Spectral Synthesis on Direct Products

TL;DR

This paper characterizes the locally compact Abelian groups where spectral synthesis is valid, building on previous work relating ideals in Fourier algebras and measure algebras to establish a comprehensive understanding of synthesisability.

Contribution

It provides a complete characterization of groups for which spectral synthesis holds, extending prior theoretical frameworks and methods.

Findings

Spectral synthesis holds on certain classes of locally compact Abelian groups.

Localisability of ideals is equivalent to synthesisability of their annihilators.

Compact elements can be neglected in the analysis of synthesisability.

Abstract

In a former paper we introduced the concept of localisation of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localisability of a closed ideal in the Fourier algebra is equivalent to the synthesisability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an effective tool to prove synthesisability of varieties on locally compact Abelian groups. In another paper we used this method to show that when investigating synthesisability of a variety, roughly speaking, compact elements of the group can be neglected. Using these results, in this paper we completely characterise those locally compact Abelian groups on which spectral synthesis holds.