We investigate the notion of conditionally positive definite in the context of Hilbert $$C^*$$ 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 $$C^*$$ C ∗ -modules. As a consequence, we show that a $$C^*$$ C ∗ -metric space (S, d) is $$C^*$$ C ∗ -isometric to a subset of a Hilbert $$C^*$$ C ∗ -module if and only if $$K(s,t)=-d(s,t)^2$$ K ( s , t ) = - d ( s , t ) 2 is a conditionally positive definite kernel. We also present a characterization of the order $$K'\le K$$ K ′ ≤ K between conditionally positive definite kernels.
Positivity – Springer Journals
Published: Nov 26, 2016
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