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

We consider group measure space II$_1$ factors $M=L^{\infty}(X)\rtimesΓ$ arising from Bernoulli actions of ICC property (T) groups $Γ$ (more generally, of groups $Γ$ containing an infinite normal subgroup with relative property (T)) and prove a rigidity result for *--homomorphisms $θ:M\to M\bar{\otimes}M$. We deduce that the action $Γ\curvearrowright X$ is W$^*$--superrigid. This means that if $Λ\curvearrowright Y$ is {\bf any other} free, ergodic, measure preserving action such that the factors $M=L^{\infty}(X)\rtimesΓ$ and $L^{\infty}(Y)\rtimesΛ$ are isomorphic, then the actions $Γ\curvearrowright X$ and $Λ\curvearrowright Y$ 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 $Γ$ is torsion free). We also prove a rigidity result for *--homomorphisms $θ:M\to M$, this time for $Γ$ in a larger class of groups than above, now including products of non--amenable groups. For certain groups $Γ$, e.g. $Γ=\Bbb F_2\times\Bbb F_2$, 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 $M$.

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