Small-2 Sets Are Riesz Sets

Abstract

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}=\{μ\in M(G):\hatμ=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.