On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III$_1$

TL;DR

This paper demonstrates the isomorphism between the relative bicentralizer flow and the relative flow of weights for certain inclusions of type III$_1$ factors, expanding understanding of their structural relationships.

Contribution

It establishes the isomorphism between the relative bicentralizer flow and the relative flow of weights for specific classes of type III$_1$ factor inclusions.

Findings

Proves the isomorphism for inclusions of injective type III$_1$ factors with finite index.

Shows the isomorphism for irreducible discrete inclusions with small algebra as an injective type III$_1$ factor.

Enhances the understanding of the structure of type III$_1$ factor inclusions.

Abstract

We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III$_1$.

Equations

$$\mathrm{AC}(\mathcal{N},\phi )=\{(x_n)\in {\ell }^{\infty }(\mathbb{N},\mathcal{N})\mid \underset{n\to \infty }{\lim }\Vert x_n\phi -\phi x_n\Vert =0\}$$

$$\mathrm{B}(\mathcal{N}\subset \mathcal{M},\phi )=\left\{a\in \mathcal{M}\mid \underset{n\to \infty }{\lim }[x_n,a]=0\text{mbox}stronglyforall(x_n)\in \mathrm{AC}(\mathcal{N},\phi ).\right\}$$

$$\Psi ((a_{ij}))=\frac{1}{1+d(\rho )}\left(a_{11}+d(\rho ){\varphi }_{\rho }(a_{22})\right),E_{\Psi }((a_{ij}))=\frac{1}{1+d(\rho )}\pi (a_{11}+d(\rho ){\varphi }_{\rho }(a_{22})).$$

$$\pi (\mathcal{N}{)}^{\prime }\cap \mathcal{Q}=\tilde{\pi }(\tilde{\mathcal{N}}{)}^{\prime }\cap \tilde{\mathcal{Q}}=\mathbb{C}{e}_{11}\oplus \mathbb{C}{e}_{22}.$$

$$\mathrm{B}(\pi (\mathcal{N})\subset \mathcal{Q},\Psi )=\pi (\mathcal{N}{)}^{\prime }\cap \mathcal{Q}=\mathbb{C}{e}_{11}\oplus \mathbb{C}{e}_{22}.$$

$${\mathfrak{H}}_{\xi }:=\{V\in \mathcal{M}\mid Vx={\rho }_{\xi }(x)V,\forall x\in \mathcal{N}\}.$$

$$\mathrm{B}(\mathcal{N}\subset \mathcal{M},\phi )=\left\{\sum _{s\in {\Xi }_0}\sum _{k=1}^{n_s}{a}_{k,s}{V}_k^s\mid a_{k,s}\in \mathbb{C}\right\}.$$

$$a_{n}V-{\lambda }^{is}V{a}_n$$

$$a_{n}V-[D\psi :D\phi {]}_{s}V{a}_n=(a_n-[D\psi :D\phi {]}_s{\sigma }_s^{\phi }(a_n))V\to 0.$$

$$d(\rho {)}^{it}[D(\psi \circ {\varphi }_{\rho }):D\chi {]}_t{\sigma }_t^{\chi }(T)=Td(\sigma {)}^{it}[D(\psi \circ {\varphi }_{\sigma }):D\chi {]}_t$$

$${\mathcal{N}}^{\prime }\cap \tilde{\mathcal{M}}=\left\{\sum _{s\in {\Xi }_0}\sum _{k=1}^{n_s}{a}_{k,s}{\lambda }^{\phi }(s{)}^{\ast }{V}_k^s\mid a_{k,s}\in \mathbb{C}\right\}$$

$${\lambda }^{\phi }(s{)}^{\ast }V{\lambda }^{\phi }(t{)}^{\ast }W={\lambda }^{\phi }(s{)}^{\ast }{\lambda }^{\phi }(t{)}^{\ast }{\sigma }_t^{\phi }(V)W={\lambda }^{\phi }(s+t{)}^{\ast }VW$$

