Hilbert C*-模扩张性质

The project contributes to the study of the property of Busby invariants corresponding to the extension sequences of Hilbert C*-modules. Following the established relationship between the Busby invariants of C*-algebra extension and K-Homology theory, we are going to study the property of the Busby invariants decided by Hilbert C*-module extensions. The main contents include the following aspects: 1. find the group, even ring structure on the set of all Busby invariants decided by Hilbert C*-module extensions. 2. study the relationship between the Busby invariants of the vector bundle extension, as a special case of Hilbert C*-module extension, and the elements in K-group which decide the bundle structure. 3. study the relationship between the connections and curvatures of each term in a vector bundle extension. 4. study the relationship between Hilbert C*-module extension and twisted K-Homology. This project is planned to generalize Hilbert C*-module extension to be an invariant which is more precise than C*-algebra extension then provide us a new tool, for the study of noncommutative geometry, to seek for extra information contained in extension sequences, which fully shows the value of extension theory.

本项目的主要研究对象是由Hilbert C*-模的扩张正合列所对应的Busby不变量的性质。期望通过已知的C*-代数的扩张正合列所对应的Busby不变量和K-同调理论之间的关系，来研究Hilbert C*-模扩张Busby不变量的性质。主要的研究内容有以下几个方面：1、找到Hilbert C*-模扩张的全体Busby不变量形成的集合上的群结构，甚至于环结构；2、研究特殊的Hilbert C*-模的扩张——向量丛的扩张所对应的Busby不变量和决定向量丛结构的K群元之间的关系；3、研究扩张正合列中各向量丛上的联络，曲率之间的关系；4、研究Hilbert C*-模扩张与扭K-同调理论之间的关系。本项目拟将Hilbert C*-模的扩张推广为一种比C*-代数扩张更细致的不变量，于是为非交换几何的研究提供一种新的不变量工具来挖掘扩张正合列所隐含的额外信息，从而更全面地体现扩张理论的价值。

本项目对于一类特殊的Hilbert C*-模——某些非紧空间上的向量丛，找到了其乘子模的几何化描述。表明了关于向量丛的上循环表达对于其乘子模同样适用。同时将这种描述应用在了Hilbert C*-模扩张上，得出了关于Hilbert C*-模扩张同伦等价性的若干结果。