The Path Space of a Directed Graph

We construct a locally compact Hausdorff topology on the path space of a directed graph \(E\), and identify its boundary-path space \(\partial E\) as the spectrum of a commutative \(C^*\)-subalgebra \(D_E\) of \(C^*(E)\). We then show that \(\partial E\) is homeomorphic to a subset of the infinite-path space of any desingularisation \(F\) of \(E\). Drinen and Tomforde showed that we can realise \(C^*(E)\) as a full corner of \(C^*(F)\), and we deduce that \(D_E\) is isomorphic to a corner of \(D_F\). Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.