$$({\lambda }^{\phi }(s{)}^{\ast }V{)}^{\ast }=V^{\ast }{\lambda }^{\phi }(s)={\lambda }^{\phi }(s){\sigma }_{-s}^{\phi }(V^{\ast })={\lambda }^{\phi }(-s{)}^{\ast }{V}^{\ast }$$

$${\mathcal{N}}^{\prime }\cap \tilde{\mathcal{M}}=\left\{\sum _{s\in {\Xi }_0}{a}_s{\lambda }^{\phi }(s{)}^{\ast }{V}^s\mid a_s\in \mathbb{C}\right\}.$$

$$\mathrm{B}(\mathcal{N}\subset \mathcal{M},\phi )=\left\{\sum _{n\in N}{a}_n{v}_n^{\ast }{V}_n\mid a_n\in \mathbb{C}\right\},$$

$${\mathcal{N}}^{\prime }\cap \mathcal{M}=\left\{\sum _{n\in \mathrm{Ker}(\nu )}{a}_n{v}_n^{\ast }{V}_n\mid a_n\in \mathbb{C}\right\},$$

Full text

1 Introduction

It is well known that the uniqueness of the injective factor of type III1 is solved by [3], [5], that is, A. Connes tried to show the uniqueness of the injective factor of type III1, and found that an injective factor of type III1 with trivial bicentralizer algebra is isomorphic to the Araki-Woods factor of type III1 [2]. Then U. Haagerup [5] solved the bicentralizer problem, and the uniqueness of the injective factor of type III1 is established. It is natural to ask if a bicentralizer algebra of any type III1 factor is trivial. Although this is still an open problem, no counterexample has been known so far. (See the introduction of [1] and reference therein about recent progress on the trivial bicentralizer problem.)

In [1], H. Ando, U. Haagerup, C. Houdayer and A. Marrakchi showed that one can associate a flow on a (relative) bicentralizer algebra canonically, and investigated its properties. On the other hand, there exists a canonical flow, i.e., the trace scaling action on the continuous core. Thus one can define the notion of a relative flow of weights for an inclusion of factors. (This notion was appeared in [16] related to the classification of subfactors.) They ask whether these two flows are conjugate, and present an example with positive answer.

In this paper, we give an affirmative answer when a small algebra of a given inclusion is an injective factor of type III1. Our proof is based on Popa’s solution of relative version of bicentralizer problem [14].

Acknowledgements. The author is grateful to Professor H. Ando, and C. Houdayer for comments on this paper. He also thanks Prof. A. Marrakchi for pointing out Corollary 3.3 to him.

2 Preliminaries and notations

Throughout this paper, we assume that all von Neumann algebras have separable predual. For a von Neumann algebra $\mathcal{M}$, we denote its continuous core by $\tilde{\mathcal{M}}$, the implementing unitary for $\sigma^{\varphi}_{s}$ by $\lambda^{\varphi}(s)\in\tilde{\mathcal{M}}$, and the trace scaling automorphism by $\theta_{t}$.

Let $\operatorname{End}(\mathcal{M})$ be the set of all unital endomorphisms of $\mathcal{M}$, and $\operatorname{End}_{0}(\mathcal{M})=\{\rho\in\operatorname{End}(\mathcal{M})\mid[\mathcal{M}:\rho(\mathcal{M})]<\infty\}$. We denote $[\mathcal{M}:\rho(\mathcal{M})]^{\frac{1}{2}}=d(\rho)$ for $\rho\in\operatorname{End}_{0}(\mathcal{M})$. Define the left inverse of $\rho$ by $\phi_{\rho}:=\rho^{-1}E_{\rho}$, where $E_{\rho}$ is the minimal conditional expectation $\mathcal{M}\rightarrow\rho(\mathcal{M})$. For $\rho\in\operatorname{End}_{0}(\mathcal{M})$, let $\tilde{\rho}\in\operatorname{End}(\tilde{\mathcal{M}})$ be the canonical extension [7]. For $\rho,\sigma\in\operatorname{End}(\mathcal{M})$, the intertwiner space $(\rho,\sigma)$ is defined by $(\rho,\sigma):=\{T\in\mathcal{M}\mid\sigma(x)T=T\rho(x),x\in\mathcal{M}\}$.

Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion of von Neumann algebras with a conditional expectation $E:\mathcal{M}\rightarrow\mathcal{N}$. When $\mathcal{N}\subset\mathcal{M}$ is a subfactor with finite index, we always choose $E$ as a minimal conditional expectation. We recall the notion of a relative bicentralizer [5], [11]. (We use the notations in [1].)

Definition 2.1

*Let $\varphi$ be a faithful normal state on $\mathcal{M}$ with $\varphi\circ E=\varphi$.

$(1)$ The asymptotic centralizer of $\varphi$ is defined by*

[TABLE]

$(2)$* The relative bicentralizer algebra of $\varphi$ is defined by*

[TABLE]

When $\mathcal{N}=\mathcal{M}$, we simply denote it by $\mathrm{B}(\mathcal{M},\varphi)$.

The following theorem due to S. Popa is essential in our argument, whose proof is an appropriate modification of that of Haagerup [5]. (Also see [11, Corollary 4.7].)

Theorem 2.2** ([14, Theorem 4.2])**

Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion of injective factors of type III1 with finite index. Assume that $\mathcal{N}^{\prime}\cap\mathcal{M}=\tilde{\mathcal{N}}^{\prime}\cap\tilde{\mathcal{M}}$. Then $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)=\mathcal{N}^{\prime}\cap\mathcal{M}$.

Let $\varphi,\psi$ be faithful normal states on $\mathcal{M}$ with $\varphi=\varphi\circ E$, $\psi=\psi\circ E$. In [1, Theorem A], the authors showed the existence of a flow $\beta:\mathbb{R}_{+}^{*}\curvearrowright\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$, and the canonical isomorphism $\beta^{\psi,\varphi}:B(\mathcal{N}\subset\mathcal{M},\varphi)\rightarrow B(\mathcal{N}\subset\mathcal{M},\psi)$, which intertwines $\beta^{\varphi}$ and $\beta^{\psi}$. We call $\beta_{\lambda}^{\varphi}$ a relative bicentralizer flow.

The flow $\beta_{\lambda}^{\varphi}$ is characterized as follows; $\lim\limits_{n\rightarrow\infty}a_{n}x-\beta_{\lambda}^{\varphi}(x)a_{n}=0$ in the strong topology for all $(a_{n})\in\ell^{\infty}(\mathbb{N},\mathcal{N})$ with $\lim\limits_{n\rightarrow\infty}\|a_{n}\varphi-\lambda\varphi a_{n}\|=0$.

In a similar way the isomorphism $\beta^{\psi,\varphi}$ is characterized as follows; $\lim\limits_{n\rightarrow\infty}a_{n}x-\beta^{\psi,\varphi}(x)a_{n}=0$ in the strong topology for all $(a_{n})\in\ell^{\infty}(\mathbb{N},\mathcal{N})$ with $\lim\limits_{n\rightarrow\infty}\|a_{n}\varphi-\psi a_{n}\|=0$.

3 Main results

We begin with the following lemma, which is key for our main results.

Lemma 3.1

Let $\mathcal{N}$ be an injective factor of type III1, and fix a faithful normal state $\varphi$ on $\mathcal{N}$. Assume that $\rho\in\operatorname{End}_{0}(\mathcal{N})$ is a non-modular endomorphism in the sense of [7], and $a\in\mathcal{N}$ satisfies $x_{n}a-a\rho(x_{n})\rightarrow 0$ in the strong topology for all $\{x_{n}\}\in\mathrm{AC}(\mathcal{N},\varphi)$. Then $a=0$.

