Conditionally positive definite kernels in Hilbert $C^*$-modules

TL;DR

This paper explores the concept of conditionally positive definite kernels within Hilbert $C^*$-modules, providing characterizations, representations, and conditions for isometric embeddings related to $C^*$-metrics.

Contribution

It introduces a characterization of conditionally positive definite kernels in Hilbert $C^*$-modules and establishes their relation to $C^*$-metric spaces and isometric embeddings.

Findings

Characterization of conditionally positive definite kernels in Hilbert $C^*$-modules

Representation of these kernels via Kolmogorov type theorem

Equivalence between $C^*$-metric spaces and subsets of Hilbert $C^*$-modules

Abstract

We investigate the notion of conditionally positive definite in the context of Hilbert $C^*$-modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert $C^*$-modules. As a consequence, we show that a $C^*$-metric space $(S, d)$ is $C^*$-isometric to a subset of a Hilbert $C^*$-module if and only if $K(s,t)=-d(s,t)^2$ is a conditionally positive definite kernel. We also present a characterization of the order $K'\leq K$ between conditionally positive definite kernels.