盖尔芬德与奈马克主题变奏曲
Variations on a theme of Gelfand and Naimark
摘要：
C*-代数在数学物理中被广泛用于表示物理系统的可观测量，有时被用作量子力学和经典统计力学严格公式的起点。然而，在许多情况下，观察量代数的天真选择不承认C*-代数结构，一些按摩是必要的。本文研究了C*-代数的哪些性质可以推广到更一般的代数，以及赋范代数的Gelfand理论的哪些修正是必要的。我们使用范畴论作为指导，通过用一个明显的函子定义代替Gelfand谱的普通定义，成功地将Gelfand-Naimark定理推广到局部凸 *-代数上.我们还记得一个鲜为人知的，但可能非常有用的推广斯通-维尔斯特拉斯定理完全正则，T2空间。
英文摘要：
C*-algebras are widely used in mathematical physics to represent the observables of physical systems, and are sometimes taken as the starting point for rigorous formulations of quantum mechanics and classical statistical mechanics. Nevertheless, in many cases the naive choice of an algebra of observables does not admit a C*-algebra structure, and some massaging is necessary. In this paper we investigate what properties of C*-algebras carry over to more general algebras and what modifications of the Gelfand theory of normed algebras are necessary. We use category theory as a guide and, by replacing the ordinary definition of the Gelfand spectrum with a manifestly functorial definition, we succeed in generalizing the Gelfand-Naimark theorem to locally convex *-algebras. We also recall a little-known but potentially very useful generalization of the Stone-Weierstrass theorem to completely regular, Hausdorff spaces.
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