarrow B_x \bar {\otimes } \mathcal {R}}$ and unitaries ${\mathcal {G} \ni (g,x) \mapsto w_{(g,x)} \in \mathcal {U}(B_{g \cdot x} \bar {\otimes } \mathcal {R})}$ , such that

(4.3) $$ \begin{align} \theta_{g \cdot x} \circ \alpha_{(g,x)} \circ \theta_{x}^{-1} &= \mathrm{Ad} \big( w_{(g,x)} \big) \circ (\alpha \otimes \delta)_{(g,x)} \text{ for all } (g,x) \in \mathcal{G} \end{align} $$

(4.4) $$ \begin{align} w_{(g,h \cdot x)} (\alpha \otimes \delta)_{(g,h \cdot x)}(w_{(h,x)}) &= w_{(gh,x)} \text{ for all } g,h \in G, x \in X. \end{align} $$

For every $x \in X$ , denote by $P_x$ the Polish space of cocycle conjugacies from $(B_x, \alpha |_{H_x})$ to $(B_x \bar {\otimes } \mathcal {R}, (\alpha \otimes \delta )|_{H_x})$ as per Remark 4.9. In this way, we get a measurable field of Polish spaces $X \ni x \mapsto P_x$ . Note that, by assumption, the sets $P_x$ are all nonempty and hence, there exists some measurable section . Defining for $g \in H_x$ , we get that — although w is not defined on all of $\mathcal {G}$ yet — the equations 4.3–4.4 are satisfied whenever they make sense. In the rest of the proof, we will show with the help of Lemma 4.8 that there exists a (potentially different) section for which there exists a well-defined map w on all of $\mathcal {G}$ obeying conditions (4.3)–(4.4).

Denote by $\mathcal {S}$ the countable nonsingular orbit equivalence relation associated to $\mathcal {G}$ , that is,

$$\begin{align*}\mathcal{S}=\{(g \cdot x,x) \mid (g,x) \in \mathcal{G}\}.\end{align*}$$

As $G \curvearrowright (X,\mu )$ is amenable, the relation $\mathcal {S}$ is amenable, and hence, it follows by the Connes–Feldman–Weiss theorem [Reference Connes, Feldman and Weiss12] that after neglecting a set of measure zero, there exists a partition of X into $\mathcal {S}$ -invariant Borel subsets $X_0\sqcup X_1$ , such that the restriction of $\mathcal {S}$ to $X_0$ has finite orbits and the restriction to $X_1$ is induced by a free nonsingular action of $\mathbb {Z}$ . This implies that the map $q \colon \mathcal {G} \rightarrow \mathcal {S}\colon (g,x) \mapsto (g \cdot x,x)$ admits a measurable section, that is, a measurable groupoid morphism $s \colon \mathcal {S} \rightarrow \mathcal {G}$ , such that $q \circ s = \mathrm {id}_{\mathcal {S}}$ . Therefore, we can view $\mathcal {G}$ as the semidirect product of the field of groups $(H_x)_{x \in X}$ and the measurable field of group isomorphisms $\phi _{(x,y)}\colon H_y \rightarrow H_x$ given by $\phi _{(x,y)}(g)= s(x,y) g s(x,y)^{-1}$ . Note that $\phi _{(x,y)} \circ \phi _{(y,z)} = \phi _{(x,z)}$ for all $(x,y), (y,z) \in \mathcal {S}$ .

This means that in order to define a measurable field $\mathcal {G} \ni (g,x) \mapsto w_{(g,x)} \in \mathcal {U}(B_{g \cdot x} \bar {\otimes } \mathcal {R})$ satisfying (4.4), it suffices to find measurable families of unitaries ${v{(x,y)} \in \mathcal {U}(B_x \bar {\otimes } \mathcal {R})}$ for $(x,y) \in \mathcal {S}$ and for $x \in X, g \in H_x$ , such that

• • v is a cocycle for the action of $\mathcal {S}$ on the field of factors $(B_x\bar {\otimes } \mathcal {R})_{x \in X}$ induced by s, that is, $v{(x,y)} (\alpha \otimes \delta )_{s(x,y)}(v{(y,z)})=v(x,z)$ for all $(x,y), (y,z) \in \mathcal {S}$ ;

• • for each $x \in X$ , the family defines a cocycle for the action $H_x \curvearrowright B_x \bar {\otimes } \mathcal {R}$ ;

• • for all $(x,y) \in \mathcal {S}$ and $g \in H_x$ .

If these conditions are met, then setting for $(x,y) \in \mathcal {S}$ and $g \in H_x$ yields condition (4.4). Moreover, in order for a measurable field of $*$ -isomorphisms ${X \ni x \mapsto \theta _x \colon B_x \rightarrow B_x \bar {\otimes } \mathcal {R}}$ to satisfy (4.3), it then suffices to check that

• • for each $x \in X$ , the pair is a cocycle conjugacy in $P_x$ ;

• • for $(x,y) \in \mathcal {S}$ , we have $\theta _x \circ \alpha _{s(x,y)} \circ \theta _y^{-1} = \mathrm {Ad}(v(x,y))\circ (\alpha \otimes \delta )_{s(y,z)}$ .

We introduce some notation to rephrase this in terms of the terminology of Lemma 4.8. Consider the natural action $\mathcal {U}(B_x \bar {\otimes } \mathcal {R}) \curvearrowright P_x$ given by composing a cocycle conjugacy with the inner one given by $\mathrm {Ad}(u)$ for $u \in \mathcal {U}(B_x \bar {\otimes } \mathcal {R})$ as per Remark 4.9. In this way, we get a measurable field $(\mathcal {U}(B_x \bar {\otimes } \mathcal {R}) \curvearrowright P_x)_{x \in X}$ of continuous actions of Polish groups on Polish spaces. Let us convince ourselves that, in the terminology of Lemma 4.8, the equivalence relation $\mathcal {S}$ acts by conjugacies on this field of actions. Firstly, we have a measurable field of group isomorphisms

$$\begin{align*}{\mathcal{S} \ni (x,y) \mapsto \gamma_{(x,y)} = (\alpha \otimes \delta)_{s(x,y)}|_{\mathcal{U}(B_y \bar{\otimes} \mathcal{R})} \colon \mathcal{U}(B_y \bar{\otimes} \mathcal{R})\rightarrow \mathcal{U}(B_x \bar{\otimes} \mathcal{R}),} \end{align*}$$

such that $\gamma _{(x,y)} \circ \gamma _{(y,z)} = \gamma _{(x,z)}$ for all $(x,y), (y,z) \in \mathcal {S}$ . The latter formula holds, as $\alpha \otimes \delta $ was a $\mathcal G$ -action and $s: \mathcal {S}\to \mathcal {G}$ is a section. Secondly, we have an action of $\mathcal {S}$ on $(P_x)_{x \in X}$ as follows. Given $(x,y) \in \mathcal {S}$ and , we define , where

This construction yields a well-defined cocycle conjugacy in $P_x$ , and we get a well-defined map $\beta _{(x,y)}\colon P_y \rightarrow P_x$ . Together, these maps combine to form a measurable field of homeomorphisms $\mathcal {S} \ni (x,y) \mapsto \beta _{(x,y)}\colon P_y \to P_x$ , such that $\beta _{(x,y)} \circ \beta _{(y,z)} = \beta _{(x,z)}$ for all $(x,y), (y,z) \in \mathcal {S}$ . This formula holds once again because $\alpha \otimes \delta $ and $\alpha $ are $\mathcal G$ -actions and $s\colon \mathcal {S}\to \mathcal {G}$ is a section. Moreover, the maps $\beta $ and $\gamma $ are compatible with the measurable field of actions ${(\mathcal {U}(B_x \bar {\otimes } \mathcal {R}) \curvearrowright P_x)_{x \in X}}$ (as required for Lemma 4.8), since for any $(x,y) \in \mathcal {S}, u \in \mathcal {U}(B_y \bar {\otimes } \mathcal {R})$ and , we may simply compare definitions and observe

