A Wiener Tauberian Theorem on discrete abelian torsion groups

Abstract

One version of the classical Wiener Tauberian Theorem states that if $G$ is a locally compact abelian group then any nonzero closed translation invariant subspace of $L^∞(G)$ contains a character. In other words, spectral analysis holds for $L^∞(G)$. In this paper we prove a similar theorem: if $G$ is a discrete abelian torsion group then spectral analysis holds for $C(G)$, the space of all complex valued functions on $G$.