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Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

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Asymptotic structure of free product von Neumann algebras

Published online by Cambridge University Press:  20 May 2016

CYRIL HOUDAYER  and
YOSHIMICHI UEDA
CYRIL HOUDAYER
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, FRANCE. e-mails: cyril.houdayer@math.u-psud.fr
YOSHIMICHI UEDA
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka, 810-8560, JAPAN. e-mails: ueda@math.kyushu-u.ac.jp
Corresponding
E-mail address:
cyril.houdayer@math.u-psud.fr
ueda@math.kyushu-u.ac.jp
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Abstract

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 ⊂ 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′ ∩ 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 1 ⊂ M in arbitrary free product von Neumann algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 489 - 516
Copyright
Copyright © Cambridge Philosophical Society 2016 

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