On maximal ideals depending on some thin sets in 𝑀(𝐺)

Abstract

Let M ( G ) M(G) be the convolution measure algebra on the LCA group G G with dual Γ \Gamma . and Δ \Delta the maximal ideal space of M ( G ) M(G) . For E ⊂ G E \subset G a compact set, let G p ( E ) Gp(E) be the subgroup of G G generated algebraically by E E . R ( E ) R(E) the measures which are carried by a countable union of translates of G p ( E ) Gp(E) . and P E {P_E} the natural projection from M ( G ) M(G) onto R ( E ) R(E) . Also let h E {h_E} be the multiplicative linear functional μ ↦ ( P E μ ) ^ ( l ) \mu \mapsto ({P_E}\mu \hat )({\text {l}}) on M ( G ) M(G) . Then we prove that if G G is an I I -group, and E E an H 1 {H_1} -set, we get h E ∈ Γ ¯ {h_E} \in \bar \Gamma (i.e. the closure of Γ \Gamma in Δ \Delta ).