#### Vol. 290, No. 2, 2017

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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 ${C}_{0}\left(K,H\right)$ 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