Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels
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| Format: | Preprint |
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2025
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| _version_ | 1866909625035522048 |
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| author | Jorgensen, Palle E. T. Tian, James |
| author_facet | Jorgensen, Palle E. T. Tian, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21037 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels Jorgensen, Palle E. T. Tian, James Operator Algebras Mathematical Physics Functional Analysis Primary: 46E22. Secondary: 43A65, 46L05, 46L07, 46L30, 47A20, 47B32, 81P15 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. |
| title | Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels |
| topic | Operator Algebras Mathematical Physics Functional Analysis Primary: 46E22. Secondary: 43A65, 46L05, 46L07, 46L30, 47A20, 47B32, 81P15 |
| url | https://arxiv.org/abs/2505.21037 |