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.
Brenden Schlader (Wed,) studied this question.