The Structure of Group $C^{\ast }$-algebras of the Generalized Dixmier Groups

Abstract

In this paper we first analyze the algebraic structure of group $C^{\ast }$-algebras of the generalized Dixmier groups, and next consider that of group $C^{\ast }$-algebras of some Lie semi-direct products with multi-diagonal or diagonal actions. As an application, we estimate the stable rank and the connected stable rank of these $C^{\ast }$-algebras in terms of groups. Also, we show that some of these group $C^{\ast }$-algebras have no nontrivial projections.