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

TL;DR

This paper characterizes KMS states for generalized gauge actions on C*-algebras associated with self-similar sets, relating their existence and uniqueness to spectral properties of Ruelle operators.

Contribution

It introduces a framework for analyzing KMS states on Kawjiwara-Watatani algebras using Ruelle operators and identifies critical inverse temperatures for these states.

Findings

No KMS states for eta<eta_c

Unique KMS state at eta=eta_c

Parametrization of KMS states for eta>eta_c

Abstract

Given a self-similar $K$ set defined from an iterated function system $\Gamma=(\gamma_1,\ldots,\gamma_n)$ 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.