High-dimensional ZZ-stable AH algebras

It is shown that a C⁎C⁎-algebra of the form C(X,U)C(X,U), where U   is a UHF algebra, is not an inductive limit of subhomogeneous C⁎C⁎-algebras of topological dimension less than that of X  . This is in sharp contrast to dimension-reduction phenomenon in (i) simple inductive limits of such algebras, where classification implies low-dimensional approximations, and (ii) when dimension is measured using decomposition rank, as the author and Winter proved that dr(C(X,U))≤2dr(C(X,U))≤2.