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Graph algebras and orbit equivalence


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.

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[1] Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographies de L’Enseignement Mathématique, 36) . L’Enseignement Mathématique, Geneva, 2000.
[2] an Huef, A. and Raeburn, I.. The ideal structure of Cuntz–Krieger algebras. Ergod. Th. & Dynam. Sys. 17 (1997), 611624.
[3] Bates, T., Hong, J., Raeburn, I. and Szymanski, W.. The ideal structure of the C -algebras of infinite graphs. Illinois J. Math. 46 (2002), 11591176.
[4] Bates, T. and Pask, D.. Flow equivalence of graph algebras. Ergod. Th. & Dynam. Sys. 24 (2004), 367382.
[5] Eilers, S., Ruiz, E. and Sørensen, A.. Amplified graph C -algebras. Münster J. Math. 5 (2012), 121150.
[6] Giordano, T., Matui, H., Putnam, I. and Skau, C.. Orbit equivalence for Cantor minimal ℤ d -systems. Invent. Math. 179 (2010), 119158.
[7] Giordano, T., Putnam, I. and Skau, C.. Topological orbit equivalence and C -crossed products. J. Reine Angew. Math. 469 (1995), 51111.
[8] Kumjian, A.. On C -diagonals. Canad. J. Math. 4 (1986), 9691008.
[9] Kumjian, A., Pask, D., Raeburn, I. and Renault, J.. Graphs, groupoids, and Cuntz–Krieger algebras. J. Func. Anal. 144 (1997), 505541.
[10] Matsumoto, K.. Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Pacific J. Math. 246 (2010), 199225.
[11] Matsumoto, K.. Full groups of one-sided topological Markov shifts. Israel J. Math. 205 (2015), 13.
[12] Matsumoto, K. and Matui, H.. Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras. Kyoto J. Math. 54 (2014), 863877.
[13] Matui, H.. Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. 104 (2012), 2756.
[14] Nagy, G. and Reznikoff, S.. Pseudo-diagonals and uniqueness theorems. Proc. Amer. Math. Soc. 142 (2014), 263275.
[15] Paterson, A. L. T.. Graph inverse semigroups, groupoids and their C -algebras. J. Operator Theory 48 (2002), 645662.
[16] Raeburn, I.. Graph Algebras. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005, vi+113.
[17] Raeburn, I. and Williams, D.. Morita Equivalence and Continuous-Trace C -Algebras (Mathematical Surveys and Monographs, 60) . The American Mathematical Society, Providence, RI, 1998, xiv+327.
[18] Renault, J. N.. A Groupoid Approach to C -Algebras (Lecture Notes in Mathematics, 793) . Springer, Berlin, 1980.
[19] Renault, J. N.. Cartan subalgebras in C -algebras. Irish Math. Soc. Bull. 61 (2008), 2963.
[20] Rørdam, M.. Classification of Cuntz–Krieger algebras. K-Theory 9 (1995), 3158.
[21] Tomiyama, J.. Topological full groups and structure of normalizers in transformation group C -algebras. Pacific J. Math. 173 (1996), 571583.
[22] Webster, S.. The path space of a directed graph. Proc. Amer. Math. Soc. 142 (2014), 213225.
[23] Yeend, T.. Groupoid models for the C -algebras of topological higher-rank graphs. J. Operator Theory 57 (2007), 95120.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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