Matrix-Test Duality: A Support-Function Characterization for $C^{\ast }$-Convex Families of CP Maps

Abstract

We develop a matrix-test dual framework for $C^{\ast }$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathcal{S}$ is an operator system and $\mathcal{T}$ is a unital $C^{\ast }$-algebra. Matrix tests $(k,f,s)$ induce evaluation functionals $\hat{I}¦\mapsto f(\hat{I}{¦}_k(s))$ and generate a natural weak topology $\ddot{I}„=\ddot{I}ƒ(\mathcal{E},\mathcal{F})$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$. Our main result provides a support-function/separation characterization of the $\ddot{I}„$-closed $C^{\ast }$-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 $\ddot{I}„$-closed $C^{\ast }$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,τ}$, an exact normalized bipolar-type reconstruction statement. We also show that $\ddot{I}„$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities.