Extreme nonArens regularity of the group algebra 

Author:  Filali, Mahmoud^{1}; Galindo, Jorge^{2} 
Organizations: 
^{1}Department of Mathematical Sciences, University of Oulu, Oulu, Finland ^{2} Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E12071, Castellón, Spain 
Format:  article 
Version:  accepted version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.4 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife2019040811483 
Language:  English 
Published: 
De Gruyter,
2018

Publish Date:  20190408 
Description: 
AbstractThe 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 nonArens 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 nonArens regular for any infinite locally compact group. We actually prove the stronger result that for any nondiscrete 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 nonArens regular for every infinite locally compact group. see all

Series: 
Forum mathematicum 
ISSN:  09337741 
ISSNE:  14355337 
ISSNL:  09337741 
Volume:  30 
Issue:  5 
Pages:  1193  1208 
DOI:  10.1515/forum20170117 
OADOI:  https://oadoi.org/10.1515/forum20170117 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
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 MTM201677143P (AEI/FEDER, UE). This support is also gratefully acknowledged. 
Copyright information: 
© 2018 Walter de Gruyter GmbH, Berlin/Boston. 