A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups

Abstract

We show that if $\Gamma \curvearrowright (X^{\Gamma },{\mu }^{\Gamma })$ is a Bernoulli action of an i.c.c. nonamenable group $\Gamma$ which is weakly amenable with Cowling-Haagerup constant $1$, and $\Lambda \curvearrowright (Y,\nu )$ is a free ergodic p.m.p. algebraic action of a group $\Lambda$, then the isomorphism $L^{\infty }(X^{\Gamma })⋊\Gamma \cong L^{\infty }(Y)⋊\Lambda$ implies that $L^{\infty }(X^{\Gamma })$ and $L^{\infty }(Y)$ are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from \cite{BIP21}.