Matrix regular operator space and operator system

We establish a relationship between Schreiner's matrix regular operator space and Werner's (nonunital) operator system. For a matrix ordered operator space $V$ with complete norm, we show that $V$ is completely isomorphic and complete order isomorphic to a matrix regular operator space if and only if both $V$ and its dual space $V^*$ are (nonunital) operator systems.Comment: 8 pages, to appear in JMA

An approximation theorem for nuclear operator systems

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 \phi_\lambda : \cl S \to M_{n_\lambda} and \psi_\lambda : M_{n_\lambda} \to \cl S such that $\psi_\lambda \circ \phi_\lambda$ converges to {\rm id}_{\cl S} in the point-norm topology. Our proof is independent of the Choi-Effros-Kirchberg characterization of nuclear $C^*$-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.Comment: 10 pages; to appear in JF