Having introduced all these data, our previous discussion can be rephrased. In order to complete the proof, it suffices to find a measurable section $X \ni x \mapsto \sigma (x) \in P_x$ and a measurable family $\mathcal {S} \ni (x,y) \mapsto v(x,y) \in \mathcal {U}(B_x \bar {\otimes } \mathcal {R})$ , such that

• • $v(x,y) \gamma _{(x,y)}(v(y,z)) = v(x,z)$ for all $(x,y), (y,z) \in \mathcal {S}$ ;

• • $v(x,y) \cdot \beta _{(x,y)}(\sigma (y)) = \sigma (x)$ for all $(x,y) \in \mathcal {S}$ .

By Lemma 4.8, such maps exist if we can merely show that there exists a measurable section , such that for all $(x,y) \in \mathcal {S}$ , the element belongs to the closure of . We claim that this is indeed the case.

Consider any measurable section . As $\delta $ is strongly self-absorbing, we can fix a cocycle conjugacy $(\Phi , \mathbb {U})$ from $(\mathcal {R}, \delta )$ to $(\mathcal {R}\bar {\otimes } \mathcal {R}, \delta \otimes \delta )$ that is approximately unitarily equivalent to $\operatorname {id}_{\mathcal R}\otimes 1_{\mathcal R}$ . For each $x \in X$ , we can define the map

Then we get a new measurable section

We claim that this section does the trick. Fix $(x,y) \in \mathcal {S}$ . If we denote , we need to convince ourselves that the cocycle conjugacy is in the closure of . First of all, we observe that the construction of can be seen as a special case of Lemma 4.7 for $M_1=M_2=B_x$ , $G=H_x$ , $\phi =\operatorname {id}_{H_x}$ and $\Delta =\operatorname {id}_{\mathcal R}$ . Hence, we can find a sequence of unitaries $y_n\in \mathcal {U}(B_x\otimes \mathcal R)$ , such that

(4.5)

in the strong- $*$ operator topology for all $b\in B_x$ and $h\in H_x$ .

Next, by our previous notation, the group isomorphism $\phi _{(x,y)}\colon H_y\to H_x$ is exactly the one so that the isomorphism of von Neumann algebras $\alpha _{s(x,y)}\colon B_y\to B_x$ can be viewed as a (genuine) conjugacy between the $H_x$ -actions $(\alpha |_{H_y})_{\phi _{(y,x)}}$ and $\alpha |_{H_x}$ . Moreover, if $s(x,y)=(k,y)$ for $k\in G$ , then $(\alpha \otimes \delta )_{s(x,y)}=\alpha _{s(x,y)}\otimes \delta _k$ and $\delta _k$ can be seen as a conjugacy between the $H_x$ -actions $(\delta |_{H_y})_{\phi _{(y,x)}}$ and $\delta |_{H_x}$ .

By definition of $\beta _{(x,y)}$ , we have

$$\begin{align*}\widetilde{\theta}_x &= (\alpha \otimes \delta)_{s(x,y)} \circ \theta_y \circ \alpha_{s(y,x)} \\&= (\alpha \otimes \delta)_{s(x,y)} \circ \big( {\theta_y'}^{-1} \otimes \mathrm{id}_{\mathcal{R}} \big) \circ \big( \mathrm{id}_{B_y} \otimes \Phi \big) \circ \theta_y' \circ \alpha_{s(y,x)} \\&= \big( ({\theta_y'}\circ\alpha_{s(y,x)})^{-1} \otimes \delta_k \big) \circ \big( \mathrm{id}_{B_y} \otimes \Phi \big) \circ (\theta_y' \circ \alpha_{s(y,x)}) \end{align*}$$

and for all $h\in H_x$ with $g=\phi _{(y,x)}(h)$ , one has

We conclude that Lemma 4.7 is applicable to the cocycle conjugacy , where we insert $G_1=H_x$ , $G_2=H_y$ , $\phi =\phi _{(y,x)}$ , $M_1=B_x$ , $M_2=B_y$ , $\Delta =\delta _k$ , and the cocycle conjugacy in place of . This allows us to find a sequence of unitaries $w_n\in \mathcal {U}(B_x\otimes \mathcal R)$ satisfying

