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The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

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  • Elói Medina Galego

Abstract

As usual denote by ℓ22$\ell _2^2$ the real two‐dimensional Hilbert space. We prove that if K$K$ and S$S$ are locally compact Hausdorff spaces and T$T$ is a linear isomorphism from C0(K,ℓ22)$C_{0}(K,\ell _2^2)$ onto C0(S,ℓ22)$C_{0}(S,\ell _2^2)$ satisfying ∥T∥∥T−1∥≤2.054208,$$\begin{equation*} \hspace*{115pt}{\Vert T\Vert} \ {\Vert T^{-1}\Vert} \le \sqrt {2.054208}, \end{equation*}$$then K$K$ and S$S$ are homeomorphic. This theorem is the strongest of all the other vector‐valued Banach–Stone theorems known so far in the sense that in none of them the distortion of the isomorphism T$T$, denoted by ∥T∥∥T−1∥${\Vert T\Vert} \ {\Vert T^{-1}\Vert}$, is as large as 2.054208$\sqrt {2.054208}$. Some remarks on the proof method developed here to prove our theorem suggest the conjecture that it is in fact very close to the optimal Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces, or in more precise words, the exact value of the Banach–Stone constant of ℓ22$\ell _2^2$ is between 2.054208$\sqrt {2.054208}$ and 2.054209$\sqrt {2.054209}$.

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  • Elói Medina Galego, 2024. "The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces," Mathematische Nachrichten, Wiley Blackwell, vol. 297(5), pages 1945-1959, May.
  • Handle: RePEc:bla:mathna:v:297:y:2024:i:5:p:1945-1959
    DOI: 10.1002/mana.202300321
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    References listed on IDEAS

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    1. Elói Medina Galego & André Luis Porto da Silva, 2019. "Isomorphisms of C0(K,X) spaces with large distortion," Mathematische Nachrichten, Wiley Blackwell, vol. 292(5), pages 996-1007, May.
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