On the classification of simple Z-stable C*-algebras with real rank zero and finite decomposition rank

摘要

We show that, if A is a separable simple unital C*-algebra which absorbs theJiang-Su algebra Z tensorially and which has real rank zero and finitedecomposition rank, then A is tracially AF in the sense of Lin, without anyrestriction on the tracial state space. As a consequence, the Elliottconjecture is true for the class of C*-algebras as above which, additionally,satisfy the Universal Coefficients Theorem. In particular, such algebras arecompletely determined by their ordered K-theory. They are approximatelyhomogeneous of topological dimension less than or equal to 3, approximatelysubhomogeneous of topological dimension at most 2 and their decomposition rankalso is no greater than 2.