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 $|c| \leqslant 1$ 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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Keywords:
LCA group,
measure algebra,
asymmetric maximal ideal,
<IMG WIDTH="34" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img34.gif" ALT="${M_0}$">-boundary

Article copyright:
© Copyright 1976
American Mathematical Society