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

DOI: 10.2140/pjm.2017.290.321

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 $C_0\left(K,H\right)$ onto $C_0\left(S,H\right)$ whose distortion $\parallel T\parallel \parallel T^{-1}\parallel$ is exactly $\sqrt{2}$, 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