Higher-Rank Graphs and Their C^*-Algebras

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexity condition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C^*-algebras generated by Cuntz–Krieger families. AMS 2000 Mathematics subject classification: Primary 46L05