Additive bijections of \(C(X)\)
type: Item
instance of: scholarly article
title: Additive bijections of \(C(X)\)
zbMATH Open document ID: 0938.46024
DOI: 10.1016/S0166-8641(99)00045-0
author: Edward Beckenstein
author: Lawrence Narici
published in: Topology and its Applications
publication date: 2000-06-09T00:00:00Z
review text: The authors consider additive bijections \(H:C(X,K)\to C(Y,K)\), where \(X,Y\) are compact Hausdorff spaces, \(K=\mathbb R,\mathbb C\), or \(\mathbb Q_p\). It is assumed that \(H\) is separating, that is the equality \(fg=0\) for some \(f,g\in C(X,K)\) implies \((Hf)(Hg)=0\). It is proved that an additive separating bijection \(H\) is automatically continuous. If \(K=\mathbb R\) or \(\mathbb Q_p\), then \(Hf=(f\circ h)H(\mathbf 1)\), where \(\mathbf 1(x)\equiv 1\), and \(h:\;Y\to X\) is a homeomorphism. A similar (a little more complicated) representation is found also for \(K=\mathbb C\). An example shows that the above result does not hold for complete nontrivially valued non-Archimedean fields, other than \(\mathbb Q_p\).
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 46H40
zbMATH DE Number: 1367554
zbMATH Keywords: isometry
zbMATH Keywords: additive bijection
zbMATH Keywords: separating bijection
zbMATH Keywords: automatically continuous
MaRDI profile type: Publication
full work available at URL: s0166-8641(99)00045-0
OpenAlex ID: W1969320998
description: scientific article; zbMATH DE number 1367554