Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

As an analogue of topological boundary of discrete groups Γ, we define the noncommutative topological boundary of tracial von Neumann algebras (M,τ) and apply it to generalize the main results of [AHO23], showing that for a trace preserving action Γ (A,τA) on an amenable tracial von Neumann algebra, a Γ-invariant measure μ(SA(Γ A)) supported on amenable intermediate subalgebras between A and Γ A is necessary supported on the subalgebras of Rad(Γ) A. By taking (A,τ)=L^∞(X,νX) for a free p.m.p. action Γ (X,νX), we obtain a similar results for the invariant random subequivalence relations of RΓ X.