Date of Award
2019
Publication Type
Master Thesis
Degree Name
M.Sc.
Department
Mathematics and Statistics
Keywords
Irreducible matrices, KMS states, Perron-Frobenius theorem
Supervisor
D. Yang
Rights
info:eu-repo/semantics/openAccess
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International 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 $.
Recommended Citation
Singh, Samandeep, "Perron-Frobenius theory and KMS states on higher-rank graph C*-Algebras" (2019). Electronic Theses and Dissertations. 7671.
https://scholar.uwindsor.ca/etd/7671