Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps
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| Format: | Preprint |
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2025
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| author | Kian, Mohsen Krnic, Mario |
| author_facet | Kian, Mohsen Krnic, Mario |
| contents | We develop a matrix-test dual framework for $C^*$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathscr S$ is an operator system and $\mathscr T$ is a unital $C^*$-algebra. Matrix tests $(k,f,s)$ induce evaluation functionals $Φ\mapsto f(Φ_k(s))$ and generate a natural weak topology $τ=σ(\mathcal E,\mathcal F)$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$.
Our main result provides a support-function/separation characterization of the $τ$-closed $C^*$-convex hull $\overline{\cconv(\mathcal K)}^{\,τ}$ of a family $\mathcal K\subseteq \CP(\mathscr S,\mathscr T)$ in terms of matrix-test inequalities. A key technical tool is a finite-dimensional folding procedure that compresses finite linear combinations of test functionals into a single higher-level matrix test. As consequences, we obtain a single-test witness for non-membership, support-function criteria for inclusion and equality of $τ$-closed $C^*$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,τ}$, an exact normalized bipolar-type reconstruction statement. We also show that $τ$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_13101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps Kian, Mohsen Krnic, Mario Operator Algebras Functional Analysis 46L07, 47L07, 46A20 We develop a matrix-test dual framework for $C^*$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathscr S$ is an operator system and $\mathscr T$ is a unital $C^*$-algebra. Matrix tests $(k,f,s)$ induce evaluation functionals $Φ\mapsto f(Φ_k(s))$ and generate a natural weak topology $τ=σ(\mathcal E,\mathcal F)$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$. Our main result provides a support-function/separation characterization of the $τ$-closed $C^*$-convex hull $\overline{\cconv(\mathcal K)}^{\,τ}$ of a family $\mathcal K\subseteq \CP(\mathscr S,\mathscr T)$ in terms of matrix-test inequalities. A key technical tool is a finite-dimensional folding procedure that compresses finite linear combinations of test functionals into a single higher-level matrix test. As consequences, we obtain a single-test witness for non-membership, support-function criteria for inclusion and equality of $τ$-closed $C^*$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,τ}$, an exact normalized bipolar-type reconstruction statement. We also show that $τ$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities. |
| title | Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps |
| topic | Operator Algebras Functional Analysis 46L07, 47L07, 46A20 |
| url | https://arxiv.org/abs/2511.13101 |