Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps

TL;DR

This paper introduces a dual matrix-test framework for $C^*$-convex families of completely positive maps, providing support-function characterizations and technical tools for analyzing their convex hulls in operator algebra.

Contribution

It develops a novel matrix-test duality approach for $C^*$-convex sets of CP maps, including support-function criteria and a finite-dimensional folding technique.

Findings

Provides a support-function/separation characterization of $C^*$-convex hulls.

Introduces a finite-dimensional folding procedure for matrix tests.

Establishes that level-1 tests generate the topology, but higher levels are needed for inequalities.

Abstract

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 $\Phi\mapsto f(\Phi_k(s))$ and generate a natural weak topology $\tau=\sigma(\mathcal E,\mathcal F)$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$. Our main result provides a support-function/separation characterization of the $\tau$-closed $C^*$-convex hull $\overline{\cconv(\mathcal K)}^{\,\tau}$ 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 $\tau$-closed $C^*$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,\tau}$, an exact normalized bipolar-type reconstruction statement. We also show that $\tau$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities.