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).

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