Isomorphisms of C 0 ( K , X ) spaces with large distortion

Let K and S be locally compact Hausdorff spaces and let X be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism T fromC 0 ( K , X )ontoC 0 ( S , X )satisfying∥ T ∥T − 1= λ ( X ) , then K and S are homeomorphic. Here λ ( X ) denotes the Schäffer constant of X . Even for the classical cases X = ℓ p n , 1 < p < ∞ and n ≥ 2 , this result is the X ‐valued Banach–Stone theorem via isomorphism with the largest distortion that is known so far, namely λ ( X ) = min { 2 1 / p , 2 1 − 1 / p } . On the other hand, it is well known that this result is not true for X = R , even though K and S are compact Hausdorff spaces.