预对偶与紧算子系统
Predual and tight operator systems
AI总结：
本文研究算子系统的预对偶构造，证明对偶算子系统V同构于完全算子系统T的对偶当且仅当V紧，且单位完全算子系统与C^*-代数均为紧的，可表示为对偶算子系统的预对偶。
AI中文摘要：
给定一个带有生成锥的（不一定是单位的）完全算子系统S，文献中已有一些关于S的算子系统对偶S^d的研究，即对偶矩阵序空间S^*，其配备矩阵范数后成为满足某种泛性质的对偶算子系统。为进一步研究S^d，需考虑算子系统预对偶构造。更准确地说，给定一个带有生成锥的对偶算子系统V，预对偶空间V_*上存在一个矩阵范数，使其成为满足某种泛性质的完全算子系统V_#。本文证明，V同构于完全算子系统T的对偶T^d当且仅当V满足称为紧性的自然性质；此时V同构于(V_#)^d。此外，本文证明若S是紧的，则S同构于对偶算子系统W的W_#；实际上S同构于(S^d)_#。由于所有单位完全算子系统和所有C^*-代数都是紧的，可见每个单位完全算子系统和每个C^*-代数都具有对偶算子系统W的W_#形式。
英文摘要：
Given a (not necessarily unital) complete operator system $S$ with a generating cone, there are some studies in literature on the operator system dual $S^\mathrm{d}$ of $S$, i.e., the dual matrix-ordered space $S^*$ equipped with a matrix norm that turns it into a dual operator system satisfying certain universal property. In order to do further study on $S^\mathrm{d}$, one needs to consider the operator system predual construction. More precisely, given a dual operator system $V$ with a generating cone, there is a matrix norm on the predual space $V_*$, that turns it into a complete operator system $V_\#$ satisfying certain universal property. In this article, we show that $V\cong T^\mathrm{d}$ for a complete operator system $T$ if and only if $V$ satisfies a natural property called tightness; in this case, $V\cong (V_\#)^\mathrm{d}$. Furthermore, we establish that if $S$ is tight, then $S\cong W_\#$ for a dual operator system $W$; in fact $S\cong (S^\mathrm{d})_\#$. Since all unital complete operator systems and all $C^*$-algebra are tight, one sees that every unital complete operator system and every $C^*$-algebra is of the form $W_\#$, for a dual operator system $W$.
