Small-2 Sets Are Riesz Sets

Let $G$ be a compact metrizable Abelian group, $L^{1}(G)$ its group algebra and $M(G)$ its measure algebra. For each proper subset $E$ of the dual group $\hat{G}$, let $L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{\mu\in M(G):\hat{\mu}=0 \text{ on }\hat{G}\setminus E\}$. If $M_{E}(G)=L^{1}_{E}(G)$ then the set $E$ is said to be a Riesz sets. If $M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G)$ then $E$ is said to be a small-2 set. The main results of this paper are the following:\vspace{2mm} 1. Every small-2 set is a Riesz set.\vspace{2mm} 2. The ideal $L^{1}_{E}(G)$ is Arens regular iff $E$ is a Riesz set.\vspace{2mm} \noindent Let $A=L_{E}(G)$ and equip $A^{**}$ with the first Arens product.\vspace{2mm} (3). The centre of $A^{**}$ is $Z(A^{**})=A+N(A^{**})$, where $N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\}$.\vspace{2mm} \noindent These results settle three long-standing open problems in this area.