Summary: 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.