Partially Positive Semidefinite Maps on $*$-Semigroupoids and Linearisations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917053796974592 |
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| author | Gheondea, Aurelian Udrea, Bogdan |
| author_facet | Gheondea, Aurelian Udrea, Bogdan |
| contents | Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphs and representations of groupoids, we obtain a generalisation of the Sz-Nagy's Dilation Theorem for operator valued partially positive semidefinite maps on $*$-semigroupoids with unit, with varying degrees of aggregation, firstly by $*$-representations with unbounded operators and then we characterise the existence of the corresponding $*$-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued partially positive semidefinite maps on $*$-algebroids with unit and then, for the special case of $B^*$-algebroids with unit, we obtain a generalisation of the Stinespring's Dilation Theorem. As an application of the generalisation of the Stinespring's Dilation Theorem, we show that some natural questions on $C^*$-algebroids are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13107 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partially Positive Semidefinite Maps on $*$-Semigroupoids and Linearisations Gheondea, Aurelian Udrea, Bogdan Operator Algebras Primary 47L75, Secondary 43A35, 47A20, 47L60, 46L99 Motivated by Cuntz-Krieger-Toeplitz systems associated to undirected graphs and representations of groupoids, we obtain a generalisation of the Sz-Nagy's Dilation Theorem for operator valued partially positive semidefinite maps on $*$-semigroupoids with unit, with varying degrees of aggregation, firstly by $*$-representations with unbounded operators and then we characterise the existence of the corresponding $*$-representations by bounded operators. By linearisation of these constructions, we obtain similar results for operator valued partially positive semidefinite maps on $*$-algebroids with unit and then, for the special case of $B^*$-algebroids with unit, we obtain a generalisation of the Stinespring's Dilation Theorem. As an application of the generalisation of the Stinespring's Dilation Theorem, we show that some natural questions on $C^*$-algebroids are equivalent. |
| title | Partially Positive Semidefinite Maps on $*$-Semigroupoids and Linearisations |
| topic | Operator Algebras Primary 47L75, Secondary 43A35, 47A20, 47L60, 46L99 |
| url | https://arxiv.org/abs/2302.13107 |