Journal article
Factorization of operator valued kernels and their applications: Factorization of operator valued
Complex analysis and operator theory, Vol.20(5), 123
06/24/2026
DOI: 10.1007/s11785-026-01984-8
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 pos- itive (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 fac- torization 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 oper- ator A ∈ πL (A)′ satisfying K (s, t)(a) = V L (s)∗πL (a)AV L (t). We further show that when πL is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representations
Details
- Title: Subtitle
- Factorization of operator valued kernels and their applications: Factorization of operator valued
- Creators
- Palle E. T. Jorgensen - University of IowaJames Tian
- Resource Type
- Journal article
- Publication Details
- Complex analysis and operator theory, Vol.20(5), 123
- DOI
- 10.1007/s11785-026-01984-8
- ISSN
- 1661-8254
- eISSN
- 1661-8262
- Publisher
- Springer International Publishing
- Grant note
No funding was received for conducting this study.
- Language
- English
- Date published
- 06/24/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985176750002771
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