Kadison duality for partially convex sets

TL;DR

This paper generalizes Kadison duality to partially convex sets, establishing a categorical duality with free order unit modules and proving related approximation and separation theorems.

Contribution

It introduces the concept of regular partially convex sets and their dual free order unit modules, extending classical dualities to a new setting.

Findings

Categorical duality between compact regular partially convex sets and free order unit modules.

Density of partially affine polynomials in continuous partially affine functions.

A Hahn-Banach-type separation theorem 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.