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

TL;DR

This paper investigates equilibrium states (KMS states) on operator algebras derived from self-similar groupoid actions on graph path spaces, revealing their structure and conditions for uniqueness at critical temperatures.

Contribution

It introduces a framework for analyzing KMS states on Cuntz-Pimsner and Toeplitz algebras associated with self-similar groupoid actions on graphs, including explicit computation methods.

Findings

KMS states above critical temperature are parametrized by traces on the groupoid's C*-algebra.

Critical temperature equals the logarithm of the graph's incidence matrix spectral radius.

Under certain conditions, there is a unique KMS state on the Cuntz-Pimsner algebra.

Abstract

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.