में बचाया:
| मुख्य लेखकों: | , , |
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| स्वरूप: | Preprint |
| प्रकाशित: |
2016
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| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/1611.07914 |
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| _version_ | 1866916845663027200 |
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| author | Houdayer, Cyril Marrakchi, Amine Verraedt, Peter |
| author_facet | Houdayer, Cyril Marrakchi, Amine Verraedt, Peter |
| contents | We show that the tensor product $M \mathbin{\overline{\otimes}} N$ of any two full factors $M$ and $N$ (possibly of type ${\rm III}$) is full and we compute Connes' invariant $τ(M \mathbin{\overline{\otimes}} N)$ in terms of $τ(M)$ and $τ(N)$. The key novelty is an enhanced spectral gap property for full factors of type ${\rm III}$. Moreover, for full factors of type ${\rm III}$ with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor $M$ and any non-type ${\rm I}$ amenable factor $P$, the tensor product factor $M \mathbin{\overline{\otimes}} P$ has a unique McDuff decomposition, up to stable unitary conjugacy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_07914 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Fullness and Connes' $\boldsymbolτ$ invariant of type III tensor product factors Houdayer, Cyril Marrakchi, Amine Verraedt, Peter Operator Algebras 46L10, 46L36, 46L40, 46L55 We show that the tensor product $M \mathbin{\overline{\otimes}} N$ of any two full factors $M$ and $N$ (possibly of type ${\rm III}$) is full and we compute Connes' invariant $τ(M \mathbin{\overline{\otimes}} N)$ in terms of $τ(M)$ and $τ(N)$. The key novelty is an enhanced spectral gap property for full factors of type ${\rm III}$. Moreover, for full factors of type ${\rm III}$ with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor $M$ and any non-type ${\rm I}$ amenable factor $P$, the tensor product factor $M \mathbin{\overline{\otimes}} P$ has a unique McDuff decomposition, up to stable unitary conjugacy. |
| title | Fullness and Connes' $\boldsymbolτ$ invariant of type III tensor product factors |
| topic | Operator Algebras 46L10, 46L36, 46L40, 46L55 |
| url | https://arxiv.org/abs/1611.07914 |