模表示理论与交换巴拿赫代数
Modular representation theory and commutative Banach algebras
摘要：
Benson和Symonds在最近的一篇论文中，为有限群的模表示引入了一个新的不变量.给出了表示环关于巴拿赫代数完备的谱半径的解释。我们在这里的目的是进一步采取这些概念，并调查的结构所产生的巴拿赫代数。该文件中的一些材料在这里重复更大的一般性，并为清晰的阐述。给出了抽象表示环和表示理想的公理化定义.完成，然后是一个交换巴拿赫代数，并从20世纪40年代的技术Gelfand适用，以研究空间的代数同态$\mathbb C$。这项研究的一个令人惊讶的结果是，雅各布森根和零根的（复杂化）表示环总是一致的。这些注释是为表征理论家准备的。因此，交换巴拿赫代数的背景材料是详细的，而表示理论的背景是更浓缩。
英文摘要：
In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to $\mathbb C$. One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
