### Citation:

Evans , G & Sims , A 2012 , ' When is the Cuntz–Krieger algebra of a higher-rank graph approximately finite-dimensional? ' Journal of Functional Analysis , vol 263 , no. 1 , pp. 183-215 . DOI: 10.1016/j.jfa.2012.03.024

### 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.

### Description:

Evans, D.G and Sims, A; When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?, preprint, arXiv:1112.4549v1