# Invariant Weakly Positive Semidefinite Kernels with Values in Topologically Ordered $*$-Spaces

## TL;DR

This paper studies invariant weakly positive semidefinite kernels valued in ordered $*$-spaces, exploring their decompositions and representations, unifying various dilation results for operator-valued kernels and maps on $*$-semigroups.

## Contribution

It introduces conditions for $*$-representations of invariant kernels on VE and VH spaces, unifying multiple existing dilation and positivity results.

## Key findings

- Established conditions for $*$-representations on VE and VH spaces.
- Unified various dilation theorems for invariant positive semidefinite kernels.
- Connected recent results on positive semidefinite maps with operator values.

## Abstract

We consider weakly positive semidefinite kernels valued in ordered $*$-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing kernel spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector Hilbert) type, more precisely, vector spaces that possess gramians (vector valued inner products). The main results refer to the case when the kernels are invariant under certain actions of $*$-semigroups and show under which conditions $*$-representations on VE-spaces, or VH-spaces in the topological case, can be obtained. Finally we show that these results unify most of dilation type results for invariant positive semidefinite kernels with operator values as well as recent results on positive semidefinite maps on $*$-semigroups with values operators from a locally bounded topological vector space to its conjugate $Z$-dual space, for $Z$ an ordered $*$-space.