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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymmetric maximal ideals in $M(G)$
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by Sadahiro Saeki PDF
Trans. Amer. Math. Soc. 222 (1976), 241-254 Request permission

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 $|c| \leqslant 1$ and each \[ \nu \in \Phi \;{\text {Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text {and}}\;|\hat \nu (f)| = r(\nu )\} \geqslant {2^{\text {c}}}.\] Here $r(\nu ) = \sup \{ |\hat \nu (f)|:f \in {\Delta _G}\backslash \hat G\}$. 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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • 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