Characterizations of Riesz sets

Let $G$ be a compact abelian group, $M(G)$ its measure algebra and $L^{1}(G)$ its group algebra. For a subset $E$ of the dual group $\widehat{G}$, let $M_{E}(G)=\{\mu\in M(G):\widehat{\mu}=0$ on $\widehat{G} \backslash E\}$ and $L_{E}^{1}(G)=\{a\in L^{1}(G):\widehat{a}=0$ on $\widehat{G}\backslash E\}$. The set $E$ is said to be a Riesz set if $M_{E}(G)=L_{E}^{1}(G)$. In this paper we present several characterizations of the Riesz sets in terms of Arens multiplication and in terms of the properties of the Gelfand transform $\Gamma :L_{E}^{1}(G)\rightarrow c_{0}(E)$.