Real rank and topological dimension of higher rank graph algebras

Abstract

We study dimension theory for the $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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.

25 pages, pictures prepared using tikz. V2: minor corrections and typos fixed; this version to appear in Indiana University Math. J

References in corpus (3)

• Simplicity of C*-algebras associated to higher-rank graphs
• The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources
• Purely infinite $C^{\ast }$-algebras associated to étale groupoids