Vol. 290, No. 2, 2017

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ISSN: 0030-8730
A vector-valued Banach–Stone theorem with distortion $\sqrt{2}$

Elói Medina Galego and André Luis Porto da Silva

Vol. 290 (2017), No. 2, 321–332
Abstract

Let K and S be locally compact Hausdorff spaces and H a real Hilbert space of finite dimension at least two. We prove that if T is an isomorphism from C0(K,H) onto C0(S,H) whose distortion TT1 is exactly 2, then K and S are homeomorphic. This is the vector-valued Banach–Stone theorem via isomorphisms with the largest distortion that is known. It improves a 1976 classical result due to Cambern.

Keywords
vector-valued Banach–Stone theorem, $C_{0}(K, X)$ spaces, finite-dimensional Hilbert space
Mathematical Subject Classification 2010
Primary: 46B03, 46E15
Secondary: 46B25, 46E40
Milestones
Received: 19 September 2016
Revised: 31 March 2017
Accepted: 4 April 2017
Published: 25 July 2017
Authors
Elói Medina Galego
Department of Mathematics, IME
University of São Paulo
05508-090 São Paulo
Brazil
André Luis Porto da Silva
Department of Mathematics, IME
University of São Paulo
05508-090 São Paulo
Brazil