Tensor product decompositions and rigidity of full factors

TL;DR

This paper establishes new rigidity results for tensor product decompositions of full factors, demonstrating limitations on their decompositions, stability of certain classes under tensor products, and providing prime factorization insights and counterexamples.

Contribution

It introduces novel rigidity theorems for full factors, including countability of tensor decompositions, stability of classes under tensor products, and prime factorization results for crossed products.

Findings

Full factors have at most countably many tensor product decompositions.

Separable full factors with countable fundamental group are stable under tensor products.

New primeness and unique prime factorization results for crossed products from higher rank lattices and Bernoulli shifts.

Abstract

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

Equations

$$\Omega \times {\Omega }_{\mathrm{full}}\to \Omega$$

$$([P],[Q])\mapsto [P\overline{\otimes }Q]$$

$$M=(R_1⋊{\Gamma }_1)\overline{\otimes }\cdots \overline{\otimes }(R_n⋊{\Gamma }_n)$$

$${\ell }^{\infty }(I,M)$$

$${\mathcal{I}}_{\omega }$$

$${\mathcal{M}}^{\omega }$$

$$\mathrm{Ad}:\mathcal{U}(M)\ni u\mapsto \mathrm{Ad}(u)\in \mathrm{Aut}(M)$$

$$\mathrm{Aut}(M)\times \mathrm{Aut}(N)\ni (\alpha ,\beta )\mapsto \alpha \otimes \beta \in \mathrm{Aut}(M\overline{\otimes }N)$$

$$\mathrm{Out}(M)\times \mathrm{Out}(N)\to \mathrm{Out}(M\overline{\otimes }N).$$

$$⟨x\xi y,\xi ⟩=⟨E(x){\phi }^{\frac{1}{2}}y,{\phi }^{\frac{1}{2}}⟩,\text{for all }x,y\in pqMpq.$$

$$⟨E(x){\phi }^{\frac{1}{2}}y,{\phi }^{\frac{1}{2}}⟩=\underset{i}{\lim }⟨E_i(x){\phi }_i^{\frac{1}{2}}y,{\phi }_i^{\frac{1}{2}}⟩\text{for all }x,y\in pqMpq.$$

$$pqr{\text{L}}^2⟨M,P⟩pqr\prec \underset{i\in I}{⨁}pq{\text{L}}^2⟨M,P_i⟩pq$$

$$\mathrm{TF}(M)\ni P\mapsto \phi {|}_P\otimes \psi {|}_{P^c}\in M_{\ast }$$

$$\underset{i}{\lim }\Vert {\sigma }_{P_i}(\phi \otimes \psi )-(\psi {|}_{P_i}\otimes \phi {|}_{P_i^c})\otimes (\phi {|}_{P_i}\otimes \psi {|}_{P_i^c})\Vert =0.$$

$$\underset{i}{\lim }(\psi {|}_{P_i}\otimes \phi {|}_{P_i^c})\otimes (\phi {|}_{P_i}\otimes \psi {|}_{P_i^c})=(\psi {|}_P\otimes \phi {|}_{P^c})\otimes (\phi {|}_P\otimes \psi {|}_{P^c})={\sigma }_P(\phi \otimes \psi ).$$

$$\psi \circ E_{D\otimes \mathbf{C}}=\underset{i}{\lim }\psi \circ {\sigma }_{P_i}\circ (E_{Q_i}\otimes E_{P_i})\circ {\sigma }_{P_i}$$

$$U_{F,\epsilon }=\{u\in \mathcal{U}(M)\mid \sum _{a\in F}\Vert ua-au{\Vert }_{\phi }<\epsilon \text{ and }\Vert u\phi -\phi u\Vert <\epsilon \}$$

$$C_{F,\epsilon }=\overline{\text{conv}}\{u\underline{x}{u}^{\ast }\mid u\in U_{F,\epsilon }\},C=\bigcap _{F,\epsilon }{C}_{F,\epsilon }.$$

$${\text{L}}^2⟨M,P⟩={\text{L}}^2(M){\otimes }_P{\text{L}}^2(M)\prec {\text{L}}^2(M)$$

$${}_{M\overline{\otimes }N}{\text{L}}^2⟨M\overline{\otimes }N,P\overline{\otimes }N{⟩}_{M\overline{\otimes }N}={}_M{\text{L}}^2⟨M,P{⟩}_M\otimes {}_N{\text{L}}^2(N{)}_N.$$

$$B^{\prime }\cap (B^{\omega }{)}_{{\phi }^{\omega }}\subset B^{\prime }\cap (B^{\omega }⋊\Gamma {)}_{{\phi }^{\omega }}\subset B^{\prime }\cap (M^{\omega }{)}_{{\phi }^{\omega }}$$

$$[B^{\prime }\cap (M^{\omega }{)}_{{\phi }^{\omega }}{]}^{\prime }\cap M\subset [B^{\prime }\cap (B^{\omega }⋊\Gamma {)}_{{\phi }^{\omega }}{]}^{\prime }\cap M\subset [B^{\prime }\cap (B^{\omega }{)}_{{\phi }^{\omega }}{]}^{\prime }\cap M$$

$$\sum _{h\in \Gamma }{x}_h{\lambda }_h=x=u_g^{\ast }{\lambda }_{g}x(u_g^{\ast }{\lambda }_g{)}^{\ast }=\sum _{h\in \Gamma }{u}_g^{\ast }{\alpha }_g(x_h){\alpha }_{gh{g}^{-1}}(u_g){\lambda }_{gh{g}^{-1}},$$

$${\text{L}}^2(P)\prec \underset{Q\in X}{⨁}{\text{L}}^2⟨M,Q⟩$$

$$\{Q\in \mathrm{TF}(M)\mid Q{\sim }_{M}P\}=\{Q\in \mathrm{TF}(M)\mid P{\prec }_{M}Q\text{ and }{P}^c{\prec }_M{Q}^c\}$$

$$\iota :\mathcal{U}(M\overline{\otimes }N)\times \mathrm{Aut}(M)\times \mathrm{Aut}(N)\to \mathrm{Aut}(M\overline{\otimes }N)$$

$$(u,\alpha ,\beta )\mapsto \mathrm{Ad}(u)\circ (\alpha \otimes \beta ).$$

$$\forall s,t\in \mathbf{R},u_{s+t}=u_s{\theta }_s(u_t),$$

$$\forall t\in \mathbf{R},{\alpha }_t\otimes {\beta }_t=\mathrm{Ad}(u_t)\circ {\theta }_t.$$

$$\iota :$$

$$(u,\alpha ,\beta )\mapsto \mathrm{Ad}(u)\circ (\alpha \otimes \beta )$$

$$\rho :$$

$$t\mapsto (u_{-t},{\alpha }_t,{\beta }_t)$$

Full text

YI is supported by JSPS KAKENHI Grant Number JP17K14201.

AM is a JSPS International Research Fellow (PE18760)

1. Introduction

A central theme in the theory of von Neumann algebras is to determine all possible tensor product decompositions of a given factor $M$. More precisely, we will say that a subfactor $P\subset M$ is a tensor factor of $M$ if $M=P\mathbin{\overline{\otimes}}P^{c}$ where $P^{c}=P^{\prime}\cap M$. We will denote by $\mathrm{TF}(M)$ the set of all tensor factors of $M$. The set $\mathrm{TF}(M)$ contains all type $\mathrm{I}$ subfactors of $M$. Moreover, if $P\in\mathrm{TF}(M)$, then $uPu^{*}\in\mathrm{TF}(M)$ for every unitary $u\in\mathcal{U}(M)$. In order to eliminate both of these trivialities, one introduces the following equivalence relation: two tensor factors $P,Q\in\mathrm{TF}(M)$ are called stably unitarily conjugate, written $P\sim Q$, if there exists type $\mathrm{I}_{\infty}$ factors $F_{1},F_{2}$ and a unitary $u\in\mathcal{U}(M\mathbin{\overline{\otimes}}F_{1}\mathbin{\overline{\otimes}}F_{2})$ such that $u(P\mathbin{\overline{\otimes}}F_{1})u^{*}=Q\mathbin{\overline{\otimes}}F_{1}$. One then wants to study the quotient space $\mathrm{TF}(M)/{\sim}$.

In many cases, one can give a complete description of $\mathrm{TF}(M)/{\sim}$. Indeed, a celebrated result of Ozawa [Oz03] says that for every ICC hyperbolic group $\Gamma$, the ${\rm II}_{1}$ factor $M=L(\Gamma)$ is prime. This means that for every tensor factor $P\in\mathrm{TF}(M)$, we have that either $P$ or $P^{c}$ is of type $\mathrm{I}$, or equivalently that $\mathrm{TF}(M)/{\sim}=\{[\mathbf{C}],[M]\}$. More generally, we say that a factor $M$ satisfies the Unique Prime Factorization (UPF) property if there exists prime factors $P_{1},\dots,P_{n}\in\mathrm{TF}(M)$ with $M=P_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}P_{n}$ such that for every $Q\in\mathrm{TF}(M)$, there exists a subset $\{i_{1},\dots,i_{m}\}\subset\{1,\dots,n\}$ such that $Q\sim P_{i_{1}}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}P_{i_{m}}$. In [OP03], Ozawa and Popa showed that if $\Gamma_{1},\dots,\Gamma_{n}$ are ICC hyperbolic groups, then the factor $M=L(\Gamma_{1}\times\dots\times\Gamma_{n})$ has the UPF property. These seminal results were later generalized to larger and larger classes of factors by using Popa’s deformation/rigidity theory and Ozawa’s $C^{*}$-algebraic techniques [Po06b, Pe06, Sa09, CSU11, SW11, Is14, CKP14, HI15, Ho15, Is16, DHI17].

The main goal of this paper is to provide new rigidity and classification results for tensor product decompositions by combining the following two approaches:

Rigidity of full factors.

A factor is called full when it has no nontrivial central sequences [Co74]. Fullness is a very weak rigidity property when compared to Kazhdan’s property (T) for example. In this paper, we use the following key bimodule characterization of fullness due to Ozawa [BMO19] (and based on [Co75, Ma18]): a factor $M$ is full if and only if for every $M$-$M$-bimodule $\mathcal{H}$ such that $\mathord{\text{\rm L}}^{2}(M)\prec\mathcal{H}$ and $\mathcal{H}\prec\mathord{\text{\rm L}}^{2}(M)$, we have $\mathord{\text{\rm L}}^{2}(M)\subset\mathcal{H}$. Note that if we remove the condition $\mathcal{H}\prec\mathord{\text{\rm L}}^{2}(M)$, this becomes precisely the definition of property (T). Therefore, in some specific situations, in particular for tensor product decompositions, full factors can behave in a very rigid way, as if they had property (T). See for instance Lemma 5.2 which shows that “relative amenability” can be automatically improved to “corner embedability” for full tensor factors. This can be seen as an instance of the spectral gap rigidity phenomenon discovered in [Po06b].

Flip automorphisms.

Let $M$ be a factor. To every $P\in\mathrm{TF}(M)$ one can associate an automorphism $\sigma_{P}\in\operatorname{Aut}(M\mathbin{\overline{\otimes}}M)$ which flips the two copies of $P$ in $M\mathbin{\overline{\otimes}}M=P\mathbin{\overline{\otimes}}P^{c}\mathbin{\overline{\otimes}}P\mathbin{\overline{\otimes}}P^{c}$ and fixes the two copies of $P^{c}$. The key point is that it is in general much easier to study the flip automorphism $\sigma_{P}$ then to study directly the mysterious tensor factor $P$. Any information obtained on $\sigma_{P}$ can then be used to locate $P$ inside $M$ (observe in particular that $P\sim Q$ if and only if $\sigma_{P}\circ\sigma_{Q}$ is an inner automorphism). As we will see, this trick combines very well with $\mathrm{W}^{*}$-rigidity results, since they generally give a good understanding of the automorphism group $\operatorname{Aut}(M\mathbin{\overline{\otimes}}M)$ in terms of the building data of $M$. This approach can be used to obtain new primeness or UPF results which do not rely on any kind of negative curvature or rank 1 assumption, but, on the other hand, cannot be used to obtain solidity or relative solidity results.

Let us now state our main theorems. We start with a very general rigidity result based on a separability argument (see [Po06c, Section 4] for a survey). Unlike the separability arguments used in [Co80] [Po86], [Oz02], [Hj02], [GP03], the rigidity in our case comes from fullness instead of property (T). Here $\mathrm{TF}_{\mathrm{full}}(M)\subset\mathrm{TF}(M)$ denotes the set of all full tensor factors of $M$. Note that $\mathrm{TF}_{\mathrm{full}}(M)=\mathrm{TF}(M)$ when $M$ itself is full.

Theorem A**.**

Let $M$ be a factor with separable predual. Then $\mathrm{TF}_{\mathrm{full}}(M)/{\sim}$ is countable. Consequently, if $\Omega$ (resp. $\Omega_{\mathrm{full}}$) denotes the set of all stable isomorphism classes of (resp. full) factors with separable predual, then the natural map

[TABLE]

is countable-to-one.

The fullness assumption in Theorem A is essential since $\mathrm{TF}(M)/{\sim}$ is uncountable whenever $M$ is an infinite tensor product of ${\rm II}_{1}$ factors (i.e. a McDuff factor). Note that if a factor $M$ satisfies the UPF property, then $\mathrm{TF}(M)/{\sim}$ is actually finite. In view of Theorem A and of all the known UPF results in the litterature, one might wonder if there exists any full factor $M$ which does not satisfy the UPF property. We answer this question affirmatively in the last section of this paper by providing the first examples of full factors which do not admit any prime factorization. For these examples, $\mathrm{TF}(M)/{\sim}$ is infinite but can still be completely described.

In our next main result, we give an application of this rigidity phenomenon to fundamental groups. Let $M$ be a ${\rm II}_{\infty}$ factor. Then every $\theta\in\operatorname{Aut}(M)$ scales the trace of $M$ by some scalar $\mathord{\text{\rm Mod}}(\theta)\in\mathbf{R}^{*}_{+}$ and the map $\mathord{\text{\rm Mod}}:\operatorname{Aut}(M)\rightarrow\mathbf{R}^{*}_{+}$ is a continuous group homomorphism. Its image is called the fundamental group of $M$ and denoted by $\mathcal{F}(M)$. The fundamental group $\mathcal{F}(M)$ is also defined when $M$ is a ${\rm II}_{1}$ factor by $\mathcal{F}(M)=\mathcal{F}(M^{\infty})$ where $M^{\infty}=M\mathbin{\overline{\otimes}}\mathbf{B}(\ell^{2})$. The invariant $\mathcal{F}(M)$ is very hard to compute in general. In fact, for a long time, the only known computation, due to Murray and von Neumann, was $\mathcal{F}(M)=\mathbf{R}^{*}_{+}$ where $M$ is the hyperfinite ${\rm II}_{1}$ factor (or more generally a McDuff factor). The first breakthrough is the rigidity result of Connes [Co80] which shows that $\mathcal{F}(M)$ is countable for $M=L(\Gamma)$ where $\Gamma$ is a countable ICC group with Kazhdan’s property (T). Voiculescu and R$\rm\breve{a}$dulescu then proved $\mathcal{F}(L\mathbf{F}_{\infty})=\mathbf{R}^{*}_{+}$ [Vo89, Ra91] by using the free entropy machinery. Since $L\mathbf{F}_{\infty}$ is full, this example shows in particular that fullness does not imply countability of the fundamental group. Later on, spectacular progress in the study of fundamental groups has been accomplished thanks to Popa’s deformation/rigidity theory [Po01], [Po03], [PV10], [PV11]. In particular, in [PV11], Popa and Vaes settled a longstanding question by giving the first example of a ${\rm II}_{\infty}$ factor $M$ with $\mathcal{F}(M)=\mathbf{R}^{*}_{+}$ but such that $M$ does not admit a trace scaling action, i.e. a continuous action $\theta:\mathbf{R}^{*}_{+}\curvearrowright M$ such that $\mathord{\text{\rm Mod}}(\theta_{\lambda})=\lambda$ for all $\lambda\in\mathbf{R}^{*}_{+}$. Moreover, they gave an example of two factors $M$ and $N$ such that $\mathcal{F}(M\mathbin{\overline{\otimes}}N)=\mathbf{R}^{*}_{+}$ but $\mathcal{F}(M)\neq\mathbf{R}^{*}_{+}$ and $\mathcal{F}(N)\neq\mathbf{R}^{*}_{+}$. This should be compared with item $(\rm ii)$ below.

Theorem B**.**

Let $M$ and $N$ be two ${\rm II}_{\infty}$ factors with separable predual and suppose that one of them is full. Then the following holds:

• $(\rm i)$

The quotient group $\mathcal{F}(M\mathbin{\overline{\otimes}}N)/\mathcal{F}(M)\mathcal{F}(N)$ is countable. In particular, if $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are countable, then $\mathcal{F}(M\mathbin{\overline{\otimes}}N)$ is also countable.

