Asymptotic structure of free product von Neumann algebras

Let (M,phi) = (M-1,phi(1)) * (M-2,phi(2)) be the free product of any sigma-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q subset of M is a von Neumann subalgebra with separable predual such that both Q and Q boolean AND M-1 are the ranges of faithful normal conditional expectations and such that both the intersection Q boolean AND M-1 and the central sequence algebra Q' boolean AND M-omega are diffuse (e.g. Q is amenable), then Q must sit inside M-1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M-1 subset of M in arbitrary free product von Neumann algebras.