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.

抽象Muhly Muhly将调查各种各样的问题，在运营商代数，可分为三组所涵盖的标题：一般运营商代数，groupoid，和Toeplitz和相关运营商。 在第一个标题下，他将追求5个战略问题，致力于理解非自伴代数的结构。 重点将放在与算子代数相关的希尔伯特模的类别上。 其灵感来自有限维纯环理论，但关键是理解所涉及的代数和度量之间的相互作用。 在Muhly的工作中，群胚提供了代表算子代数的具体坐标。 这个标题下的项目是他早期努力的自然延续。 其中最重要的，也许是问题的发展良好的，拓扑上同调理论的群胚。 问题Toeplitz运营商，他建议调查也自然来自他的早期工作和进展，他们应该有助于阐明重要的新的连接之间的经典函数理论/调和分析和运营商理论。 虽然在许多方面，问题的详细方面Muhly将调查的启发来自核心数学的问题，他们的意义，应用数学和“真实的世界”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，特别是H ∞控制理论，这反过来又是工程师设计飞机，汽车，机器人和其他需要响应外部刺激并在指定指导方针内进行控制的设备的理论框架。 他的工作群和他的一些工作Toeplitz运营商是密切相关的问题，随机控制，即，控制问题，其中涉及随机不确定性，并且人们试图引导平均最佳的路线。 他的工作算子代数，一般来说，贡献领域的多变量控制理论和非平稳控制问题，即，控制响应是输入时间的函数的问题。 这些问题在从声滤波器到数字处理器和神经网络的所有类型的设备的设计中越来越重要。