An approximation theorem for nuclear operator systems

2010/09/14 by Kyung Hoon Han, Vern I. Paulsen, Han, Kyung Hoon +1 · 2 citations
Mathematics · #46L06 #46L07 #47L07 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L06 #msc:46L07 #msc:47L07

paper · pdf · doi:10.48550/arxiv.1009.2541

10 pages; to appear in JFA

arxiv created 2011/05/05 · arxiv updated 2011/05/06

Abstract

We prove that an operator system \mathcal S is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps ϕ_λ: \cl S → M_nλ and ψ_λ: M_nλ → \cl S such that ψ_λ∘ ϕ_λ converges to \rm id_{\cl S} in the point-norm topology. Our proof is independent of the Choi-Effros-Kirchberg characterization of nuclear C^*-algebras and yields this characterization as a corollary. We give an example of a nuclear operator system that is not completely order isomorphic to a unital C^*-algebra.

Cited by

• Soft inductive limits of operator systems and a noncommutative Lazar-Lindenstrauss theorem
• Boundary representations, geometry of matrix ranges, and C^*-envelopes of finite-dimensional operator systems

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