Multiplicatively range-preserving maps between Banach function algebras
type: Item
instance of: scholarly article
title: Multiplicatively range-preserving maps between Banach function algebras
zbMATH Open document ID: 1171.46021
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-23T00:00:00Z
review text: Let \(X,Y\) be locally compact Hausdorff spaces and let \(A,B\) be two Banach function algebras on \(X,Y\) respectively. \(T:A\rightarrow B\) is said to be multiplicatively range preserving if \((TfTg)(Y)=(fg)(X)\) for all \(f,g \in A\). The study of the structure of such maps has received a lot of attention. The earliest results in this direction were obtained by \textit{L.\,Molnár} [Proc.\ Am.\ Math.\ Soc.\ 130, No.\,1, 111--120 (2002; Zbl 0983.47024)] who described such maps, in the context of compact spaces, as composition operators. Let \(c(A)\) denote the Choquet boundary of \(A\). If all the points of \(c(A),c(B)\) are strong boundary points (which is the case, for example, when \(A\) and \(B\) are both completely regular), then \(T\) has the form \(f\rightarrow h f\circ\phi\) on \(c(B)\), for an onto homeomorphism \(\phi:c(B)\rightarrow c(A)\) and a real-valued continuous map \(h\) on \(c(B)\) with \(h^2=1\).
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5568995
zbMATH Keywords: Banach function algebras
zbMATH Keywords: multiplicatively range preserving maps
zbMATH Keywords: Choquet boundary
zbMATH Keywords: spectrum preserving maps
zbMATH Keywords: 2-local maps
author: Maliheh Hosseini
author: Fereshteh Sady
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.008
OpenAlex ID: W2053186711
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DOI: 10.1016/J.JMAA.2009.04.008
description: scientific article; zbMATH DE number 5568995