On topologically finite-dimensional simple C*-algebras

Abstract

We show that, if a simple $C^{*}$-algebra $A$ is topologically finite-dimensional in a suitable sense, then not only $K_{0}(A)$ has certain good properties, but $A$ is even accessible to Elliott's classification program. More precisely, we prove the following results: If $A$ is simple, separable and unital with finite decomposition rank and real rank zero, then $K_{0}(A)$ is weakly unperforated. If $A$ has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then $A$ has stable rank one and tracial rank zero. As a consequence, if $B$ is another such algebra, and if $A$ and $B$ have isomorphic Elliott invariants and satisfy the Universal coefficient theorem, then they are isomorphic. In the case where $A$ has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered $K_{0}$-group) for $A$ to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal $C^{*}$-algebras with infinite decomposition rank.