[Paper Review] Spectral Synthesis on Varieties

László Székelyhidi|arXiv (Cornell University)|Jun 30, 2023

advanced mathematical theories4 citations

TL;DR

This paper establishes necessary and sufficient conditions for spectral synthesis on locally compact Abelian groups using abstract derivations on the Fourier algebra. By introducing the concept of localizability of ideals in the Fourier algebra, the authors generalize classical results, proving that spectral synthesis holds if and only if every closed ideal in the Fourier algebra is localizable, extending theorems of Schwartz and Lefranc and resolving the case of discrete groups with finite torsion-free rank.

ABSTRACT

In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense subspace. He studied so-called local ideals in the space of Fourier transforms, and his proof based on the observation that, on the one hand, these local ideals are completely determined by the exponential monomials in the space, and, on the other hand, these local ideals completely determine the space itself. On the other hand, Dimitri Gurevich gave counterexamples for Schwartz's theorem in higher dimension. In this paper we show that the ideas of localisation can be extended to general locally compact Abelian groups using abstract derivations on the Fourier algebra of compactly supported measures. Based on this method we present necessary and sufficient conditions for spectral synthesis for varieties on locally compact Abelian groups. Using localisation, in \cite{MR4789359} we proved 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 compact elements. This may lead to a complete characterisation of locally compact Abelian groups having spectral synthesis.

Motivation & Objective

• To extend spectral synthesis theory from the real line and discrete groups to general locally compact Abelian groups.
• To address the failure of spectral synthesis in higher-dimensional Euclidean spaces, as shown by Gurevich, by identifying the underlying algebraic obstruction.
• To characterize spectral synthesis in terms of localizability of closed ideals in the Fourier algebra.
• To unify classical results—such as those of Schwartz and Lefranc—under a single algebraic framework based on derivations and localization.
• To determine precisely when spectral synthesis holds on discrete Abelian groups, particularly in relation to their torsion-free rank.

Proposed method

• The authors use abstract derivations on the Fourier algebra of compactly supported measures to define localizability of ideals.
• They establish a connection between derivations and the vanishing of functions on exponential monomials via annihilators.
• Localization is performed via the ring-theoretic construction of the localization $ \widehat{R}_m = \widehat{S}^{-1}\widehat{R} $ at exponential maximal ideals $ \widehat{M}_m $.
• The key technical tool is Krull’s Intersection Theorem applied to localized ideals, ensuring $ \widehat{S}^{-1}\widehat{I} = \bigcap_{n=0}^\infty \widehat{S}^{-1}(\widehat{I} + \widehat{M}_m^{n+1}) $.
• They prove that if all derivations on $ \mathcal{A}(G) $ are polynomial, then every closed ideal is localizable, which implies spectral synthesis.
• The proof relies on the fact that for groups of finite torsion-free rank, all derivations are polynomial, leading to Noetherian localizations and hence localizability.

Experimental results

Research questions

• RQ1 Under what conditions does spectral synthesis hold on a general locally compact Abelian group?
• RQ2 What is the algebraic property—localizability—that characterizes synthesizable ideals in the Fourier algebra?
• RQ3 Why does spectral synthesis fail in $ \mathbb{R}^d $ for $ d \geq 2 $, and how does this relate to non-localizable ideals?
• RQ4 How can the classical results of Schwartz (on $ \mathbb{R} $) and Lefranc (on $ \mathbb{Z}^n $) be unified under a single framework?
• RQ5 What is the precise condition on a discrete Abelian group under which spectral synthesis holds?

Key findings

• Spectral synthesis holds on a locally compact Abelian group $ G $ if and only if every closed ideal in the Fourier algebra $ \mathcal{A}(G) $ is localizable.
• The failure of spectral synthesis on $ \mathbb{R}^d $ for $ d \geq 2 $ is due to the existence of non-localizable closed ideals in $ \mathcal{A}(\mathbb{R}^d) $, as shown by Gurevich.
• On the real line, spectral synthesis holds because every closed ideal in $ \mathcal{A}(\mathbb{R}) $ is localizable, as established by Schwartz.
• On $ \mathbb{Z}^n $, spectral synthesis holds because all derivations on $ \mathcal{A}(\mathbb{Z}^n) $ are polynomial, ensuring localizability.
• For a discrete Abelian group $ G $, spectral synthesis holds if and only if its torsion-free rank is finite, as otherwise $ \mathcal{A}(G) $ contains non-localizable ideals arising from infinite-dimensional generalized polynomials.
• The paper provides a new proof of the Laczkovich–Székelyhidi theorem on discrete groups by showing that finite torsion-free rank implies Noetherian localization and hence localizability of all ideals.