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Dimension theory of group \(C^*\)-algebras of connected Lie groups of type I. (English) Zbl 0971.46039

Summary: We determine isomorphism classes of connected solvable Lie groups with some conditions such that their group \(C^*\)-algebras have stable rank one, and give its applications. Also, we show that stable rank of group \(C^*\)-algebras of connected Lie groups of type I is estimated in terms of their closed normal subgroups and quotient groups.

MSC:

46L05 General theory of \(C^*\)-algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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