Problems in the Theory and Application of Operator Tensor Algebras

算子张量代数理论与应用问题

• 批准号: 0355443
• 负责人: Paul Muhly
• 金额: $ 17.59万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355443&HistoricalAwards=false
• 关键词: Problems; Theory; Application; Operator; Tensor; Problems; Theory; Application; Operator; Tensor

AbstractMuhly Muhly intends to study problems in operator algebra that may be divided into three groups. In the first are found 7 broad questions concerning the theory of general operator algebras and their interactions with quantum Markov semigroups, completely positive maps and quantum information theory. Muhly intends to use the invariants he has discovered for tensor algebras and their representations to analyze semigroups of completely positive maps and related structures. A particular focus will be "continuous tensor algebras" built from product systems. The second group (5 problems) concerns the theory and application of groupoids. First, Muhly proposes to use his recent collaborative work on the Brauer group of a groupoid to build a general cohomology theory that is based on the category of actions of a groupoid on spaces. The second two focus on the structure of operator algebras built from so-called Fell bundles over groupoids. These arise quite commonly "in nature" and a good portion of the effort will be devoted to specific examples. The fourth deals with aspects of operator algebras associated with topological quivers - graphs in which the vertex and edge spaces are topological spaces. The final problem concerns a generalization of the Brauer group that interacts with a variety of investigations of current interest. The third area is part of Muhly's long-term investigations into the foundations of general operator algebra. The focus is on so-called orthoprojective and orthoinjective Hilbert modules with an eye to understanding boundary representations for operator algebras.The projects proposed herein derive naturally from and have an impact upon dynamical systems, particularly irreversible dynamical systems that appear in a variety of settings. Of special interest to us, are certain mathematical models, based on operator algebra, that contribute to the burgeoning area of quantum computing. Also, our work has interactions with the theory of cellular automata (which are, essentially, described in terms of automorphism of shift dynamical systems that have been so thoroughly studied of late), and models for "genetic transmissions", i.e., models that describe how genetic material is passed from generation to generation. Interdisciplinary activity with colleagues in physics, computer science and biology are likely. Discussions are already under way in Iowa's joint mathematical physics seminar and with colleagues in Biology. In addition, the operator algebras associated with graphs are expected to have an impact in mathematical systems theory of the type that appears in computer aided design. They are especially well adapted to handle systems that have built in uncertainties. This projects provides projects at the frontiers of operator algebra and linear algebra for graduate and undergraduate students. Even some high school students may be able to participate. These projects will also be used in the Department's REU and minority recruiting/training efforts supported in part by AGEP funds.

AbstractMuhly Muhly打算研究操作员代数的问题，这些问题可能分为三组。在第一个中，发现了关于通用操作者代数理论及其与量子马尔可夫半群，完全正面地图和量子信息理论的相互作用的7个广泛问题。 Muhly打算使用他发现的不变性量来分析完全正面地图和相关结构的半群。特别的重点将是由产品系统构建的“连续张量代数”。第二组（5个问题）涉及类固醇的理论和应用。首先，Muhly建议使用他最近在Gromoid的Brauer小组上的合作工作来建立一种一般的共同体学理论，该理论基于集体素体在空间上的行动类别。第二个重点是由所谓的fly束束在群体界面上。这些通常是“自然界中”，大部分努力将用于特定的例子。第四个涉及与拓扑震颤相关的操作器代数的各个方面 - 顶点和边缘空间是拓扑空间的图。最后的问题涉及Brauer群体的概括，该集团与当前兴趣的各种研究相互作用。第三个领域是Muhly对普通运营商代数基础的长期调查的一部分。重点放在所谓的正面和矫正性希尔伯特模块上，着眼于理解操作员代数的边界表示。此处提出的项目自然而然地来自于动态系统，尤其是在各种环境中出现的不可逆动力系统。我们特别感兴趣的是基于运算符代数的某些数学模型，这些模型有助于量子计算的新兴领域。 同样，我们的工作与细胞自动机理论有相互作用（本质上是用移动动力学系统的自动形态描述的，这些系统已被如此彻底地研究了），以及“遗传传播”的模型，即描述遗传材料如何从产物传递到产物的模型。与同事在物理，计算机科学和生物学方面的跨学科活动可能是可能的。在爱荷华州的联合数学物理研讨会和生物学同事中，已经进行了讨论。此外，与图形相关的操作员代数有望在计算机辅助设计中出现的类型的数学系统理论产生影响。 它们特别适合处理内置不确定性的系统。 该项目为研究生和本科生的操作员代数和线性代数的前沿提供了项目。 即使是一些高中生也可以参加。 这些项目还将用于部门的REU和少数民族招募/培训工作，部分由Agep Funds支持。