Local spectral synthesis on Abelian groups

Abstract

We say that a complex valued function defined on an Abelian group G is a local polynomial, if its restriction to every finitely generated subgroup of G is a polynomial. We prove that local spectral synthesis (that is, spectral synthesis using local polynomials instead of polynomials) holds on every Abelian group having countable torsion free rank. More precisely, there is a cardinal ω1≦κ≦2ω such that local spectral synthesis holds on an Abelian group G if and only if the torsion free rank of G is less than κ.