The path space of a directed graph

By Samuel B. G. Webster. 2011

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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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8968d60384474784ffdbabbe8ef47c8c1f889b97

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In Mathematical Physics

Citations in the corpus

Partial transformation groupoids attached to graphs and semigroups (2016, background): “Following [37] (see also [14]), the partial system FE1 y ∂E can be described explicitly as follows: As a set, we identify ∂E with E∪{α ∈ E: s(α) / ∈ E 0}, where E ∞ is the set of infinite paths in E .”
Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras (2017, background): “This space generalizes the path space of a directed graph described in [31].”
Partial actions and KMS states on relative graph C⁎-algebras☆ (2013, background): “The statements (i)-(iv) in the next result can be found in [33].”
Graded Steinberg algebras and their representations (2017, unknown): “For µ ∈ E deﬁne For µ ∈ E ∗ and a ﬁnite F ⊆ s − 1 ( r (µ)) , deﬁne The sets Z (µ \ F ) constitute a basis of compact open sets for a locally compact Hausdorff topology on X = G ( 0 ) E (see [Webster 2014, Theorem 2.1]).”