Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators

Abstract

We consider positive semidefinite kernels valued in the $\ast$-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 $\ast$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $\ast$-representations of the underlying $\ast$-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 $C^{\ast }$-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 $C^{\ast }$-algebras and with values adjointable operators on Hilbert modules over locally $C^{\ast }$-algebras.

40 pages

References in corpus (1)

• Representations of $\ast$-semigroups associated to invariant kernels with values adjointable operators. I

Cited by in corpus (2)

• Isometric Representations of Calibrated Ordered Spaces on $C(X)$
• Automatic Boundedness of Adjointable Operators on Barreled VH-Spaces