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

Sato, Enji

doi:10.1090/S0002-9939-1983-0677248-4

On maximal ideals depending on some thin sets in $M(G)$
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Proc. Amer. Math. Soc. 87 (1983), 131-136 Request permission

Abstract:

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

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