C*-Algebras and Dynamical Systems

本课题的主要目标是利用核单C ~*-代数的K-理论数据对它们进行分类。 主要研究者为C*-代数引入了迹拓扑秩的概念。这一新的秩在核单C ~*-代数的分类中是非常有用的。 本文利用迹拓扑秩不大于1的单核C ~*-代数的K-理论数据对它们进行了分类。这些C*-代数包括大多数已知的核单C*-代数。 研究者还提出确定许多自然发生的单C*-代数的迹拓扑秩，非交换C*-代数最简单的例子是矩阵的集合。一般来说，C*-代数可以被认为是无限矩阵的集合。 在量子力学中，物理量由希尔伯特空间上的算子表示，即无限矩阵。C*-代数是量子物理学的产物。 但C*-代数还以具有广泛适用性的基本非交换方式为推广几何、拓扑和测度理论提供了自然框架。 简单C*-代数可以被看作是理解量子变量的关键。它们是更一般的C*-代数以及典型的非交换C*-代数的基本构建块。 本课题研究如何用少量的拓扑数据来描述这些C ~*-代数。 可以在动力系统的研究中建立直接的应用。

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.