Isometric Dilations for Representations of Product Systems
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866912829437640704 |
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| author | Barik, Sibaprasad Bhattacharjee, M. Solel, B. |
| author_facet | Barik, Sibaprasad Bhattacharjee, M. Solel, B. |
| contents | We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions) but, for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar.
Our dilation is explicitly constructed and we present some applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isometric Dilations for Representations of Product Systems Barik, Sibaprasad Bhattacharjee, M. Solel, B. Operator Algebras Functional Analysis 46L08, 46L10, 47A13, 47A20, 47A56, 47L05, 47L30, 47L55 We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions) but, for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar. Our dilation is explicitly constructed and we present some applications. |
| title | Isometric Dilations for Representations of Product Systems |
| topic | Operator Algebras Functional Analysis 46L08, 46L10, 47A13, 47A20, 47A56, 47L05, 47L30, 47L55 |
| url | https://arxiv.org/abs/2401.01775 |