Arens regularity of ideals of the group algebra of a compact Abelian group
Esmailvandi, Reza; Filali, Mahmoud; Galindo, Jorge (2023-10-27)
Cambridge University Press
27.10.2023
Esmailvandi, R., Filali, M., & Galindo, J. (2023). Arens regularity of ideals of the group algebra of a compact Abelian group. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1–17. doi:10.1017/prm.2023.110
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© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (https://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
https://creativecommons.org/licenses/by-nc-nd/4.0/
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (https://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
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Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202406244857
https://urn.fi/URN:NBN:fi:oulu-202406244857
Tiivistelmä
Abstract
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.
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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