The strongest Banach–Stone theorem for C0(K,ℓ22)C₀(K, ₂²) spaces

As usual denote by the real two‐dimensional Hilbert space. We prove that if and are locally compact Hausdorff spaces and is a linear isomorphism from onto satisfying then and 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 , denoted by , is as large as . 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 spaces, or in more precise words, the exact value of the Banach–Stone constant of is between and .