Conditionally positive definite kernels in Hilbert $C^{\ast }$-modules

Abstract

We investigate the notion of conditionally positive definite in the context of Hilbert $C^{\ast }$-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 $C^{\ast }$-modules. As a consequence, we show that a $C^{\ast }$-metric space $(S,d)$ is $C^{\ast }$-isometric to a subset of a Hilbert $C^{\ast }$-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 $K^{\prime }\le K$ between conditionally positive definite kernels.

14 pages, to appear in Positivity