Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

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