Prime II$_1$ factors arising from actions of product groups

TL;DR

This paper proves that II$_1$ factors from free ergodic actions of product groups with specific properties are prime and provides a classification of their tensor product decompositions, leading to unique prime factorization results.

Contribution

It establishes primeness of II$_1$ factors from product group actions and classifies their tensor product decompositions, advancing the understanding of their structure.

Findings

Proves primeness of II$_1$ factors from certain product group actions.

Classifies tensor product decompositions of these factors.

Establishes unique prime factorization for a broad class of II$_1$ factors.

Abstract

We prove that any II$_1$ factor arising from a free ergodic probability measure preserving action $\Gamma\curvearrowright X$ of a product $\Gamma=\Gamma_1\times\dots\times\Gamma_n$ of icc hyperbolic, free product or wreath product groups is prime, provided $\Gamma_i\curvearrowright X$ is ergodic, for any $1\leq i\leq n.$ We also completely classify all the tensor product decompositions of a II$_1$ factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II$_1$ factors. Finally, we obtain a unique prime factorization theorem for a large class of II$_1$ factors which have property Gamma.

Equations

$$P_1^t=u(L^{\infty }(X{)}^{{\Gamma }_{T_2}}⋊{\Gamma }_{T_1})u^{\ast }\text{ and }{P}_2^{1/t}=u(L^{\infty }(X{)}^{{\Gamma }_{T_1}}⋊{\Gamma }_{T_2})u^{\ast }.$$

$$P_1\prec L^{\infty }(X)⋊{\Gamma }_{T_1},\text{ and }{P}_2\prec L^{\infty }(X)⋊{\Gamma }_{T_2},$$

$$L^{\infty }(X)⋊{\Gamma }_{T_1}\prec P_1\bar{\otimes }{D}_2,\text{ and }{L}^{\infty }(X)⋊{\Gamma }_{T_2}\prec D_1\bar{\otimes }{P}_2,$$

$$D_2\sim {L^{\infty }(X)}^{{\Gamma }_{T_1}},\text{ and }{D}_1\sim {L^{\infty }(X)}^{{\Gamma }_{T_2}}.$$

$$P_1={L^{\infty }(X)}^{{\Gamma }_{T_2}}⋊{\Gamma }_{T_1},\text{ and }{P}_2={L^{\infty }(X)}^{{\Gamma }_{T_1}}⋊{\Gamma }_{T_2},$$

$$\theta (x)v=vx,\text{ for all }x\in r_1{R}_1{r}_{1}s.$$

$$\Vert E_Q(x{u}_n(v_n{r}_{1}s)y){\Vert }_2\to 0,\text{ for all }x,y\in M.$$

$$\Vert E_Q(\theta (u_n)v{v}_{n}y){\Vert }_2=\Vert E_Q(v{v}_{n}y){\Vert }_2\to 0,\text{ for all }y\in M.$$

$$\Vert E_Q(xr{v}_{n}y){\Vert }_2\to 0,\text{ for all }x,y\in M.$$

$$\Vert E_Q(x(w_{1}r{w}_1^{\ast }\vee ...\vee w_{m}r{w}_m^{\ast })v_{n}y){\Vert }_2\to 0,$$

$$x{e}_{Q_1}{u}_1{u}_2{\otimes }_M{v}_1{v}_2{e}_{Q_2}y\to x{u}_1{v}_1{\otimes }_Q{u}_2{v}_{2}y$$

$$⟨x{e}_{Q_1}{u}_1{u}_2{\otimes }_M{v}_1{v}_2{e}_{Q_2}y,e_{Q_1}{u}_1{u}_2{\otimes }_M{v}_1{v}_2{e}_{Q_2}⟩=⟨x{u}_1{v}_1{\otimes }_Q{u}_2{v}_{2}y,u_1{v}_1{\otimes }_Q{u}_2{v}_2⟩,$$

$$\begin{array}[]{rcl}\langle{xe_{Q_{1}}}u_{1}u_{2}\otimes_{M}v_{1}v_{2}e_{Q_{2}}y,{e_{Q_{1}}}u_{1}u_{2}\otimes_{M}v_{1}v_{2}e_{Q_{2}}\rangle&=&\tau(E_{Q_{2}}(v_{2}^{*}v_{1}^{*}E_{M}(u_{2}^{*}u_{1}^{*}E_{Q_{1}}(x)u_{1}u_{2})v_{1}v_{2})y)\\ &=&\tau(v_{2}^{*}u_{2}^{*}E_{Q_{2}}(v_{1}^{*}u_{1}^{*}E_{Q_{1}}(x)u_{1}v_{1})u_{2}v_{2}y)\end{array}$$

$$⟨x{u}_1{v}_1{\otimes }_Q{u}_2{v}_{2}y,u_1{v}_1{\otimes }_Q{u}_2{v}_2⟩=\tau (v_2^{\ast }{u}_2^{\ast }{E}_{Q_2}(E_{Q_1}(v_1^{\ast }{u}_1^{\ast }x{u}_1{v}_1))u_2{v}_{2}y).$$

$$\lambda =\text{inf}\{\Vert E_P(x){\Vert }_2^2\Vert x{\Vert }_2^{-2}|x\in M_{+},x\ne 0\}.$$

$$\theta (x)v=vx,\text{ for all }x\in rRr.$$

$$\psi (x)w=wx,\text{ for all }x\in nNn.$$

$$\theta (\psi (x))vw=vwx,\text{ for all }x\in nNn.$$

$$A_2^{\prime }\cap M=A⋊{\Gamma }_1.$$

$$A⋊{\Gamma }_1=Q_1\bar{\otimes }(A_2^{\prime }\cap Q_2).$$

$$a(A⋊{\Gamma }_1)a=Q_{1}a\bar{\otimes }{A}_{2}a.$$

$$P_1^t=u(L^{\infty }(X{)}^{{\Gamma }_2}⋊{\Gamma }_1)u^{\ast }\text{ and }{P}_2^{1/t}=u(L^{\infty }(X{)}^{{\Gamma }_1}⋊{\Gamma }_2)u^{\ast }.$$

$$Q_i\vee (Q_i^{\prime }\cap q_i(A⋊{\Gamma }_i)q_i)\subset q_i(A⋊{\Gamma }_i)q_i\text{ has finite index, }$$

$${\theta }_i(x)v_i=v_{i}x,\text{ for all }x\in p_i{P}_i{p}_i,\text{ and }$$

$$E_{A⋊{\Gamma }_i}(v_i{v}_i^{\ast })q_i\ge {\lambda }_i{q}_i,\text{ for some positive number }{\lambda }_i.$$

$$rRr=\mathcal{Z}(rRr)=\mathcal{Z}(r(A⋊{\Gamma }_1)r)=A^{{\Gamma }_1}r.$$

$${\psi }_i(x)w_i=w_{i}x,\text{ for all }x\in A^{{\Gamma }_i}{k}_i.$$

$$E_{P_{i+1}}(w_i{w}_i^{\ast })r_{i+1}\ge {\beta }_{i+1}{r}_{i+1},\text{ for some positive number }{\beta }_{i+1}.$$

$$\theta (x)v=vx,\text{ for all }x\in Q_1{k}_1.$$

$${\tilde{p}}_2\le r_2\text{ or }{r}_2\le {\tilde{p}}_2.$$

Full text

1. Introduction

1.1. Background

In their pioneering work [MvN36, MvN43], Murray and von Neumann found a natural way to associate a II1 factor, denoted $L(\Gamma)$, to every countable infinite conjugacy class group $\Gamma$ and a II1 factor, denoted $L^{\infty}(X)\rtimes\Gamma$, to any free ergodic probability measure preserving action $\Gamma\curvearrowright(X,\mu).$ The classification of these group and group measure space von Neumann algebras is in general a very difficult problem. Nevertheless, a plethora of remarkable results have been obtained in the last 15 years due to S. Popa’s influential deformation/rigidity theory, see the surveys [Po07, Va10a, Io12a, Io17].

A central theme is the study of tensor product decompositions. A II1 factor $M$ is called prime if it cannot be decomposed as a tensor product of II1 factors. The uncovering of primeness results has been initially explored in the group von Neumann algebra setting. In [Po83], S. Popa has discovered the first examples of prime II1 factors by showing that the von Neumann algebra of any free group on uncountable many generators is prime. Using D. Voiculescu’s free probability theory, L. Ge provided the first examples of separable prime II1 factors by proving that the free group factors $L(\mathbb{F}_{n}),$ $2\leq n\leq\infty$, are also prime [Ge96]. By providing new methods in the C∗-algebraic setting, N. Ozawa proved that any infinite conjugacy class (icc) hyperbolic group $\Gamma$ gives rise to a solid II1 factor $L(\Gamma)$, meaning that the relative commutant of any diffuse subalgebra of $L(\Gamma)$ is amenable [Oz03]; in particular it follows that $L(\Gamma)$ is prime. In [Pe06], by developing an innovative technique based on closable derivations, J. Peterson showed primeness of $L(\Gamma)$, for any icc non-amenable group $\Gamma$ which has positive first Betti number. S. Popa then used his deformation/rigidity theory and gave an alternative proof of solidity of $L(\mathbb{F}_{n})$ [Po06b]. The intense research activity over the last decade has resulted in many other primeness results, see [Oz04, Po06a, CI08, CH08, Va10b, Bo12, HV12, DI12, CKP14, Ho15].

In all these results some negative curvature condition on $\Gamma$ is needed, in the form of a geometric assumption (e.g. $\Gamma$ is a hyperbolic group), or a cohomological assumption (e.g. the existence of a certain unbounded quasi-cocycle). Any of these two conditions can be seen as a “rank one” property. Concerning the primeness problem in the framework of group measure space von Neumann algebras, the techniques presented in the aforementioned papers can be used to show that any free ergodic probability measure preserving (pmp) action of such groups gives rise to a prime II1 factor. Specifically, N. Ozawa showed that $L^{\infty}(X)\rtimes\Gamma$ is prime whenever $\Gamma\curvearrowright(X,\mu)$ is a free ergodic pmp action of a non-elementary hyperbolic group [Oz04] (see also [CS11]). By obtaining new Bass-Serre type rigidity results for II1 factors, I. Chifan and C. Houdayer showed that the II1 factor associated to any free ergodic pmp action of a free product group is prime [CH08]. Then by developing methods from [Si10, Va10b], D. Hoff proved that $L^{\infty}(X)\rtimes\Gamma$ is prime whenever $\Gamma\curvearrowright(X,\mu)$ is a free ergodic pmp action of a group which has positive first Betti number [Ho15].

1.2. Statement of the main results

The first primeness results for group von Neumann algebras arising from icc irreducible lattices in higher rank semisimple Lie groups were obtained only recently in our joint work with D. Hoff and A. Ioana [DHI16] (see also [CdSS17, dSP18]). Recall that a lattice $\Gamma$ in a product $G=G_{1}\times\dots\times G_{n}$ of locally compact second countable groups is called irreducible if the action of $G$ on the homogeneous space $G/\Gamma$ is irreducible, meaning $G_{i}\curvearrowright G/\Gamma$ is ergodic for any $1\leq i\leq n.$ More generally, a pmp action $G\curvearrowright(X,\mu)$ is called irreducible if $G_{i}\curvearrowright(X,\mu)$ is ergodic for any $1\leq i\leq n.$

Despite all these advancements, the primeness problem for II1 factors arising from arbitrary free ergodic pmp actions of groups of “higher rank type” is largely open. Our results aim in this direction by finding a large class of product groups for which all their irreducible actions give rise to prime II1 factors, see Corollary B. These examples follow from our main technical result. Before stating the result, we introduce the following class of groups and explain the terminology that will be used.

