Predual and tight operator systems

Abstract

Given a (not necessarily unital) complete operator system $S$ with a generating cone, there are some studies in literature on the operator system dual $S^{\mathrm{d}}$ of $S$, i.e., the dual matrix-ordered space $S^{\ast }$ equipped with a matrix norm that turns it into a dual operator system satisfying certain universal property. In order to do further study on $S^{\mathrm{d}}$, one needs to consider the operator system predual construction. More precisely, given a dual operator system $V$ with a generating cone, there is a matrix norm on the predual space $V_{\ast }$, that turns it into a complete operator system $V_{\#}$ satisfying certain universal property. In this article, we show that $V\cong T^{\mathrm{d}}$ for a complete operator system $T$ if and only if $V$ satisfies a natural property called tightness; in this case, $V\cong (V_{\#}{)}^{\mathrm{d}}$. Furthermore, we establish that if $S$ is tight, then $S\cong W_{\#}$ for a dual operator system $W$; in fact $S\cong (S^{\mathrm{d}}{)}_{\#}$. Since all unital complete operator systems and all $C^{\ast }$-algebra are tight, one sees that every unital complete operator system and every $C^{\ast }$-algebra is of the form $W_{\#}$, for a dual operator system $W$.