Graph algebras and orbit equivalence

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 $C^{\ast }$-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 ${\mathcal{G}}_{(C^{\ast }(E),\mathcal{D}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra $\mathcal{D}(E)$ which generalises Renault's Weyl groupoid construction applied to $(C^{\ast }(E),\mathcal{D}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),\mathcal{D}(E))}$ 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 $(C^{\ast }(E),\mathcal{D}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

27 pages

References in corpus (2)

• Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras
• The Path Space of a Directed Graph

Cited by in corpus (14)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Diagonal-preserving gauge-invariant isomorphisms of graph $C^{\ast }$-algebras
• Geometric classification of graph $C^{\ast }$-algebras over finite graphs
• Reconstruction of groupoids and C*-rigidity of dynamical systems
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Diagonal-preserving graded isomorphisms of Steinberg algebras
• Topological Full Groups of Ample Groupoids with Applications to Graph Algebras
• The dynamics of partial inverse semigroup actions
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Conjugacy of local homeomorphisms via groupoids and C*-algebras
• C*-algebras, groupoids and covers of shift spaces
• Cartan subalgebras for non-principal twisted groupoid $C^{\ast }$-algebras
• Reconstruction of twisted Steinberg algebras
• Analyzing the Weyl construction for dynamical Cartan subalgebras