Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS

Iain Raeburna1, Aidan Simsa1 and Trent Yeenda1

a1 Department of Mathematics, University of Newcastle, Callaghan, NSW 2308, Australia (iain@maths.newcastle.edu.au; aidan@maths.newcastle.edu.au; trent@maths.newcastle.edu.au)

Abstract

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

(Received July 15 2001)

Keywords

  • graph algebra;
  • Cuntz–Krieger algebra;
  • uniqueness