When is the Cuntz–Krieger algebra of a higher-rank graph approximately finite-dimensional?

Gwion Evans, Aidan Sims

Research output: Contribution to journalArticlepeer-review

26 Citations (SciVal)

Abstract

We investigate the question: when is a higher-rank graph C⁎-algebra approximately finite-dimensional? We prove that the absence of an appropriate higher-rank analogue of a cycle is necessary. We show that it is not in general sufficient, but that it is sufficient for higher-rank graphs with finitely many vertices. We give a detailed description of the structure of the C⁎-algebra of a row-finite locally convex higher-rank graph with finitely many vertices. Our results are also sufficient to establish that if the C⁎-algebra of a higher-rank graph is AF, then its every ideal must be gauge-invariant. We prove that for a higher-rank graph C⁎-algebra to be AF it is necessary and sufficient for all the corners determined by vertex projections to be AF. We close with a number of examples which illustrate why our question is so much more difficult for higher-rank graphs than for ordinary graphs.
Original languageEnglish
Pages (from-to)183-215
JournalJournal of Functional Analysis
Volume263
Issue number1
Early online date12 Apr 2012
DOIs
Publication statusPublished - Jul 2012

Keywords

  • Graph C*-algebra
  • C* -algebra
  • AF algebra
  • higher-rank graph
  • Cuntz-Krieger algebra

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