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

Abstract

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.