University of Oulu

Filali, M. and Galindo, J. (2021), On the extreme non-Arens regularity of Banach algebras. J. London Math. Soc., 104: 1840-1860.

On the extreme non-Arens regularity of Banach algebras

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Author: Filali, Mahmoud1; Galindo, Jorge2
Organizations: 1Department of Mathematical Sciences, University of Oulu, P.O. Box 8000, Oulu, FI-FI-90014 Finland
2Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, Castellón, E-12071 Spain
Format: article
Version: published version
Access: open
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Language: English
Published: London Mathematical Society, 2021
Publish Date: 2022-01-13


As is well-known, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper, we show that ℓ¹-bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai’s) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras.

These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Figà-Talamanca Herz algebra.

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Series: Journal of the London Mathematical Society
ISSN: 0024-6107
ISSN-E: 1469-7750
ISSN-L: 0024-6107
Volume: 104
Issue: 4
Pages: 1840 - 1860
DOI: 10.1112/jlms.12485
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: Research of the second-named author was supported by Research supported by Spanish AEI Project PID2019-106529GB-I00 / AEI / 10.13039/501100011033.
Copyright information: © 2021 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.