Class $\mathcal{C}$. We say that a countable group $\Gamma$ belongs to the class $\mathcal{C}$ if one of the following conditions is satisfied:

1. (1)

$\Gamma$ is an icc, weakly amenable, bi-exact group (see [PV11] for terminology), or 2. (2)

$\Gamma=\Sigma_{1}*\Sigma_{2}$ is a free product of arbitrary groups such that $|\Sigma_{1}|\geq 2$ and $|\Sigma_{2}|\geq 3$, or 3. (3)

$\Gamma=\Sigma_{0}\wr\Gamma_{0}$ is the wreath product between a non-trivial amenable group $\Sigma_{0}$ and a non-amenable group $\Gamma_{0}.$

The symbol $\prec$ stands for Popa’s intertwining-by-bimodules technique (see Section 2.2). We denote by $M^{t}$ the amplification of the II1 factor $M$ by $t>0$ and for a pmp action $\Gamma\curvearrowright(X,\mu)$ we denote by $L^{\infty}(X)^{\Sigma}$ the subalgebra of elements of $L^{\infty}(X)$ fixed by a subgroup $\Sigma$ of $\Gamma$ (see Section 2.1).

Theorem A**.**

*Let $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{n}$ be a product of $n\geq 2$ groups that belong to the class $\mathcal{C}$. Let $\Gamma\overset{}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action and denote $M=L^{\infty}(X)\rtimes\Gamma.$

Suppose that $M=P_{1}\bar{\otimes}P_{2}$, for some II1 factors $P_{1}$ and $P_{2}.$*

Then there exists a partition $T_{1}\sqcup T_{2}=\{1,\dots,n\}$ such that $L^{\infty}(X)\prec_{M}L^{\infty}(X)^{\Gamma_{T_{1}}}\vee L^{\infty}(X)^{\Gamma_{T_{2}}}$, where $\Gamma_{T_{i}}:=\times_{j\in T_{i}}\Gamma_{j}$, for any $i\in\{1,2\}$.

Moreover, if in addition the groups $\Gamma_{i}$’s have Kazhdan’s property (T), then there exist a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$, for some $t>0$, and a unitary $u\in M$ such that

[TABLE]

In particular, there exists a pmp action $\Gamma_{T_{i}}\curvearrowright(X_{i},\mu_{i})$ for any $i\in\{1,2\}$ such that $\Gamma\curvearrowright X$ is isomorphic to the product action $\Gamma_{T_{1}}\times\Gamma_{T_{2}}\curvearrowright X_{1}\times X_{2}.$

The moreover part applies if the groups $\Gamma_{i}$’s are icc, property (T), weakly amenable, bi-exact. The following classes of groups satisfy these conditions.

1. (1)

uniform lattices in $Sp(k,1)$ with $k\geq 2$ or any icc group in their measure equivalence class, 2. (2)

Gromov’s random groups with density satisfying $3^{-1}<d<2^{-1}.$

Note that the moreover part of Theorem A provides the first class of product groups for which the primeness problem for II1 factors arising from their actions is completely settled.

Corollary B**.**

Let $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{n}$ be a product of $n\geq 2$ groups111The case $n=1$ already follows from [Oz04], [CH08] and [CPS11]. that belong to the class $\mathcal{C}$. Let $\Gamma\overset{}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action.

If $\Gamma\curvearrowright(X,\mu)$ is irreducible, or if the groups $\Gamma_{i}$’s have Kazhdan’s property (T) and the action $\Gamma\curvearrowright(X,\mu)$ does not admit a direct product decomposition, then $L^{\infty}(X)\rtimes\Gamma$ is prime.

Theorem A allows us to prove a unique prime factorization theorem for any II1 factor arising from an arbitrary free ergodic pmp action of a product of groups that belong to the class $\mathcal{C}$ and have Kazhdan’s property (T). More precisely, we have:

Corollary C**.**

Let $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{n}$ be a product of $n\geq 2$ groups that belong to the class $\mathcal{C}$ and have Kazhdan’s property (T). Let $\Gamma\overset{}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action. Denote $M=L^{\infty}(X)\rtimes\Gamma.$

Then there exists a unique partition $S_{1}\sqcup\dots\sqcup S_{k}=\{1,...,n\}$, for some $1\leq k\leq n$, (up to a permutation) and a pmp action $\Gamma_{S_{i}}\curvearrowright(X_{i},\mu_{i})$, for any $1\leq i\leq k$, such that:

1. (1)

$\Gamma\curvearrowright X$* is isomorphic to the product action $\Gamma_{S_{1}}\times...\times\Gamma_{S_{k}}\curvearrowright X_{1}\times...\times X_{k}$.* 2. (2)

$M_{i}:=L^{\infty}(X_{i})\rtimes\Gamma_{S_{i}}$* is prime for any $1\leq i\leq k$.*

Moreover, the following hold:

1. (1)

If $M=P_{1}\bar{\otimes}P_{2}$, for some II1* factors $P_{1},P_{2}$, then there exist a partition $I_{1}\sqcup I_{2}=\{1,...,k\}$ and a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$, for some $t>0$, such that $P_{1}^{t}=\bar{\otimes}_{i\in I_{1}}M_{i}$ and $P_{2}^{1/t}=\bar{\otimes}_{i\in I_{2}}M_{i}$, up to unitary conjugacy in $M$.* 2. (2)

If $M=P_{1}\bar{\otimes}\dots\bar{\otimes}P_{m}$, for some $m\geq k$ and II1* factors $P_{1},...,P_{m}$, then $m=k$ and there exists a decomposition $M=P_{1}^{t_{1}}\bar{\otimes}...\bar{\otimes}P_{k}^{t_{k}}$ for some $t_{1},...,t_{k}>0$ with $t_{1}t_{2}\dots t_{k}=1$ such that after permutation of indices and unitary conjugacy we have $M_{i}=P_{i}^{t_{i}}$, for all $1\leq i\leq k$.* 3. (3)

In (2), the assumption $m\geq k$ can be omitted if each $P_{i}$ is assumed to be prime.

The first unique prime factorization results for II1 factors were obtained by N. Ozawa and S. Popa in their seminal work [OP03]. Subsequently, several other unique prime factorization results have been obtained in [Pe06, CS11, SW11, Is14, CKP14, HI15, Ho15, Is16, DHI16, De19]. Corollary C is the first unique prime factorization result that applies to II1 factors arising from arbitrary free ergodic pmp actions of product groups.

Note that all the known unique prime factorization results in the II1 factor framework are using von Neumann algebras which do not have Murray and von Neumann’s property Gamma [MvN43]. Another novel aspect of this paper is the following unique prime factorization theorem in which the factors possibly have property Gamma.

Theorem D**.**

For any $1\leq i\leq k$, let $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ be a free ergodic pmp action of a group $\Gamma_{i}$ that belongs to the class $\mathcal{C}$. For any $1\leq i\leq k$, denote $M_{i}=L^{\infty}(X_{i})\rtimes\Gamma_{i}$ and let $M=M_{1}\bar{\otimes}\dots\bar{\otimes}M_{k}$. Then the following hold:

1. (1)

If $M=P_{1}\bar{\otimes}P_{2}$, for some II1* factors $P_{1},P_{2}$, then there exists a partition $I_{1}\sqcup I_{2}=\{1,...,k\}$ such that $P_{j}$ is stably isomorphic to $\bar{\otimes}_{i\in I_{j}}M_{i}$, for any $j\in\{1,2\}.$* 2. (2)

If $M=P_{1}\bar{\otimes}\dots\bar{\otimes}P_{m}$, for some $m\geq k$ and II1* factors $P_{1},...,P_{m}$, then $m=k$ and there exists a permutation $\sigma$ of $\{1,...,k\}$ such that $P_{i}$ is stably isomorphic to $M_{\sigma(i)}$, for any $1\leq i\leq k.$* 3. (3)

In (2), the assumption $m\geq k$ can be omitted if each $P_{i}$ is assumed to be prime.

Moreover, if $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ is strongly ergodic, for any $1\leq i\leq k$, then the identifications of the von Neumann algebras in (1), (2) and (3) are implemented up to amplification by a unitary from $M$ (as in Corollary C).

Remark 1.1**.**

Assume $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ is not strongly ergodic, for any $1\leq i\leq k$. Then, the conclusion of Theorem D is optimal in the sense that it cannot be improved to deduce that the identifications in (1), (2) and (3) can be implemented up to amplification by a unitary from $M$. This follows from [Ho15, Theorem B], since all the $M_{i}$’s have property Gamma. To ilustrate this, if we assume that $k=2$, [Ho15, Theorem B] implies that $M$ admits an automorphism $\theta$ such that $\theta(M_{i}^{t})$ is not unitarily conjugate to $M_{j}$, for any $i,j\in\{1,2\}$ and $t>0.$

Comments on the proof of Theorem A. We end the introduction with some informal and brief comments on the proof of Theorem A. For simplicity, assume $\Gamma=\Gamma_{1}\times...\times\Gamma_{n}$ is a product of $n\geq 2$ icc, weakly amenable, bi-exact groups. Let $\Gamma\curvearrowright(X,\mu)$ be a free ergodic pmp action and denote $M=L^{\infty}(X)\rtimes\Gamma.$ Assume that we have the tensor product decomposition $M=P_{1}\bar{\otimes}P_{2}$ into II1 factors. We aim to show that $\Gamma\curvearrowright X$ admits a non-trivial direct product decomposition.

In order to attain this goal we will heavily use S. Popa’s deformation/rigidity theory. In the first part of the proof we use S. Popa and S. Vaes’ breakthrough work [PV11, PV12] to obtain a partition $T_{1}\sqcup T_{2}=\{1,\dots,n\}$ such that

[TABLE]

where $\Gamma_{T_{i}}=\times_{j\in T_{i}}\Gamma_{j}$, for any $i\in\{1,2\}$. Here, $P\prec Q$ denotes the fact that a corner of $P$ embeds into a corner of $Q$ inside the ambient algebra in the sense of Popa [Po03]. For ease of notation, we will write $P\sim Q$ if $Pp^{\prime}\prec Q$ and $Qq^{\prime}\prec P$, for any $p^{\prime}\in P^{\prime}\cap M$ and $q^{\prime}\in Q^{\prime}\cap M$.

Since the equality $P_{i}\vee(P_{i}^{\prime}\cap M)=M$ can be seen as a finite index inclusion of von Neumann algebras in the sense of Popa-Pimsner [PP86] for any $i\in\{1,2\}$ (see Section 2.3), we can make use of (1.1) and deduce the existence of some abelian von Neumann subalgebras $D_{1}\subset P_{1}$ and $D_{2}\subset P_{2}$ such that

[TABLE]

[TABLE]

Here, we denote by ${L^{\infty}(X)}^{\Gamma_{T_{1}}}$ and ${L^{\infty}(X)}^{\Gamma_{T_{2}}}$ the subalgebras of elements in ${L^{\infty}(X)}$ fixed by $\Gamma_{T_{1}}$ and $\Gamma_{T_{2}},$ respectively. By combining the intertwining relations (1.2) and (1.3) we show that ${L^{\infty}(X)}\sim{L^{\infty}(X)}^{\Gamma_{T_{1}}}\vee{L^{\infty}(X)}^{\Gamma_{T_{2}}}$.

