Boundary representations and pure completely positive maps

2011/10/19 by Craig Kleski, Kleski, Craig · 3 citations
Mathematics · #46L07 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L07

paper · pdf · doi:10.48550/arxiv.1110.4149

arxiv created 2011/10/19 · arxiv updated 2011/10/20

Abstract

In 2006, Arveson resolved a long-standing problem by showing that for any element x of a separable self-adjoint unital subspace S⊆ B(H), ‖x‖=sup‖π(x)‖, where π 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.

Cited by

• Boundary representations, geometry of matrix ranges, and C^*-envelopes of finite-dimensional operator systems
• A new obstruction to Arveson's hyperrigidity conjecture
• Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra

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