There's something about the diameter
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.
scientific article; zbMATH DE number 5149504
scientific article