对偶中的算子系统
Operator Systems in Duality
AI总结：
针对算子系统对偶性难题，引入相对对偶概念，证明部分算子系统及其子系统等具有对偶，推导有限层极大性与极小性对偶，建立相关闭实现。
AI中文摘要：
无限维序单位空间的通常对偶不一定带有序单位，这是算子系统的对偶性问题极其困难的主要原因。我们不尝试计算典范对偶对象，而是引入算子系统的相对对偶概念：两个系统对偶，当它们的基础向量空间的配对在每个矩阵层诱导出锥配对时成立。我们发展该概念的基本性质，表明除了有限维算子系统外，可分B(H)和带有忠实迹的C*-代数也具有对偶，而部分其他算子系统则不具备。我们证明具有对偶的系统的子系统、商系统及合适的张量积仍具有对偶。作为推论，我们证明有限层极大性与有限层极小性对偶，并建立了到矩阵代数的ℓ^∞乘积中的弱连续和弱*闭实现。
英文摘要：
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.
