Kadison duality for partially convex sets

Abstract

This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held fixed. We introduce the notion of a regular partially convex set and identify its dual as a finitely generated free module over a commutative C*-algebra endowed with a compatible Archimedean order unit structure. We call such spaces free order unit modules. We prove that for any compact regular partially convex set K, the space of continuous functions on K that are affine in the convex variables is the canonical example of such a module. Conversely, we show that the partially convex state space of a free order unit module is a compact regular partially convex set. Our main result establishes a categorical duality between compact regular partially convex sets and free order unit modules. We also establish a Stone-Weierstrass-type theorem, demonstrating that partially affine polynomials are dense in the space of continuous partially affine functions on any compact regular partially convex set. Finally, we prove a Hahn-Banach-type separation theorem of compact partially convex sets from their outer points.