C*-Algebras and Dynamical Systems

摘要：本项目的主要目的是根据核单C*-代数的K-理论数据对其进行分类。主要研究人员引入了C~*-代数的迹拓扑序的概念。这种新的秩对核单C*-代数的分类是非常有用的。作者建议用K-理论数据对迹拓扑秩数不超过1的单位单核C~*-代数进行分类。这些C*-代数包括大多数已知的核单C*-代数。研究人员还提出了确定许多自然产生的简单C*-代数的迹拓扑秩数的方法。非交换C*-代数最简单的例子是矩阵的集合。一般而言，C*-代数可以被认为是无限矩阵的集合。在量子力学中，物理量用希尔伯特空间上的算符表示，即用无限矩阵表示。C*-代数是量子物理学的产物。但C*-代数也为推广几何、拓扑和测度论提供了自然的框架，以一种广泛适用的基本非对易方式。简单的C*-代数可以被视为理解量子变量的关键。它们是更一般的C*-代数的基本构造块，也是典型的非交换C*-代数。本研究的目的是研究如何利用一些本质上具有拓扑性的数据来描述这些C*-代数。在动力系统的研究中可以立即得到应用。

AbstractLinThe main goal of this project is to classify nuclear simple C*-algebras by their K-theoretical data. The principal investigator introduced a notion of tracial topological rank for C*-algebras. It has been established that this new rank is quite useful in classification of nuclear simple C*-algebras. The investigator proposes to classify unital simple nuclear C*-algebras with tracial topological rank no more than one by their K-theoretical data. These C*-algebras include most known nuclear simple C*-algebras. The investigator also proposes to determine tracial topological ranks for many naturally occurred simple C*-algebras.The simplest examples of noncommutative C*-algebras are collections of matrices. In general, C*-algebras could be thought as collections of infinite matrices. In quantum mechanics physical quantities are represented by operators on a Hilbert space i.e. by infinite matrices. C*-algebras appeared as an outgrowth of quantum physics. But C*-algebras also provide the natural framework for generalizing geometry, topology and measure theory in a fundamental noncommutative way of wide applicability. Simple C*-algebras can be viewed as the key to understanding quantum variables. They are the fundamental building blocks of more general C*-algebras as well as the typical noncommutative C*-algebras. This research project is to study how to describe these C*-algebras via a few data which is topological in nature. Immediate applications can be established in the study of dynamical systems.