两类奇异空间上的算子代数与指标定理

This project is to study operator algebras and index theorems on two classes of singular spaces. This first type is domains with conical points.We can associate to the non-smooth boundary of a conical domain a Lie groupoi such that dilation invariant operators on the boundary of the conical domain correspond to pseudodifferential operators on the Lie groupoid that we construct.Then we can unitalize the deformation of Lie groupoids to define an analytic index, and prove that there exists a topological index. Finally ,we show an index theorem, i.e., these two indices are equal. The second type is scaleable spaces. The Baum-Connes conjecture on such spaces was proved by N.Higson and J. Roe in 1990's. The method is based on "Eilenberg Swindle" trick. We would like to use KK-theory and E-theory to write down the explicit expression of the Baum-Connes map. The we constuct an asymptotic morphism using the property of scaleable spaces. We show that this asymptotic morphism induces the inverse of the Baum-Connes map, so we obtain a constructive proof, and more geometric and operator algbebra information on such spaces from the point of view of KK-theory and E-theory.

本项目研究两类奇异空间上的算子代数及指标定理。第一类空间是带有圆锥点的区域。根据这类空间边界的奇性，构造出一个李群胚，使得区域边界上伸缩不变的拟微分算子对应于这个李群胚上的拟微分算子。利用李群胚的形变理论，我们定义一个解析指标，而且可以证明存在着一个拓扑指标；最后建立指标定理，也就是证明这两个指标相等。第二类空间是可伸缩空间。这类空间上粗Baum-Connes猜想在九十年代已经被Higon和Roe所证明，证明方法基于"Eilenberg Swindle"技巧。本项目用KK-理论和E-理论，写出粗Baum-Connes映射的具体形式，然后利用可伸缩空间本身的特性构造出一个渐进态射，证明这个渐进态射诱导的映射就是粗Baum-Connes映射的逆映射，从而得到这类空间上粗Baum-Connes猜想一个构造性的证明，从KK-理论和E-理论的角度得到这类空间上更多的几何与算子代数的信息。