Cuntz–Krieger algebras of directed graphs

Alex Kumjian, David Pask and Iain Raeburn

Vol. 184 (1998), No. 1, 161–174

DOI: 10.2140/pjm.1998.184.161

Abstract

We associate to each row-finite directed graph E a universal Cuntz-Krieger C^∗-algebra C^∗(E), and study how the distribution of loops in E affects the structure of C^∗(E). We prove that C^∗(E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C^∗(E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C^∗(E) is AF; if E has a loop, then C^∗(E) is purely infinite.

Milestones

Received: 16 October 1996

Revised: 14 April 1997

Published: 1 May 1998