Finally, if we assume in addition that the groups $\Gamma_{i}$’s have property (T), we deduce that we have the identifications

[TABLE]

up to a unitary conjugacy and amplification.

Acknowledgment. I warmly thank Ionut Chifan and Adrian Ioana for many comments and suggestions that helped improve the exposition of the paper. I am especially grateful to Adrian Ioana for valuable comments on a previous draft which helped increase the generality of the results. I also thank Martín Argerami and Remus Floricel for a useful discussion about these results. Finally, I would like to thank the referee for valuable comments. The author was partially supported by PIMS fellowship.

2. Preliminaries

2.1. Terminology

In this paper we consider tracial von Neumann algebras $(M,\tau)$, i.e. von Neumann algebras $M$ equipped with a faithful normal tracial state $\tau:M\to\mathbb{C}.$ This induces a norm on $M$ by the formula $\|x\|_{2}=\tau(x^{*}x)^{1/2},$ for all $x\in M$. We will always assume that $M$ is a separable von Neumann algebra, i.e. the $\|\cdot\|_{2}$-completion of $M$ denoted by $L^{2}(M)$ is separable as a Hilbert space. We denote by $\mathcal{U}(M)$ the unitary group of $M$ and by $\mathcal{Z}(M)$ its center.

All inclusions $P\subset M$ of von Neumann algebras are assumed unital. We denote by $e_{P}:L^{2}(M)\to L^{2}(P)$ the orthogonal projection onto $L^{2}(P)$, by $E_{P}:M\to P$ the unique $\tau$-preserving conditional expectation from $M$ onto $P$, by $P^{\prime}\cap M=\{x\in M|xy=yx,\text{ for all }y\in P\}$ the relative commutant of $P$ in $M$ and by $\mathcal{N}_{M}(P)=\{u\in\mathcal{U}(M)|uPu^{*}=P\}$ the normalizer of $P$ in $M$. We say that $P$ is regular in $M$ if the von Neumann algebra generated by $\mathcal{N}_{M}(P)$ equals $M$. For two von Neumann subalgebras $P,Q\subset M$, we denote by $P\vee Q$ the von Neumann algebra generated by $P$ and $Q$. Jones’ basic construction of the inclusion $P\subset M$ is defined as the von Neumann subalgebra of $\mathbb{B}(L^{2}(M))$ generated by $M$ and $e_{P}$, and is denoted by $\langle M,e_{P}\rangle$.

The amplification of a II1 factor $(M,\tau)$ by a positive number $t$ is defined to be $M^{t}=p(\mathbb{B}(\ell^{2}(\mathbb{Z}))\bar{\otimes}M)p$, for a projection $p\in\mathbb{B}(\ell^{2}(\mathbb{Z}))\bar{\otimes}M$ satisfying $($Tr$\otimes\tau)(p)=t$. Here Tr denotes the usual trace on $\mathbb{B}(\ell^{2}(\mathbb{Z}))$. Since $M$ is a II1 factor, $M^{t}$ is well defined. Note that if $M=P_{1}\bar{\otimes}P_{2}$, for some II1 factors $P_{1}$ and $P_{2}$, then there exists a natural identification $M=P_{1}^{t}\bar{\otimes}M_{2}^{1/t}$, for every $t>0.$

Let $\Gamma\overset{\sigma}{\curvearrowright}A$ be a trace preserving action of a countable group $\Gamma$ on a tracial von Neumann algebra $(A,\tau)$. For a subgroup $\Sigma<\Gamma$ we denote by $A^{\Sigma}=\{a\in A|\sigma_{g}(a)=a,\text{ for all }g\in\Sigma\}$, the subalgebra of elements of $A$ fixed by $\Sigma.$

Finally, for a product group $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{n}$ and a subset $T\subset\{1,\dots,n\}$, we denote $\Gamma_{T}=\times_{i\in T}\Gamma_{i}.$

2.2. Intertwining-by-bimodules

We next recall from [Po03, Theorem 2.1 and Corollary 2.3] the intertwining-by-bimodules technique of S. Popa, which gives a powerful criterion for the existence of intertwiners between arbitrary subalgebras of a tracial von Neumann algebra.

Theorem 2.1** (​​[Po03]).**

Let $(M,\tau)$ be a tracial von Neumann algebra and let $P\subset pMp,Q\subset qMq$ be von Neumann subalgebras. Let $\mathcal{G}\subset\mathcal{U}(P)$ be a subgroup such that $\mathcal{G}^{\prime\prime}=P$,

Then the following are equivalent:

• •

There exist projections $p_{0}\in P,q_{0}\in Q$, a $*$-homomorphism $\theta:p_{0}Pp_{0}\rightarrow q_{0}Qq_{0}$ and a non-zero partial isometry $v\in q_{0}Mp_{0}$ such that $\theta(x)v=vx$, for all $x\in p_{0}Pp_{0}$.

• •

There is no sequence $(u_{n})_{n}\subset\mathcal{G}$ satisfying $\|E_{Q}(xu_{n}y)\|_{2}\rightarrow 0$, for all $x,y\in M$.

If one of these equivalent conditions holds true, we write $P\prec_{M}Q$, and say that a corner of $P$ embeds into $Q$ inside $M$. If $Pp^{\prime}\prec_{M}Q$ for any non-zero projection $p^{\prime}\in P^{\prime}\cap pMp$, then we write $P\prec^{s}_{M}Q$.

Whenever the ambient algebra $(M,\tau)$ is clear from the context, we will write $P\prec Q$ instead of $P\prec_{M}Q$.

The following lemma is a consequence of [OP07, Lemma 4.11]. For completeness, we provide a short proof.

Lemma 2.2** (​​[OP07]).**

Let $\Gamma{\curvearrowright}(Y,\nu)$ be an ergodic pmp action of an icc group and denote $M=L^{\infty}(Y)\rtimes\Gamma$. If $L^{\infty}(Y)^{\prime}\cap M\prec_{M}L^{\infty}(Y)$, then $\Gamma\curvearrowright(Y,\nu)$ is free.

Proof. Let $B=L^{\infty}(Y)$. The assumption implies that there exist non-zero projections $p\in B^{\prime}\cap M,q\in B$, a non-zero partial isometry $v\in qMp$ and an injective $*$-homomorphism $\theta:p(B^{\prime}\cap M)p\to Bq$ such that $\theta(x)v=vx$, for all $x\in p(B^{\prime}\cap M)p$.

Note that a standard argument which goes back to [MvN43] shows that $M$ is a factor since $\Gamma\curvearrowright Y$ is ergodic and $\Gamma$ is icc. Since $p(B^{\prime}\cap M)p$ and $Bp$ are abelian, it follows that $p(B^{\prime}\cap M)p$ is maximal abelian in $pMp$. Using that $M$ is a II1 factor, we obtain that there exists a maximal abelian subalgebra $C\subset M$ such that $Cp=p(B^{\prime}\cap M)p$ and $p\in C.$ Hence, $C\prec_{M}B$. We can now apply [OP07, Lemma 4.11] and obtain the conclusion. $\blacksquare$

We continue by observing some elementary facts. The first result is well known and we include a short proof for the reader’s convenience.

Lemma 2.3**.**

Let $N$ be a von Neumann subalgebra of a tracial von Neumann algebra $(M,\tau)$. Let $P\subset pNp$ and $Q\subset qNq$ be von Neumann subalgebras such that $P\prec_{M}Q$ and assume that $Q\subset qMq$ is regular. Then $P\prec_{N}Q.$

Proof. Assume the contrary, that $P\nprec_{N}Q$. Thus, there exists a sequence of unitaries $(u_{n})_{n}\subset\mathcal{U}(P)$ such that $\|E_{Q}(xu_{n}y)\|_{2}\to 0$, for any $x,y\in N.$ Thus, $\|E_{Q}(u_{n}y)\|_{2}=\|E_{Q}(u_{n}E_{N}(y))\|_{2}\to 0$, for any $y\in M.$ Since $Q$ is regular in $qMq$, we obtain that $\|E_{Q}(xu_{n}y)\|_{2}\to 0$, for any $x,y\in M$, contradiction. $\blacksquare$

Lemma 2.4**.**

Let $(M,\tau)$ be a tracial von Neumann algebra and let $Q\subset M$ be a regular von Neumann subalgebra. Let $R_{1},R_{2}\subset M$ be commuting von Neumann subalgebras such that $R_{i}\prec_{M}^{s}Q$ for any $i\in\{1,2\}$. Suppose $R_{2}$ is abelian.

Then $R_{1}\vee R_{2}\prec^{s}_{M}Q.$

Proof. Take a non-zero projection $s\in(R_{1}\vee R_{2})^{\prime}\cap M$. Since $R_{1}s\prec_{M}Q,$ there exist projections $r_{1}\in R_{1}$ and $q\in Q,$ a $*$-homomorphism $\theta:r_{1}R_{1}r_{1}s\to qQq$ and a non-zero partial isometry $v\in qMr_{1}s$, satisfying

[TABLE]

We argue that $(r_{1}R_{1}r_{1}s)\vee(R_{2}r_{1}s)\prec_{M}Q.$ By supposing the contrary, there exist two sequences of unitaries $(u_{n})_{n}\subset\mathcal{U}(r_{1}R_{1}r_{1}s)$ and $(v_{n})_{n}\subset\mathcal{U}(R_{2})$ such that

[TABLE]

Note that $u_{n}r_{1}s=u_{n}$, for all $n$. By taking $x=v$ and using the intertwining relation (2.1), we get that

[TABLE]

Let $r:=v^{*}v.$ Since $Q$ is regular, we get that

[TABLE]

Denote $r^{\prime}:=\vee_{w\in\mathcal{U}(R_{2})}wrw^{*}\in R_{2}^{\prime}\cap M$. Since $R_{2}$ is abelian, (2.2) implies that $\|E_{Q}(x(wrw^{*})v_{n}y)\|_{2}\to 0,$ for any $w\in\mathcal{U}(R_{2})$ and $x,y\in M.$ Note that $p_{1}\vee p_{2}=s(p_{1}+p_{2})$, for any two projections $p_{1}$ and $p_{2}$ in $M$. Here we denote by $s(b)$ the support projection of a positive element $b\in M.$ Moreover, by using Borel functional calculus, there exist a sequence $(c_{n})_{n}\subset M$ such that $c_{n}(p_{1}+p_{2})$ converges to $s(p_{1}+p_{2})$ in the $\|\cdot\|_{2}$-norm. Therefore, it follows that $\|E_{Q}(x(w_{1}rw_{1}^{*}+w_{2}rw_{2}^{*})v_{n}y)\|_{2}\to 0$, and hence, $\|E_{Q}(x(w_{1}rw_{1}^{*}\vee w_{2}rw_{2}^{*})v_{n}y)\|_{2}\to 0$, for any $w_{1},w_{2}\in\mathcal{U}(R_{2})$ and $x,y\in M$.

Finally, by induction it follows that

[TABLE]

for any $w_{1},...,w_{m}\in\mathcal{U}(R_{2})$ and $x,y\in M$.

Hence, $\|E_{Q}(xr^{\prime}v_{n}y)\|_{2}\to 0$, for all $x,y\in M$. This shows that $R_{2}r^{\prime}\nprec_{M}Q,$ contradiction. Therefore, $(R_{1}\vee R_{2})s\prec_{M}Q$, which implies that $R_{1}\vee R_{2}\prec_{M}^{s}Q.$ $\blacksquare$

We will need the following result which is an extension of [DHI16, Lemma 2.8(2)] (see also [PV11, Proposition 2.7]).

