Real rank and topological dimension of higher rank graph algebras

TL;DR

This paper investigates the relationship between topological dimension, real rank, and pure infiniteness in $k$-graph $C^*$-algebras, establishing necessary and sufficient conditions for these properties based on graph conditions and homological criteria.

Contribution

It provides new characterizations linking strong aperiodicity, topological dimension zero, and pure infiniteness in higher-rank graph $C^*$-algebras, including homological conditions.

Findings

Strong aperiodicity is necessary and sufficient for topological dimension zero.

Purely infinite 2-graph algebras have real rank zero iff they have topological dimension zero and satisfy a homological condition.

A $k$-graph $C^*$-algebra with topological dimension zero is purely infinite iff all vertex projections are properly infinite.

Abstract

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.