Proof. Set $\mathcal{Q}:=\mathcal{N}\otimes M_{2}(\mathbb{C})$, and define an embedding $\pi:\mathcal{N}\rightarrow\mathcal{Q}$ by $\pi(x)=x\otimes e_{11}+\rho(x)\otimes e_{22}$. Define a faithful normal state $\Psi$ on $\mathcal{Q}$ and a conditional expectation $E_{\Psi}:\mathcal{Q}\rightarrow\pi(\mathcal{N})$ by

[TABLE]

Then $E_{\Psi}$ is the minimal expectation with $\mathrm{Ind}(E_{\Psi})=1+d(\rho)$, and $\Psi=\Psi\circ E_{\Psi}$. It is easy to see $\varphi(x)=\Psi(\pi(x))$ for $x\in\mathcal{N}$. Thus we have $\pi\left(\mathrm{AC}(\mathcal{N},\varphi)\right)=\mathrm{AC}(\pi(\mathcal{N}),\Psi)$.

The core inclusion for $\pi(\mathcal{N})\subset\mathcal{Q}$ is given by $\tilde{\pi}(\tilde{\mathcal{N}})\subset\tilde{\mathcal{Q}}=\tilde{\mathcal{N}}\otimes M_{2}(\mathbb{C})$, where $\tilde{\pi}(x)=x\otimes e_{11}+\tilde{\rho}(x)\otimes e_{22}$. The irreducibility and non-modularity of $\rho$ implies $(\mathrm{id},\tilde{\rho})=\{0\}$ [7, Proposition 3.4(1)], and yields

[TABLE]

By Theorem 2.2,

[TABLE]

Assume $a\in\mathcal{N}$ satisfies the condition in the statement of Lemma. Then $a\otimes e_{12}\in\mathrm{B}(\pi(\mathcal{N})\subset\mathcal{Q},\Psi)$, and hence $a=0$. $\Box$

Remark. For a type III1 factor $\mathcal{N}$, $\rho\in\operatorname{End}_{0}(\mathcal{N})$ is an irreducible modular endomorphism if and only if $\rho=\operatorname{Ad}u\circ\sigma^{\varphi}_{t}$ for some $u\in U(\mathcal{N})$.

Let $\mathcal{N}$ be an injective factor of type III1, and $\mathcal{N}\subset\mathcal{M}$ a (possibly reducible) inclusion of factors of type III1 with finite index, or a discrete irreducible inclusion. Fix a faithful normal state on $\mathcal{M}$ such that $\varphi=\varphi\circ E$. We recall results in [8] on discrete inclusions. Let $\gamma|_{\mathcal{N}}=\bigoplus_{\xi\in\Xi}n_{\xi}[\rho_{\xi}]$ be an irreducible decomposition of the canonical endomorphism $\gamma$ for $\mathcal{N}\subset\mathcal{M}$, and $\Xi_{0}:=\{t\in\mathbb{R}\mid\sigma^{\varphi}_{t}\prec\gamma|_{\mathcal{N}}\}$. We regard $\Xi_{0}$ as a subset of $\Xi$. We fix a representative of $[\rho_{\xi}]$ so that $\rho_{s}=\sigma^{\varphi}_{s}$ for $s\in\Xi_{0}$. Let

[TABLE]

Then $\mathfrak{H}_{\xi}$ is a finite dimensional Hilbert space with $\dim\mathfrak{H}_{\xi}=n_{\xi}$ by the inner product $\langle V,W\rangle=E(W^{*}V)$. Fix a CONS $\{V^{\xi}_{k}\}_{k=1}^{n_{\xi}}\subset\mathfrak{H}_{\xi}$. Then $a\in\mathcal{M}$ can be expand as $a=\sum_{\xi,k}a_{k,\xi}V^{\xi}_{k}$, $a_{k,\xi}\in\mathcal{N}$, uniquely. When $\mathcal{N}\subset\mathcal{M}$ is of infinite index, this expansion does not converge in the usual operator algebra topology. However coefficients $\{a_{k,\xi}\}_{k,\xi}$ determines $a$ uniquely.

Theorem 3.2

*Let $\mathcal{N}\subset\mathcal{M}$ be as above.

$(1)$ The relative bicentralizer algebra $B(\mathcal{N}\subset\mathcal{M},\varphi)$ is given by*

[TABLE]

*The restriction of $E$ on $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$ gives a faithful normal tracial state.

