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

Abstract

We consider positive semidefinite kernels valued in the ⁎-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of ⁎-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the ⁎-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non-topological level.