KMS states on the Toeplitz algebras of higher-rank graphs

Abstract

The Toeplitz algebra $\mathcal{T}C^{*}(Λ)$ for a finite $k$-graph $Λ$ is equipped with a continuous one-parameter group $α^{r}$ for each $ r\in \mathbb{R}^{k}$, obtained by composing the map $\mathbb{R} \ni t \to (e^{itr_{1}}, \dots , e^{itr_{k}}) \in \mathbb{T}^{k}$ with the gauge action on $\mathcal{T}C^{*}(Λ)$. In this paper we give a complete description of the $β$-KMS states for the $C^{*}$-dynamical system $(\mathcal{T}C^{*}(Λ), α^{r})$ for all finite $k$-graphs $Λ$ and all values of $β\in \mathbb{R}$ and $r\in \mathbb{R}^{k}$.