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.

Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

“…For given Archimedean order unit spaces V , there are two canonical ways to endow matrix order structures with which they are operator systems [20]. These processes are usually called quantization.…”

“…is a unital complete order isomorphism [20,Theorem 6.2]. Related with quantization, γ gives rise to the duality [29, Proposition 6.5]:…”

Representations of ⁎-semigroups associated to invariant kernels with values adjointable operators

“…see [26]. For example, L * (E) has a natural structure of matrix ordered * -algebra, see Remark 1.6.…”

Tensor products of function systems revisited

“…For the definitions of OMIN and OMAX in the following proposition, we refer to [PTT,Definition 3.3] and [PTT,Definition 3.12]. …”

“…Recently, Paulsen and Tomforde introduced the Archimedeanization [PT] and this has been applied to operator system theory in a series of papers [PTT,KPTT1,KPTT2]. In this paper, we focus on its application to function systems.…”