Journal of Operator Theory

Volume 38, Issue 1, Summer 1997 pp. 67-86.

Stable rank of the C*-algebras of solvable Lie groups of type I

Authors: Takahiro Sudo (1) and Hiroshi Takai (2)
Author institution:(1) Department of Mathematics, Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji-shi, Tokyo 192-03, JAPAN
(2) Department of Mathematics, Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji-shi, Tokyo 192-03, JAPAN

Summary: In this paper we show that the stable rank of the C*-algebras of simply-connected connected solvable Lie groups of type I is estimated by the complex dimension of the fixed point subspaces of the real dual spaces of their Lie algebras under the coadjoint actions. This result generalizes the estimation in the case of simply-connected connected nilpotent Lie groups. As corollaries, we show that the product formula of the stable rank holds for the C*-algebras of connected solvable Lie groups of type I, and estimate the real rank in the case of simply-connected connected solvable Lie groups of type I.

Keywords: Stable rank, solvable Lie group, coadjoint orbit space, polarization.

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