The path space of a higher-rank graph

Abstract

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $Λ$. We identify the boundary-path space $\partialΛ$ as the spectrum of a commutative $C^*$-subalgebra $D_Λ$ of $C^*(Λ)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\wtΛ$ with no sources in which $Λ$ is embedded, and show that $\partialΛ$ is homeomorphic to a subset of $\partial\wtΛ$ . We show that when $Λ$ is row-finite, we can identify $C^*(Λ)$ with a full corner of $C^*(\wtΛ)$, and deduce that $D_Λ$ is isomorphic to a corner of $D_{\wtΛ}$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.