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

2025/05/27 by Palle E. T. Jørgensen, Jorgensen, Palle E. T., James Tian +1
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical Analysis and Transform Methods #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2505.21037

Abstract

We introduce and study a class M of generalized positive definite kernels of the form K\colon X× X→ L(\mathfrakA,L(H)), where \mathfrakA is a unital C^*-algebra and H a Hilbert space. These kernels encode operator-valued correlations governed by the algebraic structure of \mathfrakA, 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 K\colon(X×\mathfrakA× H)²→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\inM 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≤ L in terms of a positive operator A∈π_L(\mathfrakA)' 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.

Citations

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