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 Gˆ of continuous characters of G . We study Arens regularity-related properties of the ideals L1E(G) of L1(G) that are made of functions whose Fourier transform is supported on E⊆Gˆ . Arens regularity of L1E(G) , the centre of L1E(G)∗∗ and the size of L1E(G)∗/WAP(L1E(G)) are studied. We establish general conditions for the regularity of L1E(G) and deduce from them that L1E(G) is not strongly Arens irregular if E is a small-2 set (i.e. μ∗μ∈L1(G) for every μ∈M1E(G) ), which is not a Λ(1) -set, and it is extremely non-Arens regular if E is not a small-2 set. We deduce also that L1E(G) is not Arens regular when Gˆ∖E is a Lust-Piquard set.