Simple [formula omitted]-algebras with locally finite decomposition rank

Abstract

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.