具有序单位的向量空间
Vector spaces with an order unit
摘要：
我们发展了一个具有序单位的序 *-向量空间的理论。我们证明了关于正线性泛函和状态的基本结果，我们表明，自伴元空间的序（半）规范允许多个扩展到整个空间的序（半）规范。我们挑出其中三个（半）范数作进一步的研究，并讨论它们对算子代数和算子系统的意义。此外，我们介绍了一个函子的方法，采取一个有序的空间与秩序单位，并形成阿基米德有序空间。然后，我们用这个过程来描述一个适当的概念，在阿基米德有序空间的范畴内的平行。
英文摘要：
We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple extensions to an order (semi)norm on the entire space. We single out three of these (semi)norms for further study and discuss their significance for operator algebras and operator systems. In addition, we introduce a functorial method for taking an ordered space with an order unit and forming an Archimedean ordered space. We then use this process to describe an appropriate notion of quotients in the category of Archimedean ordered spaces.
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