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.

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