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Abstract

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