Proposition 2.5**.**

Let $(M,\tau)$ be a tracial von Neumann algebra and let $Q_{1},Q_{2}\subset M$ be von Neumann subalgebras which form a commuting square, i.e. $E_{Q_{1}}\circ E_{Q_{2}}=E_{Q_{2}}\circ E_{Q_{1}}$. Assume that there exist commuting subgroups $\mathcal{N}_{1}<\mathcal{N}_{M}(Q_{1})$ and $\mathcal{N}_{2}<\mathcal{N}_{M}(Q_{2})$ satisfying $(\mathcal{N}_{1}\vee\mathcal{N}_{2})^{\prime\prime}=M$. Let $P\subset pMp$ be a von Neumann subalgebra.

If $P\prec_{M}^{s}Q_{1}$ and $P\prec_{M}^{s}Q_{2}$, then $P\prec_{M}^{s}Q_{1}\cap Q_{2}.$

Remark 2.6**.**

The proposition will be applied for the following particular case. Assume $M=P_{1}\bar{\otimes}P_{2}$ for some von Neumann subalgebras $P_{1},P_{2}\subset M$ and let $D_{i}\subset P_{i}$ be a subalgebra for any $i\in\{1,2\}.$ By taking $Q_{1}=D_{1}\bar{\otimes}P_{2}$ and $Q_{2}=P_{1}\bar{\otimes}D_{2}$ the assumptions of Proposition 2.5 are satisfied.

The proof of Proposition 2.5 follows directly by using the next lemma and adapting the proof of [DHI16, Lemma 2.8(2)] (see also [PV11, Proposition 2.7]). We leave the other details to the reader.

Lemma 2.7**.**

Let $(M,\tau)$ be a tracial von Neumann algebra and let $Q_{1},Q_{2}\subset M$ and $\mathcal{N}_{1},\mathcal{N}_{2}\subset M$ be as in Proposition 2.5. Denote $Q=Q_{1}\cap Q_{2}.$

Then the $M$-$M$-bimodule $L^{2}(\langle M,e_{Q_{1}}\rangle)\otimes_{M}L^{2}(\langle M,e_{Q_{2}}\rangle)$ is contained in a multiple of the $M$-$M$-bimodule $L^{2}(\langle M,e_{Q}\rangle)$.

Proof. The proof follows almost verbatim part of the proof of [PV11, Proposition 2.7]. However, we provide some details for the reader’s convenience.

Denote by $H$ the $M$-$M$-bimodule $L^{2}(\langle M,e_{Q_{1}}\rangle)\otimes_{M}L^{2}(\langle M,e_{Q_{2}}\rangle)$. For $u_{1},v_{1}\in\mathcal{N}_{1}$ and $u_{2},v_{2}\in\mathcal{N}_{2}$, denote by $H_{u_{1},v_{1}}^{u_{2},v_{2}}$ the closed linear span of $\{xe_{Q_{1}}u_{1}u_{2}\otimes_{M}v_{1}v_{2}e_{Q_{2}}y|x,y\in M\}.$ Note that the formulas $uE_{Q_{i}}(\cdot)u^{*}=E_{Q_{i}}(u\cdot u^{*})$, for any $u\in\mathcal{N}_{M}(Q_{i})$, combined with the commuting square property imply that the map

[TABLE]

defines an $M$-$M$-bimodular unitary operator of $H_{u_{1},v_{1}}^{u_{2},v_{2}}$ to $L^{2}(\langle M,e_{Q}\rangle)$. To show this it suffices to verify that

[TABLE]

for all $x,y\in M$. Note that the left hand side of (2.3) equals

[TABLE]

Therefore, the right hand side or (2.3) equals its left hand side since

[TABLE]

Remark that the regularity assumption on the $Q_{i}$’s implies that the closed linear span of $\{u_{1}u_{2}|u_{1}\in\mathcal{N}_{1},u_{2}\in\mathcal{N}_{2}\}$ equals to $L^{2}(M)$. Therefore, the closed linear span of $\{H_{u_{1},v_{1}}^{u_{2},v_{2}}|$ $u_{1},v_{1}\in\mathcal{N}_{1},u_{2},v_{2}\in\mathcal{N}_{2}\}$ equals to $H$. This shows that $H$ is contained in a multiple of $L^{2}(\langle M,e_{Q}\rangle)$. $\blacksquare$

2.3. Finite index inclusions of von Neumann algebras.

For an inclusion $P\subset M$ of II1 factors the Jones index is the dimension of $L^{2}(M)$ as a left $P$-module [Jo81]. In [PP86], M. Pimsner and S. Popa defined a probabilistic notion of index for an inclusion $P\subset M$ of arbitrary von Neumann algebras with conditional expectation, which in the case of inclusions of II1 factors coincides with Jones’ index. Following [PP86], we say that the inclusion $P\subset M$ of tracial von Neumann algebras has probabilistic index $[M:P]=\lambda^{-1}$, where

[TABLE]

Here we use the convention that $\frac{1}{0}=\infty.$

Lemma 2.8** (​​[PP86, Lemma 2.3]).**

Let $P\subset M$ be an inclusion of tracial von Neumann algebras such that $[M:P]<\infty$. Then the following hold:

1. (1)

If $p\in P$ is a projection, then $[pMp:pPp]<\infty.$ 2. (2)

$M\prec^{s}_{M}P.$**

For a proof, see [CIK13, Lemma 2.4]. We will need the following well known lemma and we include its proof for completeness (see also [Va08, Lemma 3.9]).

Lemma 2.9**.**

Let $(M,\tau)$ be a tracial von Neumann algebra and let $R\subset N\subset pMp$ be von Neumann subalgebras such that $[N:R]<\infty$. Then the following hold:

1. (1)

If $R^{\prime}\cap N\subset R$, then there exists a non-zero projection $z\in\mathcal{Z}(R)$ such that $\mathcal{Z}(N)z=\mathcal{Z}(R)z$. 2. (2)

Assume $R^{\prime}\cap N\subset R$ or $\mathcal{Z}(R)$ is completely atomic. If $Q\subset qMq$ is a von Neumann subalgebra such that $R\prec_{M}Q$, then $N\prec_{M}Q.$

Proof. (1) Applying Lemma 2.8(2), we get that $N\prec_{N}R$. By passing to relative commutants, we can use [Va08, Lemma 3.5] and deduce that $\mathcal{Z}(R)\prec_{N}\mathcal{Z}(N).$ Hence, we obtain that there exist projections $r\in\mathcal{Z}(R),n\in\mathcal{Z}(N)$, a non-zero partial isometry $v\in nNr$ and a $*$-homomorphism $\theta:\mathcal{Z}(R)r\to\mathcal{Z}(N)n$ such that $v\theta(x)=\theta(x)v=vx$, for all $x\in\mathcal{Z}(R)r.$ By noticing that $\mathcal{Z}(N)\subset\mathcal{Z}(R)$, we obtain that $E_{\mathcal{Z}(R)}(v^{*}v)\theta(x)=E_{\mathcal{Z}(R)}(v^{*}v)x$, for all $x\in\mathcal{Z}(R)r.$ If we denote by $p_{0}$ the support projection of $E_{\mathcal{Z}(R)}(v^{*}v)$, we get that $p_{0}\theta(x)=p_{0}x$, for all $x\in\mathcal{Z}(R)r.$ Therefore, $\mathcal{Z}(R)\prec_{\mathcal{Z}(R)}\mathcal{Z}(N)$, which clearly implies the conclusion.

(2) Assume first that $R^{\prime}\cap N\subset R$. Since $R\prec_{M}Q$, there exist projections $r\in R$, $q_{0}\in Q$, a non-zero partial isometry $v\in q_{0}Mr$ and a $*$-homomorphism $\theta:rRr\to q_{0}Qq_{0}$ such that

[TABLE]

Note that $v^{*}v\in(R^{\prime}\cap pMp)r$ and denote by $r_{1}$ the support projection of $E_{N}(v^{*}v)$. Notice that $r_{1}\in(R^{\prime}\cap N)r.$ Since $R^{\prime}\cap N\subset R$, then $r_{1}\in rRr$ and therefore, Lemma 2.8 implies that $r_{1}Nr_{1}\prec_{r_{1}Nr_{1}}r_{1}Rr_{1}.$

Thus, there exist projections $n\in r_{1}Nr_{1}$, $r_{0}\in r_{1}Rr_{1}$, a non-zero partial isometry $w\in r_{0}Nn$ and a $*$-homomorphism $\psi:nNn\to r_{0}Rr_{0}$ such that

[TABLE]

Moreover, by restricting $ww^{*}$ if necessary we can assume without loss of generality that the support projection of $E_{r_{1}Rr_{1}}(ww^{*})$ equals $r_{0}.$ Note that $\theta(\psi(\cdot)):nNn\to q_{0}Qq_{0}$ is a $*$-homomorphism which satisfies

[TABLE]

If $vw=0$, then $E_{N}(v^{*}v)ww^{*}=0$, which implies $r_{1}ww^{*}=0.$ Hence, $r_{1}E_{r_{1}Rr_{1}}(ww^{*})=0$, showing that $r_{0}=0$, contradiction. This proves that $vw\neq 0.$ By replacing $vw$ by the partial isometry from its polar decomposition, the intertwining relation (2.4) still holds. This shows that $N\prec_{M}Q.$

Assume now that $\mathcal{Z}(R)$ is completely atomic. Note that [Dr19, Lemma 2.11 and Lemma 2.4(2)] give that $N\prec_{M}Q.$ $\blacksquare$

2.4. Relative amenability

A tracial von Neumann algebra $(M,\tau)$ is amenable if there exists a positive linear functional $\Phi:\mathbb{B}(L^{2}(M))\to\mathbb{C}$ such that $\Phi_{|M}=\tau$ and $\Phi$ is $M$-central, meaning $\Phi(xT)=\Phi(Tx),$ for all $x\in M$ and $T\in\mathbb{B}(L^{2}(M))$. The famous theorem of A. Connes asserts that a von Neumann algebra $M$ is amenable if and only if it is approximately finite dimensional [Co76].

N. Ozawa and S. Popa have considered a very useful relative version of this notion[OP07]. Let $(M,\tau)$ be a tracial von Neumann algebra. Let $p\in M$ be a projection and $P\subset pMp,Q\subset M$ be von Neumann subalgebras. Following [OP07, Definition 2.2], we say that $P\subset pMp$ is amenable relative to $Q$ inside $M$ if there exists a positive linear functional $\Phi:p\langle M,e_{Q}\rangle p\to\mathbb{C}$ such that $\Phi_{|pMp}=\tau$ and $\Phi$ is $P$-central. Note that $P$ is amenable relative to $\mathbb{C}$ inside $M$ if and only if $P$ is amenable.

The following lemma is well known and it goes back to [IPV10, Lemma 10.2], but we include a proof for completeness. The arguments are essentially contained in the proof of [PV12, Proposition 3.2].

