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Published October 2011 | Published
Journal Article Open

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

Ioana, Adrian

Abstract

We consider group measure space ∥_1 factors M = L^(∞)(X) ⋊ Γ arising from Bernoulli actions of ICC property (T) groups Γ (more generally, of groups Γ containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for *-homomorphisms Θ: → M⊗(overbar)M We deduce that the action Γ ↷ X is W^*-superrigid, i.e. if Γ ↷ Y is any free, ergodic, measure preserving action such that the factors M = L^(∞)(X) ⋊ Γ and M =L^(∞)(Y) ⋊ Γ are isomorphic, then the actions Γ ↷ X and Γ ↷ Y must be conjugate. Moreover, we show that if p ∈ M\{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 → 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 = F_2 × F_2 we deduce that M does not embed into pMp, for any projection p ∈ M\{1}, and obtain a description of the endomorphism semigroup of M.

Additional Information

© 2011 American Mathematical Society. Received by editor(s): November 30, 2010; Received by editor(s) in revised form: April 20, 2011; Posted: June 8, 2011. I would like to thank Jesse Peterson for explaining [31] and Sorin Popa for many useful suggestions. I am also grateful to Vaughan Jones for bringing to my attention the question that led to Corollary G, Stefaan Vaes for providing me with an argument which simplifies the initial proof of Theorem 9.1 and the anonymous referee for several suggestions which helped improve the exposition. This paper was written while I was visiting the Department of Mathematics at UCLA. The author was supported by a Clay Research Fellowship.

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