Haagerup and Størmer's conjecture on pointwise inner automorphisms

Abstract

In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.

1. Introduction

Let $M$ be a von Neumann algebra and $\operatorname {Aut}(M)$ the set of all automorphisms on $M$. We say that $\theta \in \operatorname {Aut}(M)$ is pointwise inner [HS87] if for each positive linear functional $\varphi \in M_*^+$, there is a unitary $u\in \mathcal {U}(M)$ such that $\theta (\varphi )=u\varphi u^*$, where we used the notation $\theta (\varphi ):=\varphi \circ \theta ^{-1}$ and $x\varphi y := \varphi (y \, \cdot \, x)$ for $x,y\in M$. In other words, $\theta$ is pointwise inner if the induced map $\theta \colon M_\ast ^+ \to M_\ast ^+$ is inner at a pointwise level. This notion naturally appeared in the study of unitary equivalence relations on state spaces on von Neumann algebras and was intensively studied by Haagerup and Størmer [HS87, HS88, HS91].

Any inner automorphism is pointwise inner. Indeed, we can choose the same unitary for all elements in $M_\ast ^+$. A nontrivial and motivating example comes from Tomita–Takesaki modular theory, that is, every modular automorphism $\sigma _t^\varphi$ is pointwise inner, where $\varphi \in M_\ast ^+$ is a faithful state and $t\in \mathbb {R}$; see § 2. Hence, every composition $\operatorname {Ad}(u)\circ \sigma ^\varphi _t$ of an inner and a modular automorphism is pointwise inner because the pointwise inner property is closed under compositions. Haagerup and Størmer conjectured that the opposite phenomenon also holds for every type ${\rm III_1}$ factor [HS88].

Haagerup–Størmer conjecture Let $M$ be a type ${\rm III_1}$ factor with separable predual. Then every pointwise inner automorphism is a composition of an inner and a modular automorphism.

We note that if $M$ is a factor that is not of type ${\rm III_1}$, similar results were obtained in [HS87, HS88], hence the only remaining problem for characterizing pointwise inner automorphisms is the case for type ${\rm III_1}$ factors. We also note that the separability assumption is necessary; see [HS88, § 6] and [AH12, Remark 4.21].

In the present paper, we give a partial but satisfactory answer to this conjecture. To explain it, we prepare some terminology. For an inclusion of von Neumann algebras $N\subset M$, we say it is irreducible if $N'\cap M \subset N$, and is with expectation if there exists a faithful normal conditional expectation from $M$ onto $N$. For every faithful state, or more generally, every faithful semifinite normal weight, $\varphi$ on $M$, we denote by $M_\varphi$ the centralizer of $\varphi$, which is the fixed point algebra of $\sigma ^\varphi$. When $M$ is a type ${\rm III_1}$ factor with separable predual, the existence of a faithful state $\varphi \in M_\ast$ with $M_\varphi '\cap M=\mathbb {C}$ is equivalent to having trivial bicentralizer; see § 2. In this case, $M_\varphi \subset M$ is an irreducible subfactor of type ${\rm II_1}$ with expectation, and the existence of such a subfactor is very useful in many contexts. Connes’ bicentralizer problem, which is one of the most important problems for type ${\rm III}$ factors, asks if every type ${\rm III_1}$ factor with separable predual has trivial bicentralizer. To the best of our knowledge, all concrete examples of type ${\rm III_1}$ factors have trivial bicentralizers. Further, if $M$ is a type ${\rm III_1}$ factor, then the tensor product $M\mathbin {\overline {\otimes }} R_\infty$ has trivial bicentralizer, where $R_\infty$ is the Araki–Woods factor of type ${\rm III_1}$ [Mar18].

We now introduce our main theorem. For objects in (3) below; see § 2.

Theorem A. Let $M$ be a type ${\rm III_1}$ factor with separable predual and assume it has trivial bicentralizer. Fix a faithful state $\varphi \in M_\ast$ such that $M_\varphi '\cap M=\mathbb {C}$. Then for each $\theta \in \operatorname {Aut}(M)$, the following conditions are equivalent.

1. (1) The automorphism $\theta$ is pointwise inner.

2. (2) For each faithful state $\psi \in M_\ast$ with $M_\psi '\cap M =\mathbb {C}$, there exists $u\in \mathcal {U}(M)$ such that $\theta (x) = uxu^*$ for all $x\in M_\psi$.

3. (3) There exists $u\in \mathcal {U}(M)$ such that $\theta ^\omega (x)=uxu^*$ for all $x\in (M^\omega )_{\varphi ^\omega }$, where $\omega$ is a fixed free ultrafilter on $\mathbb {N}$.

4. (4) There exist $u\in \mathcal {U}(M)$ and $t\in \mathbb {R}$ such that $\theta = \operatorname {Ad}(u)\circ \sigma _t^\varphi$.

By the implication (1)$\Rightarrow$(4), we immediately get the following corollary.

Corollary B The Haagerup–Størmer conjecture holds for every type ${\rm III_1}$ factor with separable predual that has trivial bicentralizer. In particular, the conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer.

We emphasize that Theorem A applies to all concrete examples of type ${\rm III_1}$ factors because they have trivial bicentralizers. Although we do not know the complete answer to Connes’ bicentralizer problem, if it is solved affirmatively, then our theorem solves the Haagerup–Størmer conjecture. Even if the problem has a negative answer, we can apply the theorem to every type ${\rm III_1}$ factor up to a tensor product with $R_\infty$. Thus, in any case, Theorem A covers a large class of type ${\rm III_1}$ factors.

We explain item (2) in Theorem A. In all previous works for pointwise inner automorphisms, a specific state (or weight) $\varphi$ on a factor $M$ plays a crucial role. More precisely, the implication (1)$\Rightarrow$(2) in Theorem A for the specific $\psi \ (=\varphi )$ is the key step of the proof. For example, $\varphi$ is a trace if $M$ is of type ${\rm II}$. If $M$ is a type ${\rm III}_\lambda$ $(0\leq \lambda <1)$ factor, then $\varphi$ is a lacunary weight. We note that, in these cases, the implication (2)$\Rightarrow$(4) also follows by the fact that $\varphi$ is a specific one. Unfortunately, if $M$ is a type III$_1$ factor, there are no such specific states or weights in general, and this fact is the main difficulty for the conjecture. We do have a dominant weight, but it is not enough for our purpose because $M_\varphi \subset M$ is not with expectation. In [HS91, HI24], $\varphi$ is assumed to be an almost periodic state, but not all type ${\rm III_1}$ factors admit such a state.

