A stronger form of Banach-Stone theorem to 𝐶₀(𝐾,𝑋) spaces including the cases 𝑋=𝑙_{𝑝}², 1<𝑝<∞

Abstract

It is proven that if X X is a real strictly convex 2-dimensional space, then there exists δ > 0 \delta >0 such that if K K and S S are locally compact Hausdorff spaces and T T is an isomorphism from C 0 ( K , X ) C_{0}(K, X) onto C 0 ( S , X ) C_{0}(S, X) satisfying ‖ T ‖ ‖ T − 1 ‖ ≤ λ ( X ) + δ , \begin{equation*} \|T\| \ \|T^{-1}\| \leq \lambda (X)+ \delta , \end{equation*} then K K and S S are homeomorphic. Here λ ( X ) \lambda (X) is the Schäffer constant of X X given by λ ( X ) = inf { max { ‖ x − y ‖ , ‖ x + y ‖ } : ‖ x ‖ = 1 and ‖ y ‖ = 1 } . \begin{equation*} \lambda (X)=\inf \{\max \{\|x-y\|,\|x+y\|\}\,:\,\|x\|= 1 \text { and } \|y\|= 1\}. \end{equation*} Even for the classical cases X = ℓ p 2 X=\ell _p^2 , 1 > p > ∞ 1>p> \infty , this result is the form of Banach-Stone theorem to C 0 ( K , X ) C_{0}(K, X) spaces with the largest known distortion ‖ T ‖ ‖ T − 1 ‖ \|T\| \ \|T^{-1}\| . In particular, it shows that the Banach-Stone constant of ℓ p 2 \ell _p^2 is strictly greater than 2 1 − 1 / p 2^{1-1/p} if 1 > p ≤ 2 1>p \leq 2 and strictly greater than 2 1 / p 2^{1/p} if 2 ≤ p > ∞ 2 \leq p >\infty . Until then this theorem had only been proved for p = 2 p=2 .