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.

Invariant weakly positive semidefinite kernels with values in topologically ordered $*$-spaces

“…Theorem 5.1 (Theorem 2.8 in [3]). Let Γ be a * -semigroup acting on a nonempty set X, H be a VE-space on an ordered * -space Z, and l : X × X → L * (H) be a kernel.…”

“…The aim of this article is to develop a systematic study of invariant weakly positive semidefinite kernels with values in ordered * -spaces and to show that most of the previous dilation results as in [11], [3], [4] and hence, most of the known dilation theory, can be recovered under this setting. The main results are contained in theorems 4.3, 4.5, and 4.6, from which we then show how special cases concerning different kinds of "stronger" positive semidefiniteness can be derived.…”

“…Clear. (1)⇒(3). Assuming that Z is an admissible space, let k be positive semidefinite, let (E; V ) be the weak VE-space linearisation of k. Then, E is naturally equipped with a Hausdorff locally convex topology, see Subsection 2.3, and then completed to a VH-space H. Thus, (H; V ) is a weak VH-space linearisation of k and it is easy to see that it is topologically minimal.…”

Representations of $$\varvec{*}$$ ∗ -Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators

“…We first provide some preliminary results on positive semidefinite kernels with values adjointable operators in a VE-space, cf. [5].…”

“…The main section of this article refers to positive semidefinite kernels with values continuous and continuously adjointable linear operators on VH-spaces. Again, since we built on dilation results on VE-spaces over * -ordered spaces investigated in [5], we first review the necessary terminology, results, and constructions corresponding to positive semidefinite kernels in the nontopological case. When working with kernels, there is a paradigmatic idea that the natural approach is through reproducing kernel spaces, e.g.…”