Preprint
Factorization of positive definite kernels. Correspondences: $C^{}$-algebraic and operator valued context vs scalar valued kernels
ArXiV.org
Cornell University
05/27/2025
DOI: 10.48550/arxiv.2505.21037
Abstract
We introduce and study a class M of generalized positive definite kernels of the form K : X × X → L(A, L(H)), where A is a unital C∗-algebra and H a Hilbert space. These kernels encode operator-valued correlations governed by the algebraic structure of 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 ˜K : (X×A×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 ∈ M admits a Stinespring-type factorization CP maps, we characterize kernel domination K ≤ L in terms of a positive operator A ∈ πL(A)′ satisfying K(s, t)(a) = VL(s)∗πL(a)AVL(t). We further show that when πL is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representation
Details
- Title: Subtitle
- Factorization of positive definite kernels. Correspondences: $C^{}$-algebraic and operator valued context vs scalar valued kernels
- Creators
- Palle E. T JorgensenJames Tian
- Resource Type
- Preprint
- Publication Details
- ArXiV.org
- DOI
- 10.48550/arxiv.2505.21037
- ISSN
- 2331-8422
- Publisher
- Cornell University; Ithaca, New York
- Language
- English
- Date posted
- 05/27/2025
- Academic Unit
- Mathematics
- Record Identifier
- 9984824168502771
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