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

We show that if $\Gamma \curvearrowright (X^{\Gamma },{\mu }^{\Gamma })$ is a Bernoulli action of an i.c.c. nonamenable group Γ 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 Λ, then the isomorphism $L^{\infty }\left(X^{\Gamma }\right)⋊\Gamma \cong L^{\infty }\left(Y\right)⋊\Lambda$ implies that $L^{\infty }\left(X^{\Gamma }\right)$ and $L^{\infty }\left(Y\right)$ 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 [1].