Reduction of topological stable rank in inductive limits of<i>C</i>^∗-algebras

We consider inductive limits A of sequences A\ -> Ai - 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 is bounded. For unital A, tsr (^4) = 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 ts(Ai) < 2 for large enough / whence ts(A) < 2. The reduction of tsr from two to one requires arguments which are strongly related to this special class of C* -algebras.