Boundary and Rigidity of Nonsingular Bernoulli Actions

Abstract

Let G be a countable discrete group and consider a nonsingular Bernoulli shift action $G \curvearrowright \prod _{g\in G }(\{0,1\},\mu _g)$ with two base points. We prove the first rigidity result for Bernoulli shift actions that are not measure preserving, by proving solidity for certain non-singular Bernoulli actions, making use of a new boundary associated with such Bernoulli actions. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan–Ioana. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.

Solidity for Actions on von Neumann Algebras

Throughout this appendix, we fix a free ultrafilter $\omega$ on ${\mathbb {N}}$. We use Ocneanu’s ultraproduct von Neumann algebras $M^\omega$ in this section and we refer the reader to [AH12].

1.1 Definition and characterization

Recall that a diffuse von Neumann algebra M with separable predual is solid if for any diffuse von Neumann subalgebra $A\subset M$ with expectation, the relative commutant $A'\cap M$ is amenable [Oz03]. Any non-amenable solid factor is full (i.e. $M'\cap M^\omega ={\mathbb {C}}$) [Oz03, Proposition 7] [HU15, Theorem 3.1]. In particular, any non-amenable subfactor with expectation of a solid von Neumann algebra is full.

Let B be a diffuse von Neumann algebra with separable predual, G a countable discrete group, $G \curvearrowright ^\alpha B$ an action. Put $M:=B\rtimes _\alpha G$. We say that $\alpha$ is solid in the sense of Ozawa if for any diffuse von Neumann subalgebra $A\subset B$ with expectation, the relative commutant $A'\cap M$ is amenable. The goal of this appendix is to prove the following theorem. This is a generalization of [CI08, Proposition 6].

Theorem A.1

Let $B, G ,\alpha ,M$ be as above. Assume $\alpha$ is free in the sense that $B' \cap M\subset B$.

• 1. If $\alpha$ is solid in the sense of Ozawa, then for any intermediate von Neumann algebra $B\subset N\subset M$ with expectation, there exist mutually orthogonal projections $\{z_n\}_{n\ge 0}$ in ${\mathcal {Z}}(N)\subset {\mathcal {Z}}(B)$ such that $\sum _n z_n=1$, $Nz_0$ is amenable, and $(N'\cap B^\omega )z_n={\mathbb {C}}z_n$ for all $n\ge 1$.

• 2. Assume that B is commutative and write $B=L^\infty (X)$ and $L({\mathcal {R}}) = M$, where ${\mathcal {R}}$ is the equivalence relation arising from $\alpha$. Then $\alpha$ is solid in the sense of Ozawa if and only if for any subequivalence relation ${\mathcal {S}} \subset {\mathcal {R}}$, there exists a partition $\{X_n\}_{n\ge 0}$ of X into ${\mathcal {S}}$ invariant measurable sets such that

  • (a) ${\mathcal {S}}|_{X_0}$ is hyperfinite; and

  • (b) ${\mathcal {S}}|_{X_n}$ is strongly ergodic for all $n\ge 1$.

1.2 Proof of Theorem A.1

We keep the following setting. Let $B\subset ^{E_B} N\subset ^{E_N} M$ be inclusions of von Neumann algebras with separable predual and with expectation. We write $E_B\circ E_N$ again as $E_B$ and take a faithful normal state $\varphi \in M_*$ such that $\varphi = \varphi \circ E_B = \varphi \circ E_N$. We have inclusions with expectation

$$\begin{aligned} N'\cap (B^\omega )_{\varphi ^\omega } \subset N'\cap B^\omega \subset M^\omega . \end{aligned}$$

To prove Theorem A.1, we have to recall the following three lemmas. We introduce slightly more general ones, but the same proofs work by using the fact that B is contained in N.

The first one is by Ando and Haagerup [AH12, Theorem 5.2] (see also [HR14, Lemma 2.5]).

Lemma A.2

If $N'\cap (B^\omega )_{\varphi ^\omega }={\mathbb {C}}$, then $N'\cap B^\omega ={\mathbb {C}}$.

