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Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa). (English. French summary) Zbl 1194.46085
Séminaire Bourbaki. Volume 2005/2006. Exposés Nos. 952–966. Paris: Société Mathématique de France (ISBN 978-2-85629-230-3/pbk). Astérisque 311, 237-294, Exp. No. 961 (2007).
Summary: Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra. This is the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain $$II_{1}$$ factors with prescribed countable fundamental group.
For the entire collection see [Zbl 1115.00012].

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 28D15 General groups of measure-preserving transformations 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37A55 Dynamical systems and the theory of $$C^*$$-algebras 37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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