New Results on Spectral Synthesis

TL;DR

This paper explores spectral synthesis in locally compact Abelian groups, establishing that synthesisability depends on the quotient group by compact elements, and provides new proofs for known cases.

Contribution

It introduces a new criterion linking spectral synthesis on a group to its quotient by compact elements, simplifying proofs for compact and discrete groups.

Findings

Spectral synthesis holds on compact Abelian groups.

Spectral synthesis on a group is equivalent to that on its quotient by compact elements.

A new proof characterizes spectral synthesis on discrete Abelian groups.

Abstract

In our 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 this paper we utilise this tool to show that when investigating synthesisability of a variety, roughly speaking compact elements of the group can be neglected. Our main result is that spectral synthesis holds on a locally compact Abelian group $G$ if and only if it holds on $G/B$, where $B$ is the closed subgroup of all compact elements. In particular, spectral synthesis holds on compact Abelian groups. Also we obtain a simple proof for the characterisation theorem of spectral synthesis on discrete Abelian groups.