A generalized Banach-Stone theorem for \(C_{0}(K,X)\) spaces via the modulus of convexity of \(X^\ast\)
type: Item
instance of: scholarly article
title: A generalized Banach-Stone theorem for \(C_{0}(K,X)\) spaces via the modulus of convexity of \(X^\ast\)
zbMATH Open document ID: 1365.46008
author: Fabiano C. Cidral
author: Vinícius Morelli Cortes
author: Elói M. Galego
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-02-14T00:00:00Z
review text: Let \(X\) be a real or complex Banach space and let \(K\) be a locally compact Hausdorff space. Let \(C_0(K,X)\) denote the space of \(X\)-valued continuous functions on \(K\) that vanish at infinity, equipped with the supremum norm. Let \(T: C_0(K,X) \rightarrow C_0(S,X)\) be a surjective isomorphism such that \(\|T\|\|T^{-1}\| \leq \lambda\). Suppose that \( \sup \{ \|x_1^\ast-x_2^\ast\|: \|x_1^\ast + x_2^\ast\| =2,~ x_1^\ast,x_2^\ast \in X^\ast,~\|x_1^\ast\|,\|x_2^\ast\| \leq \lambda \} < \frac{4}{3}\). A classical result of \textit{K. Jarosz} [Stud. Math. 73, 33--39 (1982; Zbl 0501.46037)] is that in this case \(K\) and \(S\) are homeomorphic. In the present paper, the authors show that the same conclusion holds with a larger number which is approximately \(4\times 0.41883\).
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 6684623
zbMATH Keywords: isometries
zbMATH Keywords: Banach-Stone theorem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.01.009
OpenAlex ID: W2571825834
cites work: Geometric mean and triangles inscribed in a semicircle in Banach spaces
cites work: On isomorphisms of continuous function spaces
cites work: Triangles inscribed in a semicircle, in Minkowski planes, and in normed spaces
cites work: On Isomorphisms with Small Bound
cites work: Isomorphisms of \(C_ 0(Y)\) onto \(C(X)\)
cites work: Q3778482
cites work: Optimal extensions of the Banach-Stone theorem
cites work: Uniformly Convex Spaces
cites work: A Bound-Two Isomorphism Between C(X) Banach Spaces
cites work: On the moduli of convexity and smoothness
cites work: Q4205935
cites work: On two classes of Banach spaces with uniform normal structure
cites work: Q3998720
cites work: On the uniform convexity of \(L^p\) and \(l^p\)
cites work: A generalization of the Banach-Stone theorem
cites work: The space of bounded maps into a Banach space
cites work: On James and Jordan--von Neumann constants and the normal structure coefficient of Banach spaces
cites work: The modulus of convexity in normed linear spaces
cites work: When does the equality \(J(X^*)=J(X)\) hold for a two-dimensional Banach space \(X\)?
cites work: Applications of the Theory of Boolean Rings to General Topology
cites work: The James constant for the \(l_{3}\)-\(l_{1}\) space
DOI: 10.1016/J.JMAA.2017.01.009
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description: scientific article; zbMATH DE number 6684623