Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2015
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| _version_ | 1866915589910429696 |
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| author | Ay, Serdar Gheondea, Aurelian |
| author_facet | Ay, Serdar Gheondea, Aurelian |
| contents | We consider positive semidefinite kernels valued in the $*$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $*$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $*$-representations of the underlying $*$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $C^*$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally $C^*$-algebras and with values adjointable operators on Hilbert modules over locally $C^*$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_06840 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators Ay, Serdar Gheondea, Aurelian Operator Algebras Functional Analysis Primary 47A20, Secondary 43A35, 46E22, 46L89 We consider positive semidefinite kernels valued in the $*$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $*$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $*$-representations of the underlying $*$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $C^*$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally $C^*$-algebras and with values adjointable operators on Hilbert modules over locally $C^*$-algebras. |
| title | Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators |
| topic | Operator Algebras Functional Analysis Primary 47A20, Secondary 43A35, 46E22, 46L89 |
| url | https://arxiv.org/abs/1507.06840 |