On the restriction maps of the Fourier and Fourier-Stieltjes algebras over locally compact groupoids

TL;DR

This paper investigates the restriction maps of Fourier and Fourier-Stieltjes algebras over locally compact groupoids, establishing conditions for surjectivity and exploring implications for algebraic properties.

Contribution

It extends surjectivity results of restriction maps from groups to groupoids, particularly for tale groupoids and transitive groupoids, and introduces new conditions related to property FD.

Findings

Restriction map on Fourier algebra is surjective for tale groupoids.

Restriction map on Fourier-Stieltjes algebra is generally not surjective.

Surjectivity conditions relate to properties like FD and examples of non-surjective cases.

Abstract

The Fourier and Fourier-Stieltjes algebras over locally compact groupoids have been defined in a way that parallels their construction for groups. In this article, we extend the results on surjectivity or lack of surjectivity of the restriction map on the Fourier and Fourier-Stieltjes algebras of groups to the groupoid setting. In particular, we consider the maps that restrict the domain of these functions in the Fourier or Fourier-Stieltjes algebra of a groupoid to an isotropy subgroup. These maps are continuous contractive algebra homomorphisms. When the groupoid is \'{e}tale, we show that the restriction map on the Fourier algebra is surjective. The restriction map on the Fourier-Stieltjes algebra is not surjective in general. We prove that for a transitive groupoid with a continuous section or a group bundle with discrete unit space, the restriction map on the Fourier-Stieltjes algebra is surjective. We further discuss the example of an HLS groupoid, and obtain a necessary condition for surjectivity of the restriction map in terms of property FD for groups, introduced by Lubotzky and Shalom. As a result, we present examples where the restriction map for the Fourier-Stieltjes algebra is not surjective. Finally, we use the surjectivity results to provide conditions for the lack of certain Banach algebraic properties, including the (weak) amenability and existence of a bounded approximate identity, in the Fourier algebra of \'{e}tale groupoids.