Real rank and topological dimenstion of higher rank graph algebras

We study dimension theory for the C*-algebras of row-finite k-graphs with

no sources. We establish that strong aperiodicity—the higher-rank analogue of condition

(K)—for a k-graph is necessary and sufficient for the associated C*-algebra to have

topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank

zero if and only if it has topological dimension zero and satisfies a homological condition

that can be characterised in terms of the adjacency matrices of the 2-graph. We also

show that a k-graph C*-algebra with topological dimension zero is purely infinite if and

only if all the vertex projections are properly infinite. We show by example that there are

strongly purely infinite 2-graphs algebras, both with and without topological dimension

zero, that fail to have real-rank zero.