Lemma 2.10**.**

*Let $\Gamma\curvearrowright(B,\tau)$ be a trace preserving action and denote $M=B\rtimes\Gamma.$ Define the $*$-homomorphism $\Delta:M\to M\bar{\otimes}L(\Gamma)$ by letting $\Delta(bu_{g})=bu_{g}\otimes u_{g},$ for all $b\in B$ and $g\in\Gamma.$

Let $P\subset pMp$ be a von Neumann subalgebra such that there exists $p_{1}\in\Delta(P)^{\prime}\cap\Delta(p)(M\bar{\otimes}M)\Delta(p)$ with the property that $\Delta(P)p_{1}$ is amenable relative to $M\otimes 1$.*

Then there exists a non-zero projection $p_{0}\in P^{\prime}\cap pMp$ such that $Pp_{0}$ is amenable relative to $B$ inside $M$.

Proof. Define $\mathcal{M}=M\bar{\otimes}M.$ The assumption implies the existence of a positive linear functional $\Phi:p_{1}\langle\mathcal{M},e_{M\otimes 1}\rangle p_{1}\to\mathbb{C}$ such that the restriction of $\Phi$ to $p_{1}\mathcal{M}p_{1}$ equals the trace on $p_{1}\mathcal{M}p_{1}$ and $\Phi$ is $\Delta(P)p_{1}$-central. Since $E_{M\bar{\otimes}1}\circ\Delta=\Delta\circ E_{B}$, note that we can define the injective $*$-homomorphism $\Delta_{1}:\langle M,e_{B}\rangle\to\langle\mathcal{M},e_{M\otimes 1}\rangle$ by letting $\Delta_{1}(e_{B})=e_{M\otimes 1}$ and $\Delta_{1}(x)=\Delta(x)$, for all $x\in M$.

Define the positive linear functional $\Psi:p\langle M,e_{B}\rangle p\to\mathbb{C}$ by $\Psi(x)=\Phi(p_{1}\Delta_{1}(x)p_{1})$, for all $x\in p\langle M,e_{B}\rangle p.$ Note that $\Psi$ is $P$-central and its restriction to $pMp$ is normal. Therefore, [BV12, Lemma 2.9] implies that there exists a non-zero projection $p_{0}\in P^{\prime}\cap pMp$ such that $Pp_{0}$ is amenable relative to $B$. $\blacksquare$

2.5. Relatively strongly solid groups

Following [CIK13, Definition 2.7], a countable group $\Gamma$ is said to be relatively strongly solid and write $\Gamma\in\mathcal{C}_{rss}$ if for any trace preserving action $\Gamma\curvearrowright B$ the following holds: if $M=B\rtimes\Gamma$ and $A\subset pMp$ is a von Neumann algebra which is amenable relative to $B$, then either $A\prec_{M}B$ or the normalizer $\mathcal{N}_{pMp}(A)^{\prime\prime}$ is amenable relative to $B$. In their breakthrough work [PV11, PV12], S. Popa and S. Vaes proved that all non-elementary hyperbolic groups belong to $\mathcal{C}_{rss}.$ More generally, [PV12, Theorem 1.4] shows that all weakly amenable, bi-exact groups are relatively strongly solid.

A remarkable subsequent development has been made by A. Ioana [Io12] (see also [Va13]) in the context of amalgamated free products by classifying all subalgebras of $M=M_{1}*_{B}M_{2}$ that are amenable relative to $B$ and that satisfy a certain spectral gap condition.

We will make use of the following consequence for groups that belong to $\mathcal{C}_{\text{rss}}$ (see [KV15, Lemma 5.2]).

Lemma 2.11** (​​[KV15]).**

Let $\Gamma\curvearrowright B$ be a trace preserving action of a group $\Gamma\in\mathcal{C}_{\text{rss}}$. Denote $M=B\rtimes\Gamma$. Let $P_{1},P_{2}\subset pM$p be commuting von Neumann subalgebras.

Then either $P_{1}\prec_{M}B$ or $P_{2}$ is amenable relative to $B$ inside $M$.

For free product groups we will use the following consequence [CdSS17, Theorem 3.1] of [Va13, Theorem A]:

Lemma 2.12** (​​[CdSS17]).**

Let $\Gamma\curvearrowright B$ be a trace preserving action, where $\Gamma=\Sigma_{1}*\Sigma_{2}$ and $|\Sigma_{1}|\geq 2$ and $|\Sigma_{2}|\geq 3$. Denote $M=B\rtimes\Gamma$ and assume $P_{1},P_{2}\subset pMp$ are two commuting diffuse subalgebras such that $P_{1}\vee P_{2}\subset pMp$ has finite index.

Then there exists an $i\in\{1,2\}$ such that $P_{i}\prec_{M}B$.

2.6. Wreath product groups

The next lemma gives a dichotomy result for commuting subalgebras of von Neumann algebras arising from trace preserving actions of wreath product groups. The arguments rely heavily on [IPV10].

Lemma 2.13**.**

Let $\Gamma=\Sigma_{0}\wr\Gamma_{0}$ be the wreath product between a non-trivial amenable group $\Sigma_{0}$ and an infinite group $\Gamma_{0}$. Let $\Gamma\curvearrowright B$ be a trace preserving action and define $M=B\rtimes\Gamma.$ Let $P_{1},P_{2}\subset pMp$ be two commuting subalgebras such that $P_{1}\vee P_{2}\subset pMp$ has finite index.

Then there exists a non-zero projection $p_{0}\in P_{1}^{\prime}\cap pMp$ such that $P_{1}p_{0}$ is amenable relative to $B$ or $P_{2}\prec_{M}B.$

Proof. Let $\Delta:M\to M\bar{\otimes}L(\Gamma)$ be the $*$-homomorphism defined by $\Delta(bu_{g})=bu_{g}\otimes u_{g}$, for all $b\in B$ and $g\in\Gamma.$ By applying [IPV10, Corollary 4.3], one of the following possibilities occurs: (1) there exists a non-zero projection $p_{1}\in\Delta(P_{1})^{\prime}\cap\Delta(p)(M\bar{\otimes}M)\Delta(p)$ such that $\Delta(P_{1})p_{1}$ is amenable relative to $M\otimes 1$, or (2) $\Delta(P_{2})\prec M\otimes 1$, or (3) $\Delta(P_{1}\vee(P_{1}^{\prime}\cap pMp))\prec M\bar{\otimes}L(\Sigma_{0}^{(\Gamma_{0})})$, or (4) $\Delta(P_{1}\vee(P_{1}^{\prime}\cap pMp))\prec M\bar{\otimes}L(\Gamma_{0}).$

If (1) holds, by Lemma 2.10 there exists a non-zero projection $p_{0}\in P_{1}^{\prime}\cap pMp$ such that $P_{1}p_{0}$ is amenable relative to $B$. If (2) holds, [Io10, Lemma 2.9(1)] implies that $P_{2}\prec_{M}B.$

We end the proof by showing that (3) and (4) cannot hold. Indeed, if (3) or (4) is true, then Lemma 2.9 combined with [Io10, Lemma 9.2(4)] imply that $L(\Gamma)\prec_{M}L(\Sigma_{0}^{(\Gamma_{0})})$ or $L(\Gamma)\prec_{M}L(\Gamma_{0}).$ Both are in contradiction with the fact that $\Sigma_{0}$ and $\Gamma_{0}$ are infinite groups. $\blacksquare$

Combining the previous three lemmas with [DHI16, Lemma 2.6(2)], we obtain the following corollary.

Corollary 2.14**.**

Let $\Gamma$ be a group that belongs to the class $\mathcal{C}$. Let $\Gamma\curvearrowright B$ be a trace preserving action and define $M=B\rtimes\Gamma.$ Let $P_{1},P_{2}\subset pMp$ be two commuting diffuse subalgebras such that $P_{1}\vee P_{2}\subset pMp$ has finite index.

Then there exists a non-zero projection $p_{0}\in\mathcal{N}_{pMp}(P_{1})^{\prime}\cap pMp$ such that $P_{1}p_{0}$ is amenable relative to $B$ or $P_{2}\prec_{M}B.$

3. From tensor decompositions of II1 factors to decompositions of actions

The goal of this section is to prove Theorem 3.2 which is our main ingredient of the proof of Theorem A. The moreover part will provide a von Neumann algebraic criterion for pmp actions of product groups to admit a direct product decomposition. First, we need the following result.

Theorem 3.1**.**

Let $\Gamma=\Gamma_{1}\times\Gamma_{2}$ be a product of countable icc groups and let $\Gamma\overset{}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action. Denote $M=L^{\infty}(X)\rtimes\Gamma.$

Suppose that $M=P_{1}\bar{\otimes}P_{2}$ for some II1 factors $P_{1}$ and $P_{2}$ such that $P_{i}\prec_{M}^{s}L^{\infty}(X)\rtimes\Gamma_{i}$, for all $i\in\{1,2\}.$

The following hold:

1. (1)

If $L^{\infty}(X)\prec_{M}L^{\infty}(X)^{\Gamma_{1}}\vee L^{\infty}(X)^{\Gamma_{2}}$ and $\Gamma_{1}$ has property (T), then there exist a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$, for some $t>0$, and a unitary $u\in M$ such that $P_{1}^{t}=u(L^{\infty}(X)^{\Gamma_{2}}\rtimes\Gamma_{1})u^{*}\text{ and }P_{2}^{1/t}=u(L^{\infty}(X)^{\Gamma_{1}}\rtimes\Gamma_{2})u^{*}.$ 2. (2)

If $L^{\infty}(X)=L^{\infty}(X)^{\Gamma_{1}}\vee L^{\infty}(X)^{\Gamma_{2}}$, then $P_{1}$ is stably isomorphic to $L^{\infty}(X)^{\Gamma_{2}}\rtimes\Gamma_{1}$ and $P_{2}$ is stably isomorphic to $L^{\infty}(X)^{\Gamma_{1}}\rtimes\Gamma_{2}.$

Proof. Let $\{u_{g}\}_{g\in\Gamma}$ be the canonical unitaries in $M$ implementing the action $\Gamma\curvearrowright(X,\mu)$. Denote $A=L^{\infty}(X)$, $A_{1}=L^{\infty}(X)^{\Gamma_{2}},A_{2}=L^{\infty}(X)^{\Gamma_{1}}$, $B=A_{1}\vee A_{2}$ and $M_{i}=A_{i}\rtimes\Gamma_{i}$, for any $i\in\{1,2\}$. Since $B$ is a $\Gamma$-invariant subalgebra of $A$, we consider $\Gamma\overset{\rho}{\curvearrowright}B$ the natural action of $\Gamma$ on $B$. Note that [Va08, Lemma 3.5] shows that $B^{\prime}\cap(B\rtimes\Gamma)\prec_{M}A$, which implies by [Va08, Lemma 3.7] that $B^{\prime}\cap(B\rtimes\Gamma)\prec_{M}B.$ Since $B\subset M$ is regular, Lemma 2.3 shows that $B^{\prime}\cap(B\rtimes\Gamma)\prec_{B\rtimes\Gamma}B.$ An application of Lemma 2.2 gives us that $\Gamma\curvearrowright B$ is free.

Since $\Gamma\overset{\rho}{\curvearrowright}B$ is isomorphic to the product action $\Gamma_{1}\times\Gamma_{2}\curvearrowright A_{1}\bar{\otimes}A_{2}$, it follows that $\Gamma_{i}\curvearrowright A_{i}$ is free for any $i\in\{1,2\}$. Hence,

[TABLE]

Indeed, take $x=\sum_{g\in\Gamma}x_{g}u_{g}\in A_{2}^{\prime}\cap M.$ It follows that $x_{g}b=x_{g}\rho_{g}(b)$, for all $b\in A_{2}.$ Therefore, $E_{A_{2}}(x_{g}^{*}x_{g})b=E_{A_{2}}(x_{g}^{*}x_{g})\rho_{g}(b)$, for all $b\in A_{2}$. Since $\Gamma_{2}$ acts freely on $A_{2}$, we get that $E_{A_{2}}(x_{g}^{*}x_{g})=0$, for all $g\notin\Gamma_{1}$, which implies that $x_{g}=0$.

