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

TL;DR

We study generalized operator-valued positive definite kernels $K:X\times X\to L(\mathfrak{A},L(H))$ for a unital $C^{*}$-algebra $\mathfrak{A}$ and Hilbert space $H$, and show every $K$ admits a Stinespring-type factorization $K(s,t)(a)=V(s)^*\pi(a)V(t)$ via a scalar kernel $\tilde{K}$ and an associated RKHS. We develop a kernel domination theory: $K\le L$ iff there exists a positive $A$ in the commutant $\pi_L(\mathfrak{A})'$ with $K(s,t)(a)=V_L(s)^*\pi_L(a)AV_L(t)$, i.e. a Radon--Nikodym derivative $dK/dL$; irreducibility of $\pi_L$ collapses domination to scalar proportionality. The framework unifies dilation of CP maps, GNS-like constructions, and noncommutative kernel theory, and provides a scalar-valued analytic foundation for domination and Radon--Nikodym derivatives in operator-algebraic contexts with potential applications to quantum information and operator-valued learning.

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 $K(s,t)(a)=V(s)^{*}π(a)V(t)$. In analogy with the Radon--Nikodym theory for CP maps, we characterize kernel domination $K\leq L$ in terms of a positive operator $A\inπ_{L}(\mathfrak{A})'$ satisfying $K(s,t)(a)=V_{L}(s)^{*}π_{L}(a)AV_{L}(t)$. We further show that when $π_{L}$ is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representations.

Figures (1)

• Figure 2.1: Illustrations of related applications of the generalized kernels.

Theorems & Definitions (12)

• Definition 1
• Theorem 2
• proof
• Definition 3
• Theorem 4
• proof
• Definition 5
• Corollary 6
• Remark 7
• Corollary 8: Quantum Channel Simulation