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On the superrigidity of malleable actions with spectral gap. (English) Zbl 1222.46048
Summary: We prove that, if a countable group \( \Gamma \) contains infinite commuting subgroups \(H, H^{\prime}\subset \Gamma \) with \(H\) non-amenable and \( H^{\prime}\) “weakly normal” in \(\Gamma\), then any measure preserving \( \Gamma \)-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g., a Bernoulli \(\Gamma \)-action) is cocycle superrigid. If, in addition, \( H^{\prime}\) can be taken non-virtually abelian and \(\Gamma \curvearrowright X\) is an arbitrary free ergodic action, while \( \Lambda \curvearrowright Y=\mathbb{T}^{\Lambda }\) is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II\( _{1}\) factors \( L^{\infty }X \rtimes \Gamma \simeq L^{\infty }Y \rtimes \Lambda \) comes from a conjugacy of the actions.

MSC:
46L36 Classification of factors
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
28D15 General groups of measure-preserving transformations
46L55 Noncommutative dynamical systems
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