• $(\rm ii)$

$M\mathbin{\overline{\otimes}}N$* admits a trace scaling action if and only if $M$ or $N$ admits a trace scaling action.*

We point out that in many concrete cases, one can actually show that $\mathcal{F}(M\mathbin{\overline{\otimes}}N)=\mathcal{F}(M)\mathcal{F}(N)$. See [Is16] for recent results regarding this question. Nevertheless, we believe that Theorem B is optimal and that the equality $\mathcal{F}(M\mathbin{\overline{\otimes}}N)=\mathcal{F}(M)\mathcal{F}(N)$ does not hold in general, even though we do not know any counter-example.

We now move to more concrete applications. The first one is a UPF result for crossed products comming from noncommutative Bernoulli shifts. Here, by a noncommutative Bernoulli shift we always mean an action of the form $\Gamma\curvearrowright(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}$ where $B_{0}$ is a non-trivial von Neumann algebra with separable predual, $\varphi_{0}$ is a faithful normal state on $B_{0}$ and $\Gamma$ is a countable group acting by shifting the tensor components. It is known that if $\Gamma$ is non-amenable and $B_{0}$ is amenable, then the crossed product $(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}$ is prime [Po06b, SW11, Ma16]. By exploiting the fullness of $(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}\rtimes\Gamma$ (see [VV14]), we are able to remove the amenability assumption on $B_{0}$.

Theorem C**.**

Let $\Gamma\curvearrowright(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}$ be any noncommutative Bernoulli shift. Assume that $\Gamma$ is non-amenable. Then $M=(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}\rtimes\Gamma$ is prime.

Moreover, for any full factor $N$ with separable predual and any tensor product decomposition $M\mathbin{\overline{\otimes}}N=P\mathbin{\overline{\otimes}}Q$, we have $M\prec_{M\mathbin{\overline{\otimes}}N}P$ or $M\prec_{M\mathbin{\overline{\otimes}}N}Q$.

The second part of the theorem shows in particular that if $N$ is a full factor which has the UPF property, then so does $M\mathbin{\overline{\otimes}}N$. The technique used in the proof of Theorem C also allows us to prove the following rigidity result which generalizes [Io06, Corollary 0.2] where the base algebras $A_{0}$ and $B_{0}$ are assumed to be weakly rigid ${\rm II}_{1}$ factors (e.g. ${\rm II}_{1}$ factors with property (T)).

Theorem D**.**

Let $G\curvearrowright(A_{0},\psi_{0})^{\mathbin{\overline{\otimes}}G}$ and $H\curvearrowright(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}H}$ be two noncommutative Bernoulli shifts. Assume that $A_{0}$ and $B_{0}$ are diffuse full factors. If $(A_{0},\psi_{0})^{\mathbin{\overline{\otimes}}G}\rtimes G\cong(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}H}\rtimes H$, then there exists two finite normal subgroups $G_{0}\lhd G$ and $H_{0}\lhd H$ such that $G/G_{0}\cong H/H_{0}$.

In our next theorem, we present a new UPF result which shows how the flip automorphism approach can be used to study tensor factors. This result should really be considered as an application of a recent $\mathrm{W}^{*}$-rigidity result of Boutonnet-Ioana-Peterson [BIP18] for compact actions of higher rank lattices. Here, by higher rank lattice we mean a lattice $\Gamma$ in a connected semi-simple Lie group $G$ with finite center such that every simple quotient of $G$ has real rank $\geq 2$. A basic example is given by $\Gamma=\mathrm{SL}(n,\mathbf{Z})$ for $n\geq 3$. We also recall that an ergodic pmp action $\Gamma\curvearrowright(X,\mu)$ is compact if the closure of the image of $\Gamma$ in $\operatorname{Aut}(X,\mu)$ is compact. These are precisely the actions of the form $\Gamma\curvearrowright K/L$ where $K$ is a compact group, $L$ is a closed subgroup of $K$ and $\Gamma<K$ is a dense subgroup which acts by left translations.

Theorem E**.**

Let $\Gamma$ be an irreducible higher rank lattice. Let $\Gamma\curvearrowright(X,\mu)$ be a compact free ergodic pmp action. Then the crossed product $M=\mathord{\text{\rm L}}^{\infty}(X)\rtimes\Gamma$ is prime. Moreover, for any finite family of factors $M_{1},\dots,M_{n}$ of that form, the tensor product $M_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}M_{n}$ has the Unique Prime Factorization property.

We point out that the question of whether the group von Neumann algebras of irreducible higher rank lattices are prime is a well-known and notoriously difficult open problem. We mention, however, the remarkable result of [DHI17] which shows that $L(\Gamma)$ is prime whenever $\Gamma$ is an irreducible lattice in a direct product of rank one simple Lie groups.

For our last application, we consider factors of the form $M=R\rtimes\Gamma$ where $\Gamma\curvearrowright R$ is a compact minimal action of an ICC higher rank lattice $\Gamma$. Recall that an action $\Gamma\curvearrowright R$ is minimal if it is faithful and $(R^{\Gamma})^{\prime}\cap R=\mathbf{C}$. Since every compact group admits one and only one minimal action on the hyperfinite ${\rm II}_{1}$ factor [MT06], a compact minimal action $\Gamma\curvearrowright R$ is uniquely determined, up to conjugacy, by the pair $\Gamma<K$ where the compact group $K$ is the closure of $\Gamma$ in $\operatorname{Aut}(R)$. In this context, we show that all tensor factors of $M$ are McDuff so that one cannot classify them up to stable unitary conjugacy. However, we prove a unique semi-prime factorization result up to conjugacy by an automorphism. Recall that a factor $M$ is semi-prime if it is nonamenable and for every tensor product decomposition $M=P\mathbin{\overline{\otimes}}Q$, either $P$ or $Q$ is amenable.

Theorem F**.**

Let $\Gamma$ be an ICC higher rank lattice. Let $\Gamma\curvearrowright R$ be a compact minimal action on the hyperfinite ${\rm II}_{1}$ factor and put $M=R\rtimes\Gamma$. Then the following holds:

• $(\rm i)$

Every tensor factor of $M$ is either of type $\mathrm{I}$ or McDuff.

• $(\rm ii)$

$M$* admits a tensor product decomposition*

[TABLE]

where $\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$ and $R=R_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}R_{n}$ such that each $M_{i}=R_{i}\rtimes\Gamma_{i}$ is semi-prime.

• $(\rm iii)$

For every tensor product decomposition $M=P\mathbin{\overline{\otimes}}Q$ with $P$ and $Q$ nonamenable, there exists a partition $I\sqcup J=\{1,\dots,n\}$ and an automorphism $\theta$ of $M$ such that $\theta(P)=\mathbin{\overline{\otimes}}_{i\in I}M_{i}$ and $\theta(Q)=\mathbin{\overline{\otimes}}_{j\in J}M_{j}$, up to equivalence in $\mathrm{TF}(M)$.

In particular, $M$ admits a unique semi-prime factorization up to conjugacy by an automorphism.

Acknowledgments

The authors are very grateful to Adrian Ioana for allowing us to include the proof of Theorem 9.2 in our paper and for his valuable comments on an earlier draft of this paper. We also thank Narutaka Ozawa for a helpful discussion regarding Proposition 9.8.

Contents

1. 1 Introduction
2. 2 Preliminaries
3. 3 Tensor factors and flip maps
4. 4 Weakly bicentralized subalgebras
5. 5 Rigidity for full tensor factors
6. 6 Application to fundamental groups
7. 7 Noncommutative Bernoulli shifts
8. 8 Compact actions of higher rank lattices
9. 9 Full factors without Unique Prime Factorization

2. Preliminaries

Ultraproduct von Neumann algebras

Let $M$ be any von Neumann algebra. Let $I$ be any nonempty directed set and $\omega$ any cofinal ultrafilter on $I$. Define

[TABLE]

The quotient C∗-algebra $M^{\omega}:=\mathcal{M}^{\omega}/\mathcal{I}_{\omega}$ is in fact a von Neumann algebra, and we call it the ultraproduct von Neumann algebra [Oc85]. For more on ultraproduct von Neumann algebras, we refer the reader to [Oc85, AH12].

Topological groups associated to a von Neumann algebra

Let $M$ be any von Neumann algebra and let $\mathcal{U}(M)$ be its unitary group. The restrictions of the weak topology, the strong topology and the $*$-strong topology all coincide on $\mathcal{U}(M)$. Equipped with this topology, $\mathcal{U}(M)$ is a complete topological group (which is Polish if $M_{*}$ is separable). Let $\operatorname{Aut}(M)$ be the group of all automorphisms of $M$. We equip it with the topology of pointwise norm convergence on $M_{*}$, which means that a net $(\alpha_{i})_{i}$ in $\operatorname{Aut}(M)$ converges to $\alpha\in\operatorname{Aut}(M)$ if and only if $\|\phi\circ\alpha_{i}-\phi\circ\alpha\|\to 0$ for any $\phi\in M_{*}$. With this topology, $\operatorname{Aut}(M)$ is a complete topological group and it is Polish when $M_{*}$ is separable. There is continuous homomorphism

[TABLE]

where $\operatorname{Ad}(u)(x)=uxu^{*}$ for all $x\in M$. We denote by $\operatorname{Inn}(M)\subset\operatorname{Aut}(M)$ the image of $\operatorname{Ad}$, i.e. the set of all inner automorphisms. Since $\operatorname{Inn}(M)$ is a normal subgroup in $\operatorname{Aut}(M)$, we can form the quotient group $\operatorname{Out}(M):=\operatorname{Aut}(M)/\operatorname{Inn}(M)$ which we call the outer automorphism group of $M$ and we equip it with the quotient topology (which is not necessarily Hausdorf).

For any von Neumann algebras $M$ and $N$, we have a natural continuous homomorphism

[TABLE]

which also induces a continuous injective homomorphism

[TABLE]

Full factors

Following [Co74], we say that a factor $M$ is full if the map $\operatorname{Ad}\colon\mathcal{U}(M)\to\operatorname{Aut}(M)$ is open on its range. Equivalently, $M$ is full if and only if the quotient topology on $\operatorname{Inn}(M)$ comming from the surjection $\mathcal{U}(M)\rightarrow\operatorname{Inn}(M)$ coincides with the induced topology comming from the inclusion $\operatorname{Inn}(M)\subset\operatorname{Aut}(M)$. In that case $\operatorname{Inn}(M)$ is a complete topological group hence it must be closed in $\operatorname{Aut}(M)$ and the quotient group $\operatorname{Out}(M)$ is also a Hausdorf complete topological group (Polish if $M_{*}$ is separable).

We also recall [Co74] that a factor $M$ is full if and only if it satisfies the following property: every uniformly bounded net $(x_{i})_{i\in I}$ in $\ell^{\infty}(I,M)$ that is centralizing, meaning that $\lim_{i}\|x_{i}\varphi-\varphi x_{i}\|=0$ for all $\varphi\in M_{\ast}$, must be trivial, meaning that there exists a bounded net $(\lambda_{i})_{i\in I}$ in $\mathbf{C}$ such that $x_{i}-\lambda_{i}1\to 0$ strongly as $i\to\infty$. See also [Ma18] for another characterization of fullness.

Bimodules and Popa’s intertwining theory

Let $M$ and $N$ be two von Neumann algebras. An $M$-$N$-bimodule is a $*$-representation $\pi_{\mathcal{H}}\colon M\odot N^{\mathrm{op}}\rightarrow\mathbf{B}(\mathcal{H})$ that is normal on each tensor component, where $\odot$ is the algebraic tensor product and $N^{\mathrm{op}}=\{n^{\mathrm{op}}:n\in N\}$ is the opposite von Neumann algebra of $N$. When the underlying representation $\pi_{\mathcal{H}}$ is obvious, we will often use the notation ${{}_{M}}\mathcal{H}_{N}$ to specify the $M$-$N$-bimodule structure of $\mathcal{H}$. We refer the reader to the preliminary section of [AD95] for the general theory of bimodules and for the definition of the Connes’ fusion tensor product. We will simply fix some notations and recall the needed facts.

We denote by $\mathcal{L}_{N^{{}^{\mathrm{op}}}}(\mathcal{H})$ the commutant of the right $N$-action on $\mathcal{H}$. Then $\mathord{\text{\rm L}}^{2}(\mathcal{L}_{N^{{}^{\mathrm{op}}}}(\mathcal{H}))$ identifies canonically with $\mathcal{H}\otimes_{B}\overline{\mathcal{H}}$ where $\overline{\mathcal{H}}$ is the opposite $B$-$A$-bimodule of $\mathcal{H}$. Suppose that $N\subset M$ is a subalgebra of $M$. We denote by $\langle M,N\rangle$ the commutant of the right $N$-action on $\mathord{\text{\rm L}}^{2}(M)$ (namely, the restriction of the canonical right $M$-action). Then we have $\mathord{\text{\rm L}}^{2}(M)\otimes_{N}\mathord{\text{\rm L}}^{2}(M)=\mathord{\text{\rm L}}^{2}\langle M,N\rangle$ as $M$-$M$-bimodules. We view $M$ as a subalgebra of $\langle M,N\rangle$.

We will say that an $M$-$N$-bimodule $\mathcal{H}$ is contained in another $M$-$N$-bimodule $\mathcal{K}$, written abusively as $\mathcal{H}\subset\mathcal{K}$, if there exists an $M$-$N$-bimodular isometry $V\colon\mathcal{H}\rightarrow\mathcal{K}$. We will say that $\mathcal{H}$ is weakly contained in $\mathcal{K}$, written as $\mathcal{H}\prec\mathcal{K}$, if we have $\|\pi_{\mathcal{H}}(T)\|\leq\|\pi_{\mathcal{K}}(T)\|$ for all $T\in M\odot N^{\mathrm{op}}$.

We have the following very important characterizations (see [BMO19, Appendix] for item (ii)):

Theorem 2.1**.**

Let $M\subset N$ be an inclusion of von Neumann algebras. Then the following holds:

• •

${{}_{M}}\mathord{\text{\rm L}}^{2}(M)_{M}\subset{{}_{M}}\mathord{\text{\rm L}}^{2}(N)_{M}$* if and only if there exists a normal conditional expectation from $N$ to $M$.*

• •

${{}_{M}}\mathord{\text{\rm L}}^{2}(M)_{M}\prec{{}_{M}}\mathord{\text{\rm L}}^{2}(N)_{M}$* if and only if there exists a conditional expectation from $N$ onto $M$.*

We now introduce the notion of left weakly mixing bimodules and left amenable bimodules via the following propositions which are consequences of Theorem 2.1.

Proposition 2.2** (Left weakly mixing bimodules).**

Let $A$ and $B$ be two von Neumann algebras and let $\mathcal{H}$ be an $A$-$B$-bimodule. The following properties are equivalent:

• •

The $A$-$A$-bimodule $\mathcal{H}\otimes_{B}\overline{\mathcal{H}}$ is disjoint from $\mathord{\text{\rm L}}^{2}(A)$, i.e. does not contain $z\mathord{\text{\rm L}}^{2}(A)$ for any non-zero projection $z\in\mathcal{Z}(A)$.

• •

The $A$-$A$-bimodule $\mathcal{H}\otimes_{B}\mathcal{K}$ is disjoint from $\mathord{\text{\rm L}}^{2}(A)$ for every $B$-$A$-bimodule $\mathcal{K}$.

• •

There is no normal conditional expectation $E:z\mathcal{L}_{B^{{}^{\mathrm{op}}}}(\mathcal{H})z\rightarrow zA$ for any non-zero projection $z\in\mathcal{Z}(A)$.

When these properties are satisfied, we say that $\mathcal{H}$ is left weakly mixing.

Proposition 2.3** (Left amenable bimodules).**

Let $A$ and $B$ be two von Neumann algebras and let $\mathcal{H}$ be an $A$-$B$-bimodule. The following properties are equivalent:

• •

The $A$-$A$-bimodule $\mathcal{H}\otimes_{B}\overline{\mathcal{H}}$ weakly contains $\mathord{\text{\rm L}}^{2}(A)$.

• •

The $A$-$A$-bimodule $\mathcal{H}\otimes_{B}\mathcal{K}$ weakly contains $\mathord{\text{\rm L}}^{2}(A)$ for some $B$-$A$-bimodule $\mathcal{K}$.

• •

There exists a conditional expectation $E:\mathcal{L}_{B^{{}^{\mathrm{op}}}}(\mathcal{H})\rightarrow A$.

When these properties are satisfied, we say that $\mathcal{H}$ is left amenable.

We recall the following Popa’s intertwining-by-bimodule technique [Po01, Po03]. For the proof, we refer the reader to [HI15, Theorem 4.3] and [BH16, Theorem 2]. Recall that a von Neumann subalgebra $P\subset M$ is with expectation if there is a faithful normal conditional expectation from $1_{P}M1_{P}$ onto $P$.

