1 results
Arens regularity of ideals of the group algebra of a compact Abelian group
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 27 October 2023, pp. 1-17
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- Article
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Let $G$
be a compact Abelian group and $E$
a subset of the group $\widehat {G}$
of continuous characters of $G$
. We study Arens regularity-related properties of the ideals $L_E^1(G)$
of $L^1(G)$
that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$
. Arens regularity of $L_E^1(G)$
, the centre of $L_E^1(G)^{\ast \ast }$
and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$
are studied. We establish general conditions for the regularity of $L_E^1(G)$
and deduce from them that $L_E^1(G)$
is not strongly Arens irregular if $E$
is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$
for every $\mu \in M_E^1(G)$
), which is not a $\Lambda (1)$
-set, and it is extremely non-Arens regular if $E$
is not a small-2 set. We deduce also that $L_E^1(G)$
is not Arens regular when $\widehat {G}\setminus E$
is a Lust-Piquard set.
