The path space of a higher-rank graph

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