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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymmetric maximal ideals in $ M(G)$


Author: Sadahiro Saeki
Journal: Trans. Amer. Math. Soc. 222 (1976), 241-254
MSC: Primary 43A10
DOI: https://doi.org/10.1090/S0002-9947-1976-0415201-7
MathSciNet review: 0415201
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Abstract: Let G be a nondiscrete LCA group, $ M(G)$ the measure algebra of G, and $ {M_0}(G)$ the closed ideal of those measures in $ M(G)$ whose Fourier transforms vanish at infinity. Let $ {\Delta _G},{\Sigma _G}$ and $ {\Delta _0}$ be the spectrum of $ M(G)$, the set of all symmetric elements of $ {\Delta _G}$, and the spectrum of $ {M_0}(G)$, respectively. In this paper this is shown: Let $ \Phi $ be a separable subset of $ M(G)$. Then there exist a probability measure $ \tau $ in $ {M_0}(G)$ and a compact subset X of $ {\Delta _0}\backslash {\Sigma _G}$ such that for each $ \vert c\vert \leqslant 1$ and each

$\displaystyle \nu \in \Phi \;{\text{Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text{and}}\;\vert\hat \nu (f)\vert = r(\nu )\} \geqslant {2^{\text{c}}}.$

Here $ r(\nu ) = \sup \{ \vert\hat \nu (f)\vert:f \in {\Delta _G}\backslash \hat G\} $. As immediate consequences of this result, we have (a) every boundary for $ {M_0}(G)$ is a boundary for $ M(G)$ (a result due to Brown and Moran), (b) $ {\Delta _G}\backslash {\Sigma _G}$ is dense in $ {\Delta _G}\backslash \hat G$, (c) the set of all peak points for $ M(G)$ is $ \hat G$ if G is $ \sigma $-compact and is empty otherwise, and (d) for each $ \mu \in M(G)$ the set $ \hat \mu ({\Delta _0}\backslash {\Sigma _G})$ contains the topological boundary of $ \hat \mu ({\Delta _G}\backslash \hat G)$ in the complex plane.

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DOI: https://doi.org/10.1090/S0002-9947-1976-0415201-7
Keywords: LCA group, measure algebra, asymmetric maximal ideal, $ {M_0}$-boundary
Article copyright: © Copyright 1976 American Mathematical Society