Decomposition rank of Z -stable C ∗ -algebras

We show that C ∗ -algebras of the form C(X)⊗Z , where X is compact and Hausdorff and Z denotes the Jiang–Su algebra, have decomposition rank at most 2. This amounts to a dimension reduction result for C ∗ -bundles with sufficiently regular fibres. It establishes an important case of a conjecture on the fine structure of nuclear C ∗ -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.