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

## TL;DR

This paper identifies broad classes of group actions on probability spaces that exhibit strong rigidity properties, ensuring their von Neumann algebra structures uniquely determine the actions themselves, with implications for subfactor classification.

## Contribution

It provides new examples of actions satisfying the extended Neshveyev-St{
}ormer rigidity phenomenon and describes intermediate subalgebras in compact extensions, advancing rigidity theory.

## Key findings

- Actions satisfying the rigidity phenomenon are characterized.
- Complete description of intermediate subalgebras in compact extensions.
- Implications for classification of subfactors of finite Jones index.

## Abstract

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-St{\o}rmer 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.