ON THE CLASSIFICATION OF SIMPLE ${{\mathcal Z}}$-STABLE $C^{*}$-ALGEBRAS WITH REAL RANK ZERO AND FINITE DECOMPOSITION RANK

Abstract

We show that, if $A$ is a separable simple unital $C^{*}$-algebra that absorbs the Jiang–Su algebra ${{\mathcal Z}}$ tensorially and that has real rank zero and finite decomposition rank, then $A$ is tracially approximately finite-dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of $C^{*}$-algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered $K$-theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.