The Path Space of a Directed Graph

2011/02/07 by Samuel B. G. Webster · 1 citation
Mathematics · #math.OA #msc:46L05

paper · pdf · doi:10.1090/s0002-9939-2013-11755-7

published as Proc. Amer. Math. Soc. 142 (2014), 213--225 · 12 pages, all figures drawn with TikZ/PGF

arxiv created 2011/02/07 · arxiv updated 2013/11/01

Abstract

We construct a locally compact Hausdorff topology on the path space of a directed graph E, and identify its boundary-path space ∂ E as the spectrum of a commutative C^*-subalgebra D_E of C^*(E). We then show that ∂ 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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• Fibrewise compactifications and generalised limits in commutative and noncommutative topology

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