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

Abstract

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 relative property (T)) and prove a rigidity result for *--homomorphisms $\theta:M\to M\bar{\otimes}M$. We deduce that the action $\Gamma\curvearrowright X$ is W$^*$--superrigid. This means that if $\Lambda\curvearrowright Y$ is {\bf any other} 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 $\theta:M\to M$, 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. $\Gamma=\Bbb F_2\times\Bbb F_2$, 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 $M$.