  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.
