Inductive limits of subhomogeneous C∗-algebras with Hausdorff spectrum

We consider unital simple inductive limits of generalized dimension drop C∗-algebras. They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop C∗-algebras. Suppose that A is one of these C∗-algebras. We show that A⊗Q has tracial rank no more than one, where Q is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let A and B be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then A≅B if and only if they have the same Elliott invariant.