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致的数学框架的发展，其中数值函数代数由基于无限矩阵的理论所包含。无限的基质结构（例如操作员代数）能够描绘和建模基本颗粒和基础量子物理学的对称性的相互作用。本项目是一致努力的一部分，旨在将分析和几何形状的基本思想和技术扩展到操作员（Matrix）代数的非共同背景。该项目将研究运算符代数的有限维近似值的存在，这些近似值足够丰富，可以捕获初始数据的关键特征。传递到有限维模型很重要，因为它可以访问混凝土数值不变性。不可避免地，有限维模型的结构将比其无限维度对应物的对称性略有对称。精确对称的损失是大约有限模型的重要特征。 它反映了主要研究者旨在以数值形式量化的微妙拓扑特性。由此产生的不变性与基于新型材料物理学的数学（例如具有晶体对称性的拓扑绝缘子）的数学相关。第一个项目用于使用有限维矩阵模型的组，C* - 代数及其近似值。它将检查离散组和C* - 代数组的变形中的存在，该代数是由这些变形引起的不变性，以及在此过程中遇到的潜在拓扑障碍物。调查的一个基本目标是更好地理解准对态性的拓扑性质，这是一种有限的维近似特性，在C* - 代数的结构理论中起着核心作用。第二个项目涉及C* - 代数的连续领域及其对C* - 代数在一般空间上的概括。虽然C*-Algebra的原始频谱是一个基本不变的，但它具有一个重要的限制。它仅给出了如何将理想粘合在一起的低级描述。该项目的第一个目标是开发可计算的不变性，以捕获理想和本地商之间的K理论相互作用。第二个目标是进一步发展由于乌尔里希·佩尼格（Ulrich Pennig）和主要研究者而引起的强烈自我吸收的C* - 代数的连续领域的普遍性理论。