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

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.

摘要Muhly研究算子代数中的问题，可分为三类。在第一部分中发现了关于一般算子代数理论及其与量子马尔可夫半群、完全正映射和量子信息论的相互作用的7个广泛问题。Muhly打算用他所发现的张量代数的不变量及其表示来分析完全正映射和相关结构的半群。一个特别的焦点将是“连续张量代数”建立在乘积系统。第二组（5个问题）涉及类群的理论和应用。首先，Muhly建议利用他最近在群类群的Brauer群上的合作成果，建立一个基于群类群在空间上的作用范畴的广义上同调理论。第二部分关注的是由群类群上所谓的Fell束构成的算子代数的结构。这些问题在“自然界”中很常见，我们将花很大一部分精力来讨论具体的例子。第四章讨论了与拓扑颤振相关的算子代数的各个方面。拓扑颤振是指顶点和边缘空间都是拓扑空间的图。最后一个问题是关于Brauer群的泛化，它与当前感兴趣的各种调查相互作用。第三个领域是Muhly对一般算子代数基础的长期研究的一部分。重点是所谓的正射影和正射希尔伯特模块，着眼于理解算子代数的边界表示。本文提出的项目自然地来源于动力系统，并对动力系统产生影响，特别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算子代数的某些数学模型，它们有助于新兴的量子计算领域。此外，我们的工作与元胞自动机理论（本质上，这是根据移位动力系统的自同构来描述的，最近已被彻底研究）和“遗传传递”模型相互作用，即描述遗传物质如何代代相传的模型。可能与物理学、计算机科学和生物学的同事进行跨学科活动。在爱荷华州的数学物理联合研讨会上以及与生物学同事的讨论已经开始。此外，与图相关的算子代数有望对出现在计算机辅助设计中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。本项目为研究生和本科生提供算子代数和线性代数的前沿课题。甚至一些高中生也可以参加。这些项目也将用于该部的REU和少数民族征聘/培训工作，这些工作部分由AGEP资金支助。