Structure of bicentralizer algebras and inclusions of type III factors
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866909690539016192 |
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| author | Ando, Hiroshi Haagerup, Uffe Houdayer, Cyril Marrakchi, Amine |
| author_facet | Ando, Hiroshi Haagerup, Uffe Houdayer, Cyril Marrakchi, Amine |
| contents | We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, φ)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $φ\in N_*$ is a faithful state. We first construct a canonical flow $β^φ: \mathbf R^*_+ \curvearrowright {\rm B}(N \subset M, φ)$ on the relative bicentralizer algebra and we show that the W$^*$-dynamical system $({\rm B}(N \subset M, φ), β^φ)$ is independent of the choice of $φ$ up to a canonical isomorphism. In the case when $N=M$, we deduce new results on the structure of the automorphism group of ${\rm B}(M,φ)$ and we relate the period of the flow $β^φ$ to the tensorial absorption of Powers factors. For general irreducible inclusions $N \subset M$, we relate the ergodicity of the flow $β^φ$ to the existence of irreducible hyperfinite 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_05706 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Structure of bicentralizer algebras and inclusions of type III factors Ando, Hiroshi Haagerup, Uffe Houdayer, Cyril Marrakchi, Amine Operator Algebras Mathematical Physics 46L10, 46L30, 46L36, 46L37, 46L55 We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, φ)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $φ\in N_*$ is a faithful state. We first construct a canonical flow $β^φ: \mathbf R^*_+ \curvearrowright {\rm B}(N \subset M, φ)$ on the relative bicentralizer algebra and we show that the W$^*$-dynamical system $({\rm B}(N \subset M, φ), β^φ)$ is independent of the choice of $φ$ up to a canonical isomorphism. In the case when $N=M$, we deduce new results on the structure of the automorphism group of ${\rm B}(M,φ)$ and we relate the period of the flow $β^φ$ to the tensorial absorption of Powers factors. For general irreducible inclusions $N \subset M$, we relate the ergodicity of the flow $β^φ$ to the existence of irreducible hyperfinite 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. |
| title | Structure of bicentralizer algebras and inclusions of type III factors |
| topic | Operator Algebras Mathematical Physics 46L10, 46L30, 46L36, 46L37, 46L55 |
| url | https://arxiv.org/abs/1804.05706 |