Boundary representations and pure completely positive maps

TL;DR

This paper improves the understanding of boundary representations in operator systems by showing the supremum can be replaced with a maximum, confirming the classical boundary concept and enhancing the Krein-Milman theorem for matrix convex sets.

Contribution

It proves that the supremum in Arveson's boundary representation theorem can be replaced by a maximum, strengthening the connection between boundary representations and classical boundaries.

Findings

Supremum can be replaced by maximum in boundary representation theorem.

Choquet boundary for separable operator systems is a classical boundary.

Enhanced Krein-Milman theorem for matrix convex sets.

Abstract

In 2006, Arveson resolved a long-standing problem by showing that for any element $x$ of a separable self-adjoint unital subspace $S\subseteq B(H)$, $\|x\|=\sup\|\pi(x)\|$, where $\pi$ runs over the boundary representations for $S$. Here we show that "sup" can be replaced by "max". This implies that the Choquet boundary for a separable operator system is a boundary in the classical sense; a similar result is obtained in terms of pure matrix states when $S$ is not assumed to be separable. For matrix convex sets associated to operator systems in matrix algebras, we apply the above results to improve the Webster-Winkler Krein-Milman theorem.