Perron-Frobenius theory and KMS states on higher-rank graph C*-Algebras

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 $