Journal
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
Volume -, Issue -, Pages -Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/prm.2023.110
Keywords
Arens product; Arens-regular algebra; centre; extremely non-Arens regular; Lust-Piquard set; Riesz set; strongly Arens irregular; small-2 set; Sidon set
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This paper investigates the Arens regularity of ideals supported by Fourier transforms on compact Abelian groups, studying different properties under various conditions.
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.
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