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 that realize the given action by bounded operators on a Kreȋn 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 theory of hermitian maps on C * -algebras.

On L. Schwartz's Boundedness Condition for Kernels

“…Two Kolmogorov decompositions (V 1 , K 1 ) and (V 2 , K 2 ) of the same hermitian kernel K are unitarily equivalent if there exists a unitary operator Φ ∈ L(K 1 , K 2 ) such that for all x ∈ X we have V 2 (x) = ΦV 1 (x). The following result was obtained in [7]. We denote by ρ(T ) the resolvent set of the operator T .…”

“…The remaining question, especially in case K is not positive definite, is: what additional conditions on the kernel K should be imposed in order to ensure the boundedness of the operators U(a), a ∈ S? We gave a possible answer in [8], by considering an additional symmetry of the kernel.…”

“…From now on we assume that S is a unital semigroup with involution, that is, there exists a mapping I : S → S such that I 2 = the identity on S, and I(ab) = I(b)I(a) for all a, b ∈ S. The following result was obtained in [8].…”

“…We analysed in [7] conditions under which the linearization functor produces a Kreȋn space from a hermitian kernel, in the spirit of Kolmogorov type decompositions, and subsequently in [8] we generalized this construction to kernels invariant under the action of a semigroup with involution. We also related these constructions with the GNS representations of * -algebras, an issue of some recent interest in quantum field theory with indefinite metric ( [2], [14], [13], [19]).…”

Complex velocity transformations and the Bisognano–Wichmann theorem for quantum fields acting on Krein spaces

“…the topology generated by the restriction of ., . to H ± , recent results prove that there are infinitely many topologically inequivalent maximal Hilbert space structures [15,22,23]. In the generic situation in QFT the fundamental decomposition H ± can not be expected to have an intrinsically complete component.…”