Theorem 2.4** ([Po01, Po03]).**

Let $M$ be any $\sigma$-finite von Neumann algebra and $A\subset 1_{A}M1_{A}$ and $B\subset 1_{B}M1_{B}$ two von Neumann subalgebras with expectations. Then the following are equivalent:

• •

The $A$-$B$-bimodule $1_{A}\mathord{\text{\rm L}}^{2}(M)1_{B}$ is not left weakly mixing.

• •

There exists projections $e\in A$, $f\in B$, a nonzero partial isometry $v\in eMf$ and a unital normal $\ast$-homomorphism $\theta\colon eAe\to fBf$ such that $v\theta(a)=av$ for all $a\in eAe$.

When these properties hold, we write $A\prec_{M}B$.

We will also write $A\lessdot_{M}B$ when the $A$-$B$-bimodule $1_{A}\mathord{\text{\rm L}}^{2}(M)1_{B}$ is left amenable. When $1_{B}=1_{M}$, this means there is conditional expectation from $1_{A}\langle M,B\rangle 1_{A}$ onto $A$. If it is normal on $1_{A}M1_{A}$, this is equivalent to relative amenability. We will not use this normality in this paper and our notion of $A\lessdot_{M}B$ is more appropriate for our study.

Proposition 2.5**.**

Let $P\subset M$ be a von Neumann algebra and $E\colon M\to P$ a normal conditional expectation. Suppose that $E_{i}:M\rightarrow P_{i}$ is a net of normal conditional expectations on von Neumann subalgebras $P_{i}\subset M$ which converges to $E$ in the sense that $\|\phi\circ E_{i}-\phi\circ E\|\to 0$ for all $\phi\in M_{*}$. Then $\mathord{\text{\rm L}}^{2}(P)\prec\bigoplus_{i\in I}\mathord{\text{\rm L}}^{2}\langle M,P_{i}\rangle$ as $P$-$P$-bimodules. If moreover $E$ is faithful then we have $\mathord{\text{\rm L}}^{2}\langle M,P\rangle\prec\bigoplus_{i\in I}\mathord{\text{\rm L}}^{2}\langle M,P_{i}\rangle$ as $M$-$M$-bimodules.

Proof.

Let $q\in P^{\prime}\cap M$ be the support projection of $E$ and we denote by $r$ the right action of $q$ on $\mathord{\text{\rm L}}^{2}(M)$, which is contained in $\langle M,P\rangle$. Let $p\in P$ be any $\sigma$-finite projection and $\varphi\in(pPp)_{*}^{+}$ a faithful state. Let $\xi\in pr\mathord{\text{\rm L}}^{2}\langle M,P\rangle pr$ be the canonical vector such that

[TABLE]

Put $\varphi_{i}=\varphi\circ E_{i}$ and observe that $\varphi_{i}\to\varphi$ by assumption. We have

[TABLE]

This shows that

[TABLE]

as $pqMpq$-bimodules. Since the $\sigma$-finite projection $p$ is arbitrary, the case $p=1$ also holds. If $q=r=1$, then we are done. For general $q$, since $\mathord{\text{\rm L}}^{2}(P)$ is contained in $qr\mathord{\text{\rm L}}^{2}\langle M,P\rangle qr$ as $P$-$P$-bimodules, we are also done. ∎

3. Tensor factors and flip maps

Let $M$ be a factor. A tensor factor of $M$ is a subfactor $P\subset M$ such that $M=P\mathbin{\overline{\otimes}}(P^{\prime}\cap M)$. We denote by $\mathrm{TF}(M)$ the set of all tensor factors of $M$. For $P\in\mathrm{TF}(M)$, we will often denote its commutant by $P^{c}=P^{\prime}\cap M\in\mathrm{TF}(M)$ when no confusion is possible. We equip $\mathrm{TF}(M)$ with the weakest topology which makes the maps

[TABLE]

continuous for every $\varphi,\psi\in M_{*}$.

Let $\widehat{M}=M\mathbin{\overline{\otimes}}M$ be the tensor double of $M$. We will often distinguish the two copies of $M$ (and its subalgebras) by using the notation $M_{1}=M\otimes\mathbf{C}$ and $M_{2}=\mathbf{C}\otimes M$. We denote by $\sigma_{M}$ the flip automorphism of $\widehat{M}$ given by $\sigma_{M}(x\otimes y)=y\otimes x$ for every $x,y\in M$. For every $P\in\mathrm{TF}(M)$, we obtain naturally a tensor product decomposition $\widehat{M}=\widehat{P}\mathbin{\overline{\otimes}}\widehat{P^{c}}$. Therefore, we can view $\sigma_{P}$ as an automorphism of $\widehat{M}$ by identifying abusively $\sigma_{P}$ with $\sigma_{P}\otimes\text{\rm id}_{\widehat{P^{c}}}$. The map $P\mapsto\sigma_{P}$ is clearly injective since $P=\{x\in M\mid\sigma_{P}(x\otimes 1)=1\otimes x\}$ for every $P\in\mathrm{TF}(M)$.

Theorem 3.1**.**

Let $M$ be a factor. The map $\mathrm{TF}(M)\ni P\mapsto\sigma_{P}\in\operatorname{Aut}(\widehat{M})$ is a homeomorphism on its range, and its range is closed.

Proof.

Let $\iota:(M\mathbin{\overline{\otimes}}M)_{*}\rightarrow M_{*}$ be the continuous map given by $\iota(\phi)(x)=\phi(x\otimes 1)$ for all $x\in M$. Let $\varphi\in M_{*}$. Then, we have $\varphi|_{P}\otimes\psi|_{P^{c}}=\iota(\sigma_{P}(\varphi\otimes\psi))$ for all $\varphi,\psi\in M_{*}$. This shows that if $\sigma_{P_{i}}\to\sigma_{P}$ then $P_{i}\to P$.

Conversely, suppose that $P_{i}\to P$. Let $K=\{\varphi\in M_{*}\mid\varphi=\varphi|_{P}\otimes\varphi|_{P^{c}}\}$. Take $\varphi,\psi\in K$. Then we have $\|\varphi|_{P_{i}}\otimes\varphi|_{P_{i}^{c}}-\varphi\|\to 0$ and similarly for $\psi$. Thus we get

[TABLE]

But we have

[TABLE]

Thus we have the pointwise norm convergence of $\sigma_{P_{i}}$ to $\sigma_{P}$ on the set $K\odot K$ which is dense in $(M\mathbin{\overline{\otimes}}M)_{*}$. We conclude that $\sigma_{P_{i}}$ converges to $\sigma_{P}$.

Let us now show that the range is closed. Suppose that a net $(\sigma_{P_{i}})_{i\in I}$ converges to some $\alpha\in\operatorname{Aut}(\widehat{M})$. Fix $\varphi$ a normal state on $M$. For all $i\in I$, define a normal conditional expectation from $M$ to $P_{i}$ by $E_{P_{i}}=\text{\rm id}_{P_{i}}\otimes(\varphi|_{P_{i}^{c}})$. Note that we have $E_{P_{i}}(x)\otimes 1=(\text{\rm id}\otimes\varphi)(\sigma_{P_{i}}(1\otimes x))$ for all $x\in M$. Define a normal completely positive map $E:M\rightarrow M$ by $E(x)\otimes 1=(\text{\rm id}\otimes\varphi)(\alpha(1\otimes x))$. Observe that $\lim_{i}E_{P_{i}}(x)=E(x)$ in the $*$-strong topology for all $x\in M$. Also, observe that for every $\psi\in M_{*}$, we have $\|\psi\circ E_{P_{i}}-\psi\circ E\|\leq\|\sigma_{P_{i}}(\psi\otimes\varphi)-\alpha(\psi\otimes\varphi)\|\to 0$. From this, it is easy to see that in the weak∗-topology, we have $E(xE(y))=\lim_{i}E_{P_{i}}(xE_{P_{i}}(y))=\lim_{i}E_{P_{i}}(x)E_{P_{i}}(y)=E(x)E(y)$ for all $x,y\in M$. This shows that $E$ is a normal conditional expectation on a subalgebra $P\subset M$. For every unitary $u\in P$, we have $u\otimes 1=(\text{\rm id}\otimes\varphi)(\alpha(1\otimes u))$. This forces $u\otimes 1=\alpha(1\otimes u)$ because $\text{\rm id}\otimes\varphi$ is a conditional expectation and $u\otimes 1$ and $\alpha(1\otimes u)$ are unitaries. Thus we have $P=\{x\in M\mid x\otimes 1=\alpha(1\otimes x)\}$. We denote $E$ by $E_{P}$ from now on.

We can do the same with $Q_{i}=P_{i}^{\prime}\cap M$ as $\sigma_{Q_{i}}=\sigma_{M}\circ\sigma_{P_{i}}\to\sigma_{M}\circ\alpha$. We obtain a subalgebra $Q\subset M$ with a normal conditional expectation $E_{Q}:M\rightarrow Q$ given by $E_{Q}(x)\otimes 1=(\text{\rm id}\otimes\varphi)(\alpha(x\otimes 1))$ for all $x\in M$. We have $Q=\{x\in M\mid x\otimes 1=\alpha(x\otimes 1)\}$. Thus, we see that $P\otimes\mathbf{C}=\alpha(\mathbf{C}\otimes P)$ and $Q\otimes\mathbf{C}=\alpha(Q\otimes\mathbf{C})$ are in tensor product position inside $M\otimes\mathbf{C}$. It remains to show that $P$ and $Q$ generate $M$. Let $D\subset M$ be the von Neumann algebra generated by $P$ and $Q$. Observe that $D\otimes\mathbf{C}=\alpha(Q\mathbin{\overline{\otimes}}P)$. Thus $E_{D\otimes\mathbf{C}}=\alpha\circ(E_{Q}\otimes E_{P})\circ\alpha$ defines a normal conditional expectation of $M\mathbin{\overline{\otimes}}M$ onto $D\otimes\mathbf{C}$. Observe that $\psi\circ(E_{Q}\otimes E_{P})=\lim_{i}\psi\circ(E_{Q_{i}}\otimes E_{P_{i}})$ for all $\psi\in M_{*}\odot M_{*}$ hence for all $\psi\in(M\mathbin{\overline{\otimes}}M)_{*}$ by density. Thus we get

[TABLE]

for all $\psi\in(M\mathbin{\overline{\otimes}}M)_{*}$. Since $\sigma_{P_{i}}(Q_{i}\otimes P_{i})=M\otimes\mathbf{C}$ for all $i$, we have that $\sigma_{P_{i}}\circ(E_{Q_{i}}\otimes E_{P_{i}})\circ\sigma_{P_{i}}$ is a conditional expectation onto $M\otimes\mathbf{C}$ for all $i$. We conclude that $E_{D\otimes\mathbf{C}}(x\otimes 1)=x\otimes 1$ for all $x\in M$, i.e. $D=M$ as we wanted. ∎

Corollary 3.2**.**

If $M_{*}$ is separable, then $\mathrm{TF}(M)$ is a Polish space.

Proof.

By the theorem, $\mathrm{TF}(M)$ is homeomorphic to a closed subset of $\operatorname{Aut}(\widehat{M})$. ∎

The following items will be very useful in the study of $\mathrm{TF}(M)$ for a factor $M$. For this, recall that all $\sigma$-finite infinite projections in $M$ are equivalent, so that we can often reduce problems to the $\sigma$-finite case.

Lemma 3.3**.**

Let $P,Q\in\mathrm{TF}(M)$. The following conditions are equivalent:

• $(\rm i)$

$P\prec_{M}Q$;

• $(\rm ii)$

$\sigma_{Q}(P_{1})\prec_{\widehat{M}}M_{2}$;

• $(\rm iii)$

$Q^{c}\prec_{M}P^{c}$.

Proof.

By the bimodule definition of the relation $\prec_{M}$, it is easy to see that $P\prec_{M}Q$ if and only if $P_{1}\prec_{\widehat{M}}Q_{1}\mathbin{\overline{\otimes}}Q_{2}^{c}$. Then applying $\sigma_{Q}$, we get (i) $\Leftrightarrow$ (ii). For item (iii), if $M$ is $\sigma$-finite, the proof is given in [HI15, Lemma 4.9]. The general case can be reduced to the $\sigma$-finite case. ∎

Proposition 3.4**.**

Let $P,Q\in\mathrm{TF}(M)$. The following conditions are equivalent.

• $(\rm i)$

We have $P\prec_{M}Q$.

• $(\rm ii)$

There exists $D\in\mathrm{TF}(Q)\subset\mathrm{TF}(M)$ such that $P\sim D$.

Proof.

If $M$ is $\sigma$-finite, then the proof is given in [OP03], [HI15, Lemma 4.13] and [HMV16, Proposition 7.3]. The general case can be reduced to the $\sigma$-finite case. ∎

Proposition 3.5**.**

Let $P,Q\in\mathrm{TF}(M)$. The following conditions are equivalent.

• $(\rm i)$

We have $P\prec_{M}Q$ and $Q\prec_{M}P$.

• $(\rm ii)$

$P\sim Q$.

• $(\rm iii)$

$\sigma_{P}\circ\sigma_{Q}\in\operatorname{Inn}(\widehat{M})$.

Proof.

The equivalence of (i) and (ii) is an easy consequence of Proposition 3.4. Item (ii) trivially implies item (iii). Conversely, if item (iii) holds, then $P_{1}\prec_{\widehat{M}}\sigma_{Q}\circ\sigma_{P}(P_{1})$, hence $\sigma_{Q}(P_{1})\prec_{\widehat{M}}M_{2}$. By Lemma 3.4, we get $P\prec_{M}Q$. Similarly we get $Q\prec_{M}P$ and item (i) holds. ∎

4. Weakly bicentralized subalgebras

In this section, we investigate the following property which plays a key role in our deformation/rigidity arguments. It was already used in [BMO19].

Definition 4.1**.**

Let $M$ be a von Neumann algebra. We say that a subalgebra $P\subset M$ is weakly bicentralized in $M$ if ${{}_{M}}\mathord{\text{\rm L}}^{2}\langle M,P\rangle_{M}\prec{{}_{M}}\mathord{\text{\rm L}}^{2}(M)_{M}$.

The terminology is justified by the following bicentralizer criterion from [BMO19].

Proposition 4.2**.**

Let $M$ be a von Neumann algebra and let $P$ be a subalgebra of $M$. Suppose that there exists a faithful normal state $\varphi$ on $M$ such that $P$ is globally invariant by $\sigma^{\varphi}$ and $(P^{\prime}\cap(M^{\omega})_{\varphi^{\omega}})^{\prime}\cap M=P$ for some cofinal ultrafilter $\omega$. Then $P$ is weakly bicentralized in $M$.

Proof.

Let $E_{P}:M\rightarrow P$ be the unique $\varphi$-preserving conditional expectation. Let $\{x_{1},\dots,x_{n}\}$ be a finite subset of $M$. We will use the notations $\underline{M}=M^{\oplus n}$, $\underline{x}=(x_{1},\dots,x_{n})\in\underline{M}$, $\underline{\varphi}=\varphi^{\oplus n}$, $\underline{M}^{\omega}=(M^{\omega})^{\oplus n}=(M^{\oplus n})^{\omega}$. For every finite set $F\subset P$ and every $\varepsilon>0$, we define

[TABLE]

and we let

[TABLE]

Then since $(M^{\omega})_{\varphi^{\omega}}$ is finite, one can follow the proof of [BMO19, Lemma 3.4] and get that $E_{P}$ is the pointwise weak∗-limit of convex combinations of inner automorphisms of $M$. Note that the condition $\|u\varphi-\varphi u\|<\varepsilon$ above is used to make a unitary element in $M^{\omega}_{\varphi^{\omega}}$.

Now, $\mathord{\text{\rm L}}^{2}\langle M,P\rangle$ contains a natural $M$-$M$-cyclic vector $\xi$ which satisfies $\langle x\xi y,\xi\rangle=\langle E_{P}(x)\varphi^{1/2}y,\varphi^{1/2}\rangle$ for all $x,y\in M$. But we have proved that $\langle x\xi y,\xi\rangle=\langle E_{P}(x)\varphi^{1/2}y,\varphi^{1/2}\rangle$ can be approximated by convex combinations of $\langle x(u\varphi^{1/2})y,(u\varphi^{1/2})\rangle$ where $u\in\mathcal{U}(M)$. Thus ${}_{M}\mathord{\text{\rm L}}^{2}\langle M,P\rangle_{M}\prec{{}_{M}}\mathord{\text{\rm L}}^{2}(M)_{M}$. ∎

In the following two propositions, we collect basic properties for weakly bicentralized subalgebras.

Proposition 4.3**.**

Let $P,M,N$ be three von Neumann algebras.

