Abstract

We investigate the question: when is a higherrank graph C*algebra approximately finitedimensional? We prove that the absence of an appropriate higherrank analogue of a cycle is necessary. We show that it is not in general sufficient, but that it is sufficient for higherrank graphs with finitely many vertices. We give a detailed description of the structure of the C*algebra of a rowfinite locally convex higher rank graph with finitely many vertices. Our results are also sufficient to establish that if the C*algebra of a higherrank graph is AF, then its every ideal must be gaugeinvariant. We prove that for a higherrank 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 higherrank graphs than for ordinary graphs.