There's something about the diameter
type: Item
instance of: scholarly article
title: There's something about the diameter
zbMATH Open document ID: 1123.46022
author: F. Rambla
author: A. Aizpuru
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: Given a space \({\mathcal F}\) of functions with a seminorm, an interesting question in analysis is to study the structure of linear bijections \(T\) of \({\mathcal F}\) that preserve the seminorm. Using the classical Banach-Stone theorem that identifies surjective isometries of \(C(X)\), \textit{M. Györy} and \textit{L. Molnár} [Arch. Math. 71, No. 4, 301--310 (1998; Zbl 0928.46034)] studied the structure of linear bijections that preserve the seminorm diameter. A standard technique here is to extend \(T\) to a surjective isometry on an appropriate quotient space and then either to use the known form of surjective isometries or to classify surjective isometries to get the desired conclusion. Similar questions for spaces of vector-valued functions were considered by \textit{A. K. Roy} and \textit{T. S. S. R. K. Rao} [J. Austral. Math. Soc. 70, No. 3, 323--335 (2001; Zbl 1027.46009)]. In this paper, the authors carry out this program for the space of Banach space-valued continuous functions on a compact set. The authors first identify the set of extreme points of the dual unit ball of the corresponding quotient space and then use it to analyse the structure of the surjective isometries of this spaces (Theorem 2.11). Unlike the results of Roy and Rao, no additional geometric conditions are assumed on the Banach space and thus there are no crisp conclusions obtained here.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5149504
zbMATH Keywords: isometries
zbMATH Keywords: diameter
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.002
OpenAlex ID: W1971869949
cites work: Diameter-preserving maps on various classes of function spaces
cites work: Diameter preserving linear maps and isometries
cites work: Q4209059
cites work: A characterization of locally compact spaces with homeomorphic one-point compactifications
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cites work: Diameter preserving linear bijections of \(C(X)\)
cites work: Isometries between injective tensor products of Banach spaces
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DOI: 10.1016/J.JMAA.2006.08.002
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description: scientific article; zbMATH DE number 5149504