Cuntz-Krieger algebras of directed graphs

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.