Asymptotic structure of free product von Neumann algebras

Let (M, φ) = (M 1, φ1) ∗ (M 2, φ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q C M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M 1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M 1 and the central sequence algebra Q′ ∩ M^ω 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 C M in arbitrary free product von Neumann algebras.