Banach代数的非交换维数

By a classical theorem of Gelfand, every unital commutative C*-algebra is isomorphic to C(X), the algebra of all continuous functions on some compact Hausdorff space X. Hence, C*-algebras are often viewed as non-commutative topological spaces. Given this fact, people usually try to transfer concepts from topological spaces to C*-algebras, or more generally, Banach algebras. Inspired by the various definitions of covering dimension, for C*-algebras, more generally, Banach algebras, people introduced and studied different noncommutative dimensions (often called rank) and corresponding theories. They have been proved very useful in studying K-theory and some structure properties of Banach algebras. Particularly, they play a important role in Elliott's classification program. In this project, we will study several problems in noncommutative dimension theories of Banach algebras，which mainly consist of Bass stable rank of nest algebras, Rieffel's question and studying which crossed product C*-algebras have finite nuclear dimension (decomposition rank).

由经典的Gelfand定理可知，有单位元的交换C*-代数同构于某个紧空间X上复值连续函数全体构成的代数。因此，C*-代数常常被看成非交换的拓扑空间。基于此，人们经常试图将拓扑空间的概念推及到C*-代数，以及更一般地，Banach代数。受拓扑空间中覆盖维数的各种等价的刻画的启发，人们对于C*-代数，有时，对一般的Banach代数，定义了各种各样的非交换维数（常常也称为"秩" ），并建立了相应的理论。它们在研究Banach代数的结构和K-理论中发挥着非常重要的作用。特别地，推动了C*-代数的Elliott分类计划的发展。在本课题中，我们将研究Banach代数非交换维数理论中的几个问题，具体如下：套代数的Bass稳定秩的计算；Rieffel问题；研究叉积C*-代数何时具有有限单核维数(分解秩)。

本项目旨在利用Banach代数的非交换维数理论研究算子代数领域的若干问题。我们按照项目计划书开展研究，在若干关键问题上均取得了进展，基本完成了项目的研究任务。代表性的成果有：1、对于一类序型为ω、每个原子均是有限维的套代数，我们解决了（可逆元群）连通性问题。2、计算了非交换圆盘代数的拓扑稳定秩并给出了其极大理性空间的完全刻画。3、证明了有单位元的、单的、可分的、纯无限C*-代数的正规元的酉轨道的闭包是道路连通的。4、对一类典型的AF代数, 解决了Blackadar的一个问题。