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

Abstract

We consider group measure space II 1 _{1} factors M = L ∞ ( X ) ⋊ Γ M=L^{\infty }(X)\rtimes \Gamma arising from Bernoulli actions of ICC property (T) groups Γ \Gamma (more generally, of groups Γ \Gamma containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for ∗ * –homomorphisms θ : M → M ⊗ ¯ M \theta :M\rightarrow M\overline {\otimes }M . We deduce that the action Γ ↷ X \Gamma \curvearrowright X is W ∗ ^{*} –superrigid, i.e. if Λ ↷ Y \Lambda \curvearrowright Y is any free, ergodic, measure preserving action such that the factors M = L ∞ ( X ) ⋊ Γ M=L^{\infty }(X)\rtimes \Gamma and L ∞ ( Y ) ⋊ Λ L^{\infty }(Y)\rtimes \Lambda are isomorphic, then the actions Γ ↷ X \Gamma \curvearrowright X and Λ ↷ Y \Lambda \curvearrowright Y must be conjugate. Moreover, we show that if p ∈ M ∖ { 1 } p\in M\setminus \{1\} is a projection, then p M p pMp does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that Γ \Gamma is torsion free). We also prove a rigidity result for ∗ * –homomorphisms θ : M → M \theta :M\rightarrow M , this time for Γ \Gamma in a larger class of groups than above, now including products of non–amenable groups. For certain groups Γ \Gamma , e.g. Γ = F 2 × F 2 \Gamma =\mathbb {F}_{2}\times \mathbb {F}_{2} , we deduce that M M does not embed into p M p pMp , for any projection p ∈ M ∖ { 1 } p\in M\setminus \{1\} , and obtain a description of the endomorphism semigroup of M M .

W*-superrigidity for group von Neumann algebras of left-right wreath products

“…In [25], Popa introduced his spectral gap methods to prove W * -rigidity theorems for A G 0 G in the case where G is a direct product of nonamenable groups. These methods and results have been generalized in many subsequent works (see, for example, [9,13,26,33]) and were in particular extended to cover certain generalized Bernoulli actions, associated with general group actions G I. So far, the spectral gap methods could only be employed under the assumption that Stab i is amenable for all i ∈ I (see, for example, [13,Corollary 4.3]).…”

Continuous orbit equivalence rigidity

“…The idea is to find classes of actions for which orbit equivalence already implies conjugacy. Indeed, impressive orbit equivalence rigidity results have been obtained in [28,7,15,16,17,14,13]. Viewing continuous orbit equivalence as the topological analogue of orbit equivalence, a natural question is whether there are rigidity phenomena for continuous orbit equivalence.…”

W∗-superrigidity for coinduced actions

“…In Section 5, following Ioana's idea [Io10], we obtain a dichotomy theorem for certain abelian algebras. The result is a straightforward adaptation of [IPV10, Theorem 5.1] to coinduced actions and has two consequences.…”