Asymptotic structure of free product von Neumann algebras

Let $(M,\phi )=(M_1,{\phi }_1)\ast (M_2,{\phi }_2)$ be the free product of any $\sigma$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q\subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q\cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$andthecentralsequencealgebra$Q’ \cap M^\omega$arediffuse(e\mathrm{.}g\mathrm{.}$Q$isamenable),then$Q$mustsitinside$M_1$. 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 $M_1\subset M$ in arbitrary free product von Neumann algebras.