Problems in the Theory and Application of Operator Tensor Algebras

Muhly Muhly打算研究算子代数中的问题，这些问题可以分为三类。第一部分是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相互作用的7个广泛问题。Muhly打算利用他为张量代数及其表示所发现的不变量来分析完全正映射的半群及其相关结构。一个特别的焦点将是从乘积系统建立的“连续张量代数”。第二组(5个问题)涉及群胚的理论和应用。首先，Muhly建议利用他最近在群胚的Brauer群上的合作工作来建立一个基于群胚在空间上的作用范畴的一般上同调理论。后两个集中于由群上所谓的Fell丛建立的算子代数的结构。这些问题在“自然界”中很常见，很大一部分工作将致力于具体的例子。第四章讨论与拓扑箭图相关的算子代数的一些方面，其中顶点空间和边空间都是拓扑空间。最后一个问题涉及到与当前感兴趣的各种调查相互作用的布劳尔小组的推广。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。重点研究所谓的正投射和正内射Hilbert模，旨在理解算子代数的边界表示。这里提出的项目自然地源自于动力系统，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基于算符代数的数学模型，这些模型有助于新兴的量子计算领域。此外，我们的工作还与细胞自动机理论(本质上，这是根据最近被如此彻底地研究的移位动力系统的自同构来描述的)以及“遗传传递”模型，即描述遗传物质如何代代相传的模型进行了相互作用。与物理学、计算机科学和生物学领域的同事开展跨学科活动是可能的。爱荷华州的联合数学物理研讨会已经在进行讨论，并与生物学的同事进行了讨论。此外，与图相关的算子代数有望在数学系统理论中产生计算机辅助设计中出现的那种影响。它们特别适合于处理具有内置不确定性的系统。这个项目为研究生和本科生提供算符代数和线性代数前沿的项目。甚至一些高中生也可能参与其中。这些项目还将用于该部部分由AGEP资金支持的区域、区域和少数民族征聘/培训工作。

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.