Module Arens regularity for semigroup algebras
type: Item
instance of: scholarly article
title: Module Arens regularity for semigroup algebras
zbMATH Open document ID: 1161.46026
author: Reza Rezavand
author: Massoud Amini
author: Mohammad Hossein Sattari
author: D. Ebrahimi Bagha
published in: Semigroup Forum
publication date: 2008-12-02T00:00:00Z
review text: The authors introduce and study the notion of module Arens regularity. A classical result by \textit{N. J. Young} [Q. J. Math., Oxf. II. Ser. 24, 59--62 (1973; Zbl 0252.43009)] says that the group algebra \(L^1(G)\) of a locally compact group \(G\) is Arens regular if and only if \(G\) is finite. The result is not true for discrete semigroups: for example, if \(S\) is a right zero semigroup of any cardinality, the semigroup algebra \(\ell^1(S)\) is Arens regular. The paper under review presents an extension of Young's theorem to inverse semigroups \(S\) such that the subsemigroup \(E\) of the idempotents in \(S\) is totally ordered. Namely, \(\ell^1(S)\) is module Arens regular with respect to suitable actions by \(\ell^1(E)\) if and only if a certain homomorphic image of \(S\) is finite. Reviewer's remark: There is a mistake in the proof of Theorem 2.2, and the definition of module Arens regularity needs to be modified in order to recover the results. The authors are aware of the problem and have written a correction.
reviewed by: Pekka Salmi
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 5377133
zbMATH Keywords: module Arens regularity
zbMATH Keywords: semigroup algebra
zbMATH Keywords: inverse semigroup
MaRDI profile type: Publication
full work available at URL: s00233-008-9075-3
OpenAlex ID: W2053668825
cites work: Module amenability for semigroup algebras
cites work: Weak module amenability for semigroup algebras
cites work: The Adjoint of a Bilinear Operation
cites work: Banach algebras and automatic continuity
cites work: The second dual of a Banach algebra
cites work: The Convolution of Functionals on Spaces of Bounded Functions
cites work: THE IRREGULARITY OF MULTIPLICATION IN GROUP ALGEBRAS
cites work: Semigroup algebras having regular multiplication
DOI: 10.1007/S00233-008-9075-3
description: scientific article; zbMATH DE number 5377133