Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 371-435.

$C^*$-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states

Authors: Zahra Afsar (1), Nathan Brownlowe (2), Jacqui Ramagge (3), Michael F. Whittaker (4)
Author institution: (1) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(2) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(3) University of South Australia, Adelaide, SA 5001, Australia
(4) School of Mathematics and Statistics, University of Glasgow, Glasgow Q12 8QQ, U.K.

Summary: We introduce the notion of a self-similar action of a groupoid $G$ on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and thereby obtain corresponding universal Nica--Toeplitz and Cuntz--Pimsner algebras. We consider natural actions of the real numbers on both algebras and study the KMS states of the associated dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica--Toeplitz algebra in terms of traces on the full $C^*$-algebra of $G$. We prove that if the graph is $G$-aperiodic and the action satisfies a finite-state condition, then there is a unique KMS state on the Cuntz--Pimsner algebra.

DOI: http://dx.doi.org/10.7900/jot.2024iun05.2509
Keywords: self-similar, operator algebra, KMS state, k-graph, automaton

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