Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

Abstract

As an analogue of the topological boundary of discrete groups $\hat{I}“$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M,\ddot{I}„)$ and apply it to generalize the main results of [AHO23], showing that for a trace-preserving action $\hat{I}“\curvearrowright (A,\ddot{I}{„}_A)$ on an amenable tracial von Neumann algebra, any $\hat{I}“$-invariant amenable intermediate subalgebra between $A$ and $\hat{I}“⋉A$ is necessarily a subalgebra of $\mathrm{Rad}(\hat{I}“)⋉A$. By taking $(A,\ddot{I}{„}_A)=L^{\infty }(X,\hat{I}{½}_X)$ for a free pmp action $\hat{I}“\curvearrowright (X,\hat{I}{½}_X)$, we obtain a similar result for the invariant subequivalence relations of ${\mathcal{R}}_{\hat{I}“\curvearrowright X}$.

v4, 32 pages. Includes an appendix written by Tattwamasi Amrutam and Yongle Jiang, providing an alternative proof of Theorem A and B, along with an improvement of the assumptions