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arXiv · 1102.1225

The Path Space of a Directed Graph

Samuel B. G. Webster
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Abstract

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.

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Samuel B. G. Webster. 2011-02-07. The Path Space of a Directed Graph. https://doi.org/10.1090/s0002-9939-2013-11755-7

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