Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability s
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Mathematische Annalen
Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces Ionut Chifan1 · Sayan Das1 Received: 11 June 2019 / Revised: 15 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 X satisfying the extended Neshveyev–Størmer rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever Y is a free ergodic pmp action and there is a ∗-isomorphism : L ∞ (X ) →L ∞ (Y ) such that (L()) = L() then the actions X and Y are conjugate (in a way compatible with ). 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.
1 Introduction In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra to any measure preserving action X of a countable group on a probability space X . This is called the group measure space von Neumann algebra, denoted by L ∞ (X ) . The most interesting case for study is when the initial action X is free and ergodic, in which case the group measure space
Communicated by Andreas Thom.
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Ionut Chifan [email protected] Sayan Das [email protected]
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Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242, USA
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I. Chifan, S. Das
construction is in fact a type II1 factor. When X is a singleton the group measure space construction yields just the group von Neumann algebra that will be denoted by L(). The latter is a II1 factor specifically when all nontrivial conjugacy classes of are infinite (henceforth abbreviated as the icc property). A problem of central importance in von Neumann algebras is to determine how much information about the action X can be recovered from the isomorphism class of L ∞ (X ) . An unprecedented progress in this direction emerged over the last decade from Popa’s influential deformation/rigidity theory [68]. A remarkable achievement of this theory was the discovery of first classes of examples of actions that are entirely remembered by their von Neumann algebras; for some examples see [6,11,12,14,17,18,25,29,32,35,37–39,58,68,69,71–73,78]. We refer the reader to the surveys [41,77] for an overview of the recent developments. There are two distinguished subalgebras of L ∞ (X ) : the coefficient (or Cartan) subalgebra L ∞ (X ) ⊂ L ∞ (X ) and the group von Neumann subalgebra L() ⊂ L ∞ (X ) . The classificat
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