Research on the Classification of Nuclear C*-Algebras

摘要：本课题旨在发展适合于核C*-代数分类理论的k理论和局部逼近技术。特别是，研究者将为卡斯帕罗夫的kk组探索新的模型。在分析方面，他专注于局部近似性质，如拟对角性及其改进。分类程序之外的基本思想是，核C*-代数的简单性和其他不稳定的k理论条件转化为代数的某些内部动力学性质。反过来，这些性质导致了令人惊讶的刚性现象，其中代数完全由其同伦类型甚至由其k理论群决定。c *代数可以被认为是显示丰富的代数和拓扑结构的无限数字矩阵的集合。C -代数理论是量子力学的标准框架。它提供了表达和分析物理世界对称性的基本非交换方面的功能分析方法。提出的研究将试图揭示和解释核C*-代数的刚性性质。这一理论在动力系统和群的几何理论中有着深刻的影响。

AbstractDadarlatThe investigator aims at developing techniques of K-theory and local approximation suitable for the classification theory of nuclear C*-algebras. In particular, the investigator will explore new models for the KK-groups of Kasparov. On the analytical side he would focus on local approximation properties such as quasidiagonality and its refinements.The basic idea beyond the classification program is that that the simplicity and other nonstable K-theory conditions for a nuclear C*-algebra translate to certain internal dynamical properties of the algebra. In their turn, these properties lead to surprising rigidity phenomena where an algebra is completely determined by its homotopy type or even by its K-theory groups.C*-algebras can be thought as collections of infinite matrices of numbers displaying a rich algebraic and topological structure. The C*-algebra theory is the standard framework for quantum mechanics. It offers functional analytic ways of expressing and analysing fundamental noncommutative aspects of the symmetries of the physical world. The proposed research will attempt to uncover and explain rigidity properties of nuclear C*-algebras. This theory has deep ramifications in dynamical systems and geometric theory of groups.