University of Oulu

Filali, M. & Galindo, J. (2018). Extreme non-Arens regularity of the group algebra. Forum Mathematicum, 30(5), pp. 1193-1208. Retrieved 8 Apr. 2019, from doi:10.1515/forum-2017-0117

Extreme non-Arens regularity of the group algebra

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Author: Filali, Mahmoud1; Galindo, Jorge2
Organizations: 1Department of Mathematical Sciences, University of Oulu, Oulu, Finland
2 Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
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Language: English
Published: De Gruyter, 2018
Publish Date: 2019-04-08


The Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra \(L^1⁢(G)\) is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that \(L^1⁢(G)\) is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group \(G\), there is a linear isometry from \(L^\infty⁢(G)\) into the quotient space \(L^\infty⁢(G)/\mathscr{F}⁢(G)\), with \(\mathscr{F}⁢⁢(G)\) being any closed subspace of \(L^\infty⁢(G)\) made of continuous bounded functions. This, together with the known fact that \(ℓ^\infty⁢(G)/\mathscr{WAP}(G)\) always contains a linearly isometric copy of \(ℓ^\infty⁢(G)\), proves that \(L^1⁢(G)\) is extremely non-Arens regular for every infinite locally compact group.

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Series: Forum mathematicum
ISSN: 0933-7741
ISSN-E: 1435-5337
ISSN-L: 0933-7741
Volume: 30
Issue: 5
Pages: 1193 - 1208
DOI: 10.1515/forum-2017-0117
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: Parts of the article were written when the first named author was visiting Universitat Jaume I in Castellón in December 2011 and May 2012. He would like to express his warm thanks for the kind hospitality and support. Subsequently, he was partially supported by Väisälä Foundation in 2012–2014. This support is gratefully acknowledged. The second named author was supported by Ministerio de Economía y Competitividad (Spain) through project MTM2016-77143-P (AEI/FEDER, UE). This support is also gratefully acknowledged.
Copyright information: © 2018 Walter de Gruyter GmbH, Berlin/Boston.