Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911599362572288 |
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| author | Pal, Sourav Sahasrabuddhe, Prajakta Tomar, Nitin |
| author_facet | Pal, Sourav Sahasrabuddhe, Prajakta Tomar, Nitin |
| contents | It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to any family $\mathcal T$ of $q$-commuting contractions with $\|q\|=1$ by showing the equivalence of the following three statements: $(i)$ $\mathcal T$ admits a regular $q$-unitary dilation; $(ii)$ $\mathcal T$ satisfies Brehmer's positivity condition; $(iii)$ $\mathcal T$ admits a $Q$-unitary dilation for a family of $Q$-commuting unitaries. We achieve the first part of the result by an application of Stinespring's dilation theorem on a particular completely positive map acting on a quotient algebra of a group $C^*$-algebra, where the underlying group is a free group, and the second part is obtained by an application of Naimark's theorem. Next, we find several cases when $\mathcal{T}$ admits a regular $q$-unitary dilation and establish a von Neumann type inequality for such a $q$-commuting family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10232 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality Pal, Sourav Sahasrabuddhe, Prajakta Tomar, Nitin Functional Analysis Operator Algebras It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to any family $\mathcal T$ of $q$-commuting contractions with $\|q\|=1$ by showing the equivalence of the following three statements: $(i)$ $\mathcal T$ admits a regular $q$-unitary dilation; $(ii)$ $\mathcal T$ satisfies Brehmer's positivity condition; $(iii)$ $\mathcal T$ admits a $Q$-unitary dilation for a family of $Q$-commuting unitaries. We achieve the first part of the result by an application of Stinespring's dilation theorem on a particular completely positive map acting on a quotient algebra of a group $C^*$-algebra, where the underlying group is a free group, and the second part is obtained by an application of Naimark's theorem. Next, we find several cases when $\mathcal{T}$ admits a regular $q$-unitary dilation and establish a von Neumann type inequality for such a $q$-commuting family. |
| title | Theory of $q$-commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2408.10232 |