In Theorem A, we do not assume $\varphi$ is such a specific state or weight. We nevertheless prove the implication (1)$\Rightarrow$(2). This is the main observation of the paper, so we briefly explain the idea of the proof. Recall that, if $\varphi$ is almost periodic, each element in $M$ has a Fourier decomposition along the eigenspaces of $\varphi$. In [HS91, HI24], the Fourier decomposition of a unitary element $u\in \mathcal {U}(M)$ satisfying $\theta (\varphi (a\, \cdot \, )) = u \varphi (a\, \cdot \, )u^*$ for some carefully chosen $a\in M_\varphi$ is important in the proof. In our setting, we cannot use such a decomposition since $\varphi$ is not assumed to be almost periodic. Instead, we use the embedding

\[ u\mathrm{W}^*\{a^{\mathrm{i} t}\otimes \lambda_{t}\mid t\in \mathbb{R}\}u^*\subset \theta(A)\mathbin{\overline{\otimes}} L\mathbb{R},\quad A:=\mathrm{W}^*\{a\}, \]

where the inclusion holds at a continuous core of $M$. By considering the cases of $a^p$ for $0< p<1$, we deduce the condition $A\preceq _M \theta (A)$ in the sense of Popa's intertwining theory [Pop01, Pop03]. Then, by the choice of $A$, we get a contradiction and the proof is done. Proving $A\preceq _M \theta (A)$ is the key technical step and we do it by the measurable selection principle.

Once we get (2), since it is a condition for every $\psi$ with $M_\psi '\cap M=\mathbb {C}$, we can use recent results from Marrakchi's work on relative bicentralizers [Mar23] and get (4). Thus, our proof is new even for the Araki–Woods factor $R_\infty$. This is why we can avoid the use of the classification theorem of $\operatorname {Aut}(R_\infty )$ [KST89], which was necessary in all previous works [HS91, HI24] for the type ${\rm III_1}$ factor case.

By a result in [AHHM18], we can also prove the equivalence to item (3) of Theorem A, which is a condition on ultraproducts. This condition is interesting in the sense that $\varphi ^\omega$ has more information than $\varphi$. Indeed, item (3) is a condition for a single $\varphi$, while item (2) is one for many $\psi$.

2. Preliminaries

For a von Neumann algebra $M$, the $L^2$-norm with respect to $\varphi \in M_\ast ^+$ is denoted by $\| \, \cdot \, \|_{\varphi }$. We use the notation $M_p:=pMp$ for each projection $p\in M \cup M'$.

2.1 Tomita–Takesaki theory and Connes cocycles

Let $M$ be a von Neumann algebra and $\varphi \in M_\ast ^+$ a faithful functional. Throughout the paper, the modular action for $\varphi$ is denoted by $\sigma ^\varphi \colon \mathbb {R} \curvearrowright M$. The crossed product $M\rtimes _{\sigma ^\varphi }\mathbb {R}$ is called the continuous core (with respect to $\varphi$) and is denoted by $C_\varphi (M)$. We say that $M$ is a type ${\rm III_1}$ factor if $C_\varphi (M)$ is a factor. The centralizer algebra $M_\varphi$ is the fixed point algebra of the modular action $\sigma ^\varphi$. See [Tak03] for details of these objects.

Let $\alpha \colon \mathbb {R} \curvearrowright M$ be a continuous action and $p \in M$ a nonzero projection. We say that a $\sigma$-strongly continuous map $u\colon \mathbb {R} \to pM$ is a generalized cocycle for $\alpha$ (with support projection $p$) if it satisfies

\[ u_{s+t}=u_s \alpha_s(u_t),\quad u_su_s^* = p,\quad u_s^* u_s = \alpha_s(p),\quad \text{for all }s,t\in \mathbb{R}. \]

In this case, by putting $\alpha ^u_s(pxp):=u_s\alpha _s(pxp)u_s^*$ for all $x\in M$ and $s\in \mathbb {R}$, we have a continuous $\mathbb {R}$-action on $pMp$.

For $\varphi,\psi \in M_\ast ^+$ with $\varphi$ faithful and with $s(\psi )$ the support projection of $\psi$, consider the modular actions $\sigma ^\varphi$ on $M$ and $\sigma ^\psi$ on $M_{s(\psi )}$. The Connes cocycle $(u_t)_{t\in \mathbb {R}}$ (for $\psi$ with respect to $\varphi$) [Con72] is a generalized cocycle for $\sigma ^\varphi$ with support projection $s(\psi )$ such that $(\sigma ^\varphi )^{u}\colon \mathbb {R} \curvearrowright M_{s(\psi )}$ coincides with $\sigma ^\psi$. We denote it by $u_t=[D\psi :D\varphi ]_t$ for $t\in \mathbb {R}$. See [Tak03, VIII.3.19–20] for this nonfaithful version of the Connes cocycle.

Let $\theta = \sigma ^\varphi _t$ for some $t\in \mathbb {R}$ and we see that it is pointwise inner. Let $\psi \in M_\ast ^+$ be a positive functional and take a faithful $\widetilde {\psi }\in M_\ast ^+$ with $s(\psi )\widetilde {\psi }=\widetilde {\psi }s(\psi )=\psi$. Then since $s(\psi )\in M_{\widetilde {\psi }}$ and $\sigma ^{\widetilde {\psi }}_t ( \psi ) = \psi$, $u_t:=[D\varphi :D\widetilde {\psi }]_t$ satisfies

\[ \theta(\psi)= \operatorname{Ad}(u_t)\circ \sigma^{\widetilde{\psi}}_t (s(\psi)\widetilde{\psi}) = u_t\psi u_t^*. \]

Hence $\theta$ is pointwise inner.

By the uniqueness of Connes cocycles, it is straightforward to check that, if $\varphi,\psi$ are faithful, $v \in M$ a partial isometry with $e:=v v^*\in M_\psi$, $v^*v \in M_\varphi$, and $v\varphi v^* = e\psi e$, then for each $x\in M$ and $t\in \mathbb {R}$,

\[ v\sigma_t^\varphi(v^*xv)v^*=\sigma_t^\psi(exe),\quad e[D\psi : D\varphi]_t = [De\psi e: D\varphi]_t=v\sigma_t^\varphi(v^*). \]

We prove some lemmas.

Lemma 2.1 Let $N\subset M$ be an inclusion of $\sigma$-finite von Neumann algebras with expectation $E_N$. Let $\varphi _N\in N_\ast$ be a faithful tracial state and put $\varphi :=\varphi _N\circ E_N$. Let $p,q\in N$ be projections such that $N_p'\cap M_p\subset N_p$. Assume that there is a $\ast$-isomorphism $\theta \colon M_p \to M_q$ such that $\theta (N_p) = N_q$. We write $\theta _N:=\theta |_{N_p}$.

1. (1) We have $\theta ^{-1}\circ E_N = E_N\circ \theta ^{-1}$ on $M_q$. In particular,

   \[ [Dq\theta(p\varphi p)q:D\varphi]_t = [Dq\theta_N(p\varphi_N p)q :D\varphi_N ]_t \in N,\quad \text{for all }t\in \mathbb{R}. \]

2. (2) There is a unique nonsingular, positive, self-adjoint element $h$, which is affiliated with $N_q$, such that $q\theta (p\varphi p)q= \varphi _h$.

3. (3) Assume that $\theta = \operatorname {Ad}(v)\circ \theta '$ for a $\ast$-isomorphism $\theta '\colon pMp\to rMr$, a projection $r\in M$, and a partial isometry $v\in qMr$. Then $h$ in $(2)$ satisfies

   \[ h^{\mathrm{i} t} = v[D r\theta'(p\varphi p)r : D\varphi]_t \sigma_t^{\varphi}(v^*),\quad \text{for all }t\in \mathbb{R}. \]

Proof. Recall that a normal conditional expectation from $M$ onto $N$ is unique if $N\subset M$ is irreducible ([Con72, § 1.4], [Tak03, Proposition IX.4.3]). Observe that $N_p\subset M_p$ and $N_q = \theta (N_p)\subset \theta (M_p)=M_q$ are irreducible. In the proof below, for simplicity of notation, we sometimes omit $p$, for example $q\theta (\varphi )q = q\theta (p\varphi p)q$.

(1) Since $\theta \circ E_N \circ \theta ^{-1}\colon M_q \to N_q$ is a normal conditional expectation, it coincides with $E_N|_{M_q}$ by uniqueness. We get $\theta _N^{-1}\circ E_N = E_N\circ \theta ^{-1}$ on $M_q$, and hence

\[ q\theta(\varphi)q = q( \varphi_N\circ E_N\circ\theta^{-1})q = q(\varphi_N \circ \theta_N^{-1} \circ E_N )q=q( \theta_N(\varphi_N) \circ E_N)q. \]

Take a faithful $\psi _N\in N_\ast ^+$ with $q \psi _N q = q\theta _N(\varphi _N)q$. Then by [Con72, Lemma 1.4.4], for each $t\in \mathbb {R}$,

\begin{align*} [Dq\theta(\varphi)q : D\varphi]_t &= q [D\psi_N \circ E_N : D\varphi _N\circ E_N ]_t \\ &= q [D\psi_N : D\varphi_N ]_t= [Dq\theta_N(\varphi_N)q : D\varphi _N ]_t\in N. \end{align*}

(2) Since $\varphi _{N}$ is a faithful trace on $N$, there is a unique $h$, affiliated with $N_q$, such that $q\theta _N(\varphi _N)q = (\varphi _{N})_h$ on $N_q$. Combined with (1), for each $t\in \mathbb {R}$, we have

\[ [Dq\theta(\varphi)q :D\varphi]_t=[Dq\theta_N(\varphi_N)q :D\varphi_N ]_t = h^{\mathrm{i} t}. \]

Since $h$ is affiliated with $M_q$, we also have $h^{\mathrm {i} t} = [D\varphi _h:D\varphi ]_t$, hence we get $q\theta (\varphi )q =\varphi _h$.

(3) Let $\psi \in M_\ast ^+$ be a faithful element such that $r\in M_\psi$ and $r\psi r = r\theta '(\varphi )r$. Then for each $t\in \mathbb {R}$,

\begin{align*} h^{\mathrm{i} t} &= [Dq\theta(\varphi)q: D\varphi]_t= [D v\theta'(\varphi)v^*: D\varphi]_t\\ &= [D v\psi v^*: D\psi]_t[D \psi: D\varphi]_t\\ &= v\sigma_t^{\psi}(v^*)[D \psi: D\varphi]_t\\ &= v[D \psi: D\varphi]_t \sigma_t^{\varphi}(v^*)= v[D r\psi r : D\varphi]_t \sigma_t^{\varphi}(v^*), \end{align*}

and we are done.

Lemma 2.2 Let $M$ be a factor with a faithful state $\varphi \in M_\ast$ such that $M_\varphi ' \cap M =\mathbb {C}$. If $\theta \in \operatorname {Aut}(M)$ satisfies $\theta (M_\varphi )=M_\varphi$, then $\theta (\varphi )=\varphi$. In particular, $M_\varphi \subset M$ is singular in the sense that $uM_\varphi u^* = M_\varphi$ for $u\in \mathcal {U}(M)$ implies $u\in M_\varphi$.

Proof. Put $N:=M_\varphi$ and $\varphi _N:=\varphi |_N$. We apply Lemma 2.1(2) for the case $p=q=1$, and take $h$ such that $\theta (\varphi ) = \varphi _h$. Since $h$ is affiliated with $N$, we have $\varphi _h|_N =(\varphi _N)_h$. Since $N$ is a finite factor and since $\theta (\varphi )|_N$ is a trace, we have $\theta (\varphi )|_N = \varphi _N$ by the uniqueness of the trace. We get $(\varphi _N)_h = \varphi _N$ and $h=1$. This means $\theta (\varphi ) = \varphi$.

If $uM_\varphi u^* = M_\varphi$ for $u\in \mathcal {U}(M)$, we can apply the first part of the lemma to $\operatorname {Ad}(u)\in \operatorname {Aut}(M)$. We get $\operatorname {Ad}(u)(\varphi )=\varphi$ and hence $u$ is contained in $M_\varphi$.

Lemma 2.3 Let $M$ be a factor with a faithful state $\varphi \in M_\ast$ such that $M_\varphi ' \cap M =\mathbb {C}$. Let $\theta \in \operatorname {Aut}(M)$ and define an embedding

\[ \iota_{\theta}\colon M \ni x \mapsto \left[\begin{array}{@{}cc@{}} x & 0 \\ 0 & \theta (x) \end{array}\right] \in M\otimes \mathbb{M}_2. \]

If $\theta (\varphi )=\varphi$, then the following conditions are equivalent:

1. (1) $\theta |_{M_\varphi }$ is outer;

2. (2) $\iota _\theta (M_\varphi )'\cap (M\otimes \mathbb {M}_2 ) \subset M\oplus M$.

