Non-unital operator systems that are dual spaces

TL;DR

This paper characterizes non-unital operator systems that are dual spaces using matrix analogues of classical results, providing a deeper understanding of their structure and positivity properties.

Contribution

It offers an abstract characterization of self-adjoint weak* closed subspaces of bounded operators, extending Bonsall's result to matrix cones and operator systems.

Findings

Characterization of self-adjoint weak* closed subspaces of al(H)

Matrix analogues of Bonsall's result for st-operator spaces

Equivalence of complete positivity and positivity of associated linear functionals

Abstract

We will give an abstract characterization of an arbitrary self-adjoint weak$^*$-closed subspace of $\mathcal{L}(H)$ (equipped with the induced matrix norm, the induced matrix cone and the induced weak$^*$-topology). In order to do this, we obtain a matrix analogues of a result of Bonsall for $^*$-operator spaces equipped with closed matrix cones. On our way, we observe that for a $^*$-vector $X$ equipped with a matrix cone (in particular, when $X$ is an operator system or the dual space of an operator system), a linear map $\phi:X\to M_n$ is completely positive if and only if linear functional $[x_{i,j}]_{i,j}\mapsto \sum_{i,j=1}^n \phi(x_{i,j})_{i,j}$ on $M_n(X)$ is positive.