W∗-superrigidity of mixing Gaussian actions of rigid groups

We generalize W∗-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action Γ↷X is...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2013-09, Vol.244, p.69-90
Main Author: Boutonnet, Rémi
Format: Article
Language:English
Subjects:
[formula omitted] factors
[formula omitted]-superrigidity
Deformation/rigidity theory
Gaussian actions
Mathematics
Operator Algebras
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Summary:We generalize W∗-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action Γ↷X is W∗-superrigid. More precisely, if Λ↷Y is another free ergodic action such that the crossed-product von Neumann algebras are isomorphic L∞(X)⋊Γ≃L∞(Y)⋊Λ, then the actions are conjugate. We prove a similar statement whenever Γ is a non-amenable ICC product of two infinite groups.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2013.05.012