The Structure of Simple Separable Amenable C*-Algebras

算子代数的研究是由约翰·冯·诺伊曼在20世纪20年代末开始的，目的是理解量子力学。目前，算子代数的研究有两个活跃的领域——冯·诺伊曼代数和C*-代数。该项目涉及对C*-代数分类的研究。这里，对复杂对象集合的分类提供了一种更简单的方法来识别和区分更复杂类中的对象。冯诺依曼代数的分类理论是比较发达的。这个分类是由Connes（和其他人）在20世纪70年代获得的。而在C*代数方面，直到20世纪90年代，人们才开始看到分类理论的快速发展。此后，C*代数的分类成为C*代数研究的中心领域之一，并在最近几年取得了巨大的进展。本课题的研究将进一步推动分类理论的发展，并寻求分类理论与其他数学领域，特别是与动力系统研究的更多应用或相互作用。与一元C*-代数的分类相比，非一元C*-代数的分类还远远不够完整。本课题拟利用单简单C*-代数分类的策略和技术，推进非单简单C*-代数的分类规划。特别是，这涉及到证明抽象的非一元C*-代数可以被某些具体的I型C*-代数（约简定理）迹近似，为这类半具体的C*-代数发展分类理论，然后研究如何将这些可能放在一起工作以获得抽象的非一元C*-代数的完整分类。提出研究计划的另一个目的是继续研究分类计划与动力系统研究之间的相互作用。特别是，首席研究员想继续研究动力系统的零平均拓扑维数（对于整数群的作用，或更一般的可服从群的作用）是否等同于交叉积C*-代数的正则性。首席研究员还想调查最近的分类结果对潜在动力系统研究的可能影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The study of operator algebras was begun by John von Neumann in the late 1920s with the aim of understanding quantum mechanics. Currently, there are two active areas of investigation of operator algebras---von Neumann algebras and C*-algebras. The project involves investigations into the classification of C*-algebras. Here a classification of a collection of complicated objects provides a simpler way to identify and distinguish between objects in the more complicated class. The classification theory for von Neumann algebras is relatively well developed. The classification was obtained by Connes (and others) in the 1970s. While on the C*-algebra side, it was only in the 1990s that one began to see rapid developments in classification theory. Since then, the classification of C*-algebras has blossomed into one of the central areas in the study of C*-algebras, and tremendous progress has been made in the last few years. The research in this project is to push the classification theory even further, and to seek more applications or interactions of the classification theory with other areas of mathematics, in particular with the study of dynamical systems.Compared to the classification of unital C*-algebras, the classification of non-unital C*-algebras is far from complete. The principal investigator plans to use the strategy and techniques from the classification of unital simple C*-algebras to push forward the classification program for non-unital simple C*-algebras. In particular, this involves showing that an abstract non-unital C*-algebra can be tracially approximated by certain concrete type I C*-algebras (the reduction theorem), developing a classification theory for this semi-concrete class of C*-algebras, and then studying how these might be put to work together to obtain a full classification for abstract non-unital C*-algebras. The other purpose of the proposed research program is to continue the investigation on the interplay between the classification program and study of dynamical systems. In particular, the principal investigator would like to continue the study on whether the zero mean topological dimension of the dynamical system (for an action of the group of integers, or of a more general amenable group) would be equivalent to the regularity property of the crossed product C*-algebras. The principal investigator also would like to investigate possible impacts of the recent classification results on the study of the underlying dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.