Classifying C*-algebras via ordered, mod-p K-theory

We introduce an order structure on K 0 ( ) 9 K0(; Z/p ) . This group may also be thought of as Ko(; 7z @ Z/p ) . We exhibit new examples of real-rank zero C*-algebras that are inductive limits of finite dimensional and dimension-drop algebras, have the same ordered, graded K-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure on K0(; Z ~3 Z / p ) naturally distinguishes these algebras. The same invariant is used to give an isomorphism theorem for such realrank zero inductive limits. As a corollary we obtain an isomorphism theorem for all real-rank zero approximately homogeneous C*-algebras that arise from systems of bounded dimension growth and torsion-free K0 group. At the 1980 Kingston conference, Effros posed the problem of finding suitable invariants for use in studying C*-algebras that are limits of sequences of homogeneous C*-algebras. These are now called almost homogeneous (AH) C*-algebras. The classification of AH algebras is a rapidly developing field and we will not attempt to summarize all this activity. Instead, we will focus on the growth of the invariants used. Specifically, we consider an AH algebra A that is the direct limit o f a system of the form