Chi-Keung Ng
(Nankai University)
June 9, 2021
Organized by Europe GNCG Seminar.

Dual spaces of operator systems

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 $\mathcal{L}(\mathfrak{H})$, 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 $M_\infty(T)^\mathrm{sa}$ satisfies a form of bounded decomposition property.
In this case,
$$\|f\|^\mathrm{d}:= \sup \big\{\big\|[f_{ij}(x_{kl})]\big\|: x\in M_n(T)_+; \|x\|\leq 1; n\in \mathbb{N}\big\} \quad (f\in M_m(T^*); m\in \mathbb{N}),$$
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 $T^\mathrm{d}$.
It can be shown that $T^\mathrm{d}$ 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 $(S^\mathrm{d})^\mathrm{d}$ 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.

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