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 $C^*$-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.Comment: 27 page

Diagonal-preserving graded isomorphisms of Steinberg algebras

“…In particular, we show that if R is a commutative integral domain with identity and E and F are arbitrary directed graph, then there is a diagonal-preserving isomorphism between L R (E) and L R (F) if and only if the graph groupoids G E and G F are isomorphic (Corollary 4.2). By [7], the latter condition is equivalent to the existence of a diagonal-preserving isomorphism between the graph C * -algebras C * (E) and C * (F), and by [5], it is also equivalent to the existence of an orbit equivalence from E to F that preserves isolated eventually periodic points. We also show that there is a diagonalpreserving isomorphism between the stabilised Leavitt path algebras L R (E) ⊗ M ∞ (R) and L R (F) ⊗ M ∞ (R) if and only if the groupoids G E and G F are groupoid equivalent (Corollary 4.8).…”

“…• The pair (G E , c), where G E is the groupoid of a directed graph E (see for example [5,7,8,11,13,21]) and c is any continuous cocycle from G E to Γ (see the proof of Theorem 4.1); • the pair (G Λ , c), where G Λ is the groupoid of a finitely-aligned higher-rank graph Λ (see for example [20,36]) and c is any continuous cocycle from G Λ to Γ; • the pair (G tight (T ), c), where G tight (T ) is the tight groupoid constructed from a Boolean dynamical system in [10], and c is any continuous cocycle from…”

“…Diagonal-preserving graded isomorphisms of graph algebras. For a directed graph E and a unital ring R, we let G E denote the groupoid of E (see for example [5,7,8,11,13,21]), C * (E) the C * -algebra of E (see for example [2,5,7,8,11,29,35]…”

Diagonal-preserving gauge-invariant isomorphisms of graph C⁎-algebras

“…These graph C * -algebras have since attracted a lot of interest, and by using that graph C * -algebras can be constructed from groupoids, [12,Theorem 2.3] has recently been transfered to the setting of graph C * -algebras [1,2].…”

“…We recall the extended Weyl groupoid G (C * (E),D(E)) constructed in [2] from a graph C * -algebra and its diagonal subalgebra. As defined in [14], the normaliser of D(E) is the set…”