• DOI:10.1512/IUMJ.2017.66.6212
• Corpus ID: 51832574

Real rank and topological dimension of higher rank graph algebras

@article{Pask2015RealRA,
  title={Real rank and topological dimension of higher rank graph algebras},
  author={David Pask and Adam Sierakowski and Aidan Sims},
  journal={arXiv: Operator Algebras},
  year={2015}
}

• D. Pask, Adam Sierakowski, A. Sims
• Published 30 March 2015
• Mathematics
• arXiv: Operator 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.