Factorization and existence of the unit in commutative Banach algebra
type: Item
instance of: scholarly article
title: Factorization and existence of the unit in commutative Banach algebra
zbMATH Open document ID: 0967.46039
published in: Archiv der Mathematik
publication date: 2000-06-21T00:00:00Z
review text: A commutative Banach algebra \(A\) is said to satisfy the property of factorization of bounded sequences if for all bounded sequence \((x_n)\) in \(A\) there are \(a\in A\) and a bounded sequence \((y_n)\) in \(A\) with \(x_n= ay_n\) for all \(n\). The following two results are proved: (1) Let \(A\) be a commutative Banach algebra and let \(a\in A\) satisfy that \(aA\) is dense in \(A\). For all \(b\in A\) there is a bounded sequence \((b_n)\) in \(A\) such that \((ab_n)\) converges to \(b\) if and only if \(A\) has a unit. This theorem permits the authors to obtain extensions of results of \textit{P. G. Dixon} [Math. Proc. Cambridge Phil. Soc. 107, No. 3, 557-571 (1990; Zbl 0723.46034)] and \textit{V. Runde} [Arch. Math. 58, No. 2, 183-189 (1992; Zbl 0759.46045)]. (2) If a commutative Banach algebra \(A\) verifying the property of factorization of bounded sequences has an element \(a\in A\) such that \(aA\) is dense in \(A\), then \(A\) has a unit.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46J05
Mathematics Subject Classification ID: 46K99
zbMATH DE Number: 1462959
zbMATH Keywords: unitary algebras
zbMATH Keywords: commutative Banach algebra
zbMATH Keywords: property of factorization of bounded sequences
author: Rachid Choukri
author: M. Oudadess
MaRDI profile type: Publication
full work available at URL: s000130050445
OpenAlex ID: W1969482083
DOI: 10.1007/S000130050445
description: scientific article; zbMATH DE number 1462959