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Proceedings of the American Mathematical Society

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

 
 

 

Idempotent maximal ideals and independent sets


Author: Colin C. Graham
Journal: Proc. Amer. Math. Soc. 54 (1976), 133-137
DOI: https://doi.org/10.1090/S0002-9939-1976-0390645-6
MathSciNet review: 0390645
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Abstract: Let $ E$ be a compact independent subset of a nondiscrete LCA group $ G$. Let $ GpE$ be the subgroup of $ G$ generated algebraically by $ E$. If $ \mu $ is a continuous, regular, Borel measure on $ GpE$ with $ \mu (GpE) \ne 0$, then there exists a maximal ideal $ \chi $ of the algebra $ M(G)$ of regular Borel measures on $ G$ such that the restriction of $ \chi $ to $ {L^1}(\mu ) = \{ \nu \in M(G):\nu \ll \mu \} $ is a nontrivial idempotent in $ {L^\infty }(\mu )$. This result is used to give a new proof that $ GpE$ has zero Haar measure.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0390645-6
Keywords: Generalized characters on measure algebras, measures on independent sets of a LCA $ G$
Article copyright: © Copyright 1976 American Mathematical Society

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