Date of Award

6-2025

Degree Name

MS in Mathematics

Department/Program

Mathematics

College

College of Science and Mathematics

Advisor

Robert Easton

Advisor Department

Mathematics

Advisor College

College of Science and Mathematics

Abstract

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative linear functional on $L^1(\mathbb{R})$. From this, the spectrum of an element $f\in L^1(\mathbb{R})$ is precisely the closure of the range of the Fourier transform of $f$. We then generalize to the study of $L^1(G)$ for a locally compact Abelian group $G$ and establish the bijection between the dual group $\widehat{G}$ and the Gelfand space $\Delta(L^1(G))$. When considering $L^1(G)$, the Fourier transform is precisely the Gelfand transform. Lastly, we introduce Pontryagin duality and a structure theorem for the Gelfand representation of $L^1(G)$ for $G=\mathbb{R}^n\times\mathbb{T}^m\times\mathbb{Z}^j$, where $n,m,j\in\mathbb{N}$.

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