# Journal of Operator Theory

Volume 71, Issue 1, Winter 2014 pp. 45-62.

Boundary representations and pure completely positive maps**Authors**: Craig Kleski

**Author institution:**Department of Mathematics, Miami University, Oxford, OH, U.S.A.

**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.

**DOI:**http://dx.doi.org/10.7900/jot.2011oct22.1927

**Keywords:**operator system, pure completely positive map, boundary representation, peaking representation, matrix convex, $C^{\ast}$-convex, Krein-Milman theorem

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