$(2)$ The relative bicentralizer flow $\beta^{\varphi}_{\lambda}$ is given by $\beta^{\varphi}_{\lambda}(V)=\lambda^{is}V$, $V\in\mathfrak{H}_{s}$.

$(3)$ Let $\psi$ be another faithful normal state on $\mathcal{M}$ with $\psi=\psi\circ E$. The isomorphism $\beta^{\psi,\varphi}:B(\mathcal{N}\subset\mathcal{M},\varphi)\rightarrow B(\mathcal{N}\subset\mathcal{M},\psi)$ is given by $\beta_{\psi,\varphi}(V)=[D\psi:D\varphi]_{s}V$ for $V\in\mathfrak{H}_{s}$.*

Proof. (1) Take $a\in\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$. Thus $x_{n}a-ax_{n}\rightarrow 0$ in the strong topology for all $\{x_{n}\}\in\mathrm{AC}(\mathcal{N},\varphi)$. By expanding $a=\sum_{k,\xi}a_{k,\xi}V^{\xi}_{k}$, we get $x_{n}a_{k,\xi}-a_{k,\xi}\rho_{\xi}(x_{n})\rightarrow 0$ in the strong topology. By Lemma 3.1, $a_{k,\xi}=0$ for $\xi\not\in\Xi_{0}$. If $s\in\Xi_{0}$, then we can easily see $a_{k,s}\in\mathbb{C}$ by Haagerup’s theorem $\mathrm{B}(\mathcal{N},\varphi)=\mathbb{C}$ [5] and the fact $\lim\limits_{n\rightarrow\infty}\sigma_{s}^{\varphi}(x_{n})-x_{n}=0$ in the strong topology.

Take $a=\sum\limits_{s\in\Xi_{0}}\sum\limits_{k=1}^{n_{s}}a_{k,s}V^{s}_{k}\in\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$. We choose a CONS $\{V_{k}^{0}\}_{k=1}^{n_{0}}\subset\mathfrak{H}_{0}$ so that $V_{1}^{0}=1$. (If $\mathcal{N}\subset\mathcal{M}$ is irreducible, $\dim\mathfrak{H}_{0}=1$.) Then $E(a)=a_{1,0}$, and we get $E(a^{*}a)=E(aa^{*})=\sum\limits_{s\in\Xi_{0}}\sum\limits_{k=1}^{n_{s}}|a_{k,s}|^{2}$. Hence the restriction of $E$ on $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$ is tracial.

(2) The proof is similar to that of [1, Proposition 6.11]. Take $(a_{n})\in\ell^{\infty}(\mathbb{N},\mathcal{N})$ with $\lim_{n}\|a_{n}\varphi-\lambda\varphi a_{n}\|=0$. Then $\sigma^{\varphi}_{s}(a_{n})-\lambda^{-is}a_{n}\rightarrow 0$, and hence we have

[TABLE]

This shows $\beta^{\varphi}_{\lambda}(V_{k}^{s})=\lambda^{is}V_{k}^{s}$.

(3) Take $(a_{n})\in\ell^{\infty}(\mathbb{N},\mathcal{N})$ with $\|a_{n}\varphi-\psi a_{n}\|\rightarrow 0$. Then $[D\psi:D\varphi]_{s}\sigma^{\varphi}_{s}(a_{n})-a_{n}\rightarrow 0$ in the strong topology. (To see this, consider $\mathcal{N}\otimes M_{2}(\mathbb{C})$ and a state $\Psi((a_{i,j}))=\dfrac{1}{2}(\varphi(a_{11})+\psi(a_{22}))$.) For $V\in\mathfrak{H}_{s}$, we have

[TABLE]

This means $\beta_{\psi,\varphi}(V)=[D\psi:D\varphi]_{s}V$ for $V\in\mathfrak{H}_{s}$. $\Box$

Remark. Even if $\mathcal{N}\subset\mathcal{M}$ is of infinite index and reducible, the same proof works if “the Fourier expansion” holds, e.g., $\mathcal{N}\subset\mathcal{M}$ is a crossed product by a discrete group action.

