Graph algebras and orbit equivalence

Brownlowe, N., Carlsen, T. M. and Whittaker, M. (2017) Graph algebras and orbit equivalence. Ergodic Theory and Dynamical Systems, 37(2), pp. 389-417. (doi: 10.1017/etds.2015.52)

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∗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 EE we construct a groupoid G(C∗(E),D(E))G(C∗(E),D(E)) from the graph algebra C∗(E)C∗(E) and its diagonal subalgebra D(E)D(E) which generalises Renault’s Weyl groupoid construction applied to (C∗(E),D(E))(C∗(E),D(E)). We show that G(C∗(E),D(E))G(C∗(E),D(E)) recovers the graph groupoid GEGE without the assumption that every cycle in EE has an exit, which is required to apply Renault’s results to (C∗(E),D(E))(C∗(E),D(E)). We finish with applications of our results to out-splittings of graphs and to amplified graphs.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Whittaker, Professor Mike
• Authors: Brownlowe, N., Carlsen, T. M., and Whittaker, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Ergodic Theory and Dynamical Systems
• Publisher: Cambridge University Press
• ISSN: 0143-3857
• ISSN (Online): 1469-4417
• Published Online: 25 August 2015
• Copyright Holders: Copyright © 2017 Cambridge University Press
• First Published: First published in Ergodic Theory and Dynamical Systems 37(2):389-417
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher