Extreme non-Arens regularity of the group algebra

Mahmoud Filali; Jorge Galindo

doi:10.1515/forum-2017-0117

Article

Extreme non-Arens regularity of the group algebra

Mahmoud Filali and Jorge Galindo

Published/Copyright: March 1, 2018

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From the journal Forum Mathematicum Volume 30 Issue 5

Abstract

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 L1⁢(G) is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that L1⁢(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∞⁢(G) into the quotient space L∞⁢(G)/ℱ⁢(G), with ℱ⁢(G) being any closed subspace of L∞⁢(G) made of continuous bounded functions. This, together with the known fact that ℓ∞⁢(G)/𝒲⁢𝒜⁢𝒫⁢(G) always contains a linearly isometric copy of ℓ∞⁢(G), proves that L1⁢(G) is extremely non-Arens regular for every infinite locally compact group.

Keywords: Group algebra; extremely non-Arens regular; weakly almost periodic; isometry, Haar homeomorphism; metrizable groups

MSC 2010: 22D15; 43A46; 43A15; 43A60; 54H11

Communicated by Karl-Hermann Neeb

Funding source: Ministerio de Ciencia e Innovación

Award Identifier / Grant number: MTM2016-77143-P

Funding statement: 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.

Acknowledgements

We would like to thank the referee for her/his very careful reading of the paper, the corrections and for the useful comments.

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Received: 2017-06-06

Revised: 2018-01-19

Published Online: 2018-03-01

Published in Print: 2018-09-01

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https://doi.org/10.1515/forum-2017-0117

Keywords for this article

Group algebra; extremely non-Arens regular; weakly almost periodic; isometry, Haar homeomorphism; metrizable groups