Operator system structures on ordered spaces

Abstract

Given an Archimedean order unit space (V, V+, e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system 𝒮 is completely boundedly isomorphic to either OMIN(𝒮) or to OMAX(𝒮). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.