W*-superrigidity for Bernoulli actions of property (T) groups

We consider group measure space II${}_1$ factors $M=L^{\mathrm{\infty }}(X)⋊\mathrm{\Gamma }$ arising from Bernoulli actions of ICC property (T) groups $\mathrm{\Gamma }$ (more generally, of groups $\mathrm{\Gamma }$ containing an infinite normal subgroup with relative property (T)) and prove a rigidity result for *–homomorphisms $\theta :M\to M\bar{\otimes }M$. We deduce that the action $\mathrm{\Gamma }\curvearrowright X$ is W${}^{\ast }$–superrigid. This means that if $\mathrm{\Lambda }\curvearrowright Y$ is {\bf any other} free, ergodic, measure preserving action such that the factors $M=L^{\mathrm{\infty }}(X)⋊\mathrm{\Gamma }$ and $L^{\mathrm{\infty }}(Y)⋊\mathrm{\Lambda }$ are isomorphic, then the actions $\mathrm{\Gamma }\curvearrowright X$ and $\mathrm{\Lambda }\curvearrowright Y$ 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 $\mathrm{\Gamma }$ is torsion free). We also prove a rigidity result for *–homomorphisms $\theta :M\to M$, this time for $\mathrm{\Gamma }$ in a larger class of groups than above, now including products of non–amenable groups. For certain groups $\mathrm{\Gamma }$, e.g. $\mathrm{\Gamma }={\mathbb{F}}_2\times {\mathbb{F}}_2$, we deduce that $M$ does not embed in $pMp$, for any projection $p\in M\setminus \{1\}$, and obtain a description of the endomorphism semigroup of $M$.