C*-代数的近似与分类研究

本项目的第一个研究目标是研究单迹秩小于等于一C*-代数有限直和诱导极限的分类不变量、存在性结果和分类定理，并推广到其他用Elliott不变量可以处理的诱导极限型C*-代数和相应的C*-代数扩张上去（例如通过张量无限型UHF代数产生的新类）。第二个研究目标是研究有限核维数，Z稳定性以及Cuntz半群的几乎无孔性质之间的关系、Cuntz半群的分类性质以及Cuntz半群相应的不变量值域等分类问题。然后把Elliott不变量和Cuntz半群不变量的相应研究结果应用到一定的C*-代数动力系统，特别是拟自由作用图C*-代数动力系统和拟自由作用非光滑或非唯一遍历的整数群极小作用的拓扑动力系统中。第三个研究目标是研究具有CPAP、CBAP、OAP、正合性等具有一定形式有限维近似性质的C*-代数（或具有算子空间结构的代数），把分类研究概念、方法和结果推广到更一般的非核C*-代数和不定度规空间的算子代数上去。

In this research project we will first study the invariants, the existence theorem and the classification theorem of the inductive limits of the finite direct sums of simple C*-algebras with tracial rank no more than one, and extend the results to more general C*-algebra classes and their extensions (e.g. by tensor a UHF algebra with infinite type). Second we will study the relations among the finite nuclear dimension, Z-stable property and almost unperforated property of Cuntz semigroup, study more classification properties of Cuntz semigroup and the invariant ranges of of Cuntz semigroup; apply these research results to certain C*-dynamical systems, especially the graph C*-dynamical systems with quasi-free action and the minimal topological dynamical systems with the quasi-free non-smooth or non-unique ergodic action of integer. Finally we will study the C*-algebras or the algebras with the operator space structure which have also certain approximation property of finite dimension such as CPAP，CBAP，OAP and exactness , and extend some notions, methods and results of classification study to non-nuclear C*-algebras and operator algebras on the space with indefinite inner product.