Spectral Synthesis on Direct Products

Abstract In our recent work we introduced the concept of localization of ideals in the Fourier algebra of a locally compact abelian group. We showed that localizability of a closed ideal in the Fourier algebra is equivalent to the synthesizability of the variety which corresponds to the respective ideal in the measure algebra. This equivalence provides an effective tool in studying spectral synthesis on locally compact abelian groups. Recently we used localization to prove that, when investigating synthesizability of a variety, compact elements of the underlying group may essentially be disregarded. Although, in general, direct products do not preserve synthesizability, in the present paper we apply the localization method to show that if spectral synthesis holds on a locally compact abelian group, then it also holds on the direct product of this group with the group of integers. This result, combined with our earlier work, yields a complete characterization of locally compact abelian groups on which spectral synthesis holds, as presented in [9].