Non-unital operator systems that are dual spaces

Abstract

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

It is a pre-refereed version of a paper that will appear in Lin. Alg. Appl. The proof of Lemma 5 are removed in the published version. Some equation numbers and some statement numbers are also altered in the published version