Characterizations of riesz sets

Abstract

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 Ĝ, let ME(G) = {μ ∈ M(G) : μ̂ = 0 on Ĝ\E} and L 1E(G) = {a ∈ L1(G) : â = 0 on Ĝ\E}. The set E is said to be a Riesz set if ME(G) = L 1E(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 Γ : L1E(G) → c0(E).