Note that $P_{1}\prec_{M}A\rtimes\Gamma_{1}$ implies $A_{2}\prec_{M}P_{2}$ by applying [Va08, Lemma 3.5]. Since $A_{2}\subset M$ is regular, we can apply [Io06, Corollary 1.3] (see also [OP07, Proposition 12]) and obtain that there exist a unitary $u_{0}\in\mathcal{U}(M)$ and a decomposition $M=P_{1}^{1/t_{0}}\bar{\otimes}P_{2}^{t_{0}}$, for some $t_{0}>0$ such that $u_{0}A_{2}u_{0}^{*}\subset P_{2}^{t_{0}}.$ Denote $Q_{1}=u_{0}^{*}P_{1}^{1/t_{0}}u_{0}$ and $Q_{2}=u_{0}^{*}P_{2}^{t_{0}}u_{0}.$ Since $A_{2}\subset Q_{2}$ we can apply Ge’s tensor splitting theorem [Ge95, Theorem A] and derive that

[TABLE]

Since $A_{2}^{\prime}\cap Q_{2}\subset A\rtimes\Gamma_{1}$ and $Q_{2}\prec_{M}A\rtimes\Gamma_{2}$, it follows that $A_{2}^{\prime}\cap Q_{2}\prec_{M}A.$ Thus, there exists a non-zero projection $a\in A_{2}^{\prime}\cap Q_{2}$ such that $a(A_{2}^{\prime}\cap Q_{2})a$ is abelian. By cutting the von Neumann algebras of the equality (3.2) by the projection $a$ and by passing to the center, we obtain that $A_{2}a=a(A_{2}^{\prime}\cap Q_{2})a$ and hence

[TABLE]

(1) Now we can prove the first conclusion of the theorem. We first show that $L(\Gamma_{1})\prec_{M}P_{1}.$ To see this, note that since $L(\Gamma_{1})$ is a II1 factor, we obtain by [CdSS17, Lemma 4.5] that there exists a unitary $u\in A\rtimes\Gamma_{1}$ such that $b:=uau^{*}\in L(\Gamma_{1}).$ Hence, relation (3.3) gives that $b(A\rtimes\Gamma_{1})b=uQ_{1}u^{*}b\bar{\otimes}uA_{2}u^{*}b.$ Note that $bL(\Gamma_{1})b$ is a von Neumann algebra with property (T) by [Po01, Proposition 4.7(2)] and $u(A_{2}^{\prime}\cap Q_{2})u^{*}b$ is an abelian von Neumann algebra. Therefore, we obtain that $bL(\Gamma_{1})b\prec_{b(A\rtimes\Gamma_{1})b}uQ_{1}u^{*}b$ (see e.g. [HPV10, Lemma 1]). This implies that $L(\Gamma_{1})\prec_{M}P_{1}.$

By passing to relative commutants and by applying twice[Va08, Lemma 3.5], we obtain that $P_{2}\prec_{M}M_{2}$ and hence $M_{1}\prec_{M}P_{1}.$ Since $M_{1}$ and $M_{1}^{\prime}\cap M=M_{2}$ are factors, we can apply [OP03, Proposition 12] and obtain that there exist a unitary $v\in M$ and a decomposition $M=P_{1}^{s}\bar{\otimes}P_{2}^{1/s}$ for a positive $s>0$ such that $vM_{1}v^{*}\subset P_{1}^{s}.$ Therefore, $P_{2}^{1/s}\subset vM_{2}v^{*}.$ By applying [Ge95, Theorem A], we obtain that there exists a factor $D\subset P_{1}^{s}$ such that $D\bar{\otimes}P_{2}^{1/s}=vM_{2}v^{*}.$ It is easy to see that $D$ is not diffuse, which implies that $D=\mathbb{M}_{k}(\mathbb{C})$ for some integer $k\geq 1.$ By denoting $s_{0}=s/k$, we deduce that $P_{2}^{1/{s_{0}}}=vM_{2}v^{*}$ and $P_{1}^{s_{0}}=vM_{1}v^{*}.$

(2) Note that the assumption implies $A=A_{1}\bar{\otimes}A_{2}$ and the relation (3.3) shows that $a(M_{1}\bar{\otimes}A_{2})a=Q_{1}a\bar{\otimes}A_{2}a$. Therefore, by disintegrating in the above equality over the center $A_{2}a$ and by using [Ta01, Theorem IV.8.23], we deduce that $M_{1}$ is stably isomorphic to $Q_{1}$, hence to $P_{1}$. In a similar way, we obtain that $M_{2}$ is stably isomorphic to $P_{2}$. $\blacksquare$

Theorem 3.2**.**

Let $\Gamma=\Gamma_{1}\times\Gamma_{2}$ be a product of countable icc groups and let $\Gamma\overset{\sigma}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action. Suppose that $M=L^{\infty}(X)\rtimes\Gamma=P_{1}\bar{\otimes}P_{2}$ for some II1 factors $P_{1}$ and $P_{2}$ such that $P_{i}\prec_{M}^{s}L^{\infty}(X)\rtimes\Gamma_{i}$, for all $i\in\{1,2\}.$

Then $L^{\infty}(X)\prec_{M}L^{\infty}(X)^{\Gamma_{1}}\vee L^{\infty}(X)^{\Gamma_{2}}.$

Moreover, assume that $\Gamma_{1}$ has property (T). Then there exist a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$, for some $t>0$, and a unitary $u\in M$ such that

[TABLE]

In particular, there exists a pmp action $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ for any $i\in\{1,2\}$ such that the actions $\Gamma\curvearrowright X$ and $\Gamma_{1}\times\Gamma_{2}\curvearrowright X_{1}\times X_{2}$ are isomorphic.

Proof. Let $\{u_{g}\}_{g\in\Gamma}$ be the canonical unitaries in $M$ implementing the action $\Gamma\curvearrowright(X,\mu)$ and denote $A=L^{\infty}(X)$. For any $i\in\{1,2\}$, [CKP14, Proposition 2.4] implies that there exist non-zero projections $p_{i}\in P_{i}$, $q_{i}\in A\rtimes\Gamma_{i}$, a subalgebra $Q_{i}\subset q_{i}(A\rtimes\Gamma_{i})q_{i}$, a partial isometry $v_{i}\in q_{i}Mp_{i}$ and a $*$-isomorphism $\theta_{i}:p_{i}P_{i}p_{i}\to Q_{i}$ such that:

[TABLE]

[TABLE]

[TABLE]

The rest of the proof is divided between four claims.

Claim 1. We can assume, in addition to (3.4)-(3.6), that $Q_{i}$ also satisfies that $Q_{i}^{\prime}\cap q_{i}(A\rtimes\Gamma_{i})q_{i}=A^{\Gamma_{i}}q_{i}$, for any $i\in\{1,2\}.$

Proof. For simplicity we prove the claim only for $i=1$. Denote $R=Q_{1}^{\prime}\cap q_{1}(A\rtimes\Gamma_{1})q_{1}$. First, note that $R\prec_{M}A.$ Indeed, on one hand since $P_{1}\prec_{M}Q_{1}$, by considering the relative commutants, [Va08, Lemma 3.5] implies that $R\prec_{M}P_{2}.$ Applying [Va08, Lemma 3.7] we get $R\prec_{M}A\rtimes{\Gamma_{2}}$. On the other hand, $R\subset q_{1}(A\rtimes\Gamma_{1})q_{1}$. Hence, one can check that we actually have $R\prec_{M}A.$

Therefore, there exists a non-zero projection $r\in R$ such that $rRr$ is abelian. Applying Lemma 2.8(1), we have that $Q_{1}r\vee rRr\subset r(A\rtimes\Gamma_{1})r$ has finite index. Since $Q_{1}$ is a factor, Lemma 2.9(1) shows that by replacing $r$ by a smaller projection in $R$, we can assume that

[TABLE]

Lemma 2.8(1) guarantees that $Q_{1}r\vee rRr\subset r(A\rtimes{\Gamma_{1}})r$ still has finite index.

Since $E_{A\rtimes\Gamma_{1}}(v_{1}v_{1}^{*})q_{1}\geq\lambda q_{1}$ and $r\leq q_{1}$, we have that $rv_{1}\neq 0.$ Therefore, by replacing $Q_{1}$ by $Q_{1}r$, $\theta_{1}(\cdot)$ by $\theta_{1}(\cdot)r$, $q$ by $r$ and $v_{1}$ by the partial isometry from the polar decomposition of $rv_{1}$, the relations (3.4)-(3.6) are still satisfied. This proves the claim. $\square$

Let $i\in\{1,2\}$ and denote by $i+1$ the element in the set $\{1,2\}\setminus\{i\}.$ Since $P_{i}\prec_{M}Q_{i}$, by passing to relative commutants, we get that $A^{\Gamma_{i}}q_{i}\prec_{M}P_{i+1}$. Then there exist projections $k_{i}\in A^{\Gamma_{i}}q_{i}$, $r_{i+1}\in P_{i+1}$, a $*$-homomorphism $\psi_{i}:A^{\Gamma_{i}}k_{i}\to r_{i+1}P_{i+1}r_{i+1}$ and a non-zero partial isometry $w_{i}\in r_{i+1}Mk_{i}$ such that

[TABLE]

By restricting the projection $w_{i}w_{i}^{*}$ if necessary we can assume that

[TABLE]

Denote $R_{i+1}=\psi_{i}(A^{\Gamma_{i}}k_{i}).$

Claim 2. $R_{1}\prec_{M}^{s}A^{\Gamma_{2}}k_{2}\text{ and }R_{2}\prec_{M}^{s}A^{\Gamma_{1}}k_{1}.$

Proof. We prove only the second intertwining, since the first one follows in a similar way. To the end, take a non-zero projection $b\in\mathcal{N}_{r_{2}Mr_{2}}(R_{2})^{\prime}\cap r_{2}Mr_{2}\subset R_{2}^{\prime}\cap r_{2}P_{2}r_{2}$ and define the $*$-homomorphism $\psi_{b}:A^{\Gamma_{1}}k_{1}\to R_{2}b$ by $\psi_{b}(x)=\psi_{1}(x)b,$ for all $x\in A^{\Gamma_{1}}k_{1}$. Note that $\psi_{b}(x)bw_{1}=bw_{1}x,$ for all $x\in A^{\Gamma_{1}}k_{1}$. We show that $bw_{1}$ is non-zero. If this is not the case, then $0=E_{P_{2}}(bw_{1}w_{1}^{*})=bE_{P_{2}}(w_{1}w_{1}^{*})$. Since $b\leq r_{2}$, relation (3.8) implies that $b=0,$ false. Hence, $bw_{1}\neq 0.$ Thus, by replacing $bw_{1}$ by its partial isometry from its polar decomposition and by noticing that $\psi_{b}$ is a $*$-isomorphism, we get that $R_{2}b\prec_{M}A^{\Gamma_{1}}k_{1}.$ We use [DHI16, Lemma 2.4(2)] to conclude that $R_{2}\prec_{M}^{s}A^{\Gamma_{1}}k_{1}.$ $\square$

We continue with the following:

Claim 3. $Q_{1}\vee A^{\Gamma_{1}}q_{1}\prec_{M}P_{1}\bar{\otimes}R_{2}$ and $Q_{2}\vee A^{\Gamma_{2}}q_{2}\prec_{M}R_{1}\bar{\otimes}P_{2}.$

Proof. Due to symmetry we only need to show the first intertwining. First we construct a $*$-isomorphism $\theta:Q_{1}k_{1}\to p_{1}P_{1}p_{1}$ and a non-zero partial isometry $v\in p_{1}Mk_{1}$ such that

[TABLE]

Note that $\theta_{1}^{-1}:Q_{1}\to p_{1}P_{1}p_{1}$ is a $*$-isomorphism satisfying $\theta_{1}^{-1}(x)v_{1}^{*}=v_{1}^{*}x,$ for all $x\in Q_{1}.$ Since $Q_{1}$ is a factor and $k_{1}\in Q_{1}^{\prime}\cap q_{1}(A\rtimes\Gamma_{1})q_{1}$, we can define the $*$-homomorphism $\theta:Q_{1}k_{1}\to p_{1}P_{1}p_{1}$, by letting $\theta(xk_{1})=\theta_{1}^{-1}(x),$ for all $x\in Q_{1}.$ Note that $\theta(xk_{1})v^{*}_{1}k_{1}=v^{*}_{1}k_{1}xk_{1},$ for all $x\in Q_{1}.$ Let $v$ be the partial isometry obtained from the polar decomposition of $v_{1}^{*}k_{1}$. Note that $v\neq 0$ by using (3.6). Therefore, $\theta(x)v=vx$, for all $x\in Q_{1}k_{1}.$

Notice that $vv^{*}\in(P_{1}^{\prime}\cap M)p_{1}$ and $P_{1}^{\prime}\cap M=P_{2}$, so there exists a non-zero projection $\tilde{p}_{2}\in P_{2}$ such that $vv^{*}=p_{1}\otimes\tilde{p}_{2}$. Without loss of generality, we can assume that

[TABLE]

Indeed, since $P_{2}$ is a II1 factor, there exists a unitary $u\in\mathcal{U}(P_{2})$ such that $\tilde{p}_{2}\leq ur_{2}u^{*}$ or $ur_{2}u^{*}\leq\tilde{p}_{2}.$ By replacing $\psi_{1}(\cdot)$ by $u\psi_{1}(\cdot)u^{*}$, $r_{2}$ by $ur_{2}u^{*}$, $R_{2}$ by $uR_{2}u^{*}$ and $w_{1}$ by $uw_{1}$, relations (3.7) and (3.8) still hold. Therefore we can assume (3.9) to be true.

We suppose by contradiction that $Q_{1}k_{1}\vee A^{\Gamma_{1}}k_{1}\nprec_{M}P_{1}\bar{\otimes}R_{2}.$ Then there exist two sequences of unitaries $(u_{n})_{n}\subset\mathcal{U}(Q_{1})$ and $(v_{n})_{n}\subset\mathcal{U}(A^{\Gamma_{1}})$ such that

[TABLE]

By taking $x=v$, we get that

[TABLE]

We now argue that

[TABLE]

Take $y=au_{g}^{*}w_{1}^{*}$ in (3.10), for some $a\in A$ and $g\in\Gamma$. Since $A^{\Gamma_{1}}$ is normalized by $u_{g}$, we obtain

[TABLE]

tends to [math] as $n\to\infty$. This implies that $\|E_{P_{1}\bar{\otimes}R_{2}}(vau_{g}^{*}w_{1}^{*})\|_{2}=0$, for all $a\in A$ and $g\in\Gamma$. This proves (3.11), which gives us that $p_{1}E_{P_{1}\bar{\otimes}R_{2}}(\tilde{p}_{2}w_{1}w_{1}^{*})=0$. We obtain $p_{1}\otimes E_{R_{2}}(\tilde{p}_{2}w_{1}w_{1}^{*})=0$ and therefore, $p_{1}\otimes E_{R_{2}}(\tilde{p}_{2}E_{P_{2}}(w_{1}w_{1}^{*}))=0$, since $\tilde{p}_{2}\in P_{2}$. Hence, $\tau(r_{2}\tilde{p}_{2}E_{P_{2}}(w_{1}w_{1}^{*}))=0$. Using that $E_{P_{2}}(w_{1}w_{1}^{*})r_{2}\geq\beta_{2}r_{2}$, we get that $\tilde{p}_{2}E_{P_{2}}(w_{1}w_{1}^{*})r_{2}\tilde{p}_{2}\geq\beta_{2}r_{2}\tilde{p}_{2}$. Altogether it implies that $\beta_{2}\tau(\tilde{p}_{2}r_{2})=0$, which contradicts (3.9) since $\beta_{2}$ is non-zero. Thus, $Q_{1}k_{1}\vee A^{\Gamma_{1}}k_{1}\prec_{M}P_{1}\bar{\otimes}R_{2}$, ending this way the proof of the claim. $\square$

Finally, we obtain the following:

Claim 4. $A\prec_{M}^{s}A^{\Gamma_{1}}\vee A^{\Gamma_{2}}.$

Proof. Denote $D_{i}=R_{i}\oplus\mathbb{C}(1-r_{i}),$ for $i\in\{1,2\}.$ Note that Claim 1 together with relation (3.4) show that $Q_{i}\vee A^{\Gamma_{i}}q_{1}\subset q_{i}(A\rtimes\Gamma_{i})q_{i}$ has finite index, for $i\in\{1,2\}.$ Combining Claim 3 and Lemma 2.9(2), we obtain that $A\prec_{M}P_{1}\bar{\otimes}D_{2}$ and $A\prec_{M}D_{1}\bar{\otimes}P_{2}.$ Note that [DHI16, Lemma 2.4(2)] together with Proposition 2.5 imply that $A\prec^{s}_{M}D_{1}\bar{\otimes}D_{2}.$ Notice also that Claim 2 shows that $D_{i}\prec_{M}^{s}A^{\Gamma_{1}}\vee A^{\Gamma_{2}}$, for any $i\in\{1,2\}.$ Since $A^{\Gamma_{1}}\vee A^{\Gamma_{2}}$ is regular and the algebra $D_{2}$ is abelian, we can apply Lemma 2.4 and obtain that $D_{1}\bar{\otimes}D_{2}\prec_{M}^{s}A^{\Gamma_{1}}\vee A^{\Gamma_{2}}.$ By applying [Va08, Lemma 3.7], we deduce that $A\prec_{M}^{s}A^{\Gamma_{1}}\vee A^{\Gamma_{2}}$. $\square$

The moreover part follows from the first part of Theorem 3.1. In particular, we can represent $A^{\Gamma_{2}}=L^{\infty}(X_{1},\mu_{1})$ and $A^{\Gamma_{2}}=L^{\infty}(X_{2},\mu)$ for some standard probability spaces $(X_{1},\mu_{1})$ and $(X_{2},\mu_{2})$, respectively. Hence, for any $i\in\{1,2\}$ there exists a pmp action $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ such that $\Gamma\curvearrowright X$ is isomorphic to $\Gamma_{1}\times\Gamma_{2}\curvearrowright X_{1}\times X_{2}.$ $\blacksquare$

4. Proofs of the main results

We start this section by presenting another tool needed for the proof of Theorem A and we will conclude by proving the main results mentioned in the introduction.

Proposition 4.1**.**

*Let $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{n}$ be a product of $n\geq 2$ groups that belong to the class $\mathcal{C}$. Let $\Gamma\overset{}{\curvearrowright}(X,\mu)$ be a free ergodic pmp action and denote $M=L^{\infty}(X)\rtimes\Gamma.$

Suppose that $M=P_{1}\bar{\otimes}P_{2}$, for some II1 factors $P_{1}$ and $P_{2}.$*

Then there exists a partition $T_{1}\sqcup T_{2}=\{1,\dots,n\}$ into non-empty sets such that $P_{i}\prec^{s}_{M}L^{\infty}(X)\rtimes\Gamma_{T_{i}}$, for $i\in\{1,2\}.$

Proof. Denote $A=L^{\infty}(X).$ Let $T_{i}$ be the minimal subset of $\{1,\dots,n\}$ with the property that $P_{i}\prec^{s}_{M}A\rtimes\Gamma_{T_{i}}$ for all $i\in\{1,2\}.$ Notice that $T_{i}$ is non-empty since any corner of a II1 factor is non-abelian. We want to show that $T_{1}\sqcup T_{2}=\{1,\dots,n\}.$ Note that since $P_{i}$ is regular and $M$ is a factor, [DHI16, Lemma 2.4(2)] implies that $P_{i}\prec_{M}A\rtimes{\Gamma_{S}}$ if and only if $P_{i}\prec_{M}^{s}A\rtimes{\Gamma_{S}}$ for any subset $S\subset\{1,\dots,n\}$.

First we notice that $\{1,\dots,n\}=T_{1}\cup T_{2}$. Indeed, by applying [BV12, Lemma 2.3] we get that $M\prec A\rtimes\Gamma_{T_{1}\cup T_{2}}$. This shows that $\{1,\dots,n\}=T_{1}\cup T_{2}$, since $\Gamma_{t}$ is an infinite group, for any $t\in\{1,\dots,n\}.$ We will finish the proof by proving the following claim.

Claim. $T_{1}\cap T_{2}$ is empty.

Proof. For any $i\in\{1,2\},$ [CKP14, Proposition 2.4] implies that there exist non-zero projections $p_{i}\in P_{i}$, $q_{i}\in A\rtimes\Gamma_{T_{i}}$, a subalgebra $Q_{i}\subset q_{i}(A\rtimes\Gamma_{T_{i}})q_{i}$, a partial isometry $v_{i}\in q_{i}Mp_{i}$ and an onto $*$- $\theta_{i}:p_{i}P_{i}p_{i}\to Q_{i}$ such that $\theta_{i}(x)v_{i}=v_{i}x,\text{ for all }x\in p_{i}P_{i}p_{i},$ and $Q_{i}\vee(Q_{i}^{\prime}\cap q_{i}(A\rtimes\Gamma_{T_{i}})q_{i})\subset q_{i}(A\rtimes\Gamma_{T_{i}})q_{i}\text{ has finite index. }$ Moreover, the support projection of $E_{A\rtimes\Gamma_{T_{i}}}(v_{i}v_{i}^{*})$ can be assumed to equal $q_{i}.$ Denote $S_{i}:=Q_{i}^{\prime}\cap q_{i}(A\rtimes\Gamma_{T_{i}})q_{i}$.