• $(\rm i)$

If $P\subset M\subset N$, $P$ is weakly bicentralized in $M$ and $M$ is weakly bicentralized in $N$, then $P$ is weakly bicentralized in $N$.

• $(\rm ii)$

The algebra $P$ is weakly bicentralized in $M$ if and only if $P\mathbin{\overline{\otimes}}N$ is weakly bicentralized in $M\mathbin{\overline{\otimes}}N$.

• $(\rm iii)$

If $M_{*}$ is separable, then $\mathcal{Z}(M)$ is weakly bicentralized in $M$.

Proof.

$(\rm i)$ Since $P$ is weakly bicentralized in $M$, we have

[TABLE]

as $M$-$M$-bimodules. By tensoring on the left and on the right with $\mathord{\text{\rm L}}^{2}(N)\otimes_{M}$ and $\otimes_{M}\mathord{\text{\rm L}}^{2}(N)$ respectively, we get $\mathord{\text{\rm L}}^{2}\langle N,P\rangle\prec\mathord{\text{\rm L}}^{2}\langle N,M\rangle$ as $N$-$N$-bimodules. On the other hand, since $M$ is weakly bicentralized in $N$, we have $\mathord{\text{\rm L}}^{2}\langle N,M\rangle\prec\mathord{\text{\rm L}}^{2}(N)$ as $N$-$N$-bimodules. Thus, we conclude that $\mathord{\text{\rm L}}^{2}\langle N,P\rangle\prec\mathord{\text{\rm L}}^{2}(N)$ as $N$-$N$-bimodules.

$(\rm ii)$ This follows from the equality of $(M\mathbin{\overline{\otimes}}N)$-bimodules

[TABLE]

$(\rm iii)$ Let $M=\int_{\oplus}M_{x}\>\mathrm{d}\mu(x)$ be the desintegration of $M$ into factors. By item $(\rm ii)$, for any factor $N$, we have that $\mathcal{Z}(M)$ is weakly bicentralized in $M$ if and only if $\mathcal{Z}(M)=\mathcal{Z}(M\mathbin{\overline{\otimes}}N)$ is weakly bicentralized in $M\mathbin{\overline{\otimes}}N$. Thus, by taking $N=R_{\infty}$ and using [Ma18, Theorem D] if necessary, we may assume that each $M_{x}$ is a type ${\rm III}_{1}$ factor with trivial bicentralizer. Take $\varphi$ a faithful normal state on $M$. Then we get $(M^{\omega}_{\varphi^{\omega}})^{\prime}\cap M=\mathcal{Z}(M)$ and we conclude by Proposition 4.2. ∎

Recall that an action $\alpha:\Gamma\curvearrowright B$ of a discrete group $\Gamma$ on a von Neumann algebra $B$ is centrally free if for every $g\in\Gamma\setminus\{1\}$ and every nonzero $z\in\mathcal{Z}(B)$, there exists a cofinal ultrafilter $\omega$ and some $b\in B_{\omega}$ such that $\alpha_{g}(b)z\neq bz$.

Proposition 4.4**.**

Let $(B,\varphi)$ be a von Neumann algebra with a faithful normal state $\varphi$ and let $\alpha:\Gamma\curvearrowright(B,\varphi)$ be a state preserving action. Assume either that:

• $(\rm i)$

$\Gamma$* is ICC and $\alpha$ is approximately inner; or*

• $(\rm ii)$

$\alpha$* is centrally free.*

Then $B$ is weakly bicentralized in $B\rtimes\Gamma$.

Proof.

Let $\mathord{\text{\rm E}}_{B}:M\rightarrow B$ be the canonical conditional expectation and use it to extend $\varphi$ to $M$. We will use the criterion of Proposition 4.2. We trivially have $B\subset(B^{\prime}\cap(M^{\omega})_{\varphi^{\omega}})^{\prime}\cap M$, so we only have to prove the converse. Observe that

[TABLE]

and hence

[TABLE]

We have only to show that one of these sets is contained in $B$.

We first assume that $\Gamma$ is ICC and $\alpha$ is approximately inner. Fix $u_{g}\in\mathcal{U}((B^{\omega})_{\varphi^{\omega}})$ such that $\alpha_{g}=\operatorname{Ad}(u_{g})$ for all $g\in\Gamma$. Observe $u_{g}^{*}\lambda_{g}\in B^{\prime}\cap(B^{\omega}\rtimes\Gamma)_{\varphi^{\omega}}$. Fix any $x\in[B^{\prime}\cap(B^{\omega}\rtimes\Gamma)_{\varphi^{\omega}}]^{\prime}\cap M$ and let $x=\sum_{g\in\Gamma}x_{g}\lambda_{g}$ be the Fourier decomposition in $M$. We have

[TABLE]

so that $x_{ghg^{-1}}=u_{g}^{*}\alpha_{g}(x_{h})\alpha_{ghg^{-1}}(u_{g})$ for all $g,h\in\Gamma$. This implies $\|x_{ghg^{-1}}\|_{2}=\|x_{h}\|_{2}$ for all $g,h\in\Gamma$. Since $\Gamma$ is ICC, we conclude $x_{g}=0$ for all $g\neq e$ and hence $x\in B$.

Assume next that $\alpha$ is centrally free. Take any $x\in[B^{\prime}\cap(B^{\omega})_{\varphi^{\omega}}]^{\prime}\cap M$ and decompose it as $x=\sum_{g\in\Gamma}x_{g}\lambda_{g}$. For any $b\in B^{\prime}\cap(B^{\omega})_{\varphi^{\omega}}$, by comparing coefficients, it holds that $bx_{g}=x_{g}\alpha_{g}(b)=\alpha_{g}(b)x_{g}$ for all $g\in\Gamma$. This is equivalent to $(b-\alpha_{g}(b))x_{g}=0$, so one has $(b-\alpha_{g}(b))z=0$, where $z\in\mathcal{Z}(B)$ is the central left support projection of $x_{g}$. Since this holds for all $b\in B^{\prime}\cap(B^{\omega})_{\varphi^{\omega}}$, by assumption, we conclude that $x_{g}=0$ if $g\neq e$, hence $x\in B$. ∎

5. Rigidity for full tensor factors

In this section, we study the set $\mathrm{TF}(M)$ by assuming $M$ is a full factor. We particularly prove Theorem A. We start by recalling the following key property.

Proposition 5.1** ([BMO19]).**

Let $M$ be a full factor. Then every $M$-$M$-bimodule that is weakly equivalent to $\mathord{\text{\rm L}}^{2}(M)$ must contain $\mathord{\text{\rm L}}^{2}(M)$.

The next lemma applies in particular to $X=\{Q\}$ when $P\lessdot_{M}Q$.

Lemma 5.2**.**

Let $P\in\mathrm{TF}(M)$ and $X\subset\mathrm{TF}(M)$. Suppose that $P$ is full and

[TABLE]

as $P$-$P$-bimodules. Then there exists $Q\in X$ such that $P\prec_{M}Q$.

Proof.

Let $\mathcal{H}=\bigoplus_{Q\in X}\mathord{\text{\rm L}}^{2}\langle M,Q\rangle$. Since each $Q^{c}=Q^{\prime}\cap M$ is a factor, $\mathbf{C}$ is weakly bicentralized in $Q^{c}$. Combined with Proposition 4.3, each $Q$ is weakly bicentralized in $M$, hence ${{}_{M}}\mathcal{H}_{M}\prec\mathord{\text{\rm L}}^{2}(M)$. Moreover, ${{}_{P}}\mathord{\text{\rm L}}^{2}(M)_{P}$ is a multiple of ${{}_{P}}\mathord{\text{\rm L}}^{2}(P)_{P}$ because $P\in\mathrm{TF}(M)$. Hence ${{}_{P}}\mathcal{H}_{P}\prec\mathord{\text{\rm L}}^{2}(P)$. Combining this with the assumption, we get that ${{}_{P}}\mathcal{H}_{P}$ is weakly equivalent to $\mathord{\text{\rm L}}^{2}(P)$. Thus $\mathord{\text{\rm L}}^{2}(P)\subset{{}_{P}}\mathcal{H}_{P}$ by Proposition 5.1. We conclude that $\mathord{\text{\rm L}}^{2}(P)\subset{{}_{P}}\mathord{\text{\rm L}}^{2}\langle M,Q\rangle{{}_{P}}$ for some $Q\in X$ and this means that $P\prec_{M}Q$. ∎

Example 5.3**.**

Let $M$ be a factor and $P,Q\in\mathrm{TF}_{\mathrm{full}}(M)$ two full tensor factors such that $P^{c}$ and $Q^{c}$ are amenable. Since $Q^{c}$ is amenable, we have $P\lessdot_{M}Q$, hence $P\prec_{M}Q$ by Lemma 5.2. Similarly, we have $Q\prec_{M}P$. We conclude that $P\sim Q$. This provides a short proof of [Po06a, Theorem 5.1] and [HMV16, Theorem E].

Lemma 5.4**.**

Let $P\in\mathrm{TF}(M)$. Suppose that $P$ is full. Then the set $U=\{Q\in\mathrm{TF}(M)\mid P\prec_{M}Q\}$ is both closed and open.

Proof.

First, we show that $U$ is a neighborhood of $P$. Suppose, by contradiction, that there exists a net $(Q_{i})_{i\in I}$ in $\mathrm{TF}(M)$ which converges to $P$ but such that $Q_{i}\nprec_{M}P$ for all $i$. Take $\phi$ a normal state on $M$ and define a normal conditional expectation $\mathord{\text{\rm E}}_{i}=\text{\rm id}\otimes\phi|_{Q^{c}}:M\rightarrow Q_{i}$. Since $(Q_{i})_{i\in I}$ converges to $P$, we have that $\mathord{\text{\rm E}}_{i}$ converges to the normal conditional expectation $\mathord{\text{\rm E}}=\text{\rm id}\otimes\phi|_{P^{c}}:M\rightarrow P$ pointwisely in the norm of $M_{*}$. Thus $\mathord{\text{\rm L}}^{2}(P)\prec\bigoplus_{i}\mathord{\text{\rm L}}^{2}\langle M,Q_{i}\rangle$ as $P$-$P$-bimodules by Proposition 2.5. By Lemma 5.2, we conclude that $P\prec_{M}Q_{i}$ for some $i\in I$ which is a contradiction. Hence $U$ is a neighborhood of $P$.

Now, we show that $U$ is indeed closed and open. Let $(Q_{i})_{i\in I}$ be a net in $\mathrm{TF}(M)$ which converges to $Q$. Since $\sigma_{Q}\circ\sigma_{Q_{i}}\to\text{\rm id}$, we have that $(\sigma_{Q}\circ\sigma_{Q_{i}})(P_{1})$ converges to $P_{1}$ in $\mathrm{TF}(\widehat{M})$. Thus, by the first part of the proof, for $i$ large enough we have $P_{1}\prec_{\widehat{M}}(\sigma_{Q}\circ\sigma_{Q_{i}})(P_{1})$, hence $\sigma_{Q}(P_{1})\prec_{\widehat{M}}\sigma_{Q_{i}}(P_{1})$. Similarly, since $\sigma_{Q_{i}}\circ\sigma_{Q}\to\text{\rm id}$, we get $\sigma_{Q_{i}}(P_{1})\prec_{\widehat{M}}\sigma_{Q}(P_{1})$ for $i$ large enough. We conclude that for $i$ large enough we have $\sigma_{Q_{i}}(P_{1})\sim_{\widehat{M}}\sigma_{Q}(P_{1})$. Combined with Lemma 3.3, $Q\in U$ if and only if $Q_{i}\in U$ for $i$ large enough,. This means that $U$ is closed and open. ∎

Recall that $\mathrm{TF_{full}}(M)\subset\mathrm{TF}(M)$ is the set of tensor factors which are full and $\mathrm{TF}(M)$ has an equivalence relation given in Proposition 3.5. Now we prove Theorem A.

Theorem 5.5**.**

Let $M$ be any factor. Then the following holds:

• $(\rm i)$

The space $\mathrm{TF_{full}}(M)/{\sim}$ is Hausdorff and totally disconnected.

• $(\rm ii)$

If $M$ is full, the space $\mathrm{TF}(M)/{\sim}$ is discrete.

• $(\rm iii)$

If $M_{*}$ is separable, then $\mathrm{TF_{full}}(M)/{\sim}$ is countable.

Proof.

$(\rm i)$ Let $P_{1},P_{2}\in\mathrm{TF_{full}}(M)$ and suppose that $P_{1}\nsim_{M}P_{2}$. Assume, without loss of generality, that $P_{1}\nprec_{M}P_{2}$. Then, by Lemma 5.4, $\{Q\in\mathrm{TF}(M)\mid P_{1}\prec_{M}Q\}$ is an open set which contains $P_{1}$ and its complement is also an open set which contains $P_{2}$. This shows that the equivalence classes of $P_{1}$ and $P_{2}$ in $\mathrm{TF_{full}}(M)/{\sim}$ are separated by two open sets which form a partition of $\mathrm{TF_{full}}(M)/{\sim}$. Therefore, $\mathrm{TF_{full}}(M)/{\sim}$ is Hausdorff and totally disconnected.

$(\rm ii)$. Let $P\in\mathrm{TF}(M)$. Since $M$ is full, we know that $P$ and $P^{c}$ are also full. Therefore by Lemma 3.3,

[TABLE]

is an intersection of two open sets by Lemma 5.4. This shows that the equivalence classes of $\sim$ are open, which means that the quotient is discrete.

$(\rm iii)$. In this case, $\mathrm{TF}(M)$ is Polish by Theorem 3.1. When $M$ is full, the conclusion follows from $(\rm ii)$ as $\mathrm{TF}(M)/{\sim}$ is a discrete separable space, hence countable. Now, for general $M$, the space $\mathrm{TF_{full}}(M)$ is separable so we can find a dense countable subset $X\subset\mathrm{TF_{full}}(M)$. For every $P\in X$, let $U_{P}=\{Q\in\mathrm{TF_{full}}(M)\mid Q\prec_{M}P\}$. Observe that $U_{P}/{\sim}$ is in bijection with $\mathrm{TF}(P)/{\sim}$, thus it is countable because $P$ is full. Moreover, by using Lemma 5.4 and since $X$ is dense, we know that $\mathrm{TF_{full}}(M)=\bigcup_{P\in X}U_{P}$. This shows that $\mathrm{TF_{full}}(M)/{\sim}$ is a countable union of countable sets. ∎

6. Application to fundamental groups

In this section, we study tensor factors $M\mathbin{\overline{\otimes}}N$ by assuming $M$ is a full factor. For this, we will use the topological structure of $\mathrm{TF}(M\mathbin{\overline{\otimes}}N)$ discussed in the last section. We particularly prove Theorem B.

Theorem 6.1**.**

Let $M$ and $N$ be two infinite factors. Suppose that $M$ is full. Then the following map is open:

[TABLE]

In particular, $\operatorname{Out}(M)\times\operatorname{Out}(N)$ is an open subgroup of $\operatorname{Out}(M\mathbin{\overline{\otimes}}N)$.

Proof.

