Stable rank and real rank of graph $C^*$-algebras

Abstract

For any row finite directed graph $E$ there exists a universal $C^*$-algebra $C^*(E)$ ([KPR, KPRR]) generated by projections and partial isometries satisfying the Cuntz-Krieger $E$-relations. This class of graph algebras includes the Cuntz-Krieger algebras and all AF algebras up to stable isomorphisms([D]). In this paper we give conditions for $E$ under which the algebra $C^*(E)$ has stable rank one or real rank zero. A simple graph $C^*$-algebra is either AF or purely infinite, hence it is always extremally rich. We discuss the extremal richness of some graph $C^*$-algebras and present several examples of prime ones with finitely many closed ideals.