Stable rank and real rank of graph C∗-algebras

Abstract

For a row finite directed graph E, Kumjian, Pask, and Raeburn proved that there exists a universal C*-algebra C* (E) generated by a Cuntz-Krieger E-family. In this paper we consider two density problems of invertible elements in graph C * -algebras C * (E), and it is proved that C * (E) has stable rank one, that is, the set of all invertible elements is dense in C * (E) (or in its unitization when C * (E) is nonunital) if and only if no loop of E has an exit. We also prove that for a locally finite directed graph E with no sinks if the graph C * -algebra C * (E) has real rank zero (RR(C * (E)) = 0), that is, the set of invertible self-adjoint elements is dense in the set of all self-adjoint elements of C * (E) then E satisfies a condition (K) on loop structure of a graph, and that the converse is also true for C * (E) with finitely many ideals. In particular, for a Cuntz-Krieger algebra O A , RR(O A ) = 0 if and only if A satisfies Cuntz's condition (II).