Simple real rank zero algebras with locally Hausdorff spectrum

Abstract

Let A be a unital, simple, separable C*-algebra with real rank zero, stable rank one, and weakly unperforated ordered K 0 group. Suppose, also, that A can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then A is tracially approximately finite dimensional (i.e., A has tracial rank zero). Hence, A is an AH-algebra with bounded dimension growth and is determined by K-theoretic invariants. The above result also gives the first proof for the locally AH case.