Filali, M., & Galindo, J. (2022). Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann algebras. Journal of Mathematical Analysis and Applications, 512(1), 126137. https://doi.org/10.1016/j.jmaa.2022.126137

Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann algebras

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Author: Filali, M.1; Galindo, J.2
Organizations: 1Department of Mathematical Sciences, University of Oulu, Oulu, Finland
2Instituto Universitario de Matemáticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071, Castellón, Spain
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
Language: English
Published: Elsevier, 2022
Publish Date: 2022-07-05
Description:

Abstract

A Banach algebra $$\mathscr{A}$$ is Arens-regular 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 non-Arens 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 r-enAr, 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 r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr:

(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 non-discrete 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.

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Series: Journal of mathematical analysis and applications
ISSN: 0022-247X
ISSN-E: 1096-0813
ISSN-L: 0022-247X
Volume: 512
Issue: 1
Article number: 126137
DOI: 10.1016/j.jmaa.2022.126137