Choquet order and hyperrigidity for function systems

TL;DR

This paper connects the Choquet order on measures with operator algebra dilation theory, extending classical theorems to non-separable cases and linking measure maximality to hyperrigidity in C*-algebras.

Contribution

It introduces a dilation-theoretic characterization of the Choquet order, extending Cartier's theorem and relating measure maximality to hyperrigidity conjecture.

Findings

Extended Cartier's dilation theorem to non-separable settings

Linked measure maximality in Choquet order to hyperrigidity

Provided an example showing differences between orders

Abstract

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable setting, as well as a non-separable version of \v{S}a\v{s}kin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebra that have the unique extension property relative to a set of generators. This reduces the commutative case of Arveson's hyperrigidity conjecture to the question of whether measures that are maximal with respect to the classical Choquet order are also maximal with respect to this new order. An example shows that these orders are not the same in general.