Nuclear dimension, pure infiniteness and real rank for higher rank graph $C^*$-algebras

Abstract

We study the structure and regularity of higher rank graph $C^*$-algebras, with particular emphasis on their nuclear dimension. For a row-finite, locally convex $k$-graph $Λ$ with no sources, we characterise pure infiniteness of $C^*(Λ)$ in terms of generalised cycles, maximal tails, and strong aperiodicity, and we relate these conditions to topological dimension zero of the primitive ideal space and to the structure of gauge-invariant ideals. Our main application is that whenever $C^*(Λ)$ is purely infinite of topological dimension zero---in particular whenever its ideal lattice is finite---it is strongly purely infinite, $\mathcal O_\infty$-stable, and of nuclear dimension one, \emph{even when $C^*(Λ)$ is not simple}. This extends to the non-simple, higher-rank setting the nuclear-dimension-one computation known for simple UCT-Kirchberg $2$-graph algebras. Along the way we refine and correct several results in the existing graph $C^*$-algebra literature.