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

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 Γ ↷ X satisfying the extended Neshveyev–Størmer rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever Λ ↷ Y is a free ergodic pmp action and there is a ∗ -isomorphism Θ : L^∞(X) ⋊ Γ → L^∞(Y) ⋊ Λ such that Θ (L(Γ)) = L(Λ) then the actions Γ ↷ X and Λ ↷ Y are conjugate (in a way compatible with Θ). 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.