Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs

Laca, M., Raeburn, I., Ramagge, J. and Whittaker, M. F. (2018) Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs. Advances in Mathematics, 331, pp. 268-325. (doi: 10.1016/j.aim.2018.03.030)

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We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz-Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full C*-algebra of the groupoid, and we describe a program for finding such traces. The critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph, and at the critical temperature the KMS states on the Toeplitz algebra factor through states of the Cuntz-Pimsner algebra. Under a verifiable hypothesis on the self-similar action, there is a unique KMS state on the Cuntz-Pimsner algebra. We discuss an explicit method of computing the values of this KMS state, and illustrate with examples.

Item Type:Articles
Additional Information:This research was supported by the Natural Sciences and Engineering Research Council of Canada, the Marsden Fund of the Royal Society of New Zealand, and the Australian Research Council.
Glasgow Author(s) Enlighten ID:Whittaker, Professor Mike
Authors: Laca, M., Raeburn, I., Ramagge, J., and Whittaker, M. F.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
ISSN (Online):1090-2082
Published Online:25 April 2018
Copyright Holders:Copyright © 2018 Elsevier Inc.
First Published:First published in Advances in Mathematics 331: 268-325
Publisher Policy:Reproduced in accordance with the publisher copyright policy
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