Proof. If $\theta |_{M_\varphi } = \operatorname {Ad}(u)$ for some $u\in \mathcal {U}(M_\varphi )$, then $\big [{\kern-1.3pt}\begin{smallmatrix} 0 & 0 \\ u & 0 \end{smallmatrix}{\kern-1.3pt}\big ]$ is contained in $\iota _\theta (M_\varphi )'\cap M\otimes \mathbb {M}_2$.

If (2) does not hold, then there is a nonzero $a\in M$ such that $\theta (x)a=ax$ for all $x\in M_\varphi$. By $M_\varphi '\cap M=\mathbb {C}$, $a$ is a scalar multiple of a unitary, so there is a unitary $u\in \mathcal {U}(M)$ such that $\theta (x)=uxu^*$ for all $x\in M_\varphi$. Then $\operatorname {Ad}(u)$ globally preserves $M_\varphi$, hence $u$ is in $M_\varphi$ by Lemma 2.2. Thus (1) does not hold.

2.2 Ultraproduct von Neumann algebras

Let $M$ be a $\sigma$-finite von Neumann algebra and $\omega$ a free ultrafilter on $\mathbb {N}$. Put

\begin{align*} \mathcal{I}_{\omega} &= \{(x_n)_{n} \in \ell^\infty(\mathbb{N}, M) \mid x_n \to 0 \textrm{ $\ast$-strongly as } n \to \omega\},\\ \mathcal{M}^{\omega} &= \{x \in \ell^\infty(\mathbb{N}, M) \mid x \mathcal{I}_{\omega} \subset \mathcal{I}_{\omega} \text{ and } \mathcal{I}_{\omega}x \subset \mathcal{I}_{\omega}\}, \end{align*}

where $\ell ^\infty (\mathbb {N}, M)$ is the set of all norm bounded sequences in $M$. The ultraproduct von Neumann algebra [Ocn85] is defined as the quotient C$^*$-algebra $M^\omega := \mathcal {M}^\omega / \mathcal {I}_\omega$, which naturally admits a von Neumann algebra structure. The image of $(x_n)_n$ in $M^\omega$ is denoted by $(x_n)_\omega$. We have an embedding $M \subset M^\omega$ as constant sequences, and this inclusion is with expectation $E_\omega$ given by $E_\omega ( (x_n)_\omega )=\sigma \text {-weak}\lim _{n\to \omega }x_n$. For every faithful $\varphi \in M_*^+$, we can define a faithful positive functional $\varphi ^\omega :=\varphi \circ E_\omega$ on $M^\omega$.

Every $\theta \in \operatorname {Aut}(M)$ induces $\theta ^\omega \in \operatorname {Aut}(M^\omega )$ by the equation $\theta ^\omega ((x_n)_\omega ) = (\theta (x_n))_\omega$. In particular, the modular action of $\varphi ^\omega$ is given by $\sigma ^{\varphi ^\omega }_t = (\sigma ^\varphi _t)^\omega$ for all $t\in \mathbb {R}$ [AH12]. For more on ultraproduct von Neumann algebras, we refer the reader to [Ocn85, AH12].

2.3 Relative bicentralizer algebras

Let $N\subset M$ be an inclusion of von Neumann algebras with separable predual and with expectation $E_N$. Let $\varphi \in N_\ast$ be a faithful state and extend it to $M$ by $E_N$. We define the asymptotic centralizer and the relative bicentralizer algebra (with respect to $\varphi$) as

\begin{align*} \mathrm{AC}_\varphi(N)&:= \{(x_n)_n\in \ell^\infty(\mathbb{N},N) \mid \lim_{n\to \infty}\|x_n\varphi - \varphi x_n\| = 0 \},\\ \mathrm{BC}_\varphi(N\subset M)&:= \{x\in M \mid \lim_{n\to \infty} \|x_n x - x x_n\|_\varphi \to 0 \ \text{for every }(x_n)_n\in \mathrm{AC}_\varphi(N) \}. \end{align*}

This does not depend on the choice of $\varphi$ up to a canonical isomorphism, if $N$ is a type ${\rm III_1}$ factor. In the case $N=M$, we write $\mathrm {BC}_\varphi (M)=\mathrm {BC}_\varphi (N\subset M)$ and we call it the bicentralizer. Then Connes’ bicentralizer problem asks if every type ${\rm III_1}$ factor $M$ with separable predual has trivial bicentralizer, that is, $\mathrm {BC_\varphi }(M)=\mathbb {C}$. This condition is equivalent to having a faithful state $\varphi \in M_\ast$ such that $M_\varphi '\cap M=\mathbb {C}$ [Haa85]. Here is a convenient expression by ultraproducts ([HI15, Proposition 3.3], [AHHM18, Proposition 3.3]):

\[ \mathrm{BC}_\varphi(N\subset M) = (N^\omega)_{\varphi^\omega} '\cap M, \]

where $\omega$ is a fixed free ultrafilter on $\mathbb {N}$.

The next theorem explains the importance of relative bicentralizers. This should be understood as a type III counterpart of Popa's work [Pop81].

Theorem 2.4 ([Haa85], [AHHM18, Theorem C])

Assume that $N$ is a type ${\rm III_1}$ factor. If $\mathrm {BC}_\varphi (N\subset M) = N'\cap M$, then there exists an amenable subfactor $R \subset N$ of type ${\rm II_1}$ with expectation such that $R'\cap M =N'\cap M$.

Very recently, Marrakchi obtained a very general and useful criterion for computations of relative bicentralizers. We will need the following statement. Recall that the inclusion $N\subset M$ is regular if the normalizer $\mathcal {N}_M(N):=\{u\in \mathcal {U}(M)\mid uN u^*=N\}$ of $N$ in $M$ generates $M$ as a von Neumann algebra.

Theorem 2.5 [Mar23, Theorem E(5)]

Assume that $N$ is a type ${\rm III_1}$ factor. Assume that $N$ has trivial bicentralizer and that $N\subset M$ is regular. If $N'\cap C_\varphi (M)\subset C_\varphi (N)$, then we have $\mathrm {BC}_\varphi (N\subset M)=\mathbb {C}$.

