Unique prime factorization and bicentralizer problem for a class of type III factors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2015
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| _version_ | 1866918094104952832 |
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| author | Houdayer, Cyril Isono, Yusuke |
| author_facet | Houdayer, Cyril Isono, Yusuke |
| contents | We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods factors, the tensor product factor $M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m$ retains the integer $m$ and each factor $M_i$ up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [OP03, Is14] and moreover provides new UPF results in the case when $M_1, \dots, M_m$ are free Araki-Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type ${\rm III_1}$ factors in the class $\mathcal C_{(\text{AO})}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1503_01388 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Unique prime factorization and bicentralizer problem for a class of type III factors Houdayer, Cyril Isono, Yusuke Operator Algebras 46L10, 46L36 We show that whenever $m \geq 1$ and $M_1, \dots, M_m$ are nonamenable factors in a large class of von Neumann algebras that we call $\mathcal C_{(\text{AO})}$ and which contains all free Araki-Woods factors, the tensor product factor $M_1 \mathbin{\overline{\otimes}} \cdots \mathbin{\overline{\otimes}} M_m$ retains the integer $m$ and each factor $M_i$ up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [OP03, Is14] and moreover provides new UPF results in the case when $M_1, \dots, M_m$ are free Araki-Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type ${\rm III_1}$ factors in the class $\mathcal C_{(\text{AO})}$. |
| title | Unique prime factorization and bicentralizer problem for a class of type III factors |
| topic | Operator Algebras 46L10, 46L36 |
| url | https://arxiv.org/abs/1503.01388 |