The path space of a directed graph

Webster, Samuel

doi:10.1090/S0002-9939-2013-11755-7

The path space of a directed graph
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by Samuel B. G. Webster PDF

Proc. Amer. Math. Soc. 142 (2014), 213-225 Request permission

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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