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,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 T d . It can be shown that T d is again dualizable. Furthermore, we will verify that 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 (S d ) d is a completely isometric complete order isomorphism.Consequently, a nice duality framework for operator systems is obtained, which includes all C *algebras and all unital operator systems.

Tolerance relations and operator systems

“…[13, Theorem 1]) that there exist equivalent matrix norms on matrix-regular operator spaces under which they become non-unital operator systems in the sense of Definition 2.2. In this context we should also mention the recent work on dual spaces of non-unital operator systems in [23].…”

Tolerance relations and operator systems

“…[13,Theorem 1]) that matrix-regular operator spaces are non-unital operator systems. In this context we should also mention the recent work on dual spaces of non-unital operator systems in [23].…”