Duality for operator systems with generating cones

Abstract

Let $S$ be a complete operator system with a generating cone; i.e. $S_\sa = S_+ - S_+$. We show that there is a matrix norm on the dual space $S^*$, under which, and the usual dual matrix cone, $S^*$ becomes a dual operator system with a generating cone, denoted by $S^\rd$. The canonical complete order isomorphism $\iota_{S^*}: S^* \to S^\rd$ is a dual Banach space isomorphism. Furthermore, we construct a canonical completely contractive weak$^*$-homeomorphism $\beta_S: (S^\rd)^\rd\to S^{**}$, and verify that it is a complete order isomorphism. For a complete operator system $T$ with a generating cone and a completely positive complete contraction $\varphi:S\to T$, there is a weak$^*$-continuous completely positive complete contraction $\varphi^\rd:T^\rd \to S^\rd$ with $\iota_{S^*}\circ \varphi^* = \varphi^\rd \circ \iota_{T^*}$. This produces a faithful functor from the category of complete operator systems with generating cones (where morphisms are completely positive complete contractions) to the category of dual operator systems with generating cones (where morphisms are weak$^*$-continuous completely positive complete contractions). We define the notion of approximately unital operator systems, and verify that operator systems considered in \cite{CvS} and \cite{CvS2} are approximately unital. If $S$ is approximately unital, then $\iota_{S^*}:S^* \to S^\rd$ is an operator space isomorphism and $\beta_S: (S^\rd)^\rd\to S^{**}$ is a complete isometry. We will also establish that the restriction of the faithful functor $(S,T,\varphi)\mapsto (T^\rd, S^\rd, \varphi^\rd)$ to the category of approximately unital complete operator systems is both full and injective on objects.