Graph algebras and orbit equivalence

By Nathan Brownlowe, Toke Meier Carlsen, Michael F. Whittaker. Ergodic Theory and Dynamical Systems, 2014

79 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97ac2bced17cd40a71b60ab8fcf541c7bbc9f043

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Citations in the corpus

Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras (2017, background): “15 In graph C*-algebra theory it is usual to not define the shift map on elements of length zero (see for example [4]), but in symbolic dynamics this is often the case (see [18, 19, 25]).”
Reconstruction of groupoids and C*-rigidity of dynamical systems (2017, methodology): “Ad hoc results in this direction have been achieved recently for graph C∗algebras [9, 7], but there is no general theory available.”
\'Etale groupoids and their $C^*$-algebras (2017, background): “The final two equivalences, proved later, require the groupoid equivalence theorem of [12], and the results of [11].”
Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “More recently, it has been refined by Brownlowe– Carlsen–Whittaker [10] for graph C∗-algebras, by Brown–Clark–an Huef [7] for Leavitt path algebras, and by Ara–Bosa–Hazrat–Sims [4] for Steinberg algebras.”