W$^{*}$-superrigidity for Bernoulli actions of property (T) groups

We consider group measure space II$_{1}$ factors $M=L^{\infty }(X)\rtimes \Gamma$ arising from Bernoulli actions of ICC property (T) groups $\Gamma$ (more generally, of groups $\Gamma$ containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for $*$-homomorphisms $\theta:M\rightarrow M\overline{\otimes }M$.

We deduce that the action $\Gamma \curvearrowright X$ is W$^{*}$-superrigid, i.e. if $\Lambda \curvearrowright Y$ is any free, ergodic, measure preserving action such that the factors $M=L^{\infty }(X)\rtimes \Gamma$ and $L^{\infty }(Y)\rtimes \Lambda$ are isomorphic, then the actions $\Gamma \curvearrowright X$ and $\Lambda \curvearrowright Y$ must be conjugate.

Moreover, we show that if $p\in M\setminus \{1\}$ is a projection, then $pMp$ does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that $\Gamma$ is torsion free).

We also prove a rigidity result for $*$-homomorphisms $\theta :M\rightarrow M$, this time for $\Gamma$ in a larger class of groups than above, now including products of non-amenable groups. For certain groups $\Gamma$, e.g. $\Gamma =\mathbb{F}_{2}\times \mathbb{F}_{2}$, we deduce that $M$ does not embed into $pMp$, for any projection $p\in M\setminus \{1\}$, and obtain a description of the endomorphism semigroup of $M$.