Arens regularity of ideals of the group algebra of a compact Abelian group

Abstract

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$ . We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$ . Arens regularity of $L_E^1(G)$ , the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$ ), which is not a $\Lambda (1)$ -set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.