套代数的可逆元群连通性及相关问题

On the connectedness of the invertibles in nest algebras and related problems

本文综述套代数的可逆元群连通性问题进展并列出几个可考虑的相关问题.在回顾套代数的概念及基本例子的基础上,阐述迄今为止人们已知的连通性问题已得到解决的套的种类;同时以Pitts的例子为着眼点,阐述上三角代数的连通性问题的进展;最后谈到套代数稳定秩方面的结果并讨论其与连通性问题的密切联系.

In this paper, we summarize the progress of the connectedness problem of the group of invertibles in nest algebras and list some related open problems that could be further considered. After reviewing the concepts and basic examples of nest algebras, the solved cases of the connectedness problem for nest algebras are presented.Taking Pitts’ example as the starting point, we also review the progress of the connectedness problem for the upper triangular algebra. Finally, the relationships between stable ranks and the connectedness problem of nest algebras are also discussed.