Reduction of topological stable rank in inductive limits ofC∗-algebras

Abstract

We consider inductive limits A of sequences A 1 →A 2 →... 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 A i 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(A i )≤2 for large enough is whence trsr(A)≤2. The reduction of tsr from two to one requires arguments which are strongly related to this special class of C * -algebras