We have to show that the image by $\iota$ of any neighoborhood of the identity $\mathcal{V}\subset\mathcal{U}(M\mathbin{\overline{\otimes}}N)\times\operatorname{Aut}(M)\times\operatorname{Aut}(N)$ is again a neighborhood of the identity in $\operatorname{Aut}(M\mathbin{\overline{\otimes}}N)$. Let $(\theta_{i})_{i\in I}$ be a net in $\operatorname{Aut}(M\mathbin{\overline{\otimes}}N)$ which converges to the identity and let us show that $\theta_{i}\in\iota(\mathcal{V})$ for $i$ large enough. Since $\theta_{i}(M)$ converges to $M$ in $\mathrm{TF}(M\mathbin{\overline{\otimes}}N)$, we know by Lemma 5.4 that $M\prec_{M\mathbin{\overline{\otimes}}N}\theta_{i}(M)$ for $i$ large enough. By the same argument applied to $\theta_{i}^{-1}$, we also get $\theta_{i}(M)\prec_{M\mathbin{\overline{\otimes}}N}M$ for $i$ large enough. Therefore $\theta_{i}(M)\sim M$, hence also $\theta_{i}(N)\sim N$ for all $i$ large enough. This shows that $\theta_{i}$ is eventually of the form $\theta_{i}=\iota(u_{i},\alpha_{i},\beta_{i})$ for some triples $(u_{i},\alpha_{i},\beta_{i})$ in the domain of $\iota$. Next, we have to show that we can actually take $(u_{i},\alpha_{i},\beta_{i})\in\mathcal{V}$. Observe that the outer automorphism class $[\alpha_{i}\otimes\beta_{i}]$ converges to the identity in $\operatorname{Out}(M\mathbin{\overline{\otimes}}N)$. By [HMV16, Theorem A], this implies that $([\alpha_{i}],[\beta_{i}])$ converges to the identity in $\operatorname{Out}(M)\times\operatorname{Out}(N)$. Therefore, up to replacing $\alpha_{i}$ by $\operatorname{Ad}(v_{i})\circ\alpha_{i}$, $\beta_{i}$ by $\operatorname{Ad}(w_{i})\circ\beta_{i}$ and $u_{i}$ by $u_{i}(v_{i}^{*}\otimes w_{i}^{*})$ for some unitaries $v_{i}\in\mathcal{U}(M)$ and $w_{i}\in\mathcal{U}(N)$, we may simply assume that both $\alpha_{i}$ and $\beta_{i}$ converges to the identity in $\operatorname{Aut}(M)$ and $\operatorname{Aut}(N)$. Then, in that case, since $\theta_{i}=\iota(u_{i},\alpha_{i},\beta_{i})$ converges to the identity, we must also have that $\operatorname{Ad}(u_{i})$ converges to the identity in $\operatorname{Aut}(M\mathbin{\overline{\otimes}}N)$. Since $M$ is full, by [HMV16, Theorem A], there exists $v_{i}\in\mathcal{U}(N)$ such that $u_{i}(1\otimes v_{i})^{*}$ converges to $1$ strongly and $\operatorname{Ad}(v_{i})$ converges to the identity in $\operatorname{Aut}(N)$. We conclude that $\theta_{i}=\iota(u_{i}(1\otimes v_{i})^{*},\alpha_{i},\operatorname{Ad}(v_{i})\circ\beta_{i})\in\iota(\mathcal{V})$ for $i$ large enough. ∎

Theorem 6.2**.**

Let $M$ and $N$ be two infinite factors with separable predual. Suppose that $M$ is full. Let $\theta:\mathbf{R}\curvearrowright M\mathbin{\overline{\otimes}}N$ be a one-parameter group. Then there exists two one-parameter groups $\alpha:\mathbf{R}\curvearrowright M$ and $\beta:\mathbf{R}\curvearrowright N$ such that $\alpha\otimes\beta$ is a cocycle perturbation of $\theta$, i.e. there exists a continuous map $u:\mathbf{R}\rightarrow\mathcal{U}(M\mathbin{\overline{\otimes}}N)$ such that

[TABLE]

[TABLE]

Proof.

Since $\mathbf{R}$ is connected and the continuous morphism between Polish groups

[TABLE]

is open, the image of $\theta$ is contained in the image of $\iota$ and there exists a borel lift

[TABLE]

such that $\theta=\iota\circ\rho$. Observe that $\mathbf{R}:t\mapsto[\alpha_{t}]\in\operatorname{Out}(M)$ is a group morphism. By [Su80], since $M$ is infinite and the cohomology group $H^{3}(\mathbf{R},\mathbf{T})$ is trivial, we can find a borel map $\lambda\mapsto v_{t}\in\mathcal{U}(M)$ such that $t\mapsto\operatorname{Ad}(v_{t})\circ\alpha_{t}$ is a continuous group morphism. Therefore, up to replacing $u_{-t}$ by $u_{-t}(v_{t}^{*}\otimes 1)$ and $\alpha_{t}$ by $\operatorname{Ad}(v_{t})\circ\alpha_{t}$, we may assume that $\alpha:\mathbf{R}\ni t\mapsto\alpha_{t}\in\operatorname{Aut}(M)$ is a continuous group morphism. Similarly, we may assume that $\beta:t\mapsto\beta_{t}$ is a continuous group morphism. Then we have that $t\mapsto\alpha_{t}\otimes\beta_{t}=\operatorname{Ad}(u_{t})\circ\theta_{t}$ is also a group morphism. This implies that $\operatorname{Ad}(u_{s+t})=\operatorname{Ad}(u_{s})\operatorname{Ad}(\theta_{s}(u_{t}))$ for all $s,t\in\mathbf{R}$. Therefore, $u_{s+t}=\chi(s,t)u_{s}\theta_{s}(u_{t})$ where $\chi:\mathbf{R}\times\mathbf{R}\rightarrow\mathbf{T}$ is a scalar $2$-cocycle. Since $H^{2}(\mathbf{R},\mathbf{T})$ is trivial, the $2$-cocycle $\chi$ is a coboundary. Thus, we may perturb $u_{t}$ by scalars in $\mathbf{T}$ so that $t\mapsto u_{t}$ becomes a true $1$-cocycle. ∎

Proof of Theorem B.

Let $M$ and $N$ be two factors of type ${\rm II}_{\infty}$ with separable predual and suppose that one of them is full.

$(\rm i)$ There are two surjective maps

[TABLE]

They induce a surjective map from $\mathrm{Out}(M\mathbin{\overline{\otimes}}N)/\mathrm{Out}(M)\times\mathrm{Out}(N)$ onto $\mathcal{F}(M\mathbin{\overline{\otimes}}N)/\mathcal{F}(M)\mathcal{F}(N)$. By Theorem 6.1, $\mathrm{Out}(M\mathbin{\overline{\otimes}}N)/\mathrm{Out}(M)\times\mathrm{Out}(N)$ is discrete hence countable (because $M_{*}$ and $N_{*}$ are separable) and we get the conclusion.

$(\rm ii)$ Let $\theta:\mathbf{R}^{*}_{+}\rightarrow\operatorname{Aut}(M\mathbin{\overline{\otimes}}N)$ be a trace scaling action. Then, by Theorem 6.2, we can find two actions $\alpha:\mathbf{R}^{*}_{+}\rightarrow\operatorname{Aut}(M)$ and $\beta:\mathbf{R}^{*}_{+}\rightarrow\operatorname{Aut}(N)$ such that $(\alpha_{\lambda}\otimes\beta_{\lambda})\circ\theta_{\lambda}^{-1}$ is inner for all $\lambda>0$. In particular, we have $\mathrm{Mod}(\alpha_{\lambda})\mathrm{Mod}(\beta_{\lambda})=\mathrm{Mod}(\theta_{\lambda})=\lambda$ for all $\lambda>0$. Since $\lambda\mapsto\mathrm{Mod}(\alpha_{\lambda})$ is a group homomorphism, there must exist some $s\in\mathbf{R}$ such that $\mathrm{Mod}(\alpha_{\lambda})=\lambda^{s}$ hence $\mathrm{Mod}(\beta_{\lambda})=\lambda^{1-s}$ for all $\lambda>0$. We conclude that $M$ admits a trace scaling action (if $s\neq 0$) or $N$ admits a trace scaling action (if $s\neq 1$). ∎

7. Noncommutative Bernoulli shifts

In this section, we investigate the structure of full factors arising from Bernoulli actions. For this, we first observe that well-known arguments in the deformation/rigidity theory for Bernoulli actions (mostly established in [Po03]) also work in the type III setting. We will then prove Theorem C and D.

We first prove the following rigidity results for Bernoulli actions. Recall that our definition of $A\prec_{M}B$ coincides with the usual one if $M$ is $\sigma$-finite and $A,B\subset M$ are with expectation.

Theorem 7.1**.**

Let $\alpha:\Gamma\curvearrowright(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}=:(B,\varphi)$ be a noncommutative Bernoulli shift. Let $N$ be any $\sigma$-finite von Neumann algebra and put $M:=N\mathbin{\overline{\otimes}}(B\rtimes\Gamma)$. Let $p\in M$ be a projection and $P,Q\subset pMp$ von Neumann algebras with expectation such that $P$ and $Q$ are commuting and that $Q$ is finite. Then one of the following conditions holds:

• $(\rm i)$

$Q\prec_{M}N\mathbin{\overline{\otimes}}L(\Gamma)$;

• $(\rm ii)$

$Q\prec_{M}N\mathbin{\overline{\otimes}}B_{0}^{F}$* for some finite subset $F\subset\Gamma$; or*

• $(\rm iii)$

$P\lessdot_{M}N\mathbin{\overline{\otimes}}B$.

Proof.

We only give a sketch of the proof. Following [Io06] (see also [CPS11, Section 1] and [Ma16, Section 5]), we define a von Neumann algebra $\widetilde{M}$ and its deformations $(\theta_{t},\beta)$. We then apply the proof of [Ma16, Theorem 4.2] to the finite algebra $Q$, and we get either:

1. (1)

$P^{\prime}\cap p\widetilde{M}^{\omega}p\not\subset pM^{\omega}p$ for some ultrafilter $\omega\in\beta\mathbf{N}$; or 2. (2)

$(\theta_{t})_{t}$ converges uniformly on $(Q)_{1}$ (in the $\ast$-strong topology) and $Q\prec_{\widetilde{M}}\theta_{1}(Q)$.

Following the proof of [Ma16, Lemma 5.1], the second condition directly implies $(\rm i)$ or $(\rm ii)$.

We next consider the case that $P^{\prime}\cap p\widetilde{M}^{\omega}p\not\subset pM^{\omega}p$. Then, by the proof of [Ma16, Lemma 4.1], we have as $P$-$P$-bimodules

[TABLE]

for some nonzero projection $z\in\mathcal{Z}(P)$. Recall that we have a decomposition as $M$-$M$-bimodules (see [Ma16, Theorem 5.2])

[TABLE]

where each $B_{i}$ is of the form that $B_{i}=(N\mathbin{\overline{\otimes}}B_{0}^{F_{i}^{c}})\rtimes\Gamma_{i}$ for some finite $F_{i}\subset\Gamma$ and finite group $\Gamma_{i}\leq\Gamma$. Since $\Gamma_{i}$ is amenable, it holds as $M$-$M$-bimodules that

[TABLE]

Thus we obtain $z\mathord{\text{\rm L}}^{2}(P)\prec\bigoplus_{i}\mathord{\text{\rm L}}^{2}\langle M,N\mathbin{\overline{\otimes}}B_{0}^{F_{i}^{c}}\rangle$ as $P$-$P$-bimodules. This means that there exists a conditional expectation from $zLz$ onto $zP$ where $L=\bigoplus_{i}\langle M,N\mathbin{\overline{\otimes}}B_{0}^{F_{i}^{c}}\rangle$ and $P$ is embedded diagonally in $L$. Since $\langle M,N\mathbin{\overline{\otimes}}B\rangle$ embeds diagonally in $L$, we can restrict it to a conditional expectation from $z\langle M,N\mathbin{\overline{\otimes}}B\rangle z$ on $Pz$. We conclude that $P\lessdot_{M}N\mathbin{\overline{\otimes}}B$. ∎

The following two lemma are useful to control normalizers in Bernoulli shift von Neumann algebras.

Lemma 7.2**.**

Keep the notation $M=(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma$ as in Theorem 7.1. Let $C_{0}\subset B_{0}$ be a von Neumann subalgebra (possibly trivial) with expectation which is globally preserved by $\sigma^{\varphi_{0}}$ and put $C:=\mathbin{\overline{\otimes}}_{\Gamma}(C_{0},\varphi_{0})\subset B$. Let $p\in M$ be a projection and $P\subset pMp$ a von Neumann subalgebra with expectation. The following assertions hold true.

• $(\rm i)$

Let $F\subset\Gamma$ be a finite set and assume that $p\in N\mathbin{\overline{\otimes}}B_{0}^{F}$, $P\subset p(N\mathbin{\overline{\otimes}}(C\vee B_{0}^{F}))p$, and $P\not\prec_{N\mathbin{\overline{\otimes}}(C\vee B_{0}^{F})}N\mathbin{\overline{\otimes}}(C\vee B_{0}^{E})$ for all genuine subsets $E\subset F$. Then any $x\in pMp$ such that $xa=\beta(a)x$ for all $a\in P$ for some $\beta\in\operatorname{Aut}(P)$, is contained in $(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma_{F}$, where $\Gamma_{F}:=\{g\in\Gamma\mid gF=F\}$.

• $(\rm ii)$

Assume that $P\prec_{M}N\mathbin{\overline{\otimes}}(C\vee B_{0}^{F})$ for a finite set $F\subset\Gamma$ and that $P\not\prec_{M}N\mathbin{\overline{\otimes}}C$. Then we have $\mathcal{N}_{qMq}(Pq)^{\prime\prime}\prec_{M}N\mathbin{\overline{\otimes}}B$ for some projection $q\in P^{\prime}\cap pMp$.

Proof.

For simplicity, we will write $D^{F}:=C\vee B_{0}^{F}$ for all $F\subset\Gamma$.

$(\rm i)$ Take $x\in pMp$ as in the statement and let $x=\sum_{g\in\Gamma}x_{g}\lambda_{g}\in(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma$ be the Fourier decomposition. By comparing coefficients, it holds that

[TABLE]

Fix $g\in\Gamma$ and we prove that if $x_{g}\neq 0$, then $F=gF$. If $x_{g}\neq 0$, then one has $P=\beta(P)\prec_{N\mathbin{\overline{\otimes}}B}\alpha_{g}(P)$. Since $P\subset N\mathbin{\overline{\otimes}}D^{F}$, this implies $P\prec_{N\mathbin{\overline{\otimes}}B}N\mathbin{\overline{\otimes}}D^{gF}$. Thus by our definition of $\prec$, we have

[TABLE]

By the assumption of $P$, this implies $F\cap gF=F$, hence $F=gF$. This finished the proof of item $(\rm i)$.

$(\rm ii)$ Fix a finite set $F\subset\Gamma$ such that $P\prec_{M}N\mathbin{\overline{\otimes}}D^{F}$ and take $(H,f,\pi,w)$ as in [Is19, Lemma 2.6]. Write $\mathbf{B}=\mathbf{B}(H)$ for simplicity. We may assume the support of $E_{N\mathbin{\overline{\otimes}}D^{F}\mathbin{\overline{\otimes}}\mathbf{B}}(w^{*}w)$ is $f$. Assume that there is a genuine subset $E\subset F$ such that $\pi(P)\prec_{N\mathbin{\overline{\otimes}}D^{F}\mathbin{\overline{\otimes}}\mathbf{B}}N\mathbin{\overline{\otimes}}D^{E}\mathbin{\overline{\otimes}}\mathbf{B}$. Then by the choice of $f$, this implies $P\prec_{M}N\mathbin{\overline{\otimes}}D^{E}$. In this case, we can replace $F$ by the smaller set $E$. By continuing this procedure, we can finally find $F$ such that $P\prec_{M}N\mathbin{\overline{\otimes}}D^{F}$ with $(H,f,\pi,w)$ such that $\pi(P)\not\prec_{N\mathbin{\overline{\otimes}}D^{F}\mathbin{\overline{\otimes}}\mathbf{B}}N\mathbin{\overline{\otimes}}D^{E}\mathbin{\overline{\otimes}}\mathbf{B}$ for all genuine subsets $E\subset F$.

In this setting we can apply item $(\rm i)$ to the inclusion $\pi(P)\subset N\mathbin{\overline{\otimes}}D^{F}\mathbin{\overline{\otimes}}\mathbf{B}$ (by regarding $N\mathbin{\overline{\otimes}}\mathbf{B}$ as $N$). Write $f_{0}=w^{*}w\in\pi(P)^{\prime}\cap f(M\mathbin{\overline{\otimes}}\mathbf{B})f$ and $e_{0}\otimes e_{1,1}=ww^{*}\in({P}^{\prime}\cap pMp)\otimes\mathbf{C}e_{1,1}$ (where $e_{1,1}\in\mathbf{B}$ is a minimal projection), and observe that $\operatorname{Ad}(w^{*}):e_{0}(M\mathbin{\overline{\otimes}}\mathbf{B})e_{0}\rightarrow f_{0}(M\mathbin{\overline{\otimes}}\mathbf{B})f_{0}$ sends $Pe_{0}$ onto $\pi(P)f_{0}$. Therefore, we have

[TABLE]

Using item $(\rm i)$ , it is easy to see that the right hand side of this equation is contained in $(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma_{F}\mathbin{\overline{\otimes}}\mathbf{B}$. We obtain that

[TABLE]

Finally, since $\Gamma_{F}$ is a finite group by assumption, we conclude that

[TABLE]

∎

Lemma 7.3**.**

Keep the notation $M=(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma$ as in Lemma 7.1. Let $p\in M$ be a projection and $P\subset pMp$ von Neumann subalgebras with expectations. The following assertions hold true.

• $(\rm i)$

Assume that $p\in N\rtimes\Gamma$ and $P\subset p(N\rtimes\Gamma)p$. If $P\nprec_{N\rtimes\Gamma}N$, then one has $\mathcal{N}_{pMp}(P)\subset N\rtimes\Gamma$.

• $(\rm ii)$

If $P\prec_{M}N\rtimes\Gamma$ and $P\nprec_{M}N$, then one has $\mathcal{N}_{qMq}(Pq)^{\prime\prime}\prec_{M}N\rtimes\Gamma$ for some projection $q\in P^{\prime}\cap pMp$.

Proof.

