Orbit equivalence of graphs and isomorphism of graph groupoids
We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the (topological) isolated points of the boundary path space of a graph. As a result, we are able to show that the groupoids of two directed graphs with finitely many vertices and no sinks are isomorphic if and only if the two graphs are orbit equivalent, and that the groupoids of the stabilisations of two such graphs are isomorphic if and only if the stabilisations of the graphs are orbit equivalent.
Ara, P., Bosa, J., Hazrat, R., and Sims, A., Reconstruction of graded groupoids from graded Steinberg algebras, Forum Math. 29 (2017), no. 5, 1023–1037. https://doi.org/10.1515/forum-2016-0072
Arklint, S. E., Eilers, S., and Ruiz, E., A dynamical characterization of diagonal preserving $*$-isomorphisms of graph $C^*$-algebras, preprint arXiv:1605.01202 [math.OA].
Brown, J. H., Clark, L. O., and an Huef, A., Diagonal-preserving ring $*$-isomorphisms of Leavitt path algebras, J. Pure Appl. Algebra 221 (2017), no. 10, 2458–2481. https://doi.org/10.1016/j.jpaa.2016.12.032
Brownlowe, N., Carlsen, T. M., and Whittaker, M. F., Graph algebras and orbit equivalence, Ergodic Theory Dynam. Systems 37 (2017), no. 2, 389–417. https://doi.org/10.1017/etds.2015.52
Carlsen, T. M., $ast $-isomorphism of Leavitt path algebras over ℤ, Adv. Math. 324 (2018), 326–335. https://doi.org/10.1016/j.aim.2017.11.018
Carlsen, T. M., Eilers, S., Ortega, E., and Restorff, G., Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids, preprint arXiv:1610.09945 [math.DS].
Carlsen, T. M. and Rout, J., Diagonal-preserving gauge-invariant isomorphisms of graph $C^ast $-algebras, J. Funct. Anal. 273 (2017), no. 9, 2981–2993. https://doi.org/10.1016/j.jfa.2017.06.018
Carlsen, T. M., Ruiz, E., and Sims, A., Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1581–1592. https://doi.org/10.1090/proc/13321
Matsumoto, K., Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Pacific J. Math. 246 (2010), no. 1, 199–225. https://doi.org/10.2140/pjm.2010.246.199
Renault, J., Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. (2008), no. 61, 29–63.
Tomforde, M., Stability of $C^ast $-algebras associated to graphs, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1787–1795. https://doi.org/10.1090/S0002-9939-04-07411-8
Webster, S. B. G., The path space of a directed graph, Proc. Amer. Math. Soc. 142 (2014), no. 1, 213–225. https://doi.org/10.1090/S0002-9939-2013-11755-7