Projects In Operator Algebra: Tensor Algebras, Coordinates, and Toeplitz Operators

算子代数中的项目：张量代数、坐标和 Toeplitz 算子

• 批准号: 0070405
• 负责人: Paul Muhly
• 金额: $ 14.23万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070405&HistoricalAwards=false
• 关键词: Projects; Operator; Algebra; Tensor; Algebras

ABSTRACTDMS-0070405Muhly will study a variety of problems in operator algebra that may bedivided into three groups: General Operator Algebras, Groupoids, andToeplitz and Related Operators. Under the first heading are found a numberof problems pertaining to the structure and representation theory ofnon-self-adjoint operator algebras. The theory of these algebras has beenreinvigorated in recent years through advances in what has come to be knownas quantized functional analysis, i.e., the theory of operator spaces. Thissubject coupled with the recent work of Muhly and Solel that identifies theC*-envelopes of various very general operator algebras allows one to makegood progress on the program proposed more than thirty years ago by Wm.Arveson as a generalization of the Sz.-Nagy - Foias model theory ofcontraction operators on Hilbert space: study the representations of anoperator algebra in terms of the C*-representations of its C*-envelope. Inthe area of groupoids, emphasis will be placed on the study of Fell bundlesover groupoids. The impetus for this study comes from the representationtheory of groupoids and from efforts to understand co-actions of groupoids.Further, these bundles (more accurately, generalizations of them) occur inthe theory of product systems and the structure of non-self-adjoint operatoralgebras just mentioned. Toeplitz operators have long been a subject ofintense interest in operator algebra/theory. In recent years Muhly and Xiahave identified the automorphisms of the C*-algebra generated by allToeplitz operators. The identification is described in terms of constructsthat have played important roles in function theory and harmonic analysis onthe disc. A particular importance is played by commutator singularintegrals. Muhly proposes to extend these results, to the extent possible,to multivariable settings. Again, commutator singular integrals should playa decisive role, but the jump to dimensions greater than one presentsdifficult challenges.Muhly's work on non-self-adjoint operator algebras, while inspired primarilyby developments in pure mathematics, has connections with problems of anapplied nature, most particularly, mathematical systems theory - the theoryof so-called H-infinity control. This, basically, is a body of knowledgededicated to the problem of building machines, such as cars and airplanes,so that they meet certain performance criteria. Among these are such thingsas the responsiveness of the steering wheel, in the case of cars, or therudder, in the case of airplanes. In short, Muhly's work will contribute totheoretical underpinnings of design problems ensuring the safety of allsorts of conveyances. Much of Muhly's work on tensor algebras overC*-correspondences, which has been inspired by the operator theory thatunderwrites this kind of control theory, seems to be usable in multivariablecontrol problems. Indeed, quite serendipitously, it appears to beready-made for multivariable problems. Muhly expects his research toprovide a theoretical underpinning of multivariable control problems thathere-to-fore have been handled by essentially ad hoc methods. Muhly's workin non-self-adjoint algebras and the theory of groupoids has made unexpectedconnections with quantum Markov processes - the theory of so-called opensystems that arise in a variety of places such as the theory of lasers andmaterials science devoted to super conductivity. They also arise naturallyin quantum field theory through the constructs known as sectors. It isanticipated that Muhly's work will help to provide the mathematicalunderpinnings of quantum field theory and advances in laser and materialsscience. Finally, Muhly's work on Toeplitz operators has made connectionswith perturbation theories that arise in areas of relativistic quantumtheory and in quantum chemistry. It is anticipated that his projects onhigher dimensional Toeplitz operators will have an increased impact in thisarena.

Muhly将研究算子代数中的各种问题，这些问题可分为三组：一般算子代数、群胚和Toeplitz及相关算子。 在第一个标题下发现了一些与非自伴算子代数的结构和表示论有关的问题。 这些代数的理论近年来通过量子化泛函分析的进展而重新焕发了活力，即，算子空间理论 这一主题加上Muhly和Solel最近的工作，确定了各种非常一般的算子代数的C *-包络，使人们能够在30多年前由Wm.Arveson提出的作为Sz. Hilbert空间上压缩算子的Nagy-Foias模型理论：研究算子代数的C ~*-包络的C ~*-表示。在广群领域，重点研究广群上的Fell代数。 这一研究的动力来自于群胚的表示理论和理解群胚的余作用的努力。此外，这些丛（更准确地说，是它们的推广）出现在刚才提到的乘积系统理论和非自伴算子代数的结构中。 Toeplitz算子一直是算子代数/理论研究的热点。 近年来，Muhly和Xia确定了由所有Toeplitz算子生成的C ~*-代数的自同构. 识别描述的constructsthat发挥了重要作用的函数理论和谐波分析的光盘。 一个特别重要的是发挥了换向器奇异积分。 Muhly建议将这些结果尽可能地扩展到多变量设置。 同样，换位子奇异积分应该发挥决定性的作用，但跳跃到大于1的维度提出了困难的挑战。Muhly关于非自伴算子代数的工作，虽然主要受到纯数学发展的启发，但与应用性质的问题有关，特别是数学系统理论-所谓的H ∞控制理论。基本上，这是一个知识体系，致力于制造机器的问题，比如汽车和飞机，使它们满足一定的性能标准。 其中包括方向盘的响应性，就汽车而言，或乳房，就飞机而言。 简而言之，Muhly的工作将有助于设计问题的理论基础，以确保各种安全性。 Muhly的大部分工作张量代数overC*-对应，这已经受到启发的运营商理论，承保这种控制理论，似乎是可用的多变量控制问题。 事实上，非常偶然的是，它似乎是为多变量问题准备的。 Muhly希望他的研究能为多变量控制问题提供理论基础，而这些问题在以前基本上都是用特别的方法来处理的。 Muhly在非自伴代数和群胚理论方面的工作与量子马尔可夫过程--所谓的开放系统理论--产生于各种地方，如激光理论和致力于超导的材料科学--建立了意想不到的联系。 在量子场论中，它们也通过被称为扇区的结构自然出现。 预期Muhly的工作将有助于提供量子场论的理论基础以及激光和材料科学的进展。 最后，Muhly关于Toeplitz算子的工作与相对论量子理论和量子化学领域中出现的微扰理论建立了联系。 预计他的高维Toeplitz算子项目将在这一领域产生越来越大的影响。