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Rigidity of free product von Neumann algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  17 November 2016

Cyril Houdayer  and
Yoshimichi Ueda
Cyril Houdayer
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email cyril.houdayer@math.u-psud.fr
Yoshimichi Ueda
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan email ueda@math.kyushu-u.ac.jp
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Abstract

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Let $I$ be any nonempty set and let $(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$ of (possibly type $\text{III}$) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. For the free product $(M,\unicode[STIX]{x1D711})=\ast _{i\in I}(M_{i},\unicode[STIX]{x1D711}_{i})$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_{i}$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type $\text{II}_{1}$ factors and is new for free product type $\text{III}$ factors. It moreover provides new rigidity phenomena for type $\text{III}$ factors.

Keywords

free product von Neumann algebrasPopa’s deformation/rigidity theoryproperty Gammatype III factorsultraproduct von Neumann algebras

MSC classification

Primary: 46L10: General theory of von Neumann algebras 46L54: Free probability and free operator algebras 46L36: Classification of factors
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2461 - 2492
Copyright
© The Authors 2016 

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