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

摘要

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 curved right arrow X satisfying the extended Neshveyev-Stormer rigidity phenomenon Neshveyev and Stormer (J Funct Anal 195(2):239-261, 2002): wheneve Lambda curved right arrow Y is a free ergodic pmp action and there is a*-somorphism Theta:L-infinity(X) (si) Gamma -> L-infinity(Y) (sic) Lambda such that Theta(L(Gamma))=L(Lambda) then the actions Gamma curved right arrow X and Lambda curved right arrow 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.