Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

Abstract

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