Relatively weak closed ideals of A(G), sets of synthesis and sets of uniqueness

Abstract

Let G be a locally compact amenable group, and A(G) and B(G) the Fourier and Fourier-Stieltjes algebras of G. For a closed subset E of G, let J(E) and k(E) be the smallest and largest closed ideals of A(G) with hull E, respectively. We study sets E for which the ideals J(E) or/and k(E) are sigma(A(G),C*(G))-closed in A(G). Moreover, we present, in terms of the uniform topology of C-0(G) and the weak* topology of B(G), a series of characterizations of sets obeying synthesis. Finally, closely related to the above issues, we present a series of results about closed sets of uniqueness (i.e. closed sets E for which <(J(E))over bar>w* = B(G)).