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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 QM is a von Neumann subalgebra with separable predual such that both Q and QM 1 are the ranges of faithful normal conditional expectations and such that both the intersection QM 1 and the central sequence algebra Q′Mω 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 1M in arbitrary free product von Neumann algebras.



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Asymptotic structure of free product von Neumann algebras



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