Abstract
The overall goal of this thesis is to study spectral and Gelfand theory as it relates to unital Banach and C∗-algebras. In the first part, we develop the necessary algebraic, analytic, and topological background relevant to the content of this work. We also discuss concrete examples of algebras frequently used in the subsequent sections. In the second part of this thesis, we develop spectral theory by first defining the spectrum of an algebra through the characterization of the invertible and noninvertible elements. In particular, we establish properties of the commutative unital Banach algebra ℓ1(Z). We also establish fundamental results such as Gelfand’s spectral radius formula and prove the Gelfand-Mazur theorem. In the last part of this thesis, we study the spectrum, also known as the maximal ideal space, of commutative unital Banach algebras. Through this, we develop the Gelfand transform, which maps elements of the algebra to continuous functions on its spectrum.
Advisor
Kris Hollingsworth
Committee Member
Dat Cao
Committee Member
Wook Kim
Date of Degree
2025
Language
english
Document Type
Thesis
Degree
Master of Arts (MA)
Program of Study
Mathematics
Department
Mathematics and Statistics
College
Science, Engineering and Technology
Recommended Citation
Schlader, B. (2025). Spectral Theory and the Gelfand Theory [Master’s thesis, Minnesota State University, Mankato]. Cornerstone: A Collection of Scholarly and Creative Works for Minnesota State University, Mankato. https://cornerstone.lib.mnsu.edu/etds/1551/
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Rights Statement
