On the reduction theory of $W^{*}$-algebras by Hilbert modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916407083532288 |
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| author | Fidaleo, Francesco Zsido, Laszlo |
| author_facet | Fidaleo, Francesco Zsido, Laszlo |
| contents | We deal with the reduction theory of a $W^*$-algebra $M$ along a $W^*$-subalgebra $Z$ of the centre of $M$. This is done by using Hilbert modules naturally constructed by suitable spatial representations of the abelian $W^*$-algebra $Z$. We start with an exhaustive investigation of such kind of Hilbert modules, which is also of self-contained interest. After explaining the notion of the reduction in this framework, we exhibit the reduction of the standard form of a $W^*$-algebra $M$ along any $W^*$-subalgebra of its centre, containing the unit of $M$. In a forthcoming paper, this result is applied to study the structure of the standard representation of the $W^*$-tensor product $M_1\ots_Z M_2$ of two $W^*$-algebras $M_1$ and $M_2$ over a common $W^*$-subalgebra $Z$ of the centres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the reduction theory of $W^{*}$-algebras by Hilbert modules Fidaleo, Francesco Zsido, Laszlo Operator Algebras 46L10, 46L08 We deal with the reduction theory of a $W^*$-algebra $M$ along a $W^*$-subalgebra $Z$ of the centre of $M$. This is done by using Hilbert modules naturally constructed by suitable spatial representations of the abelian $W^*$-algebra $Z$. We start with an exhaustive investigation of such kind of Hilbert modules, which is also of self-contained interest. After explaining the notion of the reduction in this framework, we exhibit the reduction of the standard form of a $W^*$-algebra $M$ along any $W^*$-subalgebra of its centre, containing the unit of $M$. In a forthcoming paper, this result is applied to study the structure of the standard representation of the $W^*$-tensor product $M_1\ots_Z M_2$ of two $W^*$-algebras $M_1$ and $M_2$ over a common $W^*$-subalgebra $Z$ of the centres. |
| title | On the reduction theory of $W^{*}$-algebras by Hilbert modules |
| topic | Operator Algebras 46L10, 46L08 |
| url | https://arxiv.org/abs/2409.15029 |