Proof. By [Mar23, Theorem E(5)], we have

\[ \mathrm{BC}_\varphi(N\subset C_\varphi(M)) = L\mathbb{R}\vee (N'\cap C_\varphi(M))\subset C_\varphi(N), \]

where the relative bicentralizer algebra is defined for arbitrary inclusions. This implies

\[ \mathrm{BC}_\varphi(N\subset M) =M\cap \mathrm{BC}_\varphi(N\subset C_\varphi(M)) \subset M\cap C_\varphi(N) = N. \]

We get $\mathrm {BC}_\varphi (N\subset M) \subset \mathrm {BC}_\varphi (N) = \mathbb {C}$.

2.4 Popa's intertwining theory

We recall Popa's intertwining theory [Pop01, Pop03]. For this, we fix a $\sigma$-finite von Neumann algebra $M$ and a finite von Neumann subalgebra $B\subset M$ with expectation $E_B$. Fix a trace $\tau _B$ on $B$. We can define a canonical trace $\mathord {\text { Tr}}$ on the basic construction $\langle M, B\rangle$ satisfying $\mathord {\text { Tr}}( xe_By) = \tau _B\circ E_B(yx)$ for $x,y\in M$ (see for example, [HI15, § 4]).

In this setting, we have the following equivalence. For the proof of this theorem, we refer the reader to [HI15, Theorem 4.3]. Since $B$ is finite, the canonical operator-valued weight $T_M$ from $\langle M, B\rangle$ to $M$, which appears in [HI15, Theorem 4.3 (6)], is unnecessary.

Theorem 2.6 [Pop01, Pop03]

Retain the setting of this section. For a finite von Neumann subalgebra $A\subset M$ with expectation, the following conditions are equivalent.

1. (1) We have $A \preceq _M B$. This means that there exist projections $e\in A$, $f\in B$, a partial isometry $v\in eMf$, and a unital normal $\ast$-homomorphism $\theta \colon eAe \to fBf$ such that $v\theta (a)= av$ for all $a\in eAe$.

2. (2) There exists no net $(u_i)_{i}$ of unitaries in $\mathcal {U}(A)$ such that for every $a,b\in M$,

   \[ \|E_B(b^* u_i a )\|_{\tau_B} \rightarrow 0,\quad \text{as}\ i\to \infty. \]

3. (3) There exists a nonzero positive element $d\in A' \cap \langle M, B\rangle$ such that $\mathord {\text { Tr}}(d)<\infty$.

The next three lemmas are well known to experts. We include short proofs for the reader's convenience.

Lemma 2.7 Retain the setting as in Theorem 2.6. Let $A_1,A_2\subset M$ be finite von Neumann subalgebras with expectation such that $A_1$ and $A_2$ are commuting with each other and that $A_1\vee A_2\subset M$ is a finite von Neumann subalgebra with expectation.

Suppose that there are nonzero positive elements $d_i\in A_i' \cap \langle M,B\rangle$ satisfying the condition of item $(3)$ in Theorem 2.6 for $i=1,2$ such that $d_1d_2\neq 0$. Then we have $A_1\vee A_2\preceq _M B$.

Proof. Consider $\mathord {\text { Tr}}$ on $\langle M,B\rangle$ as in Theorem 2.6. Put

\[ \mathcal{K} := \overline{\mathrm{conv}}^{\textrm{$\sigma$-weak}} \{ u d_1 u^* \mid u\in \mathcal{U}(A_2) \} \subset A_1'\cap \langle M,B\rangle. \]

By the normality of $\mathord {\text { Tr}}$, every element in $\mathcal {K}$ has finite value in $\mathord {\text { Tr}}$. By regarding $\mathcal {K}$ as a closed subset of $L^2(\langle M,B\rangle,\mathord {\text { Tr}})$ (see [HI15, Lemma 4.4]), take the unique element $d\in \mathcal {K}$ which has the minimum distance from $0$. The uniqueness condition implies $udu^*=u$ for all $u\in \mathcal {U}(A_2)$, hence $d$ is contained in $(A_1\vee A_2)'\cap \langle M,B\rangle$. Then observe that $d_1,d_2\in L^2(\langle M,B\rangle,\mathord {\text { Tr}})$, and for each $u\in \mathcal {U}(A_2)$,

\[ \langle ud_1u^*,d_2 \rangle_{{\rm Tr}} = {\rm Tr} ( d_2 ud_1u^*) = {\rm Tr} ( u^*d_2 ud_1)= {\rm Tr}(d_2d_1) = {\rm Tr}(d_1^{1/2} d_2 d_1^{1/2})>0. \]

This implies that $\mathcal {K}$ does not contain $0$, hence $d\neq 0$. This implies $A_1\vee A_2\preceq _M B$.

Lemma 2.8 Let $M$ be a von Neumann algebra, $\varphi \in M_\ast$ a faithful state, and $A,B\subset M_\varphi$ von Neumann subalgebras. If $A\not \preceq _M B$, then $A\mathbin {\overline {\otimes }} L\mathbb {R} \not \preceq _{C_\varphi (M)} B\mathbin {\overline {\otimes }} L\mathbb {R}$.

Proof. Let $E_B\colon M\to B$ be the unique $\varphi$-preserving conditional expectation. Then since $E_B$ commutes with $\sigma ^\varphi$, we can extend $E_B$ to one from $C_\varphi (M)$ onto $B\mathbin {\overline {\otimes }} L\mathbb {R}$. Take a net $(u_i)_i$ of $\mathcal {U}(A)$ as in item (2) in Theorem 2.6. Then it is easy to see that $(u_i\otimes 1_{L\mathbb {R}})_i$ works for $A\mathbin {\overline {\otimes }} L\mathbb {R}\not \preceq _{C_\varphi (M)} B\mathbin {\overline {\otimes }} L\mathbb {R}$.

In the lemma below, we need the case that $A,B \subset M$ are possibly nonunital finite von Neumann subalgebras, where subalgebras $A\subset 1_A M 1_A$ and $B\subset 1_B M1_B$ with units $1_A,1_B$ are assumed to be with expectation. In this case, we can use the same definition for $A\preceq _M B$, and item (2) in Theorem 2.6 works by replacing $a,b\in M$ by $a,b\in M1_B$. Item (3) is slightly more complicated, but we do not need it. See [HI15, Theorem 4.3] for more on the nonunital case.

