Simple C∗-algebras with locally finite decomposition rank

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C∗-algebras. The concept is particularly relevant for Elliott's program to classify nuclear C∗-algebras by K-theory data. We study some of its properties and show that a simple unital C∗-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang–Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence of our result and of a theorem of Elliott and Gong, any such C∗-algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homogeneous of topological dimension at most 3. Our result in particular confirms the Elliott conjecture for the class of simple unital Z-stable ASH algebras with real rank zero. Moreover, it implies that simple unital Z-stable AH algebras with real rank zero not only have slow dimension growth in the ASH sense, but even in the AH sense.