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Motivated by Popa's seminal work \cite{Po04}, in this paper, we provide a fairly large class of examples of group actions $\Gamma \curvearrowright X$ satisfying the extended Neshveyev-Stormer rigidity phenomenon \cite{NS03}: whenever $\Lambda \curvearrowright Y$ is a free ergodic pmp action and there is a $\ast$-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 \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.

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