Mathematical Sciences: Problems in Operator Algebra

摘要Muhly Muhly将研究算子代数中的各种问题，这些问题可以分为三组：一般算子代数、群组和Toeplitz及其相关算子。在第一个标题下，他将探讨5个致力于理解非自伴代数结构的策略性问题。重点将放在与算子代数相关的Hilbert模范畴上。这方面的灵感来自有限维纯环理论，但关键是要理解代数和所涉及的度量之间的相互作用。在Muhly的工作中，群胚为表示算子代数提供了具体的坐标。这个题目下的项目是他早期工作的自然延续。其中最重要的可能是为群胚发展一个良好的拓扑上同调理论的问题。他提议研究的关于Toeplitz算子的问题也自然地源于他早期的工作和进展，应该有助于阐明经典函数论/调和分析和算子理论之间的重要新联系。尽管在许多方面，Muhly将研究的问题的详细方面都受到核心数学问题的启发，但它们对应用数学和“现实世界”的意义是实质性的。Muhly的许多工作为控制理论，特别是H无穷控制理论奠定了数学基础，而H无穷控制理论又是工程师设计飞机、汽车、机器人和其他需要对外部刺激做出反应并被控制在特定指导方针内执行的设备的理论框架。他在群胚上的工作和他在Toeplitz算子上的一些工作与随机控制问题密切相关，即存在随机不确定性的控制问题，人们试图引导一条平均而言最好的路线。一般而言，他在算子代数方面的工作贡献于多变量控制理论领域和非平稳控制问题，即控制响应是输入时间的函数的问题。这样的问题在所有类型的设备设计中都变得越来越重要，从声学过滤器到数字处理器和神经网络。

Abstract Muhly Muhly will investigate a variety of problems in operator algebra that may be divided into three groups covered by the headings: General Operator Algebras, Groupoids, and Toeplitz and Related Operators. Under the first heading, he will pursue 5 strategic problems dedicated to understanding the structure of non-self-adjoint algebras. The focus will be on the categories of Hilbert modules associated with operator algebras. Inspiration for this derives from finite dimensional pure ring theory, but the key is to understand the interplay between the algebra and the metrics involved. In Muhly's work, groupoids provide concrete coordinates for representing operator algebras. The projects under this rubric are natural continuations of his earlier efforts. Most important among them, perhaps, is the problem of developing a good, topological cohomology theory for groupoids. The problems on Toeplitz operators that he proposes to investigate also derive naturally from his earlier work and progress on them should help shed light on important new connections between classical function theory/harmonic analysis and operator theory. Although in many respects, the detailed aspects of the problems Muhly will investigate are inspired by questions from core mathematics, their significance to applied mathematics and the "real world" is substantial. Much of Muhly's work contributes to the mathematical underpinning of control theory, H-infinity control theory, in particular, which, in turn, is the theoretical framework on which engineers base the design of aircraft, automobiles, robots, and other devices that need to respond to external stimuli and to be controlled to perform within specified guidelines. His work on groupoids and some of his work on Toeplitz operators is closely connected to issues of stochastic control, i.e., control problems where there are random uncertainties involved and one tries to steer a course that is best on the average. His work on operator algebras, generally, c ontributes to the field of multivariable control theory and to non-stationary control problems, i.e., problems where the control response is a function of the time of the input. Such problems are of increasing importance in the design of all types of devices, from acoustical filters, to digital processors, and neural networks.