Orthogonal ℓ_{1}sets and extreme nonArens regularity of preduals of von Neumann algebras 

Author:  Filali, M.^{1}; Galindo, J.^{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:  published version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.5 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife2022050933814 
Language:  English 
Published: 
Elsevier,
2022

Publish Date:  20220705 
Description: 
AbstractA Banach algebra \(\mathscr{A}\) is Arensregular when all its continuous functionals are weakly almost periodic, in symbols when \(\mathscr{A^{∗}} = \mathscr{WAP}(\mathscr{A})\). To identify the opposite behaviour, Granirer called a Banach algebra extremely nonArens regular (enAr, for short) when the quotient \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\) contains a closed subspace that has \(\mathscr{A^{∗}}\) as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra \(\mathscr{A}\) is renAr, with \(r ≥ 1\), when there is an isomorphism with distortion \(r\) of \(\mathscr{A^{∗}}\) into \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\). When \(r = 1\), we obtain an isometric isomorphism and we say that \(\mathscr{A}\) is isometrically enAr. We then identify sufficient conditions for the predual \(\mathfrak{V_{∗}}\) of a von Neumann algebra \(\mathfrak{V}\) to be renAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be renAr: (i) the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), the algebra is isometrically enAr, (ii) the weighted group algebra of any nondiscrete locally compact infinite group and for any weight, (iii) the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), the algebra is isometrically enAr. The Fourier algebra \(A(G)\) of a locally compact infinite group \(G\) is shown to be isometrically enAr provided that (1) the local weight of \(G\) is greater or equal than its compact covering number, or (2) \(G\) is countable and contains an infinite amenable subgroup. see all

Series: 
Journal of mathematical analysis and applications 
ISSN:  0022247X 
ISSNE:  10960813 
ISSNL:  0022247X 
Volume:  512 
Issue:  1 
Article number:  126137 
DOI:  10.1016/j.jmaa.2022.126137 
OADOI:  https://oadoi.org/10.1016/j.jmaa.2022.126137 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Copyright information: 
© 2022 The Author(s). This is an open access article under the CC BYNCND license (http://creativecommons.org/licenses/byncnd/4.0/). 
https://creativecommons.org/licenses/byncnd/4.0/ 