Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

Abstract

As an analogue of 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, a $\Gamma$-invariant measure $\mu\in\mathrm{Prob}(\mathrm{SA}(\Gamma\ltimes A))$ supported on amenable intermediate subalgebras between $A$ and $\Gamma\ltimes A$ is necessary supported on the subalgebras of $\mathrm{Rad}(\Gamma)\ltimes A$. By taking $(A,\tau)=L^\infty(X,\nu_X)$ for a free p.m.p. action $\Gamma \curvearrowright(X,\nu_X)$, we obtain a similar results for the invariant random subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.