Combining Theorem 3.2 and [6, Theorem 3.5], we get the following. (This is pointed out by A. Marrakchi to the author.)

Corollary 3.3

Let $\mathcal{N}\subset\mathcal{M}$ be an irreducible discrete inclusion of factors of type III1 such that $\mathcal{N}$ is injective. Then the bicentralizer algebra $\mathrm{B}(\mathcal{M},\varphi)$ of $\mathcal{M}$ is trivial.

Proof. By definition, it is trivial $\mathrm{B}(\mathcal{M},\varphi)\subset\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$. Since $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$ is finite, [6, Theorem 6.5] implies $\mathrm{B}(\mathcal{M},\varphi)=\mathbb{C}$. $\Box$

In [1], the authors ask whether the relative bicentralizer flow is conjugate to the relative flow of weights $\{\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}},\theta_{t}\}$, and they exhibit examples in [1, Proposition 6.11]. Our case also provides a positive answer to their question.

Theorem 3.4

Let $\mathcal{N}\subset\mathcal{M}$ be as in Theorem 3.2. Then two flows $\{\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi),\beta^{\varphi}_{\lambda}\}$ and $\{\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}},\theta_{t}\}$ are conjugate via a map $\alpha$ given by $\alpha(V)=\lambda^{\varphi}(s)^{*}V$, $s\in\Xi_{0}$, $V\in\mathfrak{H}_{s}$.

Proof. At first we treat the case $[\mathcal{M}:\mathcal{N}]<\infty$, and show the following claim.

Claim. When $\mathcal{N}\subset\mathcal{M}$ is of finite index, we have $\sigma^{\varphi}_{t}(V)=d(\rho_{\xi})^{-it}[D\varphi\circ\phi_{\xi}:D\varphi]^{*}_{t}V$ for any $V\in\mathfrak{H}_{\xi}$. where $\phi_{\xi}$ is the left inverse of $\rho_{\xi}$. In particular, $\sigma^{\varphi}_{t}(V)=V$ for $V\in\mathfrak{H}_{s}$, $s\in\Xi_{0}$.

This statement is proved in [8, p.45]. We present a proof for readers convenience.

We recall the following general fact. For injective unital homomorphisms $\rho,\sigma:\mathcal{N}\rightarrow\mathcal{M}$ with finite index, take $T\in\mathcal{M}$ satisfying $\rho(x)T=T\sigma(x)$, $x\in\mathcal{N}$. Then we have

[TABLE]

for faithful normal states $\psi\in\mathcal{N}_{*}$, $\chi\in\mathcal{M}_{*}$, cf. [13, Lemma A.2]. (We can define $d(\rho)$ and $\phi_{\rho}$ for $\rho:\mathcal{N}\rightarrow\mathcal{M}$ similarly.)

Let $\iota:\mathcal{N}\rightarrow\mathcal{M}$ an inclusion map, and apply the above result for $\rho=\iota\rho_{\xi}$, $\sigma=\iota$, $T=V\in\mathfrak{H}_{\xi}$, $\psi=\varphi|_{\mathcal{N}}$, $\chi=\varphi$. Since left inverses are given by $\phi_{\iota\rho_{\xi}}=\phi_{\xi}\circ E$, $\phi_{\iota}=E$, we get the desired result.

Claim implies that $Va=\tilde{\rho_{\xi}}(a)V$ for $a\in\tilde{\mathcal{N}}$, $V\in\mathfrak{H}_{\xi}$. Hence $\{V_{k}^{\xi}\}_{\xi,k}$ is a basis for $\tilde{\mathcal{N}}\subset\tilde{\mathcal{M}}$, and every $a\in\tilde{\mathcal{M}}$ can be expanded $a=\sum_{k,\xi}a_{k,\xi}V_{k}^{\xi}$ uniquely. Then we get

[TABLE]

by Connes-Takesaki relative commutant theorem [4] and [7, Proposition 3.4(1)]. (This also provides a proof of the relative commutant theorem [14, Theorem 4.3] for subfactors with finite index.)

