Dual spaces of operator systems

Abstract

The aim of this article is to give an infinite dimensional analogue of a result of Choi and Effros concerning dual spaces of finite dimensional unital operator systems. An (not necessarily unital) operator system is a self-adjoint subspace of L(H), equipped with the induced matrix norm and the induced matrix cone. We say that an operator system T is dualizable if one can find an equivalent dual matrix norm on the dual space T⁎ such that under this dual matrix norm and the canonical dual matrix cone, T⁎ becomes a dual operator system. We show that an operator system T is dualizable if and only if the ordered normed space M∞(T)sa satisfies a form of bounded decomposition property. In this case,‖f‖d:=sup⁡{‖[fi,j(xk,l)]‖:x∈Mn(T)+;‖x‖≤1;n∈N}(f∈Mm(T⁎);m∈N), is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on T⁎ that turns it into a dual operator system, denoted by Td. It can be shown that Td is again dualizable. For every completely positive completely bounded map ϕ:S→T between dualizable operator systems, there is a unique weak-⁎-continuous completely positive completely bounded map ϕd:Td→Sd which is compatible with the dual map ϕ⁎. From this, we obtain a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that if S is either a C⁎-algebra or a unital operator system, then S is dualizable and the canonical weak-⁎-homeomorphism from the unital operator system S⁎⁎ to the operator system (Sd)d is a completely isometric complete order isomorphism. Furthermore, via the duality functor above, the category of C⁎-algebras and that of unital operator systems (both equipped with completely positive complete contractions as their morphisms) can be regarded as full subcategories of the category of dual operator systems (with weak-⁎-continuous completely positive complete contractions as its morphisms). Consequently, a nice duality framework for operator systems is obtained, which includes all C⁎-algebras and all unital operator systems.