(4.6)

in the strong- $*$ operator topology for all $b\in B_x$ and $h\in H_x$ . If we consider both of the conditions (4.5) and (4.6) and keep in mind that G is countable and $B_x$ is separable and semifinite, we can apply Lemma 2.1 and find an increasing sequence of natural numbers $m_n$ , such that the resulting sequence of unitaries $z_n=w_ny_{m_n}^*$ satisfies

in the strong- $*$ operator topology for all $b\in B_x$ and $h\in H_x$ . Then it follows from Remark 4.9 that indeed belongs to the closure of . This finishes the proof.

Proof. Adopt the notation from Example 4.6 and Theorem 4.10. We identify $\alpha $ with an action of ${\mathcal {G}:= G \ltimes X}$ on a measurable field of semifinite factors with separable predual $(B_x)_{x \in X}$ . Denote by ${X \ni x \mapsto H_x}$ the measurable field of isotropy groups. Amenability of the action $\alpha $ implies that the action on $(X,\mu )$ is amenable in the sense of Zimmer, which, in turn, implies amenability of the associated transformation groupoid. In particular, almost all isotropy groups $H_x$ are amenable.

By assumption on H and [Reference Masuda and Tomatsu26, Theorem 9.14], it follows for every $g\in G\setminus H$ and $\mu $ -almost all $x\in X$ that either $g\notin H_x$ or the automorphism $(\alpha |_{H_x})_g$ on the McDuff factor $B_x$ is centrally nontrivial. In other words, after discarding a null set from X, we may assume for all $x\in X$ that for all $h\in H_x\setminus (H_x\cap H)$ , the automorphism $(\alpha |_{H_x})_g$ on $B_x$ is centrally nontrivial. By Theorem 3.11, we get that $(\alpha |_{H_x})$ is cocycle conjugate to $(\alpha \otimes \delta _\pi )|_{H_x}$ . The claim then follows via Theorem 4.10.

Proof. Assume M is represented faithfully on a Hilbert space $\mathcal {H}$ , and consider the canonical inclusion $\pi \colon M \rtimes _\alpha G \rightarrow \mathcal {B}(L^2(G)) \bar {\otimes } M \subset \mathcal {B}(L^2(G, \mathcal {H}))$ , which on $x \in M \subset M \rtimes _\alpha G$ is given by

$$\begin{align*}[\pi(x) \xi](g) = \alpha_g^{-1}(x)\xi(g), \quad \xi \in L^2(G, \mathcal{H}), g \in G.\end{align*}$$

If $(x_n)_{n \in \mathbb {N}}$ is any bounded sequence in M representing an element $x \in (M_{\omega , \alpha })^{\alpha _\omega }$ , then the invariance of x will guarantee that $\pi (x_n) - (1 \otimes x_n) \rightarrow 0$ in the strong- $*$ operator topology as $n \rightarrow \omega $ . Since $(1 \otimes x_n)_{n \in \mathbb {N}}$ represents an element in $(\mathcal {B}(L^2(G)) \bar {\otimes } M)_\omega $ , it follows that $(\pi (x_n))_{n \in \mathbb {N}}$ represents an element in $(\pi (M\rtimes _\alpha G))_\omega $ , so $x \in (M\rtimes _\alpha G)_\omega $ . This finishes the proof.

