An approximation theorem for nuclear operator systems

We prove that an operator system S is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps φλ:S→Mnλ and ψλ:Mnλ→S such that ψλ∘φλ converges to idS 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 explicit example of a nuclear operator system that is not completely order isomorphic to a unital C⁎-algebra.