Lemma 2.9 Let $A,B_1,\ldots,B_n\subset M$ be (possibly nonunital) inclusions of $\sigma$-finite von Neumann algebras with expectation. Let $E_{B_k}\colon M_{1_{B_k}}\to B_k$ be faithful normal conditional expectations for all $k$ and assume that $A,B_1,\ldots,B_n$ are finite with trace $\tau _k\in (B_k)_\ast$ for all $k$. If $A\not \preceq _M B_k$ for all $k$, then there exists a net $(u_i)_{i}$ in $\mathcal {U}(A)$ such that for all $k$,

\[ \|E_{B_k}(b^* u_i a ) \|_{\tau_k}\rightarrow 0,\quad \textrm{for all $a,b\in M1_{B_k}$.} \]

Proof. Consider $M_n:=M\otimes \mathbb {C}^n$ and embeddings

\[ \widetilde{A}:=A\otimes \mathbb{C}1_{\mathbb{C}^n}\subset M_n,\quad \widetilde{B}:=B_1\oplus\cdots\oplus B_n \subset M_n. \]

Observe that $A\not \preceq _M B_k$ implies $\widetilde {A}\not \preceq _{M_n} B_k$ for all $k$ (use item (2) of Theorem 2.6). Then by [HI15, Remark 4.2(2)], we have that $\widetilde {A}\not \preceq _{M_n} \widetilde {B}$. Then a net of unitaries in item $(2)$ of Theorem 2.6 for $\widetilde {A}\not \preceq _{M_n} \widetilde {B}$ works.

3. Proof of Theorem A: (1)$\Rightarrow$(2)

We need two lemmas. The first one uses the measurable selection principle.

Lemma 3.1 Let $M$ be a von Neumann algebra with separable predual, $\varphi$ a faithful normal state, and $\theta \in \operatorname {Aut}(M)$ a pointwise inner automorphism. Then for each positive element $a\in M_\varphi$, there exist a compact subset $K\subset (0,1)$ with positive Lebesgue measure, and a continuous map $K \ni p\mapsto u_p\in \mathcal {U}(M)$ such that

\[ \theta(\varphi_{a^p}) = u_p \varphi_{a^p} u^*_p ,\quad \text{for all}\ p\in K, \]

where $\mathcal {U}(M)$ is equipped with the $\ast$-strong topology.

Proof. Throughout the proof, we consider the $\ast$-strong topology for $\mathcal {U}(M)$ and we regard it as a Polish space. Set

\[ \mathcal{S}:= \{ (p,u)\in (0,1)\times \mathcal{U}(M)\mid \theta(\varphi_{a^p}) = u \varphi_{a^p} u^* \} . \]

Then it is easy to see that $\mathcal {S}$ is a closed subset. Since $\theta$ is pointwise inner, we have $(0,1)=\pi _1(\mathcal {S})$, where $\pi _1$ is the projection onto the first coordinate. Hence, with the Lebesgue measure on $(0,1)$, we can apply the measurable selection principle (e.g. [KR97, Theorem 14.3.6]) and find a Lebesgue measurable map

\[ \eta \colon (0,1)=\pi_1(\mathcal{S}) \to \mathcal{U}(M) \]

such that $(p,\eta (p)) \in \mathcal {S}$ for all $p\in (0,1)$. Since $\mathcal {U}(M)$ is a Polish space, we can apply Lusin's theorem to $\eta$ and find a compact subset $K\subset (0,1)$ such that $\eta |_{K}$ is continuous and that $(0,1)\setminus K$ has arbitrarily small Lebesgue measure. The conclusion follows.

The second lemma uses the first. This lemma is the key observation of the paper.

Lemma 3.2 Let $M$ be a von Neumann algebra with separable predual, $\varphi \in M_\ast$ a faithful state, and $\theta \in \operatorname {Aut}(M)$ a pointwise inner automorphism such that $\theta (\varphi )=\varphi$. Then, for each positive invertible element $a\in M_\varphi$, we have $A\preceq _M \theta (A)$, where $A:=\mathrm {W}^*\{a\}\subset M_\varphi$.

Proof. Observe that $\theta (M_\varphi )=M_\varphi$ and $\theta (A)\subset M_\varphi$. Take $u\in \mathcal {U}(M)$ such that $\varphi _{\theta (a)} = \theta (\varphi _a) = u \varphi _a u^*$. Then since

\[ [D\varphi_{\theta(a)} :D\varphi_a]_t = [D\varphi_{\theta(a)} :D\varphi]_t[D\varphi :D\varphi_a]_t = \theta(a^{\mathrm{i} t})a^{-\mathrm{i} t} \]

and

\[ [Du \varphi_a u^* :D\varphi_a]_t= u\sigma_t^{\varphi_a} (u^*) = u a^{\mathrm{i} t} \sigma_t^{\varphi} (u^*) a^{-\mathrm{i} t}, \]

we get $\theta (a^{\mathrm {i} t}) = u a^{\mathrm {i} t} \sigma _t^{\varphi } (u^*)$ for all $t\in \mathbb {R}$. Define

\[ \widetilde{A}:=\mathrm{W^*}\{ a^{\mathrm{i} t}\otimes \lambda_t \mid t\in \mathbb{R} \}\subset C_\varphi(M), \]

where $\lambda _t$ denotes the left regular representation on $L^2(\mathbb {R})$, and observe that $\widetilde {A}\subset C_\varphi (M)$ is with expectation. Then it is easy to see that $\operatorname {Ad}(u)$ restricts to a $\ast$-homomorphism from $\widetilde {A}$ into $\theta (A)\mathbin {\overline {\otimes }} L\mathbb {R}$, so that we get $\widetilde {A}\preceq _{C_\varphi (M)} \theta (A)\mathbin {\overline {\otimes }} L\mathbb {R}$. More precisely, if we denote by $e$ the Jones projection for a fixed faithful normal conditional expectation from $C_\varphi (M)$ onto $\theta (A)\mathbin {\overline {\otimes }} L\mathbb {R}$, then the projection $u^* e u$ satisfies the condition of item $(3)$ in Theorem 2.6.

Fix $0< p<1$, and we consider the positive invertible element $a^p$. Then, by the same reasoning as above, we have that

\[ \widetilde{A}_p\preceq_{C_\varphi(M)} \theta(A)\mathbin{\overline{\otimes}} L\mathbb{R}, \quad \text{where}\ \widetilde{A}_p:=\mathrm{W^*}\{ a^{\mathrm{i} pt}\otimes \lambda_t \mid t\in \mathbb{R} \}, \]

together with the projection $u_p^* e u_p$, where $u_p$ is a unitary satisfying $\varphi _{\theta (a^p)} = u_p \varphi _{a^p} u_p^*$. We note that the algebra $\theta (A)\mathbin {\overline {\otimes }} L\mathbb {R}$ does not depend on $p$.

