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 \cite{Po04}, in this paper, we provide a fairly large class of examples of group actions $\hat{I}“\curvearrowright X$ satisfying the extended Neshveyev-Størmer rigidity phenomenon \cite{NS03}: whenever $\hat{I}›\curvearrowright Y$ is a free ergodic pmp action and there is a $\ast$-isomorphism $\hat{I}˜:L^{\infty }(X)⋊\hat{I}“\to L^{\infty }(Y)⋊\hat{I}›$ such that $\hat{I}˜(L(\hat{I}“))=L(\hat{I}›)$ then the actions $\hat{I}“\curvearrowright X$ and $\hat{I}›\curvearrowright Y$ are conjugate (in a way compatible with $\hat{I}˜$). 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 \cite{Suzuki}. 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.

second version; 39 pages; few references added; stylistic changes