Diameter preserving linear maps and isometries. II
type: Item
instance of: scholarly article
title: Diameter preserving linear maps and isometries. II
zbMATH Open document ID: 0956.46012
DOI: 10.1007/BF02829491
author: Félix Cabello Sánchez
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2001-03-02T00:00:00Z
review text: Let \({\mathcal A}(S)\) be the space of all (real or complex) affine continuous functions on a compact convex set \(S\) in a locally convex Hausdorff space. A linear bijection \(T\) of \({\mathcal A}(S)\) is called diameter preserving if \(\rho (Tf) =\rho (f)\), where, for \(g\in {\mathcal A}(S)\), \(\rho (g) = \sup \{|g(x)-g(y)|: x,y, \in S\}\) denotes the diameter of the range of \(g\). The main result of the paper asserts that if \(S\) is a simplex then a linear bijection \(T\) of \({\mathcal A}(S)\) is diameter preserving iff there are an affine automorphism \(\phi :S\to S\), a linear functional \(\mu :{\mathcal A}(S)\to \mathbb K\), and a number \(\tau\) with \(|\tau|=1\) and \(\mu(1_S) + \tau \neq 0,\) such that \(Tf =\tau f\circ \phi +\mu (f)1_S, \) for \(f\in {\mathcal A}(S)\). Similar results for the space \(C(X)\) have been obtained by \textit{M. Györy} and \textit{L. Molnár} [Arch. Math. 71, No.~4, 301-310 (1998; Zbl 0928.46034)], and by the author in the first part of the present paper [Arch. Math. (Basel) 73, No. 5, 373-379 (1999)].
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 1471244
zbMATH Keywords: spaces of affine functions
zbMATH Keywords: simplexes
zbMATH Keywords: diameter preserving maps
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
cites work: Q5611281
cites work: Diameter preserving linear maps and isometries
cites work: Q4266789
cites work: Diameter preserving linear bijections of \(C(X)\)
cites work: Linear preserver problems: A brief introduction and some special techniques
cites work: Q3796197
cites work: A simplex with dense extreme points
cites work: Diameter-preserving linear bijections of function spaces
cites work: Q5631608
Recommended article: Diameter preserving linear maps and isometries
Recommended article: ON DIAMETER PRESERVING LINEAR MAPS
Recommended article: Linear extensions, almost isometries, and diameter two
Recommended article: Linear bijections which preserve the diameter of vector-valued maps
Recommended article: Diameter-preserving linear bijections of function spaces
Recommended article: Diameter preserving linear bijections and \({\mathcal C}_0(L)\) spaces
Recommended article: Diameter preserving linear bijections of \(C(X)\)
Recommended article: Q4258704
Recommended article: Isometry of mappings that almost preserve certain distances
Recommended article: Diameter preserving maps on function spaces
description: scientific article; zbMATH DE number 1471244