$(\rm i)$ Up to replacing $P$ by $\widetilde{P}=P\oplus p^{\perp}(N\rtimes\Gamma)p^{\perp}$, we may assume that $p=1$. We only have to show that the $P$-$P$-bimodule $\mathord{\text{\rm L}}^{2}(M)\ominus\mathord{\text{\rm L}}^{2}(N\rtimes\Gamma)$ is weakly mixing. Obseve that the $L(\Gamma)$-bimodule $\mathord{\text{\rm L}}^{2}(B\rtimes\Gamma)\ominus\mathord{\text{\rm L}}^{2}(L(\Gamma))$ is a multiple of the coarse $L(\Gamma)$-bimodule. Thus the $(N\rtimes\Gamma)$-bimodule $\mathord{\text{\rm L}}^{2}(M)\ominus\mathord{\text{\rm L}}^{2}(N\rtimes\Gamma)$ is a multiple of the $(N\rtimes\Gamma)$-bimodule $\mathord{\text{\rm L}}^{2}\langle N\rtimes\Gamma,N\rangle$. Since $P\nprec_{N\rtimes\Gamma}N$, the $P$-$P$-bimodule $\mathord{\text{\rm L}}^{2}\langle N\rtimes\Gamma,N\rangle$ is weakly mixing. Thus the $P$-$P$-bimodule $\mathord{\text{\rm L}}^{2}(M)\ominus\mathord{\text{\rm L}}^{2}(N\rtimes\Gamma)$ is also weakly mixing.

$(\rm ii)$ This follows in a similar way to the proof of Lemma 7.2.$(\rm ii)$. Take $(H,f,\pi,w)$ as in [Is19, Lemma 2.6] for $P\preceq_{M}N\rtimes\Gamma$, and we may assume that $\pi(P)\nprec_{N\rtimes\Gamma\mathbin{\overline{\otimes}}\mathbf{B}}N\mathbin{\overline{\otimes}}\mathbf{B}$, where $\mathbf{B}=\mathbf{B}(H)$. By item $(\rm i)$, we have

[TABLE]

Since $w^{*}w\in\pi(P)^{\prime}\cap f({M}\mathbin{\overline{\otimes}}\mathbf{B})f\subset f(N\rtimes\Gamma\mathbin{\overline{\otimes}}\mathbf{B})f$, we can assume $w^{*}w=f$. Putting $e_{0}\otimes e_{1,1}=ww^{*}\in(P^{\prime}\cap pMp)\otimes\mathbf{C}e_{1,1}$, one has

[TABLE]

This implies the conclusion. ∎

We next show that the sufficient condition in Proposition 4.4 is easily verified for Bernoulli actions.

Proposition 7.4**.**

Let $\alpha:\Gamma\curvearrowright(B_{0},\varphi_{0})^{\mathbin{\overline{\otimes}}\Gamma}=:(B,\varphi)$ be a noncommutative Bernoulli shift where $\Gamma$ is infinite and $B_{0}$ is nontrivial. Then $\alpha$ is centrally free. In particular, for any subset $F\subset\Gamma$, the subalgebra $\mathcal{Z}(B)\vee B_{0}^{F}$ is weakly bicentralized in $B\rtimes\Gamma$.

Proof.

The central freeness of $\alpha$ is obvious if $(B_{0})_{\varphi_{0}}\neq\mathbf{C}$. Suppose now that $(B_{0})_{\varphi_{0}}=\mathbf{C}$ (this forces $B_{0}$ to be a type ${\rm III}_{1}$ factor). Take $g\in\Gamma\setminus\{1\}$. Let $(h_{n})_{n\in\mathbf{N}}$ a sequence in $\Gamma$ which goes to infinity, then we can find a sequence of unitaries $u_{n}\in\alpha_{h_{n}}(B_{0})$ such that $\varphi(u_{n})=0$ and $\|u_{n}\varphi-\varphi u_{n}\|\leq\frac{1}{n}$. Then $u=(u_{n})^{\omega}\in B_{\omega}$ for $\omega\in\beta\mathbf{N}\setminus\mathbf{N}$ and we have $\varphi(\alpha_{g}(u)u^{*})=0$. Thus $\alpha$ is centrally free. By Proposition 4.4, we then have that $B$ is weakly bicentralized in $M$ and thanks to Proposition 4.3, we conclude that $\mathcal{Z}(B)\vee B_{0}^{F}$ is weakly bicentralized in $M$ for every subset $F\subset\Gamma$. ∎

Lemma 7.5**.**

Let $M$ and $N$ be factors with separable predual. Assume that $M$ is a type ${\rm III}_{1}$ factor with trivial bicentralizer. Then $M\mathbin{\overline{\otimes}}N$ is a type ${\rm III}_{1}$ factor with trivial bicentralizer.

Proof.

By [AHHM18, Proposition 7.1], the bicentralizer flow of $M\mathbin{\overline{\otimes}}N$ is trivial. By [Ma18, Theorem D], we conclude that $M\mathbin{\overline{\otimes}}N$ has trivial bicentralizer. ∎

Now we can prove Theorem C.

Proof of Theorem C.

Let $M=(N\mathbin{\overline{\otimes}}B)\rtimes\Gamma$ be as in the last statement in Theorem C. Suppose that $M=P\mathbin{\overline{\otimes}}Q$ and observe that $P$ and $Q$ are full. We have to show that $B\rtimes\Gamma\prec_{M}P$ or $B\rtimes\Gamma\prec_{M}Q$. Let $K$ be a full type ${\rm III}_{1}$ factor with trivial bicentralizer (e.g. a free Araki-Woods factor). By Lemma 7.5, up to replacing $P,Q,N$ by $P\mathbin{\overline{\otimes}}K,Q\mathbin{\overline{\otimes}}K,N\mathbin{\overline{\otimes}}K\mathbin{\overline{\otimes}}K$ in $M\mathbin{\overline{\otimes}}K\mathbin{\overline{\otimes}}K$ respectively, we may assume that $P$ and $Q$ have trivial bicentralizers. Then we can find irreducible finite subfactors with expectation $P_{0}\subset P$ and $Q_{0}\subset Q$. By Theorem 7.1, we know that one of the following conditions holds:

• $(\rm i)$

$Q_{0}\prec_{M}N\mathbin{\overline{\otimes}}L(\Gamma)$;

• $(\rm ii)$

$Q_{0}\prec_{M}N\mathbin{\overline{\otimes}}B_{0}^{F}$ for some finite subset $F\subset\Gamma$; or

• $(\rm iii)$

$P\lessdot_{M}N\mathbin{\overline{\otimes}}B$.

If $P_{0}\prec_{M}N$ or $Q_{0}\prec_{M}N$ then, by taking the commutants, we get $B\rtimes\Gamma\prec_{M}P$ or $B\rtimes\Gamma\prec_{M}Q$ and we are done. So we may assume, for the sake of a contradiction, that $P_{0}\nprec_{M}N$ and $Q_{0}\nprec_{M}N$. Then, if one of the conditions $(\rm i)$ or $(\rm ii)$ is satisfied, we can apply Lemma 7.2 or Lemma 7.3 to get $P\prec_{M}N\mathbin{\overline{\otimes}}L(\Gamma)$ or $P\prec_{M}N\mathbin{\overline{\otimes}}B$. If $P\prec_{M}N\mathbin{\overline{\otimes}}L(\Gamma)$, then by Lemma 7.3, we get $M\prec_{M}N\mathbin{\overline{\otimes}}L(\Gamma)$ which is not possible because $B_{0}$ is non-trivial. If $P\prec_{M}N\mathbin{\overline{\otimes}}B$, then a fortiori, we have $P\lessdot_{M}N\mathbin{\overline{\otimes}}B$. Thus it only remains to deal with the case where condition $(\rm iii)$ holds.

Now we assume condition (iii) holds. Since $B$ is the increasing union of $\mathcal{Z}(B)\vee B_{0}^{F}(=:D^{F})$ over finite subsets $F\subset\Gamma$, Proposition 2.5 shows, as $M$-$M$-bimodules

[TABLE]

Since $P\prec_{M}^{w}N\mathbin{\overline{\otimes}}B$, we get as $P$-$P$-bimodules

[TABLE]

But thanks to Proposition 7.4, we also have as $M$-$M$-bimodules

[TABLE]

Since $P$ is full, we conclude that $P\prec_{M}N\mathbin{\overline{\otimes}}D^{F}$ for some finite subset $F\subset\Gamma$ (e.g. the proof of Lemma 5.2). Therefore, by Lemma 7.2, we must have $P\prec_{M}N\mathbin{\overline{\otimes}}\mathcal{Z}(B)$. Since $\mathcal{Z}(B)$ is amenable, we get $P\lessdot_{M}N$. Finally, since $P$ and $N$ are tensor factors, Lemma 5.2 is applied and we conclude $P\prec_{M}N$. ∎

We next prove Theorem D. We prepare a lemma.

Lemma 7.6**.**

Let $A$ and $B$ be two $\sigma$-finite factors and let $G\curvearrowright A$ and $H\curvearrowright B$ be two outer actions of discrete groups $G$ and $H$. Suppose that $M=A\rtimes G=B\rtimes H$ with $A\prec_{M}B$. Then there exists a unitary $u\in\mathcal{U}(M)$, a normal subgroup $G_{0}\lhd G$ and a finite normal subgroup $H_{0}\lhd H$ such that $u(A\rtimes G_{0})u^{*}=B\rtimes H_{0}$. If we also have $B\prec_{M}A$, then $G_{0}$ is also finite.

Proof.

By [Is19, Proposition 4.4], we can find a unitary $u\in\mathcal{U}(M)$ such that $uAu^{*}\subset B\rtimes H_{0}$ for some finite normal subgroup $H_{0}\lhd H$. Since $A\subset A\rtimes G$ has the intermediate subfactor property, we know that $u^{*}(B\rtimes H_{0})u=A\rtimes G_{0}$ for some subgroup $G_{0}<G$. Since $B\rtimes H_{0}$ is regular in $M$, the subgroup $G_{0}$ must be normal in $G$. If we also assume that $B\prec_{M}A$, then $B\rtimes H_{0}\prec_{M}A$ because $H_{0}$ is finite. Thus we get $A\rtimes G_{0}\prec_{M}A$ and this forces $G_{0}$ to be finite. ∎

Proof of Theorem D.

Let $M=A\rtimes G=B\rtimes H$. Let $Q$ be a diffuse finite subalgebra with expectation in $A_{0}^{\prime}\cap A$. By, Theorem 7.1, we have either

• $(\rm i)$

$Q\prec_{M}L(H)$;

• $(\rm ii)$

$Q\prec_{M}B_{0}^{F}$ for some finite subset $F\subset H$; or

• $(\rm iii)$

$A_{0}\lessdot_{M}B$.

In the case $(\rm i)$, by applying Lemma 7.3 three times we get $A_{0}\prec_{M}L(H)$, then $A\prec_{M}L(H)$ and finally $M\prec_{M}L(H)$ which is not possible. In the case $(\rm ii)$, by applying Lemma 7.2, we get $A_{0}\prec_{M}B$ which implies condition $(\rm iii)$. Finally, assume $(\rm iii)$ holds. Then, since $B$ is the increasing union of $B_{0}^{F}$ over finite subsets $F\subset H$, Proposition 2.5 implies that $\mathord{\text{\rm L}}^{2}(A_{0})\prec\mathcal{H}=\bigoplus_{F}\mathord{\text{\rm L}}^{2}\langle M,B_{0}^{F}\rangle$ as $A_{0}$-$A_{0}$-bimodules.

On the other hand, by Proposition 7.4, we have $\mathord{\text{\rm L}}^{2}\langle M,B_{0}^{F}\rangle\prec\mathord{\text{\rm L}}^{2}(M)$ as $M$-$M$-bimodules. Moreover, ${}_{A_{0}}\mathord{\text{\rm L}}^{2}(A)_{A_{0}}$ is a multiple of $\mathord{\text{\rm L}}^{2}(A_{0})$ while ${}_{A_{0}}(\mathord{\text{\rm L}}^{2}(M)\ominus\mathord{\text{\rm L}}^{2}(A))_{A_{0}}$ is a multiple of the coarse bimodule $\mathord{\text{\rm L}}^{2}(A_{0})\otimes\mathord{\text{\rm L}}^{2}(A_{0})$. This shows that ${}_{A_{0}}\mathord{\text{\rm L}}^{2}(M)_{A_{0}}\prec\mathord{\text{\rm L}}^{2}(A_{0})$. Thus we have $\mathord{\text{\rm L}}^{2}\langle M,B_{0}^{F}\rangle\prec\mathord{\text{\rm L}}^{2}(A_{0})$ as $A_{0}$-$A_{0}$-bimodules for every finite subset $F\subset H$.

Therefore we have showed that $\bigoplus_{F}\mathord{\text{\rm L}}^{2}\langle M,B_{0}^{F}\rangle$ is weakly equivalent to $\mathord{\text{\rm L}}^{2}(A_{0})$ as an $A_{0}$-$A_{0}$-bimodule. Since $A_{0}$ is a full factor, Proposition 5.1 implies that $A_{0}\prec_{M}B_{0}^{F}$ for some finite subset $F\subset H$. By Lemma 7.2, we conclude that $A\prec_{M}B$. Similarly, we have $B\prec_{M}A$ and we can therefore apply Lemma 7.6. ∎

8. Compact actions of higher rank lattices

In this section, we prove Theorem E and F. We first translate the unique prime factorization property, using the flip map $\sigma_{P}$ on the double $\widehat{M}$.

Proposition 8.1**.**

Let $\mathcal{C}$ be a set of factors. Then the following are equivalent:

• $(\rm i)$

Every $P\in\mathcal{C}$ is prime and for every finite family $P_{1},\dots,P_{n}\in\mathcal{C}$, the factor $M=P_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}P_{n}$ has the Unique Prime factorization property.

• $(\rm ii)$

For every finite family $P_{1},\dots,P_{n}\in\mathcal{C}$, and every automorphism $\alpha$ of the factor $M=P_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}P_{n}$, there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $\alpha(P_{i})\sim_{M}P_{\sigma(i)}$ for all $i\in\{1,\dots,n\}$.

Proof.

$(\rm i)\Rightarrow(\rm ii)$ is obvious. Let us prove the other direction. Assume that $(\rm ii)$ holds. Let $P\in\mathcal{C}$ and take $Q\in\mathrm{TF}(P)$. Then by $(\rm ii)$, the automorphism $\sigma_{Q}$ of $P\mathbin{\overline{\otimes}}P$ satisfies $\sigma_{Q}(P\otimes 1)\sim_{\widehat{P}}1\otimes P$ or $\sigma_{Q}(P\otimes 1)\sim_{\widehat{P}}P\otimes 1$. Applying Lemma 3.3, in the first case, we get $Q^{c}\prec_{P}\mathbf{C}$ and in the second case we get $Q\prec_{P}\mathbf{C}$. Thus, $Q$ or $Q^{c}$ is of type $\mathrm{I}$. This shows that $P$ is prime. Now, consider $P_{1},\dots,P_{n}\in\mathcal{C}$ and $M=P_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}P_{n}$ and take $Q\in\mathrm{TF}(M)$. By $(\rm ii)$, for every $i$, we must have $\sigma_{Q}(P_{i})\sim_{M\mathbin{\overline{\otimes}}M}P_{j}\otimes 1$ or $\sigma_{Q}(P_{i})\sim_{M\mathbin{\overline{\otimes}}M}1\otimes P_{j}$ for some $j\in\{1,\dots,n\}$. In the first case, we get $P_{i}\prec_{M}Q^{c}$ and in the second case we get $P_{i}\prec_{M}Q$. We conclude that $M$ has the UPF property. ∎

In what follows, by higher rank irreducible lattice we mean an irreducible lattice $\Gamma<G$ where $G$ is a connected semisimple Lie group with finite center such that every simple quotient of $G$ has real rank $\geq 2$. It is known that such a lattice $\Gamma$ has property (T) and satisfies the conclusion of Margulis’ normal subgroup theorem, i.e. any normal subgroup $N<\Gamma$ is either finite (and contained in the center) or has finite index in $\Gamma$.

We will need the following elementary lemma. Recall that two subgroups $H_{1},H_{2}$ of a same group $H$ are commensurable if $H_{1}\cap H_{2}$ has finite index in both $H_{1}$ and $H_{2}$.

Lemma 8.2**.**

Let $L_{1},\dots,L_{n},R_{1},\dots,R_{n}$ be irreducible higher rank lattices. Let $H<L_{1}\times\cdots\times L_{n}$ and $K<R_{1}\times\dots\times R_{n}$ be two finite index subgroups and $\phi:H\rightarrow K$ an isomorphism. Then there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $\phi(L_{i}\cap H)$ and $R_{\sigma(i)}$ are commensurable for all $i\in\{1,\dots,n\}$.

Proof.

