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

Abstract

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.