Banach algebraic properties of the Fourier and Fourier-Stieltjes algebras over locally compact groupoids
Date
2024
Authors
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Publisher
University of Delaware
Abstract
The Fourier and Fourier-Stieltjes algebras are two Banach algebras that can be associated to a locally compact group and were introduced in the paper L’alg`ebre de Fourier d’un groupe localement compact by Eymard. These algebras formed connections between other widely studied algebraic objects defined for groups- the group C∗-algebra is the continuous linear predual to the Fourier-Stieltjes algebra, and the group von Neumann algebra is the continuous linear dual to the Fourier algebra. Since their introduction, many group properties as well as Banach algebraic properties have been linked to these algebras, allowing one to study these as opposed to the groups or properties themselves. ☐ Separately, groups have been generalized to an object known as a groupoid. Groupoids have received much attention as of recent due to their wide array of applications, from the study of discrete graphs, to dynamical systems, to topology. The Fourier and Fourier-Stieltjes algebras are built out of the harmonic analysis of groups, so it is natural to ask whether there are sufficient tools to make analogous definitions over groupoids. While the answer is yes, there is no unique way to extend these definitions to the groupoid setting in a way that will reduce to the usual notion when the groupoid is a group. ☐ We study the continuous Fourier and Fourier-Stieltjes algebras, particularly their structure and norms. Our main motivating question was to explicitly determine these algebras for specific families of groupoids. We also study restriction style results, asking what objects we obtain when we restrict the domain of functions in Fourier or Fourier-Stieltjes algebras to subgroups of the base groupoid. We establish connections between these restricted algebras and the Fourier/Fourier-Stieltjes algebras of the subgroups. In line with this study, we examine the behavior of the restriction maps to particular classes of groupoids, including transitive groupoids, group bundles, and HLS groupoids. ☐ In the transitive case, we show that the restriction map on the Fourier-Stieltjes algebra is surjective, as well as for group bundles with a particular topology. We also identify the Gelfand spectrum of these group bundles and find an explicit description for their Fourier and Fourier-Stieltjes algebras. In the HLS case, we explicitly construct pre-images for the restriction map on the Fourier algebra, and demonstrate that it is not always surjective on the Fourier-Stieltjes algebra. We further apply these restriction results to obtain a number of Banach algebraic corollaries based on the idea of hereditary properties; those which are preserved under continuous homomorphism with dense range. We also present a preliminary probe into Fourier coefficient spaces for groupoids.
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Keywords
Fourier algebra, Functional analysis, Harmonic analysis, Banach algebraic properties, Homomorphism