Date of Award

2019

Publication Type

Master Thesis

Degree Name

M.Sc.

Department

Mathematics and Statistics

First Advisor

D. Yang

Keywords

Irreducible matrices, KMS states, Perron-Frobenius theorem

Rights

info:eu-repo/semantics/openAccess

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Abstract

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 $.

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