Proof. Denote $\alpha '=\operatorname {id}_{\mathcal {B}(L^2(\mathbb R))}\otimes \alpha $ . We apply Takesaki–Takai duality [Reference Takesaki50, Chapter X, Theorem 2.3(iii)] to understand the G-action $\widetilde {\alpha }$ . If M is represented faithfully on a Hilbert space $\mathcal H$ , then the natural isomorphism

$$\begin{align*}\Theta: \widetilde{M}\rtimes_{\hat{\sigma}^\varphi}\mathbb R = (M\rtimes_{\sigma^\varphi}\mathbb R)\rtimes_{\hat{\sigma}^\varphi}\mathbb R \to \mathcal{B}(L^2(\mathbb R)) \bar{\otimes} M\subseteq \mathcal{B}(L^2(\mathbb R,\mathcal H)) \end{align*}$$

has the following properties. Let $\xi \in L^2(\mathbb R,\mathcal H)$ . For $x\in M$ and $s,t\in \mathbb R$ , we have

$$\begin{align*}[\Theta(x)\xi ](s) = \sigma^\varphi_{-s}(x)\xi(s) \quad\text{and}\quad [\Theta(\lambda^{\sigma^\varphi}_t)\xi](s) = \xi(s-t). \end{align*}$$

Furthermore, if the dual flow is given via the convention $\hat {\sigma }^\varphi _t(\lambda _s^{\sigma ^{\varphi }}) = e^{its}\lambda ^{\sigma ^{\varphi }}_s$ , then we also have

$$\begin{align*}[\Theta(\lambda^{\hat{\sigma}^\varphi}_t)\xi](s)=e^{its}\xi(s). \end{align*}$$

We consider a continuous family of unitaries $G\ni g\mapsto \mathbb {W}_g\in \mathcal {B}(L^2(\mathbb R)) \bar {\otimes } M$ given by

$$\begin{align*}[\mathbb{W}_g\xi](s) = (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}\xi(s),\quad \xi\in L^2(\mathbb R,\mathcal H),\quad s\in\mathbb R. \end{align*}$$

We claim that this defines an $\alpha '$ -cocycle. Indeed, by using the chain rule for the Connes cocycle ([Reference Takesaki50, Theorem VIII.3.7]), we observe for all $g,h\in G$ , $\xi \in L^2(\mathbb R,\mathcal H)$ and $s\in \mathbb R$ that

$$\begin{align*}[\mathbb{W}_g\alpha^{\prime}_g(\mathbb W_h)\xi](s) &= (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}[\alpha^{\prime}_g(\mathbb W_h)\xi](s) \\&= (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}\alpha_g\big( (D\varphi: D\varphi\circ\alpha_h^{-1})_{-s} \big)\xi(s) \\&= (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}\cdot (D\varphi\circ\alpha_g^{-1}: D\varphi\circ\alpha_{gh}^{-1})_{-s} \big)\xi(s) \\&= (D\varphi: D\varphi\circ\alpha_{gh}^{-1})_{-s}\xi(s) \ = \ [\mathbb{W}_{gh}\xi](s). \end{align*}$$

Given how $\widetilde {\alpha }$ acts on the domain of $\Theta $ , we can observe (using [Reference Takesaki50, Corollary VIII.1.4]) for any $g\in G$ , $x \in M$ , $\xi \in L^2(\mathbb {R}, \mathcal {H})$ , and $s \in \mathbb {R}$ that

$$\begin{align*} [\Theta(\widetilde{\alpha}_g(x))\xi ](s) &= \sigma^\varphi_{-s}(\alpha_g(x))\xi(s) \\&= \operatorname{Ad}(D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}\Big( \sigma_{-s}^{\varphi\circ\alpha_g^{-1}}(\alpha_g(x)) \Big)\xi(s) \\&= \Big(\operatorname{Ad}(D\varphi: D\varphi\circ\alpha_g^{-1})_{-s}\circ\alpha_g\Big) \big( \sigma^\varphi_{-s}(x) \big)\xi(s) \\&= \Big[ \Big( \operatorname{Ad}(\mathbb{W}_g)\circ\alpha^{\prime}_g\circ\Theta\Big)(x)\, \xi \Big](s). \end{align*}$$

