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 Λ . We identify the boundary-path space ∂Λ 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 Λ ˜ with no sources in which Λ is embedded, and show that ∂Λ is homeomorphic to a subset of ∂Λ ˜ . We show that when Λ is row-finite, we can identify C ∗ (Λ) with a full corner of C ∗ (Λ ˜ ) , and deduce that D Λ is isomorphic to a corner of D Λ ˜ . Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.