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

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 $C^{\ast }$-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.

48 pages

Cited by in corpus (12)

• Equilibrium states on right LCM semigroup C*-algebras
• $C^{\ast }$-algebras of self-similar graphs over arbitrary graphs
• Self-Similar $k$-Graph C*-Algebras
• Preferred traces on C*-algebras of self-similar groupoids arising as fixed points
• Groupoid actions on $C^{\ast }$-correspondences
• Left regular representations of Garside categories II. Finiteness properties of topological full groups
• On groupoids and $C^{\ast }$-algebras from self-similar actions
• The bicategory of groupoid correspondences
• Higman-Thompson groups from self-similar groupoid actions
• KMS States of Self-Similar $k$-Graph C*-Algebras
• Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies
• Simplicity of algebras and $C^{\ast }$-algebras of self-similar groupoids