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

As usual denote by $$\ell _2^2$$ the real two-dimensional Hilbert space. We prove that if $$K$$ and $$S$$ are locally compact Hausdorff spaces and $$T$$ is a linear isomorphism from $$C_{0}(K,\ell _2^2)$$ onto $$C_{0}(S,\ell _2^2)$$ satisfying

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 $$T$$, denoted by $${\Vert T\Vert} \ {\Vert T^{-1}\Vert}$$, is as large as $$\sqrt {2.054208}$$.

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 $$C_{0}(K, \ell _2^2)$$ spaces, or in more precise words, the exact value of the Banach–Stone constant of $$\ell _2^2$$ is between $$\sqrt {2.054208}$$ and $$\sqrt {2.054209}$$.