Mathematical Sciences: Noncommutative Geometric Methods in Operator Algebras

小行星9500886 本计画主要是利用非交换微分几何的方法来研究与离散群及C*-动力系统相关的算子代数。本文的主要工作是构造相应的交叉积C ~*-代数的光滑稠密子代数，并对光滑子代数进行相关的分析。采用的主要技术是康纳斯的循环理论的结合代数和施瓦茨上同调离散群介绍的调查员。完成本提案中描述的项目有助于阐明算子代数，代数拓扑和微分几何的深层性质，例如，进一步理解非交换Toeplitz指数理论;解决Baum-Connes映射的有理内射性，Novikov猜想和Kadison-Kaplansky猜想快速衰减离散群。 这项研究处于几何学、分析学和代数学之间的交界处。Connes的现代方法使用非交换代数结构研究几何不变量。 非交换结构的一个很好的例子是固定大小的方阵集合上的乘法结构。这个项目将采用康纳斯机制来阐明其他重要的非交换代数称为算子代数，可以被视为无限矩阵的集合。结果将影响代数，几何和分析。 ***

9500886 Ji This project involves the study of operator algebras associated with discrete groups and C*-dynamical systems by the method of noncommutative differential geometry. Central to this study is the construction of smooth and dense subalgebras of the corresponding crossed product C*-algebras and the related analysis to the smooth subalgebras. The main techniques employed are Connes' cyclic theory for associative algebras and the Schwartz cohomology for discrete groups introduced by the investigator. Completion of the projects described in this proposal help elucidate deep properties of operator algebras, algebraic topology and differential geometry, such as, a further understanding of noncommutative Toeplitz index theory; a solution to the rational injectivity of the Baum-Connes map, the Novikov conjecture and the Kadison-Kaplansky conjecture for the class of rapidly decaying discrete groups. This research lies at the interface between geometry, analysis and algebra. A modern approach of Connes studies geometrical invariants using non-commutative algebraic structures. A good example of a noncommutative structure is the multiplicative structure on the collection of square matrices of fixed size. This project will employ Connes machinery to elucidate other important non-commutative algebras called operator algebras which can be viewed as collections of infinite matrices. The results will impact algebra, geometry and analysis. ***