Reduction of the dimension of nuclear C*-algebras

Abstract

We show that for a large class of C*-algebras $\mathcal{A}$, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If $A\in \mathcal{A}$ and $B$ is a simple projectionless C*-algebra with trivial K-groups that can be written as a direct limit of a system of (nonunital) recursive subhomogeneous algebras with no dimension growth then the stable rank of $A\otimes B$ is one. As a consequence we show that if $A\in \mathcal{A}$ then the stable rank of $A\otimes \mathcal{W}$ is one. We also prove the following stronger result: If $A$ is separable C*-algebra that can be written as a direct limit of C*-algebras of the form ${\mathrm{C}}_0(X)\otimes {\mathrm{M}}_n$, where $X$ is locally compact and Hausdorff, then $A\otimes \mathcal{W}$ can be written as a direct limit of a sequence of 1-dimensional noncommutative CW-complexes.