Graph C*-Algebras with Real Rank Zero

Given a row-finite directed graph E, a universal C*-algebra C*(E) generated by a family of partial isometries and projections subject to the relations determined by E is associated to the graph E. The Cuntz–Krieger algebras are those graph C*-algebras associated to some finite graphs. We prove that a graph C*-algebra C*(E) has real rank zero in the sense that the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements in C*(E) (or in the unitization of C*(E) if C*(E) is nonunital) if and only if E satisfies a loop condition (K) that is analogous to the condition for a finite {0, 1} matrix A under which Cuntz analyzed the ideal structure of the Cuntz–Krieger algebra OA.