Graph algebras and orbit equivalence

TL;DR

This paper explores the concept of orbit equivalence in directed graphs, establishing conditions under which their associated $C^*$-algebras are isomorphic, and introduces a new groupoid construction that generalizes previous frameworks.

Contribution

It introduces a new notion of orbit equivalence for directed graphs, relates it to $C^*$-algebra isomorphisms, and constructs a generalized groupoid that extends Renault's Weyl groupoid to arbitrary graphs.

Findings

Orbit equivalence characterized by diagonal-preserving isomorphisms of $C^*$-algebras.

Constructed a generalized groupoid that recovers the graph groupoid without exit cycle assumptions.

Applied results to out-splittings and amplified graphs.

Abstract

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^*$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ from the graph algebra $C^*(E)$ and its diagonal subalgebra $\mathcal{D}(E)$ which generalises Renault's Weyl groupoid construction applied to $(C^*(E),\mathcal{D}(E))$. We show that $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ recovers the graph groupoid $\mathcal{G}_E$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault's results to $(C^*(E),\mathcal{D}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.