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Structure of bicentralizer algebras and inclusions of type \(\mathrm{III}\) factors

  • Hiroshi Ando
  • Uffe Haagerup
  • Cyril HoudayerEmail author
  • Amine Marrakchi
Article
  • 20 Downloads

Abstract

We investigate the structure of the relative bicentralizer algebra \(\mathrm{B}(N \subset M, \varphi )\) for inclusions of von Neumann algebras with normal expectation where N is a type \(\mathrm{III}_1\) subfactor and \(\varphi \in N_*\) is a faithful state. We first construct a canonical flow \(\beta ^\varphi : \mathbf {R}^*_+ \curvearrowright \mathrm{B}(N \subset M, \varphi )\) on the relative bicentralizer algebra and we show that the W\(^*\)-dynamical system \((\mathrm{B}(N \subset M, \varphi ), \beta ^\varphi )\) is independent of the choice of \(\varphi \) up to a canonical isomorphism. In the case when \(N=M\), we deduce new results on the structure of the automorphism group of \(\mathrm{B}(M,\varphi )\) and we relate the period of the flow \(\beta ^\varphi \) to the tensorial absorption of Powers factors. For general irreducible inclusions \(N \subset M\), we relate the ergodicity of the flow \(\beta ^\varphi \) to the existence of irreducible AFD subfactors in M that sit with normal expectation in N. When the inclusion \(N \subset M\) is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison’s problem when N is amenable.

Mathematics Subject Classification

46L10 46L30 46L36 46L37 46L55 

Notes

Acknowledgements

Cyril Houdayer is grateful to Sorin Popa for thought-provoking discussions that took place in Orsay in the Spring of 2017, and later inspired the idea of the relative bicentralizer theorem for discrete inclusions (Theorem E). He also thanks him for mentioning the reference [35] in connection with Theorem C. We thank Toshihiko Masuda for his comment regarding Application 2. We finally thank Yoshimichi Ueda for kindly sharing his unpublished personal notes with us regarding Kadison’s problem.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsChiba UniversityInageJapan
  2. 2.University of CopenhagenOdenseDenmark
  3. 3.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud, CNRS,Université Paris-SaclayOrsayFrance
  4. 4.Institut Universitaire de FranceOrsayFrance
  5. 5.CNRS-ENS Lyon, UMPA UMR 5669Lyon Cedex 7France

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