## Electronic Theses and Dissertations

2019

Master Thesis

M.Sc.

#### Department

Mathematics and Statistics

D. Yang

#### Keywords

Irreducible matrices, KMS states, Perron-Frobenius theorem

#### Rights

info:eu-repo/semantics/openAccess

In this thesis, we study the Perron-Frobenius theory for irreducible matrices and irreducible family of commuting matrices in detail. We then apply it to study the KMS states of the $C^*$-algebras of $k$-graphs. To be more precise, we define the Toeplitz algebra $\mathcal{T}C^{*}(\Lambda)$ and $C^*$-algebra $C^{*}(\Lambda)$ for a $k$-graph $\Lambda$. For $r\in(0,\infty)^{k}$, there is a natural one-parameter $C^{*}$-dynamical system $(\mathcal{T}C^{*}(\Lambda),\alpha^{r},\mathbb{R})$ induced from the gauge action of $\mathbb{T}^{k}$ on $\mathcal{T}C^{*}(\Lambda)$. We study the KMS states on the dynamical system $(\mathcal{T}C^{*}(\Lambda),\alpha^{r},\mathbb{R})$. With suitable selections of $r\in(0,\infty)^{k}$ and $\beta\in(0,\infty)$, with emphasis on $\Lambda$ being strongly connected, it is shown that the KMS states are closely related to the unimodular Perron-Frobenius eigenvector of $\Lambda$.