Choquet order and hyperrigidity for function systems

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.