Intermediate subalgebras of cross-product von Neumann algebras arising from weakly compact actions

arXiv:2610.33320v1 Announce Type: new Abstract: In this short paper, we establish the following Galois-type property for all measure-preserving, weakly compact actions $G \curvearrowright (X,\mu) $ of a non-inner amenable group $G$ on a standard probability space $X$: for any intermediate von Neumann subalgebra $\mathcal L(G)\subseteq \mathcal N \subseteq L^\infty (X)\rtimes G$, there is a factor $G\curvearrowright Y$ of $G \curvearrowright X$ such that $\mathcal N= L^\infty(Y)\rtimes G$. In particular, this holds for all pmp distal actions $G\curvearrowright X$, thereby answering positively an open problem from \cite{BCCMSW24} (which can also be traced back to \cite{CD20}). We also provide an example showing our result is optimal.

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