Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces

Abstract

Motivated by Popa’s seminal work Popa (Invent Math 165:409-45, 2006), in this paper, we provide a fairly large class of examples of group actions \(\Gamma \curvearrowright X\) satisfying the extended Neshveyev–Størmer rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever \(\Lambda \curvearrowright Y\) is a free ergodic pmp action and there is a \(*\)-isomorphism \(\Theta :L^\infty (X)\rtimes \Gamma {\rightarrow }L^\infty (Y)\rtimes \Lambda \) such that \(\Theta (L(\Gamma ))=L(\Lambda )\) then the actions \(\Gamma \curvearrowright X\) and \(\Lambda \curvearrowright Y\) are conjugate (in a way compatible with \(\Theta \)). We also obtain a complete description of the intermediate subalgebras of all (possibly non-free) compact extensions of group actions in the same spirit as the recent results of Suzuki (Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm Math Phy. ArXiv Preprint: arXiv:1805.02077, 2020). This yields new consequences to the study of rigidity for crossed product von Neumann algebras and to the classification of subfactors of finite Jones index.

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Notes

  1. 1.

    For every diffuse \(A\subseteq L(\Gamma )\) the relative commutant \(A'\cap L(\Gamma )\) is amenable

  2. 2.

    For every diffuse amenable \(A\subseteq L(\Gamma )\) the normalizer \(\mathcal N_{L(\Gamma )}(A)''\) is amenable

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Acknowledgements

The authors are grateful to Adrian Ioana and Jesse Peterson for many helpful discussions related to this project. The authors are extremely grateful to Yuhei Suzuki for carefully reading a first draft of this paper, for his helpful comments and suggestions, and for correcting numerous typos and minor inaccuracies. The authors are also grateful to Rahel Brugger for her helpful comments on our paper, and for correcting a few typos. The second author would like to thank Vaughan Jones for several suggestions and comments regarding the results of this paper. The second author would also like to thank Krishnendu Khan and Pieter Spaas for stimulating conversations regarding the contents of this paper. The first author was partially supported by NSF Grant DMS #1600688. Finally, the authors would like to thank the anonymous referee for many helpful comments and suggestions that greatly improved the exposition of the paper.

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Chifan, I., Das, S. Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces. Math. Ann. 378, 907–950 (2020). https://doi.org/10.1007/s00208-020-02064-8

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