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

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.

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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 We finally thank Yoshimichi Ueda for kindly sharing his unpublished personal notes with us regarding Kadison’s problem.

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Correspondence to Cyril Houdayer.

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HA is supported by JSPS KAKENHI 16K17608.

CH and AM are supported by ERC Starting Grant GAN 637601.

Uffe Haagerup: Deceased (19 December 1949 - 5 July 2015).

Communicated by Andreas Thom.

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Ando, H., Haagerup, U., Houdayer, C. et al. Structure of bicentralizer algebras and inclusions of type \(\mathrm{III}\) factors. Math. Ann. 376, 1145–1194 (2020). https://doi.org/10.1007/s00208-019-01939-9

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Mathematics Subject Classification

  • 46L10
  • 46L30
  • 46L36
  • 46L37
  • 46L55