An approximation theorem for nuclear operator systems

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 $ϕ_λ: \cl S \to M_{n_λ}$ and $ψ_λ: M_{n_λ} \to \cl S$ such that ${\psi }_{\lambda }\circ {\varphi }_{\lambda }$ converges to ${\rm id}_{\cl S}$ in the point-norm topology. Our proof is independent of the Choi-Effros-Kirchberg characterization of nuclear $C^{\ast }$-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 $C^{\ast }$-algebra.

10 pages; to appear in JFA