Factorization of positive definite kernels. Correspondences: $C^{*}$-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 $K\colon X\times X\to L(\mathfrak{A},L(H))$, where $\mathfrak{A}$ is a unital $C^{*}$-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 $C^{*}$-algebras. Our approach is based on a scalar-valued kernel $\tilde{K}\colon(X\times\mathfrak{A}\times H)^{2}\to\mathbb{C}$ 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 $K(s,t)(a)=V(s)^{*}\pi(a)V(t)$. In analogy with the Radon--Nikodym theory for CP maps, we characterize kernel domination $K\leq L$ in terms of a positive operator $A\in\pi_{L}(\mathfrak{A})'$ satisfying $K(s,t)(a)=V_{L}(s)^{*}\pi_{L}(a)AV_{L}(t)$. We further show that when $\pi_{L}$ is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representations.