KMS states for generalized gauge actions on C*-algebras associated with self-similar sets

Abstract

Given a self-similar $K$ set defined from an iterated function system $\Gamma =({\gamma }_1,\dots ,{\gamma }_n)$ and a set of function $H=\{h_i:K\to \mathbb{R}{\}}_{i=1}^d$ satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras ${\mathcal{O}}_{\Gamma }$ and their Toeplitz extensions ${\mathcal{T}}_{\Gamma }$. We then characterize the KMS states for this action. For each $\beta \in (0,\infty )$, there is a Ruelle operator ${\mathcal{L}}_{H,\beta }$ and the existence of KMS states at inverse temperature $\beta$ is related to this operator. The critical inverse temperature ${\beta }_c$ is such that ${\mathcal{L}}_{H,{\beta }_c}$ has spectral radius 1. If $\beta <{\beta }_c$, there are no KMS states on ${\mathcal{O}}_{\Gamma }$ and ${\mathcal{T}}_{\Gamma }$; if $\beta ={\beta }_c$, there is a unique KMS state on ${\mathcal{O}}_{\Gamma }$ and ${\mathcal{T}}_{\Gamma }$ which is given by the eigenmeasure of ${\mathcal{L}}_{H,{\beta }_c}$; and if $\beta >{\beta }_c$, including $\beta =\infty$, the extreme points of the set of KMS states on ${\mathcal{T}}_{\Gamma }$ are parametrized by the elements of $K$ and on ${\mathcal{O}}_{\Gamma }$ by the set of branched points.

References in corpus (2)

• A construction of $C^{\ast }$-algebras from $C^{\ast }$-correspondences
• C^*-algebras associated with self-similar sets