Reduction of topological stable rank in inductive limits of C*-algebras

Abstract

We consider inductive limits A of sequences A1 → A2 → … of finite direct sums of C*-algebras of continuous functions from compact Hausdorff spaces into full matrix algebras. We prove that A has topological stable rank (tsr) one provided that A is simple and the sequence of the dimensions of the spectra of Ai is bounded. For unital A, tsr (A) = 1 means that the set of invertible elements is dense in A. If A is infinite dimensional, then the simplicity of A implies that the sizes of the involved matrices tend to infinity, so by general arguments one gets tsr(Ai) ≤ 2 for large enough i whence tsr(A) ≤ 2. The reduction of tsr from two to one requires arguments which are strongly related to this special class of C*-algebras. © 1992 by Pacific Journal of Mathematics.