### 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 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.

Original language | English |
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Pages (from-to) | 489-516 |

Number of pages | 28 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 161 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*161*(3), 489-516. https://doi.org/10.1017/S0305004116000396

**Asymptotic structure of free product von Neumann algebras.** / Houdayer, Cyril; Ueda, Yoshimichi.

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 161, no. 3, pp. 489-516. https://doi.org/10.1017/S0305004116000396

}

TY - JOUR

T1 - Asymptotic structure of free product von Neumann algebras

AU - Houdayer, Cyril

AU - Ueda, Yoshimichi

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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′ ∩ 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 C M in arbitrary free product von Neumann algebras.

AB - 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′ ∩ 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 C M in arbitrary free product von Neumann algebras.

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UR - http://www.scopus.com/inward/citedby.url?scp=84969745116&partnerID=8YFLogxK

U2 - 10.1017/S0305004116000396

DO - 10.1017/S0305004116000396

M3 - Article

AN - SCOPUS:84969745116

VL - 161

SP - 489

EP - 516

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -