Banach-Stone theorems for vector valued functions on completely regular spaces
type: Item
instance of: scholarly article
title: Banach-Stone theorems for vector valued functions on completely regular spaces
zbMATH Open document ID: 1253.46036
author: Ngai-Ching Wong
author: Lei Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-08T00:00:00Z
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 6064301
zbMATH Keywords: spaces of vector-valued continuous functions
zbMATH Keywords: Banach-Stone theorems
zbMATH Keywords: nonvanishing preservers
zbMATH Keywords: biseparating maps
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.05.033
OpenAlex ID: W2075761203
cites work: The space of bounded maps into a Banach space
cites work: A representation theorem for isometries of C(X,E)
cites work: On the Banach–Stone problem
cites work: M-structure and the Banach-Stone theorem
cites work: On the Banach-Stone theorem
cites work: How to obtain vector-valued Banach-Stone theorems by using M-structure methods
cites work: Q3270120
cites work: Banach-Stone theorems and separating maps
cites work: Biseparating maps and homeomorphic real-compactifications
cites work: Separating maps and linear isometries between some spaces of continuous functions
cites work: Q3308615
cites work: Q3889571
cites work: Biseparating linear maps between continuous vector-valued function spaces
cites work: Into isometries of \(C_ 0(X,E)s\)
cites work: Realcompactness and spaces of vector-valued functions
cites work: Realcompactness and Banach-Stone theorems
cites work: Q3416478
cites work: Banach-Stone theorems and Riesz algebras
cites work: A Banach-Stone theorem for Riesz isomorphisms of Banach lattices
cites work: The Banach--Stone theorem revisited
cites work: The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions
cites work: Banach-Stone theorems for maps preserving common zeros
cites work: Kaplansky Theorem for completely regular spaces
cites work: Q4074053
cites work: SEPARATING LINEAR MAPS OF CONTINUOUS FIELDS OF BANACH SPACES
cites work: Maps preserving common zeros between subspaces of vector-valued continuous functions
cites work: Composition operators between algebras of uniformly continuous functions
cites work: Q4252796
cites work: Local automorphisms of operator algebras
DOI: 10.1016/J.JMAA.2012.05.033
Recommended article: A Banach-Stone theorem for completely regular spaces
Recommended article: On Banach spaces of vector valued continuous functions
Recommended article: Q3580739
Recommended article: Banach-Stone theorem for isometries on spaces of vector-valued differentiable functions
Recommended article: Banach spaces of vector-valued functions
Recommended article: A weak vector-valued Banach-Stone theorem
Recommended article: Q4283126
Recommended article: Compactness in vector-valued Banach function spaces
Recommended article: Banach-Stone theorems for spaces of vector bundle continuous sections
Recommended article: The cancellation law for compact Hausdorff spaces and vector-valued Banach-Stone theorems
review text: This paper is an interesting contribution to the area of functional analysis that deals with the variations on the theme of the classical Banach-Stone theorem. For realcompact topological spaces \(X,Y\) and for locally convex spaces \(E,F\), and for subspaces \(A \subset C(X,E)\), \(B \subset C(Y,X)\), let \(\phi: A \rightarrow B\) be a linear bijection such that \(z(f) \subset z(g)\) if and only if \(z(\phi(f)) \subset z(\phi(g))\) (\(z(f)\) stands for the zero-set). Then a natural question to ask is whether \(\phi(f)(y) = J_y(f(\tau(y))\) for all \(y \in Y\), \(f \in A\), where \(\tau:Y \rightarrow X\) is a homeomorphism and \(J_y:E \rightarrow F\) is a linear bijection.NEWLINENEWLINEThe authors give a positive answer for the entire space of continuous functions and for certain special subspaces. When the domain spaces are metric and range spaces are normed linear spaces, spaces of uniformly continuous or Lipschitz continuous functions fall in this category. Under the assumption that \(\phi\) is continuous with respect to the topology of uniform convergence, one also gets that the family of fiber linear maps \(\{J_y\}\) is equicontinuous.
description: scientific article; zbMATH DE number 6064301