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Journal of Operator Theory

Volume 46, Issue 1, Summer 2001  pp. 25-38.

Structure of group $C^*$-algebras of Lie semi-direct products ${\bbb C}^n\rtimes {\bbb R}$

Authors:  Takahiro Sudo
Author institution: Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa 903--0213, Japan

Summary:  In this paper we analyze the structure of group $C^*$-algebras of Lie semi-direct products of $\Bbb C^n$ by $\Bbb R$ to show that these $C^*$-algebras have finite composition series with their subquotients $C^*$-tensor products involving commutative $C^*$-algebras or the $C^*$-algebra of compact operators or noncommutative tori. As an application, we estimate stable rank and connected stable rank of these group $C^*$-algebras in terms of groups, and we deduce that group $C^*$-algebras of Lie semi-direct products of $\Bbb R^n$ by $\Bbb R$ have a similar structure as in the complex cases.

Keywords:  Group $C^*$-algebras, solvable Lie groups, stable rank

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