Asymptotic structure of free product von Neumann algebras

Abstract

AbstractLet (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. 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 M1 ⊂ M in arbitrary free product von Neumann algebras.