An approximation theorem for nuclear operator systems

TL;DR

This paper establishes a new approximation characterization for nuclear operator systems using nets of unital completely positive maps, independent of existing $C^*$-algebra characterizations, and provides a counterexample to their isomorphism with $C^*$-algebras.

Contribution

It introduces an approximation theorem for nuclear operator systems that does not rely on the Choi-Effros-Kirchberg characterization, expanding understanding of their structure.

Findings

Characterization of nuclear operator systems via nets of unital completely positive maps.

Proof independent of existing $C^*$-algebra characterizations.

Example of a nuclear operator system not isomorphic to a $C^*$-algebra.

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 $\phi_\lambda : \cl S \to M_{n_\lambda}$ 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.