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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