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

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及其相关算子。在第一个标题下，我们发现了一些与非自伴算子代数的结构和表示理论有关的问题。近年来，通过量化泛函分析的进展，即算子空间理论，这些代数的理论重新焕发了生机。这一主题，再加上Muhly和Solel最近识别各种非常一般的算子代数的C*-包络的工作，使得人们能够在30多年前由Wm.Arveson提出的程序上取得良好的进展，作为Hilbert空间上压缩算子的Sz.-Nagy-FOIas模型理论的推广：根据算子代数的C*-包络的C*-表示来研究算子代数的表示。在群胚领域，重点将放在对群胚的Fall丛的研究上。这项研究的动力来自于群群的表示理论和对群群相互作用的理解。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算子长期以来一直是算子代数/理论研究的热点。近年来，Muhly和夏证明了由所有Toeplitz算子生成的C*-代数的自同构。识别是用在函数理论和圆盘调和分析中起重要作用的结构来描述的。换位子奇异积分起着特别重要的作用。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起决定性作用，但跳到大于1的维度带来了困难的挑战。穆利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联系，最特别的是数学系统理论-所谓的H-无穷控制理论。基本上，这是一门专门研究汽车和飞机等机器制造问题的知识，以使它们满足特定的性能标准。其中包括方向盘的响应性，汽车的反应能力，飞机的方向舵反应能力。简而言之，穆利的工作将有助于为确保各种运输工具的安全的设计问题提供理论基础。Muhly关于*-对应上的张量代数的许多工作都受到了支撑这种控制理论的算子理论的启发，似乎可以用于多变量控制问题。事实上，非常巧合的是，它似乎已经为多变量问题做好了准备。Muhly希望他的研究能为多变量控制问题提供理论基础，这些问题目前基本上是用特别的方法处理的。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建立了意想不到的联系--所谓的开放系统理论出现在各种地方，如激光理论和致力于超导的材料科学。它们也通过被称为扇区的结构自然地出现在量子场论中。预计Muhly的工作将有助于提供量子场论的数学基础，以及激光和材料科学的进步。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。预计他在高维Toeplitz算子上的项目将在这一领域产生更大的影响。

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.