Now we claim that $A\mathbin {\overline {\otimes }} L\mathbb {R} \preceq _{C_\varphi (M)} \theta (A)\mathbin {\overline {\otimes }} L\mathbb {R}$. To see this, we first apply Lemma 3.1 and find a compact $K\subset (0,1)$ with positive measure and a continuous map $K\ni p \mapsto u_p$. Then we take a sequence $p_n\in K$ such that $p_n$ converges to $p\in K$ and $p\not \in \{p_n\}_n$. Then $u_{p_n} \to u_p$ in the $\ast$-strong topology, hence

\[ d_{p_n}:=u_{p_n}^* e u_{p_n}\to u_{p}^* e u_{p}=:d_p \]

as well. In particular, we can find $p_n=:q$ such that $q\neq p$ and $d_q d_p\neq 0$. Observe that $\widetilde { A}_p\vee \widetilde {A}_q=A\mathbin {\overline {\otimes }} L\mathbb {R}$, which is a finite von Neumann subalgebra in $C_\varphi (M)$ with expectation. Then by Lemma 2.7, we have that

\[ A\mathbin{\overline{\otimes}} L\mathbb{R}=\widetilde{ A}_p\vee \widetilde{A}_q \preceq_{C_\varphi(M)} \theta(A)\mathbin{\overline{\otimes}} L\mathbb{R}. \]

Hence the claim is proven.

Finally, this condition implies $A\preceq _M \theta (A)$ by Lemma 2.8.

Now we prove (1)$\Rightarrow$(2) of Theorem A. For the proof, we need Popa's $\mathcal {G}$-singular maximal abelian $\ast$-subalgebra (MASA) technique for type ${\rm III_1}$ factors; see Theorem A.1.

Proof of Theorem A: (1)other$\Rightarrow$(2) Let $\psi \in M_\ast ^+$ be a faithful state such that $M_\psi '\cap M=\mathbb {C}$. Since $\theta$ is pointwise inner, there is $v\in \mathcal {U}(M)$ such that $\operatorname {Ad}(v)\circ \theta (\psi )=\psi$. So up to replacing $\operatorname {Ad}(v)\circ \theta$ by $\theta$, we may assume $\theta (\psi )=\psi$. Then we show that $\theta |_{M_\psi }$ is inner.

Suppose by contradiction that $\theta$ is not inner on $M_\psi$. Then, putting $\mathcal {G}:=\{{\text { id}}_M,\ \theta ^{-1}\}$, we apply Theorem A.1 and take a $\mathcal {G}$-singular MASA $A\subset M_\psi$. Take a generator $a$ of $A$ which is positive and invertible. We can apply Lemma 3.2 to $a$ and get $A\preceq _M \theta (A)$. Since $A$ and $\theta (A)$ are MASAs in $M$, by [HV12, Theorem 2.5], one can find a nonzero partial isometry $v\in M$ such that

\[ v^*v \in A,\quad vv^* \in \theta(A),\quad v A v^* = \theta(A)vv^* . \]

Thus $\operatorname {Ad}(v)$ restricts to a $\ast$-isomorphism from $Av^*v$ onto $\theta (A)vv^*$. Since $A$ is diffuse, up to exchanging $v^*v$ with a smaller projection in $A$ (or $vv^*$ with one in $\theta (A)$) if necessary, we may assume $v^*v\neq 1$ and $vv^*\neq 1$. Then since $M$ is a type III factor, $1-v^*v$ and $1-vv^*$ are equivalent, so we can take a partial isometry $w\in M$ such that $w^*w=1-v^*v$ and $ww^* = 1-vv^*$. Then $\widetilde {v}:= v+w\in M$ is a unitary element satisfying $\widetilde {v}p = v$ where $p:=v^*v$, so that

\[ \widetilde{v} A p \widetilde{v}^* = v A v^*\subset \theta(A). \]

This means $\operatorname {Ad}(\theta ^{-1}(\widetilde {v})) \circ \theta ^{-1} (A p) \subset A.$ By $\mathcal {G}$-singularity, we have that $\theta ^{-1}={\text { id}}_M$, a contradiction.

4. Proof of Theorem A: (2)$\Rightarrow$(3) and (3)$\Rightarrow$(4)

Proof of Theorem A: (2)$\Rightarrow$(3) We may assume that $\theta$ is outer on $M$. Set $M_2:=M\otimes \mathbb {M}_2$ and consider an embedding $\iota _\theta \colon M\to M_2$ with expectation as in Lemma 2.3. Since $\theta$ is outer, we have $M'\cap M_2 = \mathbb {C} \oplus \mathbb {C}$. We consider the relative bicentralizer

\[ \mathrm{BC}_\varphi(M\subset M_2) = (M^\omega)_{\varphi^\omega}' \cap M_2. \]

We claim $\mathrm {BC}_\varphi (M\subset M_2)\neq M'\cap M_2$.

To see this, suppose by contradiction that $\mathrm {BC}_\varphi (M\subset M_2)=M'\cap M_2$. Then by Theorem 2.4, there is an amenable ${\rm II_1}$ factor $R\subset M$ with expectation $E_R$ such that $R'\cap M_2 = M'\cap M_2 = \mathbb {C}\oplus \mathbb {C}$. Put $\psi :=\tau _R\circ E_R$, where $\tau _R$ is the trace on $R$, and observe that $R\subset M_\psi \subset M$. Then since $M_\psi '\cap M\subset R'\cap M=\mathbb {C}$ (this follows by $R'\cap M_2=\mathbb {C}\oplus \mathbb {C}$), by assumption, there is $u\in \mathcal {U}(M)$ such that $\operatorname {Ad}(u)\circ \theta |_{M_\psi } = {\text { id}}_{M_\psi }$. Then $\big[{\kern-1.3pt}\begin{smallmatrix} 0 & u \\ 0 & 0 \end{smallmatrix}{\kern-1.3pt}\big ]$ is contained in $M_\psi '\cap M_2 \subset R'\cap M_2 = \mathbb {C}\oplus \mathbb {C}$. This is a contradiction and the claim is proven.

We take $\big [{\kern-1.3pt}\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}{\kern-1.3pt}\big ]\in (M^\omega )_{\varphi ^\omega }' \cap M_2$, which is not contained in $\mathbb {C}\oplus \mathbb {C}$. Then since $(M^\omega )_{\varphi ^\omega }' \cap M = \mathrm {BC}_\varphi (M)=\mathbb {C}$, we have $a,d\in \mathbb {C}$, hence $b\neq 0$ or $c\neq 0$. By taking the adjoint if necessary, we may assume $c\neq 0$. Then observe that $\theta ^\omega (x)c = cx$ for all $x\in (M^\omega )_{\varphi ^\omega }$. Since $(M^\omega )_{\varphi ^\omega }' \cap M =\mathbb {C}$, by the polar decomposition, we can replace $c$ by a unitary element. This means (3) holds.

