LAUSR

Abstract It is shown that every Jiang–Su stable approximately subhomogeneous C * {{\mathrm{C}^{*}}} -algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous C * {{\mathrm{C}^{*}}} -algebras. A key step in the proof is that subhomogeneous C * {{\mathrm{C}^{*}}} -algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product C * {{\mathrm{C}^{*}}} -algebras associated to minimal homeomorphisms with mean dimension zero.