Spectra of Translation-Invariant Function Algebras of Compact Groups

Abstract

Let G be a compact group and let Trig(G) denote the algebra of trigonometric polynomials of G. For a translation-invariant subalgebra A of Trig(G), one can consider the completions of A under the uniform norm and the Fourier norm. We show in Chapter 2 using techniques developed by Gichev that both completions have the same Gelfand spectrum, answering a question posed in a paper of Spronk and Stokke. In the same paper, a theorem describing of the Gelfand spectrum of the Fourier completion of finitely-generated such algebras A was given. In Chapter 3, we extend this theorem to the case of countably-generated, translation-invariant subalgebras, A. In Chapter 4, we give a brief overview of the Beurling--Fourier algebra, a weighted variant of the classical Fourier algebra studied by Ludwig, Spronk, and Turowska. The addition of a weight for these particular algebras invites new spectral data in contrast to its classical counterpart. In Chapter 5, we show for Beurling--Fourier algebras of compact abelian groups G that its weight can be used to construct a seminorm on tensor product of the real numbers with the Pontryagin dual of G that remembers the spectral data of the algebra.