Asymmetric maximal ideals in $M(G)$

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.