Proof Theorem A: (3)other$\Rightarrow$(4) By assumption, take $u\in \mathcal {U}(M)$ such that $\operatorname {Ad}(u)\circ \theta ^\omega = {\text { id}}$ on $(M^\omega )_{\varphi ^\omega }$. Up to replacing $\operatorname {Ad}(u)\circ \theta$ by $\theta$, we may assume $u=1$. Then by Lemma 2.2, we have that $\theta (\varphi )=\varphi$. In particular, $\theta$ commutes with the modular action of $\varphi$.

We set

\[ H(\theta):= \{ n\in \mathbb{Z}\mid \theta^n = \operatorname{Ad}(u)\circ \sigma^\varphi_t \ \text{for some }u\in \mathcal{U}(M), \ t\in \mathbb{R}\}. \]

Observe that by $M_\varphi '\cap M=\mathbb {C}$ and $\theta |_{M_\varphi }={\text { id}}$, the equality $\theta ^n = \operatorname {Ad}(u)\circ \sigma ^\varphi _t$ implies $u\in \mathbb {C}$, so we can actually remove $u\in \mathcal {U}(M)$ in the definition of $H(\theta )$. Since $H(\theta )$ is a subgroup in $\mathbb {Z}$, we can write $H(\theta )= k\mathbb {Z}$ for some $k\geq 0$.

Assume that $k>0$ and $\theta ^k= \sigma ^\varphi _t$. Then, putting $\widetilde {\theta }:=\sigma ^\varphi _{-t/k}\circ \theta$, observe that $H(\widetilde {\theta })=H(\theta )$, $\widetilde {\theta }^k={\text { id}}$, and $\widetilde {\theta }^\omega ={\text { id}}$ on $(M^\omega )_{\varphi ^\omega }$. Hence up to replacing $\widetilde {\theta }$ by $\theta$, we may assume $\theta ^k ={\text { id}}$. If $k=0$, we do not need this replacement.

For each $k\geq 0$, set $G:=\mathbb {Z}/k\mathbb {Z}$ and define an outer action $G\curvearrowright M$ given by $\{\theta ^n\}_{n\in \mathbb {Z}}$. We denote this action again by $\theta$. Put $\widetilde {M}:=M\rtimes _\theta G$ with canonical expectation $E_M\colon \widetilde {M}\to M$ and note that $M'\cap \widetilde {M}=\mathbb {C}$ by the outerness of $\theta$. We extend $\varphi$ to $\widetilde {M}$ by $E_M$. Since $\theta$ commutes with the modular action of $\varphi$, the continuous core of $\widetilde {M}$ is such that

\[ C_\varphi(\widetilde{M}) = M\rtimes_{\theta\times \sigma^\varphi} (G\times \mathbb{R}) = C_\varphi(M)\rtimes_\theta G, \]

where $\theta \colon G\curvearrowright C_\varphi (M)$ on the right-hand side is the canonical extension.

The next claim is important to us.

Claim We have $M'\cap C_\varphi (\widetilde {M}) = \mathbb {C}$.

Proof of Claim Since the canonical extension $\theta \colon G\curvearrowright C_\varphi (M)$ is still outer (e.g. [Tak03, Lemma XII.6.14]), the crossed product $C_\varphi (M)\rtimes _\theta G$ is a factor. Let $E:= \varphi \otimes {\text { id}}\colon \mathbb {B}(L^2(M)) \mathbin {\overline {\otimes }} \mathbb {B}(L^2(G\times \mathbb {R})) \to \mathbb {B}(L^2(G\times \mathbb {R}))$, where $\varphi$ is extended to a normal state on $\mathbb {B}(L^2(M))$, and observe that it restricts to a conditional expectation $M\rtimes (G\times \mathbb {R}) \to L(G\times \mathbb {R})$ (because $\theta$ and $\sigma ^\varphi$ preserve $\varphi$).

Since the $G\times \mathbb {R}$ action is trivial on $M_{\varphi }$, we have that

\[ M_{\varphi}'\cap C_\varphi(\widetilde{M})\subset (M_{\varphi}' \cap M) \mathbin{\overline{\otimes}} \mathbb{B}(L^2(G\times \mathbb{R})) = \mathbb{C} \mathbin{\overline{\otimes}} \mathbb{B}(L^2(G\times \mathbb{R})) . \]

We have $E(x)=x$ for all $x\in M_{\varphi }'\cap C_\varphi (\widetilde {M})$, so that $M_{\varphi }'\cap C_\varphi (\widetilde {M})=L(G\times \mathbb {R})$. Then since $G\times \mathbb {R}$ is abelian,

\[ M'\cap C_\varphi(\widetilde{M}) = M'\cap L(G\times \mathbb{R}) \subset \mathcal{Z}(C_\varphi(\widetilde{M})). \]

Since $C_\varphi (\widetilde {M})$ is a factor, the conclusion follows.

To finish the proof, we have to show $k=1$. For this, suppose by contradiction that $k\neq 1$ and $G=\mathbb {Z}/k\mathbb {Z}$ is a nontrivial group. Then observe that the inclusion $M\subset \widetilde {M}$ is with expectation, regular, $\mathrm {BC}_\varphi (M)=\mathbb {C}$, and $M'\cap C_\varphi (\widetilde {M})\subset C_\varphi (M)$. Hence we can apply Theorem 2.5 and get that $\mathrm {BC}_\varphi (M\subset \widetilde {M}) =\mathbb {C}$. Since $G$ is nontrivial, $1\in G$ (the image of $1\in \mathbb {Z}$ in $G$) is nontrivial. We denote by $\lambda _1^G\in LG$ the canonical unitary corresponding to $1\in G$. Then since $\theta ^\omega ={\text { id}}$ on $(M^\omega )_{\varphi ^\omega }$ and since $\operatorname {Ad}(\lambda _1^G)\in \operatorname {Aut}(\widetilde {M})$ coincides with $\theta$ on $M$, we get

\[ \lambda_1^G \in (M^\omega)_{\varphi^\omega}' \cap \widetilde{M} = B_\varphi(M\subset \widetilde{M}) =\mathbb{C}. \]

This is a contradiction because $1\in G$ is nontrivial. Thus we get $k=1$ and this finishes the proof.