Duality for operator systems with generating cones
Yu-Shu Jia, Chi-Keung Ng
TL;DR
The paper develops a broad duality framework for operator systems with generating cones, extending duality beyond dualizable (unital) cases by constructing a dual operator system $S^d$ from the dual space $S^*$ and a universal, weak$^*$-continuous theory of CP maps between dual objects. It introduces approximately unital operator systems, showing that their duals behave particularly well: $\iota_{S^*}$ is an operator-space isomorphism and $\beta_S$ is a complete isometry, with the dual functor acting fully and faithfully on these objects. A general dual functor is defined for all operator systems, relating biduals and double-duals via $\tau_S$, and its faithfulness/injectivity properties hold on the approximately unital subcategory. The development relies on semi-matrix ordered spaces (SMOS) and matrix-convex techniques (including Effros–Winkler separations), enabling a robust, functorial passage to duals and biduals, with applications to tolerance-relations operator systems and related nonunital constructions. Collectively, the results provide a comprehensive, mostly categorical, duality theory for operator systems with generating cones and their approximate-unital generalizations.
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 $ι_{S^*}: S^* \to S^\rd$ is a dual Banach space isomorphism. Furthermore, we construct a canonical completely contractive weak$^*$-homeomorphism $β_S: (S^\rd)^\rd\to S^{**}$, 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 $ι_{S^*}\circ \varphi^* = \varphi^\rd \circ ι_{T^*}$. 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 $ι_{S^*}:S^* \to S^\rd$ is an operator space isomorphism and $β_S: (S^\rd)^\rd\to S^{**}$ 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.