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

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$\sqrt {2.054209}$.