Representations of hermitian kernels by means of Krein spaces II. Invariant kernels

Abstract

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels which realize the given action by bounded operators on a Krein space. Applications to the GNS representation of *-algebras associated to hermitian functionals are given. We explain the key role played by the Kolmogorov decomposition in the construction of Weyl exponentials associated to an indefinite inner product and in the dilation thoery of hermitian maps on C*-algebras.