Representations of $*$-semigroups associated to invariant kernels with values adjointable operators. I

TL;DR

This paper develops a general dilation theorem for positive semidefinite kernels valued in adjointable operators on VE-spaces, invariant under $*$-semigroup actions, unifying various dilation results in a non-topological setting.

Contribution

It introduces a broad dilation theorem for invariant kernels valued in adjointable operators, connecting dilation theory with reproducing kernel VE-spaces.

Findings

Established a general dilation theorem for invariant kernels.

Unified multiple dilation results in a non-topological framework.

Demonstrated the reproducing kernel structure's role in dilation theory.

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.