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. © 2011 Elsevier Inc. All rights reserved.