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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Enji Sato
Journal: Proc. Amer. Math. Soc. 87 (1983), 131-136
MSC: Primary 43A46; Secondary 43A10
MathSciNet review: 677248
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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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Keywords: Measure algebra on LCA group, maximal ideal, Helson set
Article copyright: © Copyright 1983 American Mathematical Society