Operator Models for Hilbert Locally $C^*$-Modules
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866918180661755904 |
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| author | Gheondea, Aurelian |
| author_facet | Gheondea, Aurelian |
| contents | We single out the concept of concrete Hilbert module over a locally $C^*$-algebra by means of locally bounded operators on certain strictly inductive limits of Hilbert spaces. Using this concept, we construct an operator model for all Hilbert locally $C^*$-modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally $C^*$-modules. These are obtained as consequences of a general dilation theorem for positive semidefinite kernels invariant under an action of a $*$-semigroup with values locally bounded operators. As a by-product, we obtain two Stinespring type theorems for completely positive maps on locally $C^*$-algebras and with values locally bounded operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_07643 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Operator Models for Hilbert Locally $C^*$-Modules Gheondea, Aurelian Operator Algebras Primary 47A20, Secondary 46L89, 46E22, 43A35 We single out the concept of concrete Hilbert module over a locally $C^*$-algebra by means of locally bounded operators on certain strictly inductive limits of Hilbert spaces. Using this concept, we construct an operator model for all Hilbert locally $C^*$-modules and, as an application, we obtain a direct construction of the exterior tensor product of Hilbert locally $C^*$-modules. These are obtained as consequences of a general dilation theorem for positive semidefinite kernels invariant under an action of a $*$-semigroup with values locally bounded operators. As a by-product, we obtain two Stinespring type theorems for completely positive maps on locally $C^*$-algebras and with values locally bounded operators. |
| title | Operator Models for Hilbert Locally $C^*$-Modules |
| topic | Operator Algebras Primary 47A20, Secondary 46L89, 46E22, 43A35 |
| url | https://arxiv.org/abs/1507.07643 |