Decomposition rank of $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras

Abstract

We show that $C^{\ast }$-algebras of the form $C\left(X\right)\otimes \mathcal{Z}$, where $X$ is compact and Hausdorff and $\mathcal{Z}$ denotes the Jiang–Su algebra, have decomposition rank at most $2$. This amounts to a dimension reduction result for $C^{\ast }$-bundles with sufficiently regular fibres. It establishes an important case of a conjecture on the fine structure of nuclear $C^{\ast }$-algebras of Toms and Winter, even in a nonsimple setting, and gives evidence that the topological dimension of noncommutative spaces is governed by fibres rather than base spaces.