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

TL;DR

This paper explores conditions under which higher-rank graph C*-algebras are approximately finite-dimensional, identifying necessary and sufficient conditions, especially for graphs with finitely many vertices, and analyzing their ideal structures.

Contribution

It provides a detailed characterization of AF properties in higher-rank graph C*-algebras, extending known results from ordinary graphs and highlighting complexities unique to higher-rank structures.

Findings

Absence of higher-rank cycles is necessary for AF-ness.

For finitely many vertices, this condition is also sufficient.

AF C*-algebras have gauge-invariant ideals and AF corners.

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.