Now we can show $\alpha$ is a $*$-isomorphism. Take $V\in\mathfrak{H}_{s}$ and $W\in\mathfrak{H}_{t}$ for $s,t\in\Xi_{0}$. On one hand, we have $VW\in\mathfrak{H}_{s+t}$ and $V^{*}\in\mathfrak{H}_{-s}$. (Hence $VW=0$ if $s+t\not\in\Xi_{0}$.) On the other hand, we have

[TABLE]

and

[TABLE]

by Claim. This shows $\alpha$ is a $*$-isomorphism.

Since $\theta_{t}(\lambda^{\varphi}(s)^{*}V_{k}^{s})=e^{ist}\lambda^{\varphi}(s)^{*}V_{k}^{s}$, $\alpha\circ\beta_{\lambda}^{\varphi}\circ\alpha^{-1}=\theta_{\log\lambda}$ holds, and hence $\alpha$ intertwines two flows $\beta^{\varphi}$ and $\theta$.

Let us assume that $\mathcal{N}\subset\mathcal{M}$ is an irreducible discrete inclusion with $[\mathcal{M}:\mathcal{N}]=\infty$. One should note that the statement of Claim may fail in this case, i.e., the operator $a_{\xi}$ defined in [8, p.39] is not trivial in general. (See [8, p.45, Appendix] for such an example. Crossed product inclusions by a discrete quantum group of non-Kac type also provide such examples.) However, one has $a_{s}=1$ and the statement of Claim holds for $s\in\Xi_{0}$, since $d(\rho_{s})=d(\sigma^{\varphi}_{s})=1$. Moreover we have $\dim\mathfrak{H}_{s}=1$, $s\in\Xi_{0}$.

In this case, $\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}}$ is described in [1, Theorem 6.9], which can be regard as a generalization of computation in [10], [15]. Namely, fix a unitary $V^{s}\in\mathfrak{H}$. Then we have

[TABLE]

Thus the proof of finite index case also works in this case.

One should note that the restriction of a conditional expectation on $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$ (resp. on $\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}}$) gives a faithful normal tracial state, and $\alpha$ preserves these states. $\Box$

We present two examples of inclusions $\mathcal{N}\subset\mathcal{M}$ such that $\mathcal{N}$ is an injective factor of type III1, but $\mathcal{M}$ is not injective, and $\mathcal{N}^{\prime}\cap\mathcal{M}\neq B(\mathcal{N}\subset\mathcal{M},\varphi)$.

Example 3.5

This example is a modification of [1, Proposition 6.11]. Let $N$ be an abelian discrete group such that there exist an injective homomorphism $\nu:N\rightarrow\mathbf{R}$, and a 2-cocycle $\mu\in Z^{2}(N,\mathbf{T})$ such that $\mu(g,h)\overline{\mu(h,g)}$ is a non-degenerate bicharactor. (For example, $N=\mathbf{Z}\oplus\mathbf{Z}$, $\nu(a,b)=a+\sqrt{2}b$, $\mu((a,b),(c,d))=e^{2\pi i\theta bc}$, $\theta\in\mathbb{R}\backslash\mathbb{Q}$.)

Let $K$ be a non-amenable discrete group, and $\gamma$ a free action of $K$ on an injective factor of type II1 $\mathcal{R}_{0}$. Then $\mu$ can be regard as an element of $Z^{2}(N\times K,\mathbb{T})$ naturally.

Let us define an action of $\alpha$ of $N\times K$ on $\mathcal{N}\otimes\mathcal{R}_{0}$ by $\alpha_{(n,g)}=\sigma^{\varphi}_{\nu(n)}\otimes\gamma_{g}$, and $\mathcal{M}:=(\mathcal{N}\otimes\mathcal{R}_{0})\rtimes_{\alpha,\mu}(N\times K)$. Since $\alpha$ is a free action of a non-amenable discrete group, $\mathcal{M}$ is not injective and $\mathcal{N}^{\prime}\cap\mathcal{M}=\mathbb{C}$. Let $V_{g}\in\mathcal{M}$ be the implementing unitary. By Theorem 3.2, $B(\mathcal{N}\subset\mathcal{M},\varphi)=\{\sum_{n\in N}a_{n}V_{n}\mid a_{n}\in\mathbb{C}\}$, and this algebra is a factor of type II1 by the choice of $\mu$.

