Spectral Synthesis on Direct Products

In a former paper we introduced the concept of localization of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localizability of a closed ideal in the Fourier algebra is equivalent to the synthesizability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an effective tool to prove synthesizability of varieties on locally compact Abelian groups. In another paper we used this method to show that when investigating synthesizability of a variety, roughly speaking, compact elements of the group can be neglected. Here we show, using localization, that the extension of a synthesizable locally compact Abelian group $G$ to the direct product $G\times \Z$, where $\Z$ is the integers, is synthesizable as well. These results are used to provide a complete characterization of synthesizable locally compact Abelian groups in \cite{Sze23d}.