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

Authors

D Gwion Evans, University of AberystwythFollow
Aidan Sims, University of WollongongFollow

RIS ID

55510

Publication Details

Evans, D. Gwion., Sims, A. (2012). When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?. Journal of Functional Analysis, 263 (1), 183-215.

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.