Mathematical Sciences: Crossed Product Algebras, Non-Commutatuve Duality, and Applications to Group Representations

本课题是关于算子代数理论的数学研究。这些物体出现在量子力学的一些公式中，其中它们(或者更确切地说，它们的自伴部分)代表了被建模的系统的可观性。这一领域的基本结构之一是由给定代数上的群作用形成交叉积。(在物理学中，这个群很可能是实数，通过时间转换作用于可观察到的代数，得到系统的动力学。)最近，一种被称为交叉余积的对偶结构被证明是非常有用的，特别是当所讨论的群是非交换的时候。古特曼教授将继续他对交叉积和余积的研究，特别是着眼于研究这些代数的各种性质(理想结构、对偶拓扑、类型等)。通过将非对偶理论、非阿贝尔调和分析和表象理论的使用相互联系起来。

This project is mathematical research in the theory of operator algebras. These objects turn up in some formulations of quantum mechanics, where they (or rather their selfadjoint part) represent the observables of the system being modeled. One of the basic constructions in this area is the formation of the crossed product from a group action on a given algebra. (In physics, the group would most likely be the real numbers, acting on the algebra of observables by time translation to give the dynamics of the system.) Recently, a dual construction called the crossed coproduct has turned out to be very useful, especially when the group in question is nonabelian. Professor Gootman will continue his investigations of crossed products and coproducts, particularly with an eye to studying various properties of these algebras (ideal structure, dual topology, type, etc.) by interrelating the use of non- commutative duality theory, non-abelian harmonic analysis, and representation theory.