Operator system structures on ordered spaces

By Vern I. Paulsen, Ivan G. Todorov, M. Tomforde. 2009

99 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/621e81e23fedc5c7e63294157d452975a174ce9e

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In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Quotients, exactness, and nuclearity in the operator system category (2010, methodology): “n+Jis a matrix order unit of Mn/J,but it is not Archimedean. Let (Q,{Dn}∞ n=1,1) be any matrix ordered space with matrix order unit 1, which is not assumed to be Archimedean. It was shown in [20] and [19] that Qgives rise to an operator system through an Archimedeanization process, which we wish to ”
Minimal and maximal matrix convex sets (2017, methodology): “were described in several places; we follow the conventions of [6, Section 4] (see also [8], [12] and [21], noting that analogous constructions in the theory of operator spaces have appeared as far back as [19]).”
Characterisations of the weak expectation property (2013, background): “The Archimedanisation [26] of the family (Dn)n∈N of cones is an operator system structure on S⊗T ; the corresponding operator system is denoted by S ⊗max T .”
Operator system quotients of matrix algebras and their tensor products (2011, methodology): “The operator systems M n and Mn are completely order isomorphic via the map Sij → Eij , where {Sij }1≤i,j≤n is the basis of M n that is dual to the basis {Eij }1≤i,j≤n of Mn [16].”
Tensor products of operator systems (2009, background): “-algebraic tensor product A⊗ C*min B. For the next proposition, we recall that if (V,V+) is an AOU space, OMIN(V) denotes the minimal operator system whose underlying ordered ∗-vector space is (V,V+) [18]. Proposition 4.11. Let V and Wbe AOU spaces. Equip the tensor product V⊗W with the cone Q min =”