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

2010/02/24 by Ioana, Adrian · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1002.4595

Abstract

We consider group measure space II₁ factors M=L^∞(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→ M⊗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^∞(X)\rtimesΓ and L^∞(Y)\rtimesΛ are isomorphic, then the actions Γ\curvearrowright X and Λ\curvearrowright 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=\Bbb F₂×\Bbb F₂, we deduce that M does not embed in pMp, for any projection p∈ M∖\1\, and obtain a description of the endomorphism semigroup of M.

Cited by

• Property (T) group factors whose Jones index set equals all positive integers
• The impact of Schur multipliers in harmonic analysis and operator algebras
• Relative solidity results and their applications to computations of some II₁ factor invariants

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