Diameter preserving mappings between function algebras
type: Item
instance of: scholarly article
title: Diameter preserving mappings between function algebras
zbMATH Open document ID: 1241.47030
author: Juan J. Font
author: Maliheh Hosseini
published in: Taiwanese Journal of Mathematics
publication date: 2011-12-01T00:00:00Z
review text: The authors consider function algebras \(A\subset C_0(X)\) satisfying that the extreme points of the unit ball of the quotient space \(A/C\), where \(C\) stands for the constant functions endowed with the diameter norm, are given by \(\alpha(\delta_{x}-\delta{x'})\) for some \(|\alpha|=1\) and \(x,x'\in Ch(A)\) (the Choquet boundary of \(A\)). They show that, for any diameter preserving map \(T:A\to B\) between function algebras \(A\subset C(X)\) and \(B\subset C(Y)\), with compact spaces \(X\) and \(Y\) such that \(A\) satisfies the above condition, there exist a linear functional \(L:A\to \mathbb C\), \(|\lambda|=1\), a set \(Y_0\subset Ch(B)\) and a continuous bijection \(\psi:Y_0\to Ch(A)\) such that \(Tf(y)= \lambda f(\psi(y))+L(f)\) for all \(y\in Y_0\) and \(f\in A\). Moreover, in the case when also \(B\) satisfies the above condition, then \(Y_0\) coincides with \(Ch(B)\) and the mapping \(\phi\) is an homomorphism. Results in the locally compact case are also provided and some applications are presented.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5984243
zbMATH Keywords: diameter preserving maps
zbMATH Keywords: function algebras
zbMATH Keywords: extreme points
zbMATH Keywords: Choquet boundary
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500406360
OpenAlex ID: W4205581013
DOI: 10.11650/TWJM/1500406360
description: scientific article; zbMATH DE number 5984243