Classification of Amenable C*-Algebras and Applications

Gong/LiSimple C*-代数是C*-代数理论中的基本构件。主要研究者龚桂华和李良清建议继续研究简单的、可分的、可服从的C*-代数的分类。他们还计划应用分类结果和分类项目中发展的技术来研究C*-代数和微分拓扑上的群作用(例如关于高签名同伦不变性的Novikov猜想)。量子物理学中从有限个自由度到无限个自由度的转变导致了某些无限维代数的数学理论，称为C*-代数。C*-代数是一个代数系统，类似于数的代数系统，具有加法、减法、乘法和除法运算。但与数的乘法不同，C*-代数中的乘法是不可交换的-也就是说，一般而言，X乘以Y并不等于Y乘以X。这一重要特征对应于量子力学中的海森伯格测不准原理。简单C*-代数是那些不能被分成更小的片段的代数，在某种意义上，所有的C*-代数都是在它们的基础上构造出来的。研究人员建议研究简单可服从C*-代数的一个完全计数(或分类)。他们预计，拟议研究的进展将在几个数学领域做出重要贡献，包括算子代数、微分拓扑学，以及对量子物理无限维世界的理解。

AbstractGong/LiSimple C*-algebras are basic building blocks in the theory of C*-algebras. The principal investigators, Guihua Gong and Liangqing Li, propose to continue their research on the classification of simple, separable, amenable C*-algebras. They also plan to apply the classification results and the techniques developed in the classification project to study group actions on C*-algebras and differential topology (e.g. the Novikov conjecture on the homotopy invariance of higher signature). The passage from a finite to an infinite number of degrees of freedom in quantum physics led to the mathematical theory of certain infinite dimensional algebras, called C*-algebras. A C*-algebra is an algebraic system, similar to that of numbers, with its operations of addition, subtraction, multiplication, and division. But unlike the multiplication for numbers, the multiplication in a C*-algebra is not commutative ---- that is, in general, X times Y is not as same as Y times X. This important feature corresponds to Heisenberg uncertainty principle in Quantum Mechanics. The simple C*-algebras are those that cannot be broken into smaller pieces, and in some sense all C*-algebras are built out of them. The investigators propose to work on a complete enumeration (or classification) of simple amenable C*-algebras. They expect that progress on the proposed research will result in important contributions to several mathematical fields including operator algebras, differential topology, and also to the understanding of the infinite dimensional world of quantum physics.