Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)

By S. Vaes. 2006

107 citations recorded by the source corpus.

We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. 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, yielding 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c89773cd67ff736dc3de198691ef0382de016c7c

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In Algebra and Number Theory

1. A. Furman
2. Sorin Popa
3. S. Vaes
4. Y. Kida
5. Adrian Ioana
6. Bachir Bekka
7. Damien Gaboriau
8. J. Peterson

In Mathematical Physics

1. A. Furman
2. S. Vaes
3. Sorin Popa
4. Adrian Ioana
5. Y. Kida
6. Damien Gaboriau
7. Bachir Bekka
8. J. Peterson

Citations in the corpus

• Measure equivalence rigidity of the mapping class group (2006, background): “The first magnificent result about orbit equivalence is due to Ornstein-Weiss [33] following Dye [5], [6] and it can be stated in terms of measure equivalence as follows: a discrete group is measure equivalent to Z if and only if it is an infinite amenable group.”
• Operator-algebraic superridigity for SLn(ℤ), n≥3 (2006, background): “ algebra L(Γ).In recent years, a series of remarkable results concerning this question, with applications to ergodic theory, have been obtained by S. Popa ([Popa06-a], [Popa06-b]; for an account, see [Vaes06]). Other relevant work includes [CoHa89] and [Furm99]. The purpose of this paper is to discu”
• A random matrix approach to the Peterson-Thom conjecture (2020, unknown): “See [64, 67, 68, 38, 39] for further results, including resolutions”
• On Popa's Cocycle Superrigidity Theorem (2006, methodology): “The reader is also referred to Vaes’ Seminare Bourbaki [32] for an overview and more references.”
• A SURVEY OF MEASURED GROUP THEORY (2009, background): “The above mentioned assumption of essential freeness of an action Γ y (X ,µ) means that, up to a null set, the action is free; equivalently, for µ-a.e. x ∈ X the stabilizer {γ ∈ Γ : γ.x = x} is trivial.”
• Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups (2006, methodology): “7 in [44], using the measure theoretic framework developed in [17], Theorem 3.”