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In this paper we first analyze the algebraic structure of group C∗-algebras of the generalized Dixmier groups, and next consider that of group C∗-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∗-algebras in terms of groups. Also, we show that some of these group C∗-algebras have no nontrivial projections.

The structure of group $C\sp *$-algebras of the generalized Dixmier groups