The Path Space of a Directed Graph

TL;DR

This paper develops a topological framework for the path space of directed graphs, linking boundary-path spaces with spectra of subalgebras and desingularisations, enhancing understanding of graph $C^*$-algebras.

Contribution

It constructs a topology on the path space, identifies the boundary-path space as a spectrum, and relates it to desingularisations and corners of $C^*$-algebras.

Findings

Boundary-path space is homeomorphic to a subset of the desingularisation's infinite-path space.

The subalgebra $D_E$ is isomorphic to a corner of $D_F$.

The isomorphism induces a homeomorphism between boundary-path spaces.

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.