Simple C*-algebras with locally finite decomposition rank
Abstract
Description
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, 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.
30 pages, no figures
30 pages, no figures