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、类群和toeplitz及相关算子。在第一个标题下，我们发现了一些关于非自伴随算子代数的结构和表示理论的问题。近年来，这些代数的理论通过被称为量子化泛函分析的进展，即算子空间理论，重新焕发了活力。这个主题加上Muhly和Solel最近的工作，即识别各种非常一般的算子代数的c *包络，使得人们在三十多年前由Wm提出的计划上取得了良好的进展。Arveson作为Sz的推广。- nagy - Foias Hilbert空间上的收缩算子模型理论：用C*包络的C*表示研究一个算子代数的表示。在群类群中，重点研究了群类群上的费尔束。本研究的动力来自类群的表征理论和对类群相互作用的理解。此外，这些束（更准确地说，是它们的推广）出现在刚刚提到的积系统理论和非自伴随算子代数的结构中。在算子代数/理论中，Toeplitz算子一直是一个备受关注的课题。近年来，Muhly和夏已经确定了由所有toeplitz算子生成的C*-代数的自同构。该识别描述了在圆盘上的函数理论和谐波分析中发挥重要作用的结构。对易子奇异积分具有特别重要的意义。Muhly建议尽可能将这些结果扩展到多变量设置。再一次，换易子奇异积分应该起到决定性的作用，但是跳跃到大于1的维度提出了困难的挑战。Muhly在非自伴随算子代数上的工作，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，尤其是数学系统理论——所谓的h -∞控制理论。基本上，这是一个专门研究制造机器问题的知识体系，比如汽车和飞机，使它们达到一定的性能标准。其中包括汽车方向盘的反应能力，飞机的方向舵的反应能力。简而言之，Muhly的工作将为确保各种交通工具安全的设计问题提供理论基础。Muhly关于c *-对应上的张量代数的大部分工作，受到了支撑这种控制理论的算子理论的启发，似乎可用于多变量控制问题。事实上，非常偶然的是，它似乎已经为多变量问题准备好了。Muhly希望他的研究能为多变量控制问题提供一个理论基础，而这些问题到目前为止基本上都是由特别的方法来处理的。Muhly在非自伴代数和类群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领域，如激光理论和致力于超导的材料科学。在量子场论中，它们也通过被称为扇区的构造自然产生。预计Muhly的工作将有助于为量子场论提供数学基础，并在激光和材料科学方面取得进展。最后，Muhly对Toeplitz算子的研究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运营商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.