Moreover, using the cocycle identity and the chain rule again, we can see for any $g \in G$ , $\xi \in L^2(\mathbb {R},\mathcal {H})$ , and $s,t \in \mathbb {R}$ that

$$\begin{align*} [\Theta(\widetilde{\alpha}_g(\lambda^{\sigma^\varphi}_t))\xi](s) &= [\Theta\big( (D \varphi\circ\alpha_g^{-1} : D\varphi)_t\cdot\lambda^{\sigma^\varphi}_t \big)\xi](s) \\&= \sigma^{\varphi}_{-s}(D \varphi\circ\alpha_g^{-1} : D\varphi)_t [ \Theta(\lambda^{\sigma^\varphi}_t)\xi ](s) \\&= \sigma^{\varphi}_{-s}(D \varphi\circ\alpha_g^{-1} : D\varphi)_t \xi(s-t) \\&= (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s} (D\varphi: D\varphi\circ\alpha_g^{-1})_{t-s}^* \xi(s-t) \\&= (D\varphi: D\varphi\circ\alpha_g^{-1})_{-s} \Big[ \Theta(\lambda_t^{\sigma^\varphi}) \mathbb{W}_g^* \xi \Big](s) \\&= \Big[ \mathbb{W}_g \Theta(\lambda_t^{\sigma^\varphi}) \mathbb{W}_g^* \xi \Big](s) \\&= \Big[ \Big( \operatorname{Ad}(\mathbb{W}_g)\circ\alpha^{\prime}_g\circ\Theta\Big)(\lambda_t^{\sigma^\varphi}) \xi \Big](s).\end{align*}$$

Here, we used that $\alpha ^{\prime }_g$ fixes the shift operator given by $\Theta (\lambda ^{\sigma ^\varphi }_t)$ for all $g\in G$ and $t\in \mathbb {R}$ . Lastly, it is trivial to see that $\alpha '$ fixes operators of the form $\Theta (\lambda ^{\hat {\sigma }^\varphi }_t)$ , which, in turn, also commute pointwise with the cocycle $\mathbb {W}$ . In conclusion, all of these observations culminate in the fact that the isomorphism $\Theta $ extends to the cocycle conjugacy $(\Theta ,\mathbb {W})$ between $\widetilde {\alpha }$ and $\alpha '$ .

Proof. The ‘only if’ part follows exactly as in the proof of Corollary 3.9, so we need to show the ‘if’ part. Since Corollary 3.9 already covers the case when M is finite, we may split off the largest finite direct summand of M and assume without loss of generality that M is properly infinite.

Let $\varphi $ be a faithful normal state on M. As in Remark 5.1, we consider the (semifinite) continuous core $\widetilde {M}$ and the extended G-action $\widetilde {\alpha }: G\curvearrowright \widetilde {M}$ . Since the image of $\widetilde {\alpha }$ commutes with the dual flow $\hat {\sigma }^{\varphi }$ , we have a continuous action $\beta =\widetilde {\alpha }\times \hat {\sigma }^\varphi : G\times \mathbb {R}\curvearrowright \widetilde {M}$ via $\beta _{(g,t)}=\widetilde {\alpha }_g\circ \hat {\sigma }^\varphi _t$ for all $g\in G$ and $t\in \mathbb {R}$ . Let us also consider the action $\delta ^{\mathbb R}: G\times \mathbb {R}\curvearrowright \mathcal R$ given by $\delta ^{\mathbb R}_{(g,t)}=\delta _g$ for all $g\in G$ and $t\in \mathbb {R}$ , which is evidently also strongly self-absorbing.

We apply Proposition 5.2 to $\mathbb {R}$ in place of G and to the modular flow in place of $\alpha $ . In this context, we note that by [Reference Ando and Haagerup4, Theorem 4.1, Proposition 4.11], we have that the $M^\omega $ agrees with the $(\sigma ^\varphi ,\omega )$ -equicontinuous part $M^\omega _{\sigma ^\varphi }$ . In particular, the induced ultrapower flow $(\sigma ^\varphi )^\omega $ on it is continuous and its restriction to $M_\omega $ is trivial. So, $M_\omega =(M_{\omega ,\sigma ^\varphi })^{(\sigma ^\varphi )_\omega }$ and Proposition 5.2 implies that the inclusion $M\subset \widetilde {M}$ induces an embedding $M_\omega \to \widetilde {M}_\omega $ . Since, by definition, one has $\widetilde {\alpha }|_M=\alpha $ as G-actions, it is clear that bounded $(\alpha ,\omega )$ -equicontinuous sequences in M become $(\widetilde {\alpha },\omega )$ -equicontinuous sequences in $\widetilde {M}$ . Keeping in mind that the dual flow $\hat {\sigma }^\varphi $ acts trivially on M by definition, the aforementioned inclusion therefore induces an equivariant unital $*$ -homomorphism

