Operator Systems in Duality
The usual dual of an infinite-dimensional order-unit space need not carry an order unit, and this is the main reason why duality for operator systems is notoriously hard. Rather than trying to compute a canonical dual object, we introduce a relative notion of duality for operator systems: Two systems are in duality when a pairing of their underlying vector spaces induces conic pairings at every matrix level. We develop basic properties of this notion, showing that beyond finite-dimensional operator systems, also separable $B(H)$ and $C^*$-algebras with a faithful trace admit duals, while some other operator systems do not. We prove that subsystems, quotients and suitable tensor products of systems with duals again admit duals. As consequences, we show that finite-level maximality is dual to finite-level minimality, and we establish weakly continuous and weak$^*$ closed realizations into $\ell^\infty$-products of matrix algebras.