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.

AbstractMuhly Muhly打算研究算子代数中的问题，可以分为三组。在第一个被发现的7个广泛的问题有关的理论一般算子代数和它们的相互作用与量子马尔可夫半群，完全正映射和量子信息理论。Muhly打算使用他发现的张量代数及其表示的不变量来分析完全正映射和相关结构的半群。一个特别的重点将是“连续张量代数”建立从产品系统。第二组（5个问题）涉及群胚的理论和应用。首先，Muhly建议使用他最近的合作工作的Brauer群的groupoid建立一个一般上同调理论，是基于一类行动的groupoid的空间。第二个两个重点放在结构的运营商代数建成所谓的费尔丛广群。这些现象在“自然界”中非常普遍，我们将用大量的精力来研究具体的例子。第四部分讨论了与拓扑箭图有关的算子代数的一些方面，其中顶点和边空间是拓扑空间。最后一个问题涉及的推广的布劳尔组，与各种调查的当前利益。第三个领域是穆利的长期调查的基础一般运营商代数。重点是所谓的正交投射和正交内射希尔伯特模，着眼于理解算子代数的边界表示。本文提出的项目自然来自动力系统，并对动力系统产生影响，特别是出现在各种环境中的不可逆动力系统。我们特别感兴趣的是某些数学模型，基于算子代数，有助于量子计算的新兴领域。 此外，我们的工作与元胞自动机理论（基本上是根据最近已经深入研究的移位动力系统的自同构来描述的）和“遗传传递”模型（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。讨论已经在爱荷华州的联合数学物理研讨会上进行，并与生物学的同事进行。此外，与图相关的算子代数有望对计算机辅助设计中出现的数学系统理论产生影响。 它们特别适合处理具有不确定性的系统。 该项目为研究生和本科生提供算子代数和线性代数前沿的项目。 甚至一些高中生也可以参加。 这些项目还将用于该部的REU和少数族裔征聘/培训工作，这些工作部分得到AGEP基金的支助。