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

2025/11/17 by Mohsen Kian, Kian, Mohsen, Mario Krnic
#math.OA #math.FA

paper · pdf · doi:10.48550/arxiv.2511.13101

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 Φ↦ f(Φ_k(s)) and generate a natural weak topology τ=σ(\mathcal E,\mathcal F) on \mathcal E=span_{\mathbb C}(\CP(\mathscr S,\mathscr T)). Our main result provides a support-function/separation characterization of the τ-closed C^*-convex hull \cconv(\mathcal K) ^τ of a family \mathcal K⊆ \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∈\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.

Citations

• C*-extreme entanglement breaking maps on operator systems
• Noncommutative Choquet theory
• C*-Convexity of Norm Unit Balls
• Tensor Products of Operator Systems
• Vector spaces with an order unit
• Completely positive linear maps on complex matrices

Related

• Duality of cones of positive maps
• A bipolar theorem for L⁰₊(\Om, \Cal F, \P)
• Construction of exposed indecomposable positive linear maps between matrix algebras
• Linear preservers of copositive and completely positive matrices
• C^*-extreme contractive completely positive maps