Assume by contradiction that there exist a non-zero projection $z\in\mathcal{N}_{q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}}(S_{1})^{\prime}\cap q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}$ and an index $j\in T_{1}$ such that $S_{1}z_{0}$ is non-amenable relative to $A\rtimes\Gamma_{T_{1}\setminus\{j\}},$ for all $z_{0}\in(Q_{1}\vee S_{1})^{\prime}\cap z(A\rtimes\Gamma_{T_{1}})z$. Note that $z,z_{0}\in S_{1}$ and that the inclusion $z_{0}(Q_{1}\vee S_{1})z_{0}\subset z_{0}(A\rtimes\Gamma_{T_{1}})z_{0}$ has finite index by Lemma 2.8. Therefore, Corollary 2.14 implies that $Q_{1}z_{0}\prec_{A\rtimes{{\Gamma_{T_{1}}}}}A\rtimes\Gamma_{T_{1}\setminus\{j\}}.$ By applying [DHI16, Lemma 2.4(3)], we get that $Q_{1}z\prec^{s}_{A\rtimes{{\Gamma_{T_{1}}}}}A\rtimes\Gamma_{T_{1}\setminus\{j\}}.$ It is easy to see that the moreover part of the previous paragraph shows that $P_{1}\prec_{M}Q_{1}z.$ Hence, by applying [Va08, Lemma 3.7] we get that $P_{1}\prec_{M}A\rtimes\Gamma_{T_{1}\setminus\{j\}},$ which contradicts the minimality of $T_{1}.$

Therefore, for any $j\in T_{1}$ and $z\in\mathcal{N}_{q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}}(S_{1})^{\prime}\cap q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}$, there exists a non-zero projection $z_{0}\in(Q_{1}\vee S_{1})^{\prime}\cap z(A\rtimes\Gamma_{T_{1}})z$ such that $S_{1}z_{0}$ is amenable relative to $A\rtimes\Gamma_{T_{1}\setminus\{j\}}.$ [DHI16, Lemma 2.6(2)] shows that we can assume $z_{0}\in\mathcal{N}_{z(A\rtimes\Gamma_{T_{1}})z}(S_{1}z)^{\prime}\cap z(A\rtimes\Gamma_{T_{1}})z$. By applying [PV11, Proposition 2.7] finitely many times, we obtain that there exists a non-zero projection $z_{1}\in\mathcal{N}_{q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}}(S_{1})^{\prime}\cap q_{1}(A\rtimes\Gamma_{T_{1}})q_{1}$ such that $S_{1}z_{1}$ is amenable. In a similar way, there exists a non-zero projection $z_{2}\in\mathcal{N}_{q_{2}(A\rtimes\Gamma_{T_{2}})q_{2}}(S_{2})^{\prime}\cap q_{2}(A\rtimes\Gamma_{T_{2}})q_{2}\subset Q_{2}$ such that $S_{2}z_{2}$ is amenable.

Since $P_{1}\prec_{M}Q_{1}z_{1}$ and $P_{2}\prec_{M}Q_{2}z_{2}$, [Va08, Lemma 3.5] implies that $S_{1}z_{1}\prec_{M}P_{2}$ and $S_{2}z_{2}\prec_{M}P_{1}.$ By proceeding as in the proof of Theorem 3.2 (Claims 3 and 4), we obtain that there exist amenable subalgebras $D_{1}\subset P_{1}$ and $D_{2}\subset P_{2}$ such that $L(\Gamma_{T_{1}}\cap\Gamma_{T_{2}})\prec_{M}D_{1}\bar{\otimes}D_{2}.$ This implies that $\Gamma_{T_{1}}\cap\Gamma_{T_{2}}$ is amenable, hence $T_{1}\cap T_{2}$ is empty. $\blacksquare$

Remark 4.2**.**

We provide the following shorter argument for proving Proposition 4.1 if the groups $\Gamma_{i}$’s are only weakly amenable, bi-exact groups or free products. We only need to prove the claim. First, for any $t\in\{1,\dots,n\}$, denote $\hat{t}=\{1,\dots,n\}\setminus\{t\}$. Suppose by contradiction that there exists an element $t\in T_{1}\cap T_{2}.$ Then $P_{1}\nprec_{M}A\rtimes\Gamma_{T_{1}\setminus\{t\}}$ based on the minimality of $T_{1}$. This is equivalent to $P_{1}\nprec_{M}A\rtimes\Gamma_{\hat{t}}$ using [DHI16, Lemma 2.8(2)]. If $\Gamma_{t}$ is a free product, by applying Lemma 2.12, we must have $P_{2}\prec_{M}A\rtimes\Gamma_{\hat{t}}$. Using [DHI16, Lemma 2.8(2)] this shows that $P_{2}\prec_{M}A\rtimes\Gamma_{T_{2}\setminus\{t\}}$, which contradicts the minimality of $T_{2}$. On the other hand, if $\Gamma_{t}\in\mathcal{C}_{rss}$, by applying Lemma 2.11 we obtain that $P_{2}$ is amenable relative to $A\rtimes\Gamma_{\hat{t}}$, which implies that $P_{2}\prec A\rtimes\Gamma_{T_{2}\setminus\{t\}}$ or $M$ is amenable relative to $A\rtimes\Gamma_{\hat{t}}.$ The former contradicts the minimality of $T_{2}$, while the last one contradicts the non-amenability of $\Gamma_{t}$ by [OP07, Proposition 2.4].

The proof of Theorem A follows by combining Proposition 4.1 with Theorem 3.2. Corollary B is obtained directly from Theorem A. We continue now with the proofs of Corollary C and Theorem D.

Proof of Corollary C. By applying Theorem A finitely many times, we can find an integer $1\leq k\leq n$, a partition $S_{1}\sqcup...\sqcup S_{k}=\{1,...,n\}$ and pmp actions $\Gamma_{S_{i}}\curvearrowright(X_{i},\mu_{i})$ such that $\Gamma\curvearrowright X\text{ is isomorphic to }\Gamma_{S_{1}}\times...\times\Gamma_{S_{k}}\curvearrowright X_{1}\times...\times X_{k},$ and $M_{i}=L^{\infty}(X_{i})\rtimes\Gamma_{S_{i}}$ is prime for all $i\in\{1,...,k\}$. Note that the following holds:

Claim. If $\Gamma\curvearrowright X$ is isomorphic to $\Gamma_{T_{1}}\times\Gamma_{T_{2}}\curvearrowright Y_{1}\times Y_{2}$ for a partition $T_{1}\sqcup T_{2}=\{1,...,n\}$, then there exists a partition $J_{1}\sqcup J_{2}=\{1,...,k\}$ such that $T_{1}=\sqcup_{i\in J_{1}}S_{i}$ and $T_{2}=\sqcup_{i\in J_{2}}S_{i}.$

Proof. First note that it is enough to show that if $S_{i}\cap T_{j}\neq\emptyset$, for some $i\in\{1,...,k\}$ and $j\in\{1,2\}$, then $S_{i}\subset T_{j}.$ To this end, take $i$ and $j$ as before. The assumption implies that the actions $\Gamma_{S_{i}}\curvearrowright X$ and $\Gamma_{S_{i}\cap T_{1}}\times\Gamma_{S_{i}\cap T_{2}}\curvearrowright Y_{1}\times Y_{2}$ are isomorphic. This shows that

[TABLE]

where we denote $S_{i}^{c}:=\{1,...,n\}\setminus S_{i}$, for any $1\leq i\leq k.$ Since the algebra $M_{i}$ is prime, we must have $S_{i}\cap T_{j}=S_{i},$ which implies that $S_{i}\subset T_{j}.$ $\square$

The Claim shows that the partition $S_{1}\sqcup...\sqcup S_{k}=\{1,...,n\}$ is unique up to a permutation of the sets. We continue now by proving the moreover part.

(1) Let $M=P_{1}\bar{\otimes}P_{2}$ for some II1 factors $P_{1}$ and $P_{2}$. If we apply Theorem A we obtain a partition $T_{1}\sqcup T_{2}=\{1,...,n\}$, a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$ for some positive $t$, and a unitary $u\in M$ such that

[TABLE]

The Claim shows that there exists a partition $J_{1}\sqcup J_{2}=\{1,...,k\}$ such that $T_{1}=\sqcup_{i\in J_{1}}S_{i}$ and $T_{2}=\sqcup_{i\in J_{2}}S_{i}.$ This exactly implies the conclusion.

(2) Assume that $M=P_{1}\bar{\otimes}\dots\bar{\otimes}P_{m}$, for some $m\geq k$. Part (1) combined with induction implies that $m\leq k$ and that there exist a partition $J_{1}\sqcup\dots\sqcup J_{m}=\{1,\dots,k\}$, a decomposition $M=P_{1}^{t_{1}}\bar{\otimes}\dots\bar{\otimes}P_{m}^{t_{m}}$ with $t_{1}\dots t_{m}=1$, and a unitary $u\in M$ such that $P_{s}^{t_{j}}=u(\bar{\otimes}_{j\in J_{s}}M_{j})u^{*}$, for any $s\in\{1,...,m\}.$ Therefore $m=k$ and the conclusion holds.

(3) For proving this last part, we proceed as in (2). Since each $P_{j}$ is prime, $J_{s}$ has only one element. This shows once again that $m=k$ and the conclusion holds. $\blacksquare$

Proof of Theorem D. (1) Denote $X=X_{1}\times...\times X_{k}$. Let $M=P_{1}\bar{\otimes}P_{2}$ for some II1 factors $P_{1}$ and $P_{2}$. By applying Proposition 4.1 and Theorem 3.1(2), we obtain that there exists a partition $I_{1}\sqcup I_{2}=\{1,...,n\}$ such that $P_{1}$ is stably isomorphic to $L^{\infty}(X)^{\Gamma_{I_{2}}}\rtimes\Gamma_{I_{1}}=\bar{\otimes}_{i\in I_{1}}M_{i}$ and $P_{2}$ is stably isomorphic to $L^{\infty}(X)^{\Gamma_{I_{1}}}\rtimes\Gamma_{I_{2}}=\bar{\otimes}_{i\in I_{2}}M_{i}.$

(2) & (3) Assume that $M=P_{1}\bar{\otimes}\dots\bar{\otimes}P_{m}$ for some integer $m$ and II1 factors $P_{1},\dots,P_{m}.$ Part (1) combined with induction implies that $m\leq k$ and that there exists a partition $J_{1}\sqcup\dots\sqcup J_{m}=\{1,\dots,k\}$ such that $P_{s}$ is stably isomorphic to $\bar{\otimes}_{j\in J_{s}}M_{j}$, for any $1\leq s\leq m$. If $m\geq k$ or if each $P_{s}$ is prime, we obtain that $m=k$ and each $J_{s}$ has only one element.

For proving the moreover part, assume that $\Gamma_{i}\curvearrowright(X_{i},\mu_{i})$ is strongly ergodic for any $1\leq i\leq k.$ Note that [CSU13, Theorem C] combined with [CSU13, Examples 1.4,1.5] imply that the class $\mathcal{C}$ is contained in the class of non-inner amenable groups. In combination with [Ch82], we obtain that $M$ does not have property Gamma.

Let $M=P_{1}\bar{\otimes}P_{2}$ for some II1 factors $P_{1}$ and $P_{2}$. We will show only part (1) from the moreover part since part (2) and (3) can be deduced as in Corollary C. The previous paragraph implies that $P_{1}$ and $P_{2}$ does not have property Gamma. As before, we can apply Proposition 4.1 and obtain a partition $I_{1}\sqcup I_{2}=\{1,...,n\}$ such that $P_{j}\prec_{M}L^{\infty}(X)\rtimes I_{j}$, for any $j\in\{1,2\}$. Applying [Ho15, Proposition 6.3], we obtain that $P_{j}\prec_{M}\bar{\otimes}_{i\in I_{j}}M_{i}$, for any $j\in\{1,2\}.$ By proceeding as in the proof of Theorem 3.2, we obtain that there exist a unitary $u\in M$ and a decomposition $M=P_{1}^{t}\bar{\otimes}P_{2}^{1/t}$, for some $t>0$, such that $P_{1}^{t}=u(\bar{\otimes}_{i\in I_{1}}M_{i})u^{*}$ and $P_{2}^{1/t}=u(\bar{\otimes}_{i\in I_{2}}M_{i})u^{*}$. This ends the proof. $\blacksquare$