$$\begin{align*}(M_{\omega,\alpha},\alpha_\omega) \to ( (\widetilde{M}_{\omega,\beta})^{(\hat{\sigma}^\varphi)_\omega},\widetilde{\alpha}_\omega ). \end{align*}$$

If we compose this $*$ -homomorphism with a given unital equivariant $*$ -homomorphism $(\mathcal R,\delta )\to (M_{\omega ,\alpha },\alpha _\omega )$ , we can view the resulting map as a $(G\times \mathbb {R})$ -equivariant unital $*$ -homomorphism $(\mathcal R,\delta ^{\mathbb R}) \to (\widetilde {M}_{\omega ,\beta },\beta _\omega )$ . Since $\widetilde {M}$ is semifinite, it follows from Corollary 3.9 that $\beta $ and $\beta \otimes \delta ^{\mathbb R}$ are cocycle conjugate as $(G\times \mathbb R)$ -actions, say via $(\Psi ,\mathbb {V}): (\widetilde {M},\beta )\to (\widetilde {M}\bar {\otimes }\mathcal {R},\beta \otimes \delta ^{\mathbb R})$ . Remembering $\beta =\widetilde {\alpha }\times \hat {\sigma }^\varphi $ , we consider the $(\hat {\sigma }^\varphi \otimes \mathrm {id}_{\mathcal {R}})$ -cocycle and the $\widetilde {\alpha }\otimes \delta $ -cocycle . The cocycle identity for $\mathbb {V}$ then implies the relation

(5.1)

for all $g\in G$ and $t\in \mathbb R$ . The cocycle conjugacy of flows induces an isomorphism

via

With slight abuse of notation (as in Proposition 5.3), we also denote by $\widetilde {\alpha }$ the obvious induced G-action on the crossed product $\widetilde {M}\rtimes _{\hat {\sigma }^\varphi }\mathbb R$ . Using that $(\Psi ,\mathbb {V})$ was a cocycle conjugacy, we observe for all $g\in G$ and $t\in \mathbb {R}$ that

and

In conclusion, the pair defines a cocycle conjugacy between the G-actions $\widetilde {\alpha }$ on $\widetilde {M}\rtimes _{\hat {\sigma }^\varphi }\mathbb R$ and $\widetilde {\alpha }\otimes \delta $ on $\big (\widetilde {M}\rtimes _{\hat {\sigma }^\varphi }\mathbb R\big )\otimes \mathcal R$ . By Proposition 5.3, the action $\widetilde {\alpha }$ is cocycle conjugate to $\operatorname {id}_{\mathcal B(\ell ^2(\mathbb {N}))}\otimes \alpha $ . Since we assumed M to be properly infinite, it furthermore follows that $\operatorname {id}_{\mathcal B(\ell ^2(\mathbb {N}))}\otimes \alpha $ is cocycle conjugate to $\alpha $ .Footnote ⁴ Combining all these cocycle conjugacies yields one between $\alpha $ and $\alpha \otimes \delta $ . This finishes the proof.

Proof. Choose a faithful normal state $\varphi $ on M. Recall that the induced faithful normal state $\varphi ^\omega $ on $M^\omega $ restricts to a tracial state on $M_\omega $ . We denote by $\|\cdot \|_2=\|\cdot \|_{\varphi ^\omega }$ the induced tracial 2-norm on $M_\omega $ . Since we assumed M to be McDuff, it follows that $M_\omega $ is locally McDuff in the following sense: Given any $\|\cdot \|_2$ -separable $*$ -subalgebra $B\subset M_\omega $ , there exists a unital $*$ -homomorphism $\mathcal R\to M_\omega \cap B'$ .