Invariant Weakly Positive Semidefinite Kernels with Values in Topologically Ordered $*$-Spaces
Serdar Ay, Aurelian Gheondea
TL;DR
The paper develops a unified dilation theory for weakly positive semidefinite kernels valued in topologically ordered $*$-spaces, realized on VE-spaces and VH-spaces. It establishes invariant linearisations and reproducing kernel spaces for such kernels, bridging non-topological and topological (boundedly/continuously adjointable) settings via $*$-representations. Through a series of unifying theorems, it shows that many existing dilation results for operator-valued kernels and maps on $*$-semigroups are instances of a single framework, and extends these ideas to novel topological contexts, including maps into the $Z$-dual space. The work provides a comprehensive, structurally unified view of dilation theory, with broad implications for operator-valued kernels, reproducing kernel spaces, and invariant positive semidefinite maps on $*$-semigroups.
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.