We proceed by induction. The result is obvious for $n=1$. Let $n\geq 2$ and suppose that we have proved the result for $n-1$. For each $i$, let $\pi_{i}$ be the projection on $R_{i}$. Observe that $L_{n}\cap H$ has finite index in $L_{n}$. In particular, $L_{n}\cap H$ is infinite. Thus there exists $i$ such that $\pi_{i}(\phi(L_{n}\cap H))$ is infinite. Assume, without loss of generality, that $i=n$. Note that $\phi(L_{n}\cap H)$ is a normal subgroup of $K$. Thus $\pi_{n}(\phi(L_{n}\cap H))$ is a normal subgroup of $\pi_{n}(K)\subset R_{n}$. But $\pi_{n}(K)$ is an irreducible lattice because it has finite index in $R_{n}$. Therefore, since $\pi_{n}(\phi(L_{n}\cap H))$ is infinite, it must actually have finite index in $\pi_{n}(K)$, hence also in $R_{n}$. But, if we let $H^{\prime}=(L_{1}\times\dots\times L_{n-1})\cap H$, then $\pi_{n}(\phi(H^{\prime}))$ is also a normal subgroup of $\pi_{n}(K)$ which commutes with $\pi_{n}(\phi(L_{n}\cap H))$. Thus $\pi_{n}(\phi(H^{\prime}))$ is finite. Let $H^{\prime\prime}\subset H^{\prime}$ be the kernel of $\pi_{n}\circ\phi|_{H^{\prime}}$. We have that $H^{\prime\prime}$ is a finite index subgroup of $L_{1}\times\dots\times L_{n-1}$ and $\phi(H^{\prime\prime})\subset R_{1}\times\dots\times R_{n-1}$. Therefore, we can apply the induction hypothesis and wet get that $\phi(H^{\prime\prime}\cap L_{i})$ and $R_{\sigma(i)}$ are commensurable for some permutation $\sigma$ of $\{1,\dots,n-1\}$. Since $H^{\prime\prime}$ has finite index in $H$, we actually have that $\phi(H\cap L_{i})$ and $R_{\sigma(i)}$ are commensurable. It only remains to show that $\phi(H\cap L_{n})$ and $R_{n}$ are commensurable.

We know that $\pi_{i}(\phi(H\cap L_{n}))$ is finite for all $i\leq n-1$ because it is a normal subgroup of $\pi_{i}(K)$ and it commutes with $\phi(H\cap L_{i})\cap R_{i}$ which has finite index in $R_{i}$. Thus the kernel of $\pi_{i}|_{\phi(H\cap L_{n})}$ has finite index in $\phi(H\cap L_{n})$ for all $i\leq n-1$. We deduce that the intersection of all this kernels, which is precisely $\phi(H\cap L_{n})\cap R_{n}$, has finite index in $\phi(H\cap L_{n})$. It also has finite index in $R_{n}$ because it is an infinite normal subgroup of $K\cap R_{n}$. We conclude that $\phi(L_{n}\cap H)$ and $R_{n}$ are commensurable. ∎

Proof of Theorem E.

Let $M_{i}=A_{i}\rtimes\Gamma_{i}$, $A=A_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}A_{n}$, $\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$ and $M=M_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}M_{n}=A\rtimes\Gamma$. Let $\alpha$ be an automorphism of $M$. By Proposition 8.1, we have only to show that there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $\alpha(M_{i})\sim_{M}M_{\sigma(i)}$ for all $i\in\{1,\dots,n\}$. By [BIP18, Theorem 1.4], up to composing $\alpha$ by an inner automorphism, we may assume that $\alpha(A)=A$. By [Io08, Theorem A] (see also [Fu09, Theorem 5.21]), up to composing $\alpha$ by $\operatorname{Ad}(u)$ for some $u\in\mathcal{N}_{M}(A)$, we may further assume that $\alpha$ induces a virtual self-conjugacy of the action $\Gamma\curvearrowright A$. In particular, there exists finite index subgroups $H,K\subset\Gamma_{1}\times\dots\times\Gamma_{n}$ and an isomorphism $\theta:H\rightarrow K$ such that $\alpha(u_{g})\in\mathbf{T}u_{\theta(g)}$ for all $g\in H$. By Lemma 8.2, there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $\theta(H\cap\Gamma_{i})$ and $\Gamma_{\sigma(i)}$ are commensurable for all $i\in\{1,\dots,n\}$. Let $F_{i}=\theta(H\cap\Gamma_{i})\cap\Gamma_{\sigma(i)}$ which has finite index in $\Gamma_{\sigma(i)}$. Then $L(\Gamma_{\sigma(i)})\prec_{M}L(F_{i})\subset\alpha(L(\Gamma_{i}))$. By taking relative commutants, we obtain $\alpha(M_{i})^{\prime}\cap M\prec_{M}(L(\Gamma_{\sigma(i)})^{\prime}\cap M_{\sigma(i)})\mathbin{\overline{\otimes}}(M_{\sigma(i)}^{\prime}\cap M)$. Since $L(\Gamma_{i})^{\prime}\cap M_{i}=L(\mathcal{Z}(\Gamma_{i}))$ is a finite dimensional abelian algebra, we get $\alpha(M_{i})^{\prime}\cap M\prec_{M}M_{\sigma(i)}^{\prime}\cap M$. Finally, by taking relative commutants again, we conclude that $M_{\sigma(i)}\prec_{M}\alpha(M_{i})$ for all $i\in\{1,\dots,n\}$ as we wanted. ∎

Now, we will prove Theorem F. We will need the following intertwining lemma.

Lemma 8.3**.**

Let $\Gamma\curvearrowright N$ be a minimal action of an ICC group $\Gamma$ on a ${\rm II}_{1}$ factor $N$ and let $M=N\rtimes\Gamma$. Let $P\in\mathrm{TF}(M)$ and suppose that $L(\Gamma)\prec_{M}P$. Then there exists a tensor product decomposition $N=A\mathbin{\overline{\otimes}}B$ with $\Gamma$ acting trivially on $B$ and a unitary $u\in\mathcal{U}(M)$ such that $uPu^{*}=A\rtimes\Gamma$.

Proof.

Since $\Gamma$ is ICC and acts minimally on $N$, we have that $L(\Gamma)$ and $L(\Gamma)^{\prime}\cap M=N^{\Gamma}$ are ${\rm II}_{1}$ factors. Since by assumption $L(\Gamma)\prec_{M}P$, the proof of [OP03, Proposition 12], shows that we have $L\Gamma\subset P_{0}$ for some $P_{0}\in\mathrm{TF}(M)$ with $P_{0}\sim_{M}P$. Put $B=P_{0}^{\prime}\cap M$ and observe that $B\subset L(\Gamma)^{\prime}\cap M=N^{\Gamma}$. Since $B\in\mathrm{TF}(M)$, we can write $N=A\mathbin{\overline{\otimes}}B$ by putting $A=B^{\prime}\cap N\subset P_{0}$ and we get $P_{0}=A\rtimes\Gamma$ as we wanted. Finally, since $P\sim P_{0}$, we have $P=uP_{0}^{t}u^{*}=u(A^{t}\rtimes\Gamma)u^{*}$ for some $u\in\mathcal{U}(M)$ and $t>0$ where $N=A^{t}\mathbin{\overline{\otimes}}B^{1/t}$. ∎

Proof of Theorem F.

The main step is to prove the following fact.

Claim**.**

For every tensor product decomposition $M=P\mathbin{\overline{\otimes}}Q$, there exists a unitary $u\in\mathcal{U}(M)$, a direct product decomposition $\Gamma=G\times H$ and a tensor product decomposition $R=A\mathbin{\overline{\otimes}}B$ with $G$ and $H$ acting trivially on $B$ and $A$ respectively, such that $uPu^{*}=A\rtimes G$ and $uQu^{*}=B\rtimes H$.

Proof of the claim.

Consider the double action of $\widehat{\Gamma}=\Gamma_{1}\times\Gamma_{2}$ on $\widehat{R}=R_{1}\mathbin{\overline{\otimes}}R_{2}$ and let $\widehat{M}=M_{1}\mathbin{\overline{\otimes}}M_{2}=\widehat{R}\rtimes\widehat{\Gamma}$. Since the action of $\Gamma$ on $R$ is compact, then so is the action of $\widehat{\Gamma}$ on $\widehat{R}$. Thus, we have $\sigma_{P}(\widehat{R})\prec_{\widehat{M}}\widehat{R}$ by [BIP18, Theorem 4.16]. By Lemma 7.6, we can find a unitary $u\in\mathcal{U}(\widehat{M})$ such that $\alpha(\widehat{R})=\widehat{R}$ where $\alpha=\operatorname{Ad}(u)\circ\sigma_{P}$. Then there exists $\theta\in\operatorname{Aut}(\Gamma_{1}\times\Gamma_{2})$ such that $\alpha(u_{g})\in\widehat{R}u_{\theta(g)}$ for all $g\in\Gamma_{1}\times\Gamma_{2}$ (where $u_{g}$ is a canonical unitary in $\widehat{R}\rtimes\widehat{\Gamma}$ for $g\in\widehat{\Gamma}$). Since $\Gamma$ is ICC, we can find a direct product decomposition $\Gamma=G\times H$ such that $\theta(\Gamma_{2})=G_{1}\times H_{2}$. This implies that

[TABLE]

Here $G_{1}$ and $H_{2}$ have property (T) so that every central sequence of $L$ lies in $R_{1}^{G_{1}}\mathbin{\overline{\otimes}}R_{2}^{H_{2}}$. But $\alpha(R_{1})\in\mathrm{TF}(L)$ is amenable and therefore we get $\alpha(R_{1})\prec_{L}R_{1}^{G_{1}}\mathbin{\overline{\otimes}}R_{2}^{H_{2}}$ (if not, one can construct a central sequence from $\alpha(R_{1})$ which is away from $R_{1}^{G_{1}}\mathbin{\overline{\otimes}}R_{2}^{H_{2}}$). By taking commutants inside $L$, we get $L(G_{1})\mathbin{\overline{\otimes}}L(H_{2})\prec_{L}\alpha(M_{2})=u(P_{1}\mathbin{\overline{\otimes}}Q_{2})u^{*}$. We conclude that $L(G)\prec_{M}P$ and $L(H)\prec_{M}Q$. Now, we will show that, up to exchanging $P,Q$ with equivalent ones in $\mathrm{TF}(M)$ and up to unitary conjugating in $M$, we actually have $L(G)\subset P$ and $L(H)\subset Q$.

By applying Lemma 8.3 to $G$ acting on $N:=R\rtimes H$, we may assume that $L(G)\subset P$, $Q\subset L(G)^{\prime}\cap N=R^{G}\rtimes H$ and $Q\in\mathrm{TF}(N)$. We still have $L(H)\prec_{M}Q$. We claim that we actually have $L(H)\prec_{N}Q$. Indeed, as an $N$-$N$-bimodule, we have $\mathord{\text{\rm L}}^{2}(M)=\bigoplus_{g\in G}\mathord{\text{\rm L}}^{2}(N)u_{g}$. But $L(H)$ and $Q$ are both fixed by $G$. Thus, we have that ${{}_{L(H)}}\mathord{\text{\rm L}}^{2}(M)_{Q}$ is unitarily equivalent to a mutiple ${{}_{L(H)}}\mathord{\text{\rm L}}^{2}(N)_{Q}$. This shows that $L(H)\prec_{N}Q$. Now, by applying Lemma 8.3 again to $H$ acting on $R$, up to same equivalences as before, we have a tensor product decomposition $R=C\mathbin{\overline{\otimes}}D$ with $H$ acting trivially on $C$ such that $Q=D\rtimes H$ (but a priori, we no longer have $L(G)\subset P$). Put

[TABLE]

and observe that $Z=P\mathbin{\overline{\otimes}}(L(H)^{\prime}\cap Q)=P\mathbin{\overline{\otimes}}D^{H}$ (because $L(H)\subset Q\in\mathrm{TF}(M)$). Then since $D^{H}\in\mathrm{TF}(Z)$ is amenable and $G$ has property (T), the same reasoning as above shows that $D^{H}\prec_{Z}L(G)^{\prime}\cap Z=R^{\Gamma}$. By taking the commutants inside $Z$, we get $L(G)\prec_{Z}P$. Now, Lemma 8.3 implies that there exists a unitary $u\in\mathcal{U}(Z)$ such that $L(G)\subset uPu^{*}$. Since $Z$ commutes with $L(H-$, we still have $L(H)\subset uQu^{*}$ and therefore we may assume that $L(G)\subset P$ and $L(H)\subset Q$.

Since $\Gamma$ is ICC, hence also $G$ and $H$, we have $Q\subset L(G)^{\prime}\cap M=R^{G}\rtimes H$. Thus $R\rtimes H=A\mathbin{\overline{\otimes}}Q$ for some $A\subset R\rtimes H$. Since $A$ commutes with $L(H)\subset Q$, we have $A\subset R^{H}$. In particular, $R=A\mathbin{\overline{\otimes}}B$ for some $B\subset Q$. Then $G$ acts trivially on $B$ and $H$ acts trivially on $A$ and $P=A\rtimes G$ and $Q=B\rtimes H$ as we wanted. ∎

We can now prove items $(\rm i)$, $(\rm ii)$ and $(\rm iii)$ of Theorem F.

$(\rm i)$ Let $P$ be a tensor factor of $M$ which is not of type $\mathrm{I}$. By the claim, $P$ is unitarily conjugate to a factor of the form $A\rtimes G$ where $A$ must be a hyperfinite ${\rm II}_{1}$ factor. Note that $G\curvearrowright A$ is a compact minimal action. By the uniqueness of minimal actions of compact groups on the hyperfinite ${\rm II}_{1}$ factor [MT06], we know that $\sigma:G\curvearrowright A$ is conjugate to $\sigma\otimes 1:G\curvearrowright A\mathbin{\overline{\otimes}}R$. Thus $P\cong A\rtimes G\cong(A\rtimes G)\mathbin{\overline{\otimes}}R$ is McDuff.

$(\rm ii)$ Suppose that $M$ is not semi-prime and write $M=P\mathbin{\overline{\otimes}}Q$ where $P$ and $Q$ are nonamenable. Then by the claim, we can assume that $P=A\rtimes G$ and $Q=B\rtimes H$ where $\Gamma=G\times H$ is a nontrivial direct product decomposition and $R=A\mathbin{\overline{\otimes}}B$. If $P$ or $Q$ is not semi-prime, we can repeat this procedure. This must stop at some point because the length of a direct product decomposition of the lattice $\Gamma$ is bounded by its rank.

$(\rm iii)$ Suppose that we have a tensor product decomposition $M=M_{1}\mathbin{\overline{\otimes}}\cdots\mathbin{\overline{\otimes}}M_{n}$ with $M_{i}=R_{i}\rtimes\Gamma_{i}$ as in item $(\rm ii)$ of the theorem. Let $M=P\mathbin{\overline{\otimes}}Q$ be another tensor product decomposition with $P$ and $Q$ nonamenable. By the claim, we may assume that $P=A\rtimes G$ and $Q=B\rtimes H$ with $\Gamma=G\times H$. Since $\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$ is ICC, we have a decomposition $\Gamma_{i}=G_{i}\times H_{i}$ for all $i\leq n$ with $G=G_{1}\times\cdots\times G_{n}$ and $H=H_{1}\times\cdots\times H_{n}$. Let $K=\overline{\Gamma}$ be the closure of $\Gamma$ in $\operatorname{Aut}(R)$. Then we have $K=\overline{\Gamma_{1}}\times\cdots\times\overline{\Gamma_{n}}=\overline{G}\times\overline{H}$. Thus $\overline{\Gamma_{i}}=\overline{G_{i}}\times\overline{H_{i}}$. By the uniqueness of the minimal action of $\overline{\Gamma_{i}}$ on the hyperfinite ${\rm II}_{1}$ factor, we know that the action $\overline{\Gamma_{i}}\curvearrowright A_{i}$ is conjugate to a tensor product of a minimal action of $\overline{G_{i}}$ and a minimal action of $\overline{H_{i}}$. Since $R_{i}\rtimes\Gamma_{i}$ is semi-prime, this forces either $G_{i}$ or $H_{i}$ to be amenable, hence finite because it has property (T), hence trivial because $\Gamma$ is ICC. Since this holds for all $i\in\{1,\cdots,n\}$, we conclude that $G=\times_{i\in I}\Gamma_{i}$ and $H=\times_{j\in J}\Gamma_{j}$ for some partition $I\sqcup J=\{1,\dots,n\}$. Now, since $R$ is hyperfinite, it has only one nontrivial tensor product decomposition up to conjugacy by an automorphism. Thus, we can find $\theta\in\operatorname{Aut}(R)$ such that $\theta(A)=\mathbin{\overline{\otimes}}_{i\in I}R_{i}$ and $\theta(B)=\mathbin{\overline{\otimes}}_{j\in J}R_{j}$. By the uniqueness of the minimal action of $\overline{G}$ and $\overline{H}$ on the hyperfinite ${\rm II}_{1}$ factor, we may assume that $\theta|_{A}$ is $G$-equivariant and that $\theta|_{B}$ is $H$-equivariant, hence that $\theta$ is $\Gamma$-equivariant. We conclude that $\theta$ extends to an automorphism of $M$ such that $\theta(P)=\mathbin{\overline{\otimes}}_{i\in I}M_{i}$ and $\theta(Q)=\mathbin{\overline{\otimes}}_{j\in J}M_{j}$. ∎

9. Full factors without Unique Prime Factorization

The goal of this section is to provide examples of full factors which do not satsify the UPF property. We first need to recall some definitions and make some general observations about the notion of spectral gap.