Since the canonical extension $\tilde{\alpha}_{g}$ is inner if and only if $g\in N$, we can see that $\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}}=\{\sum_{n\in N}a_{n}\lambda^{\varphi}(\nu(n))^{*}V_{n}\mid n\in N\}$ as in [10], [15], [1, Proposition 6.11].

Example 3.6

Let $K$ be a non-amenable discrete group, and $\Gamma$ a wreath product of $\mathbb{Z}$ and $K$. Namely, let $N=\bigoplus_{g\in K}\mathbb{Z}$, and define an action of $K$ on $N$ by shift. Then $\Gamma$ is defined by $\Gamma=N\rtimes K$.

Let $\nu:N\ni(n(g))_{g\in K}\mapsto\sum_{g\in K}n(g)\in\mathbb{R}$. (Here we write the group operation of $N$ additively, but will write it multiplicatively in what follows.) Then $\nu$ is a homomorphism with $\nu(g^{-1}ng)=\nu(n)$ for $g\in\Gamma$.

We can construct an action $\alpha$ of $\Gamma$ on $\mathcal{N}$ with following properties.

(1) Let $\tilde{\alpha}_{g}$ be the canonical extension of $\alpha_{g}$. Then $\{g\in\Gamma\mid\tilde{\alpha}_{g}\in\operatorname{Int}(\tilde{\mathcal{N}})\}=N$.

(2) There exists a unitary $u_{n}\in\tilde{\mathcal{N}}$, $n\in N$, such that $\tilde{\alpha}_{n}=\operatorname{Ad}u_{n}$, $u_{n}u_{m}=u_{nm}$, and $\tilde{\alpha}_{g}(u_{g^{-1}ng})=u_{n}$, $g\in\Gamma$.

(3) $\theta_{t}(u_{n})=e^{-it\nu(n)}u_{n}$.

(See [9, Proposition 22], [12, Section 6.1] for existence of such actions. Note that what we actually need is the existence of a free action of $\Gamma$ on the injective factor of type II1 and the amenability of $N$ in the construction, and $\Gamma$ does not need to be amenable.)

Let $\mathcal{M}=\mathcal{N}\rtimes_{\alpha}\Gamma$, and $V_{g}\in\mathcal{M}$ be the implementing unitary. Property (3) means that $v_{n}:=u_{n}\lambda^{\varphi}(\nu(n))^{*}\in\tilde{\mathcal{N}}^{\theta}=\mathcal{N}$, and $\alpha_{n}=\operatorname{Ad}v_{n}\circ\sigma^{\varphi}_{\nu(n)}$. By Theorem 3.2, we have

[TABLE]

which is isomorphic to the group von Neumann algebra of $N$. By [10], [15], we can see that the center of $\tilde{\mathcal{M}}$ is trivial, i.e., $\mathcal{M}$ is of type III1. Indeed, we can see that $\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}}=\{\sum_{n\in N}a_{n}u_{n}^{*}V_{n}\mid a_{n}\in\mathbb{C}\}$ by property (1). Then it is easy to see $\mathcal{M}^{\prime}\cap\tilde{\mathcal{M}}=\mathbb{C}$ by using property (2). Since $\Gamma$ is non amenable, $\mathcal{M}$ is not injective.

The inner part of $\alpha$ is $\mathrm{Ker}(\nu)$, and it is not a trivial subgroup. Thus we have

[TABLE]

and hence $\mathcal{N}\subset\mathcal{M}$ is not irreducible. But we can see that Theorem 3.4 holds in this case by the description of $\mathrm{B}(\mathcal{N}\subset\mathcal{M},\varphi)$ and $\mathcal{N}^{\prime}\cap\tilde{\mathcal{M}}$.