We next introduce Ioana’s lemma [Io12, Lemma 2.7]. For the proof, see its generalization [HR14, Theorem 2.3] by Houdayer and Raum and use the above Lemma A.2.

Lemma A.3

If there is a projection $z\in {\mathcal {Z}}(N'\cap B^\omega )$ such that $(N'\cap B^\omega ) e$ is discrete, then $e\in {\mathcal {Z}}(N'\cap B)$ and $(N'\cap B^\omega ) e=(N'\cap B)e$.

The last lemma is due to Popa [Oz03, Proposition 7]. For the proof, see its generalization [HU15, Theorem 3.1] by Houdayer and Raum and use Lemma A.2.

Lemma A.4

The following conditions are equivalent.

• 1. $N'\cap B^\omega$ is diffuse.

• 2. There exists a decreasing sequence $\{A_{n}\}_{n\ge 0}$ of diffuse abelian von Neumann subalgebras in B with expectation such that $N=\bigvee _n (A_n'\cap N)$.

Summarizing these lemmas, we get the following useful proposition.

Proposition A.5

Let $B\subset N\subset M$ be as above. Take the unique projection $z\in {\mathcal {Z}}(N'\cap B^\omega )$ such that $(N'\cap B^\omega )z$ is discrete and $(N'\cap B^\omega )z^\perp$ is diffuse. Then we have:

• $z\in {\mathcal {Z}}(N'\cap B)\subset {\mathcal {Z}}(B)$ and $(N'\cap B^\omega ) z=(N'\cap B) z$;

• there is a decreasing sequence $\{A_{n}\}_{n\ge 0}$ of diffuse abelian von Neumann subalgebras in $B z^\perp$ with expectation such that $Nz^\perp =\bigvee _n (A_n'\cap Nz^\perp )$.

Proof of Theorem A.1

1. Observe first that for any projection $p\in {\mathcal {Z}}(B)$ and any diffuse von Neumann subalgebra $A\subset Bp$ with expectation, the relative commutant $A'\cap pMp$ is amenable (indeed $(A\oplus Bp^\perp )'\cap M = (A'\cap pMp)\oplus {\mathcal {Z}}(B)p^\perp$ is amenable).

We see that for any projection $p\in {\mathcal {Z}}(B)$ and any intermediate von Neumann algebra $Bp\subset Q\subset pMp$ with expectation, if Q has no amenable direct summand, then $Q'\cap (B^\omega p)$ is discrete. Indeed, if not, then we can apply Proposition A.5 to $Bp\subset Q\subset pMp$, and there is $p\ne z\in {\mathcal {Z}}(Q'\cap Bp)\subset {\mathcal {Z}}(B)$ and $\{A_n\}_{n\ge 0}$ such that $Qz^\perp = \bigvee _n A_n'\cap Qz^\perp$. Since $Qz^\perp$ is not amenable, $A_n'\cap Qz^\perp$ is not amenable for n large enough. This implies that $A_n\subset Bz^\perp$ is diffuse while $A_n'\cap z^\perp Mz^\perp$ is not amenable. This contradicts the solidity of $\alpha$.

Let $B\subset N\subset M$ be as in the statement and take the unique projection $z_0\in {\mathcal {Z}}(N)\subset {\mathcal {Z}}(B)$ such that $Nz_0$ is amenable and $Nz_0^\perp$ has no amenable direct summand. We may assume $z_0^\perp \ne 0$. Put $p:=z_0^\perp$ and $Q:=Np$. By the observation in the previous paragraph, $Q'\cap (B^\omega p)$ is discrete, hence $Q'\cap (B^\omega p) = Q'\cap Bp = {\mathcal {Z}}(Q)$ by Proposition A.5. Let $\{z_n\}_{n\ge 1}$ be a family of mutually orthogonal minimal projections in ${\mathcal {Z}}(Q)$ such that $z_0^\perp = \sum _{n\ge 1}z_n$. Then since $z_n$ is minimal in $Q'\cap (B^\omega p)$, the conclusion follows.

2. We have only to prove the ‘if’ direction. For this, we can follow the proof of (3)$\Rightarrow$(2)$\Rightarrow$(1) in [CI08, Proposition 6]. $\quad \square$