C*-algebras, Groups, and Topological Invariants

C*-代数、群和拓扑不变量

• 批准号: 1362824
• 负责人: Marius Dadarlat
• 金额: $ 24万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362824&HistoricalAwards=false
• 关键词: algebras; Groups; Topological; Invariants

The study of physical laws has led to the development of a refined mathematical framework where the algebra of numerical functions is subsumed by a theory based on infinite matrices. Infinite matriceal structures such as operator algebras are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The present project is part of a concerted effort to extend fundamental ideas and techniques of analysis and geometry to the noncommutative context of operator (matrix) algebras. The project will investigate the existence of finite-dimensional approximations of operator algebras that are sufficiently rich to capture key features of the initial data. Passing to finite-dimensional models is important since it gives access to concrete numerical invariants. Inevitably, the structure of the finite-dimensional models will be somewhat less symmetric than their infinite-dimensional counterparts. The loss of exact symmetries is an essential feature of the approximant finite models. It reflects subtle topological properties that the principal investigator aims to quantify in numerical form. The resulting invariants are related to those arising in the mathematics underpinning the physics of novel materials, such as topological insulators with crystalline symmetry.The research concerns two projects in operators algebras that have analytical and topological aspects. The first project is devoted to groups, C*-algebras, and their approximations by finite-dimensional matrix models. It will examine the existence of deformations of discrete groups and group C*-algebras into matrix algebras, the invariants that arise from these deformations, and potential topological obstructions encountered in the process. One underlying goal of the investigation is to develop a better understanding of the topological nature of quasidiagonality, a finite-dimensional approximation property that plays a central role in the structure theory of C*-algebras. The second project concerns the theory of continuous fields of C*-algebras and their generalizations to C*-algebras over general spaces. While the primitive spectrum of a C*-algebra is a fundamental invariant, it has one important limitation. It gives only a low-level description of how the ideals are glued together. A first goal of the project is to develop computable invariants that capture K-theoretical interactions between ideals and local quotients. A second goal is to further develop the generalized Dixmier-Douady theory of continuous fields of strongly self-absorbing C*-algebras due to Ulrich Pennig and the principal investigator.

对物理定律的研究导致了一个精确的数学框架的发展，其中数值函数的代数被基于无限矩阵的理论所包含。无限矩阵结构，如算子代数，能够描述和模拟基本粒子的相互作用和量子物理学基础的对称性。目前的项目是一个共同努力的一部分，扩展的基本思想和技术的分析和几何的非交换上下文的运营商（矩阵）代数。该项目将研究算子代数的有限维近似的存在，这些近似足够丰富，可以捕获初始数据的关键特征。传递到有限维模型是重要的，因为它可以访问具体的数值不变量。不可避免的是，有限维模型的结构将比无限维模型的结构更不对称。精确对称性的损失是近似有限模型的一个基本特征。 它反映了微妙的拓扑性质，主要研究者的目的是量化的数字形式。由此产生的不变量与新材料物理学的数学基础中出现的不变量有关，例如具有晶体对称性的拓扑绝缘体。研究涉及两个具有分析和拓扑方面的算子代数项目。第一个项目致力于群，C*-代数，和他们的近似有限维矩阵模型。它将研究离散群和群C*-代数到矩阵代数的变形的存在性，从这些变形中产生的不变量，以及在这个过程中遇到的潜在拓扑障碍。调查的一个基本目标是发展一个更好的理解的拓扑性质的拟对角性，有限维近似属性，在C*-代数的结构理论中起着核心作用。第二个项目是关于C*-代数的连续域理论及其在一般空间上的推广。虽然C*-代数的本原谱是一个基本不变量，但它有一个重要的限制。它只给出了一个低层次的描述如何理想是粘在一起。该项目的第一个目标是开发可计算的不变量，捕获理想和局部不变量之间的K理论相互作用。第二个目标是进一步发展Ulrich Pennig和首席研究员的强自吸收C*-代数连续场的广义Dixmier-Douady理论。