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

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.