We say that a a unitary representation $\pi:G\rightarrow\mathcal{U}(H)$ has spectral gap if it has no almost invariant vectors, or equivalently, if there exists a finite set $K\subset G$ such that $\|\frac{1}{|K|}\sum_{g\in K}\pi(g)\|<1$. Such a set $K$ will be called a critical set.

Following [Po06b, Section 3], we say that a representation $\pi:G\rightarrow\mathcal{U}(H)$ has stable spectral gap if $\pi\otimes\rho$ has spectral gap for any other representation $\rho:G\rightarrow\mathcal{U}(K)$. It is known that $\pi$ has stable spectral gap if and only if the representation $\pi\otimes\bar{\pi}$ of $G$ on $H\otimes\overline{H}$ has spectral gap [Po06b, Lemma 3.2].

Proposition 9.1**.**

Let $G\curvearrowright I$ be an action of a group $G$ on a set $I$ and let $\pi:G\curvearrowright\ell^{2}(I)$ be the associated representation. Then $\pi$ has spectral gap if and only if it has stable spectral gap.

Proof.

This is the idea of [Ch81]. Suppose that $\pi$ has spectral gap and let $\rho:G\curvearrowright H$ be any unitary representation of $G$. Let $\xi_{n}\in\ell^{2}(I)\otimes H$ be a sequence of $\pi\otimes\rho$ almost invariant vectors. View each $\xi_{n}$ as a function $\xi_{n}:I\ni i\mapsto\xi_{n}(i)\in H$. Define a sequence $\eta_{n}\in\ell^{2}(I)$ by $\eta_{n}(i)=\|\xi_{n}(i)\|$ for all $i\in I$. Then $(\eta_{n})_{n\in\mathbf{N}}$ is a sequence of almost invariant vectors for $\pi$. Thus $\|\xi_{n}\|=\|\eta_{n}\|\to 0$ when $n\to\infty$. This shows that $\pi\otimes\rho$ has spectral gap. ∎

When the representation $\pi$ in Proposition 9.1 has (stable) spectral gap, we will simply say that the action $G\curvearrowright I$ has spectral gap. Recall that an ICC group $G$ is non-inner amenable [Ef73] if and only if its action on itself by conjugation $\operatorname{Ad}:\Gamma\curvearrowright\Gamma\setminus\{1\}$ has spectral gap.

Similarly, we say that an action of a group $G$ on a ${\rm II}_{1}$ factor $M$ has (stable) spectral gap if the associated Koopman representation of $G$ on $\mathord{\text{\rm L}}^{2}(M)\ominus\mathbf{C}$ has (stable) spectral gap. We denote by $\operatorname{Ad}:\mathcal{U}(M)\curvearrowright M$ the canonical action of $\mathcal{U}(M)$ on $M$ by inner automorphisms. By [Co74], $M$ is full if and only if the action $\operatorname{Ad}$ has spectral gap. The next result shows that it actually has stable spectral gap, like in the group case. The proof is essentially due to A. Ioana. We warmly thank him for allowing us to reproduce it here.

Theorem 9.2**.**

Let $M$ be a full ${\rm II}_{1}$ factor. Then the adjoint action $\operatorname{Ad}:\mathcal{U}(M)\curvearrowright M$ has stable spectral gap.

Lemma 9.3**.**

Let $M$ be a ${\rm II}_{1}$ factor. Then there exists unitaries $u_{1},\dots,u_{n}\in\mathcal{U}(M)$ such that

[TABLE]

Proof.

If not, then there exists a net of unit vectors $(\xi_{i})_{i\in I}$ in $\mathord{\text{\rm L}}^{2}(M)\mathbin{\overline{\otimes}}\mathord{\text{\rm L}}^{2}(M)$ such that $(u\otimes u^{*})\xi_{i}-\xi_{i}\to 0$ for all $u\in\mathcal{U}(M)$, or equivalently $(a\otimes 1-1\otimes a)\xi_{i}\to 0$ for all $a\in M$. It is easy to see that this implies that $(ab\otimes 1-ba\otimes 1)\xi_{i}\to 0$ for all $a,b\in M$. But since $M$ is a ${\rm II}_{1}$ factor, we can find $a,b\in M$ such that $ab-ba$ is invertible. ∎

Proof of Theorem 9.2.

Let $H=\mathord{\text{\rm L}}^{2}(M)\ominus\mathbf{C}$ and we will show that the action $\operatorname{Ad}\otimes\overline{\operatorname{Ad}}$ on $H\otimes\overline{H}$ has spectral gap. Since $\overline{\operatorname{Ad}}$ is canonically identified with $\operatorname{Ad}$, we consider the action $\operatorname{Ad}\otimes\operatorname{Ad}$ on $H\otimes H$. Let $(\xi_{i})_{i}$ be a net of vectors in $H\otimes H$ such that

[TABLE]

for all $u\in\mathcal{U}(M)$. Let $I=\{T\in\mathbf{B}(\mathord{\text{\rm L}}^{2}(M\mathbin{\overline{\otimes}}M))\mid\lim_{i}\|T\xi_{i}\|=0\}$. Observe that $I$ is a norm closed left ideal. Thus we have $a-Ja^{*}J\in I$ for every $a$ in the $C^{*}$-algebra $A$ generated by $\{u\otimes u\mid u\in\mathcal{U}(M)\}\subset M\mathbin{\overline{\otimes}}M$, where $J$ is the canonical conjugation on $\mathord{\text{\rm L}}^{2}(M)\otimes\mathord{\text{\rm L}}^{2}(M)$. For every $h=h^{*}\in M$, we have

[TABLE]

By taking linear combinations, we get $x\otimes 1+1\otimes x\in A$ for all $x\in M$. Let $u\in\mathcal{U}(M)$ and $a=u\otimes 1+1\otimes u$. Since $a\in A$, we have $a^{*}-JaJ\in I$ hence

[TABLE]

By Lemma 9.3, by regarding $\mathcal{U}(M)$ as a discrete group, we can find a probability distribution $\mu_{1}\in\ell^{1}(\mathcal{U}(M))^{+}$ with finite support such that

[TABLE]

for some $\varepsilon>0$, where we used the notation

[TABLE]

Note that this implies that

[TABLE]

for any other $\nu\in\ell^{1}(\mathcal{U}(M))^{+}$. Since $M$ is full, the representation $\operatorname{Ad}$ has spectral gap [Co75], so we can find $\mu_{2}\in\ell^{1}(\mathcal{U}(M))^{+}$ such that

[TABLE]

Since $\mathcal{U}(M)\ni u\mapsto uJuJ$ is a representation, if we replace $\mu_{2}$ by $\mu_{2}^{\ast n}$ for some $n$ large enough, we can actually assume that

[TABLE]

Similarly, since $M$ is nonamenable, the representation $\mathcal{U}(M)\ni u\mapsto u\otimes JuJ$ has spectral gap, so we can find $\mu_{3}$ such that

[TABLE]

Finally, by letting $\mu=\mu_{3}\ast\mu_{2}\ast\mu_{1}$ and $f(u)=u\otimes u^{*}-uJuJ\otimes 1-u\otimes JuJ$ for $u\in\mathcal{U}(M)$, we obtain

[TABLE]

Thus $1+\mathbb{E}_{\mu}(f(u))$ is invertible. But $1+\mathbb{E}_{\mu}(f(u))\in I$ because $1+f(u)\in I$ for all $u\in\mathcal{U}(M)$. Since $I$ is a left ideal, we conclude that $1\in I$, i.e. $\lim_{i}\|\xi_{i}\|=0$ as we wanted. ∎

The next theorem provides a class of full ${\rm II}_{1}$ factors without the Unique Prime Factorization property. In fact, these factors have infinitely many tensor product decompositions up to stable unitary conjugacy. Under some assumptions (see [Is16] for the definition of strong primeness and examples), these tensor product decompositions can still be completely classified.

Theorem 9.4**.**

Let $M$ be a ${\rm II}_{1}$ factor and $G$ an ICC group. Let $\sigma_{0}:G\curvearrowright M$ be an action by inner automorphisms and consider the diagonal action $\sigma=\sigma_{0}^{\mathbf{N}}:G\curvearrowright M^{\mathbin{\overline{\otimes}}\mathbf{N}}$. Let $N=M^{\mathbin{\overline{\otimes}}\mathbf{N}}\rtimes_{\sigma}G$. Then the following properties are satisfied:

• $(\rm i)$

For every finite subset $F\subset\mathbf{N}$, $M^{\mathbin{\overline{\otimes}}F}\subset M^{\mathbin{\overline{\otimes}}\mathbf{N}}$ is a tensor factor of $N$ whose relative commutant is isomorphic to $N$. The tensor factors $M^{\mathbin{\overline{\otimes}}F}$ are pairwise not stably unitarily conjugate.

• $(\rm ii)$

If $G$ is non-inner amenable and $\sigma_{0}$ has stable spectral gap then $N$ is a full factor.

• $(\rm iii)$

If moreover $M$ is strongly prime and $G$ is a hyperbolic group, then for every $P\in\mathrm{TF}(N)$, there exists a finite subset $F\subset\mathbf{N}$ such that $P\sim M^{\mathbin{\overline{\otimes}}F}$ or $P^{c}\sim M^{\mathbin{\overline{\otimes}}F}$. In particular, $N$ does not admit any prime factorization.

Proof.

$(\rm i)$ This follows from the assumption that $\sigma_{0}$ is inner, which implies that $\sigma$ is also inner on every $M^{\mathbin{\overline{\otimes}}F}$ where $F\subset\mathbf{N}$ is a finite subset.

$(\rm ii)$ By [Ch81], it is enough to show that $\sigma$ has spectral gap. Let $H=\mathord{\text{\rm L}}^{2}(M)\ominus\mathbf{C}$. Let $\pi_{0}:G\rightarrow H$ the representation associated to $\sigma_{0}$. Then the representation associated to $\sigma$ is $\pi=\bigoplus_{F\subset\mathbf{N}}\pi_{0}^{\otimes F}$ on $\bigoplus_{F\subset\mathbf{N}}H^{\otimes F}$. In particular, $\pi$ is equivalent to $\pi_{0}\otimes\pi^{\prime}$ for some representation $\pi^{\prime}$. Since $\pi_{0}$ has stable spectral gap, we conclude that $\pi$ has spectral gap.

$(\rm iii)$ Let $B=M^{\mathbin{\overline{\otimes}}\mathbf{N}}$. By the proof of [Is16, Theorem C and Proposition 7.1.(1)], we have $P\prec_{N}B$ or $P^{c}\prec_{N}B$. Assume without loss of generality that $P\prec_{N}B$. Then $\mathord{\text{\rm L}}^{2}(P)$ is contained in ${{}_{P}}\mathord{\text{\rm L}}^{2}\langle N,B\rangle_{P}$. Since $B$ is the increasing union of $M^{\mathbin{\overline{\otimes}}F}$ over finite subsets $F\subset\mathbf{N}$, Proposition 2.5 implies

[TABLE]

as $P$-$P$-bimodules. By Lemma 5.2, we get $P\prec_{N}M^{\mathbin{\overline{\otimes}}F}$ for some finite subset $F\subset\mathbf{N}$. Since $M$ is strongly prime, we get $P\sim_{N}M^{\mathbin{\overline{\otimes}}K}$ for some subset $K\subset F$ (see the proof of [Is16, Proposition D]). ∎

Example 9.5**.**

Let $G$ be any non inner amenable ICC group. Let $M=L(G)$ and $\sigma_{0}:G\curvearrowright M$ the action by inner conjugation. Then the assumptions of $(\rm ii)$ are satisfied thanks to Proposition 9.1. If $G$ is hyperbolic, then $(\rm iii)$ is also satisfied [Is16].

Example 9.6**.**

Let $M$ be any full ${\rm II}_{1}$ factor. By Theorem 9.2, we can find a critical family of unitaries $u_{1},\dots,u_{n}\in\mathcal{U}(M)$ witnessing the stable spectral gap of $\operatorname{Ad}:\mathcal{U}(M)\curvearrowright M$. Define an action $\sigma_{0}:\mathbf{F}_{n}\curvearrowright M$ of the free group $\mathbf{F}_{n}=\langle a_{1},\dots,a_{n}\rangle$ by letting $\sigma_{0}(a_{k})=\operatorname{Ad}(u_{k})$ for all $k\in\{1,\dots,n\}$. Then $\sigma_{0}$ has stable spectral gap and $\mathbf{F}_{n}$ is non-inner amenable. Thus $N=M^{\mathbin{\overline{\otimes}}\mathbf{N}}\rtimes_{\sigma}\mathbf{F}_{n}$ is a full ${\rm II}_{1}$ factor which satisfies $N\cong N\mathbin{\overline{\otimes}}M$. Note that $\mathbf{F}_{n}$ is hyperbolic, so if $M$ is strongly prime, then property $(\rm iii)$ is also satisfied.

Corollary 9.7**.**

For every full (separable) ${\rm II}_{1}$ factor $M$, there exists a full (separable) ${\rm II}_{1}$ factor $N$ such that $N\cong N\mathbin{\overline{\otimes}}M$.

Our next result provides an example of a full ${\rm II}_{1}$ factor $M$ such that $M\cong M\mathbin{\overline{\otimes}}M$. In fact, we will use a remarkable construction due to Meier [Me82] of a finitely generated group $G$ such that $G\cong G\times G$. Let us recall the construction of $G$. Consider first the following group

[TABLE]

Note that $\{a,t\}$ and $\{b,s\}$ generate two copies of the Baumslag-Solitar group

[TABLE]

Moreover, $t$ and $[a,tat^{-1}]$ freely generate a free subgroup of rank $2$ inside $\mathrm{BS}(2,3)$. Therefore, we can think of $T$ as an amalgamated free product of two copies of $\mathrm{BS}(2,3)$ with amalgamation over $\mathbf{F}_{2}$. Next, we consider $T^{\mathbf{N}}$ the infinite product group (which is uncountable) and we embed $T\subset T^{\mathbf{N}}$ diagonally. We define an element $h=(a,a^{2},a^{3},a^{4},\dots)\in T^{\mathbf{N}}$ and finally we let $G$ be the subgroup of $T^{\mathbf{N}}$ generated by $T$ and $h$. Then one can show that the isomorphism

[TABLE]

sends $G$ onto $G\times G$. In particular, $G\cong G\times G$.

Proposition 9.8**.**

Meier’s group $G$ is non-inner amenable. In particular $M=L(G)$ is a full separable ${\rm II}_{1}$ factor which satisfies $M\cong M\mathbin{\overline{\otimes}}M$.

Proof.

First, we observe that $T$ is non-inner amenable. Indeed, $T$ is an amalgamated free product with amalgamation over a free group of rank $2$. By [DTW19, Theorem 1.1], we know that any conjugacy invariant mean on $T$ must be supported on the amalgam hence trivial because free groups of rank $2$ are non-inner amenable. Thus $T$ is not inner amenable.

Now, consider the action of $T$ by conjugation on $T^{\mathbf{N}}\setminus\{1\}$ diagonally. We claim that this action has spectral gap and this will imply a fortiori that $G$ is non-inner amenable because $T\subset G\subset T^{\mathbf{N}}$. Let $\pi:T\curvearrowright\ell^{2}(T^{\mathbf{N}}\setminus\{1\})$ be the associated representation. Observe that a sequence $(t_{n})_{n\in\mathbf{N}}\in T^{\mathbf{N}}$ is non-trivial if and only if there exists at least one $n\in\mathbf{N}$ such that $t_{n}\neq 1$. This shows that $\pi$ is contained in a multiple of $\pi_{0}\otimes\rho$ where $\pi_{0}:T\curvearrowright\ell^{2}(T\setminus\{1\})$ and $\rho:T\curvearrowright\ell^{2}(T^{\mathbf{N}})$. Since $T$ is non-inner amenable, $\pi_{0}$ has spectral gap. Thus $\pi_{0}\otimes\rho$ also has spectral gap by Proposition 9.1. We conclude that $\pi$ has spectral gap. ∎