Stable rank and real rank of graph C^∗-algebras

J.A. Jeong, G.H. Park and D.Y. Shin

Vol. 200 (2001), No. 2, 331–343

DOI: 10.2140/pjm.2001.200.331

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).

Milestones

Received: 30 May 1999

Revised: 6 October 1999

Published: 1 October 2001