Small isomorphisms of C(X,E) spaces
type: Item
instance of: scholarly article
title: Small isomorphisms of C(X,E) spaces
zbMATH Open document ID: 0698.46033
author: Krzysztof Jarosz
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A linear map T between Banach spaces A and B is called an \(\epsilon\)- isometry if \((1-\epsilon)\| f\| \leq \| Tf\| \leq (1+\epsilon)\| f\|\) for all \(f\in A\). If \(\epsilon >0\) is small, is there an isometry S of A into (onto) B with \(\| T-S\|\) small? In this paper A and B are subspaces of various \(C_ 0(X,E):\) the continuous E-valued functions which vanish at \(\infty\) on X, E a Banach space, X a locally compact Hausdorff space. The circle of ideas represents several directions in which the classical Banach-Stone theorem has been generalized. That theorem says that for \(X_ 1\), \(X_ 2\) compact and E either \({\mathbb{R}}\) or \({\mathbb{C}}\), every linear isometry T of \(C(X_ 1,E)\) onto \(C(X_ 2,E)\) has the form \(Tf=u\cdot f\circ \tau\), where \(\tau\) is a homoeomorphism of \(X_ 2\) onto \(X_ 1\) and u is a unimodular function in \(C(X_ 2,E).\) We can relax the hypothesis on T to (1) its being a linear bijection satisfying \(\| T\| \| T^{- 1}\| <1+\epsilon\) [with \(\epsilon\) not too large, since \(\exists\) non- homeomorphic compact metric \(X_ 1\), \(X_ 2\) and surjective linear T: C(X\({}_ 1)\to C(X_ 2)\) with \(\| T\| \| T^{-1}\| =2]\) or to (2) its being only an ``into'' linear isometry. We can keep T a surjective linear isometry but (3) defined between subspaces (or subalgebras) A, B of \(C(X_ j,E)\). [Here the following example of the author's is instructive: for \(A=B=H^{\infty}=:H^{\infty}({\mathbb{D}})\) and any \(\delta >0\), \(\exists\) a surjective linear map T: \(H^{\infty}\to H^{\infty}\) with \(\| T\| \leq 1+\delta,\) \(\| T^{-1}\| \leq 1+\delta\) and \(\| S- T\| \geq 2-\delta\) for every linear isometry S: \(H^{\infty}\to H^{\infty}.]\) Finally, in the original Banach-Stone set-up, as well as in (1), (2) and (3), \(E={\mathbb{R}}\) or \({\mathbb{C}}\) can be replaced by other Banach spaces. Since the original formulation fails in the case of certain C(X) in the role of E, for positive results when E is a general Banach space hypot…
reviewed by: Robert Burckel
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 4144733
zbMATH Keywords: almost isometry
zbMATH Keywords: \(\epsilon\)-isometry
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: Banach-Mazur distance
zbMATH Keywords: completely injective tensor products
MaRDI profile type: Publication
full work available at URL: pjm.1989.138.295
OpenAlex ID: W2045088190
DOI: 10.2140/PJM.1989.138.295
description: scientific article; zbMATH DE number 4144733