Factorization of positive definite kernels. Correspondences: $C^{\ast }$-algebraic and operator valued context vs scalar valued kernels

Abstract

We introduce and study a class $\mathcal{M}$ of generalized positive definite kernels of the form $K:X\times X\to L(\mathfrak{A},L(H))$, where $\mathfrak{A}$ is a unital $C^{\ast }$-algebra and $H$ a Hilbert space. These kernels encode operator-valued correlations governed by the algebraic structure of $\mathfrak{A}$, and generalize classical scalar-valued positive definite kernels, completely positive (CP) maps, and states on $C^{\ast }$-algebras. Our approach is based on a scalar-valued kernel $\tilde{K}:(X\times \mathfrak{A}\times H{)}^2\to \mathbb{C}$ associated to $K$, which defines a reproducing kernel Hilbert space (RKHS) and enables a concrete, representation-theoretic analysis of the structure of such kernels. We show that every $K\in \mathcal{M}$ admits a Stinespring-type factorization $K(s,t)(a)=V(s{)}^{\ast }\ddot{I}€(a)V(t)$. In analogy with the Radon--Nikodym theory for CP maps, we characterize kernel domination $K\le L$ in terms of a positive operator $A\in \ddot{I}{€}_L(\mathfrak{A}{)}^{\prime }$ satisfying $K(s,t)(a)=V_L(s{)}^{\ast }\ddot{I}{€}_L(a)A{V}_L(t)$. We further show that when $\ddot{I}{€}_L$ is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representations.