Mathematical Sciences: Problems in Operator Algebra

数学科学：算子代数问题

• 批准号: 9706713
• 负责人: Paul Muhly
• 金额: $ 11.4万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706713&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Operator; Algebra

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.

摘要Muhly Muhly将研究操作员代数中的各种问题，这些问题可能分为标题涵盖的三组：总操作员代数，Groupoids和Toeplitz及相关操作员。 在第一个标题下，他将提出5个致力于理解非自我支持代数的结构的战略问题。 重点将放在与操作员代数相关的希尔伯特模块的类别上。 此的灵感来自有限的维纯环理论，但关键是要了解代数与所涉及的指标之间的相互作用。 在Muhly的工作中，类固醇为代表操作员代数的具体坐标提供了混凝土坐标。 这个标题下的项目是他早期努力的自然延续。 也许其中最重要的是开发良好的，拓扑的共同体学理论的问题。 他提议调查的Toeplitz运营商的问题也自然而然地从他的早期工作中得出，并在他们身上的进步应该有助于阐明经典功能理论/谐波分析与操作员理论之间的重要新联系。 尽管在许多方面，Muhly会调查的问题的详细方面是受到核心数学的问题的启发，它们对应用数学的意义和“现实世界”是实质性的。 Muhly的大部分工作都有助于控制理论的数学基础，尤其是H-赋值控制理论，这又是理论上的框架，工程师以飞机，汽车，机器人和其他设备的设计为基础，这些框架以对外部刺激进行响应并在指定的指南中进行控制。 他在群体方面的工作以及他在Toeplitz运营商方面的一些工作与随机控制问题的问题密切相关，即控制问题，其中涉及随机不确定性并试图引导一门最好的课程。 通常，他在运算符代数方面的工作通常归类于多变量控制理论和非平稳控制问题的领域，即控制响应是输入时间的函数的问题。 从声学过滤器到数字处理器和神经网络的所有类型的设备的设计，此类问题在设计中越来越重要。