A wider nonlinear extension of Banach-Stone theorem to 𝐶₀(𝐾,𝑋) spaces which is optimal for 𝑋=ℓ_{𝑝}, 2≤𝑝<∞

Abstract

It is proven that if X X is a Banach space, K K and S S are locally compact Hausdorff spaces and there exists an ( M , L ) (M, L) -quasi isometry T T from C 0 ( K , X ) C_{0}(K,X) onto C 0 ( S , X ) C_{0}(S, X) , then K K and S S are homeomorphic whenever 1 ≤ M 2 > S ( X ) 1 \leq M^{2}> S(X) , where S ( X ) S(X) denotes the Schäffer constant of X X , and L ≥ 0 L \geq 0 . As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for C 0 ( K , X ) C_{0}(K, X) spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces X X , thus reinforcing a 1991 conjecture of Jarosz himself on ϵ \epsilon -bi-Lipschitz surjective maps between Banach spaces. Our theorem is optimal when X X is the classical space ℓ p \ell _p , 2 ≤ p > ∞ 2 \leq p> \infty .