Representations of $$\varvec{*}$$ ∗ -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.