Isomorphisms of spaces with large distortion

Abstract

Letand be locally compact Hausdorff spaces and let be a strictly convex Banach space of finite dimension at least 2. In this paper, we prove that if there exists an isomorphism from 0 ( , ) onto 0 ( , ) satisfyingthen and are homeomorphic. Here ( ) denotes the Schäffer constant of . Even for the classical cases = , 1 < < ∞ and ≥ 2, this result is the -valued Banach-Stone theorem via isomorphism with the largest distortion that is known so far, namely ( ) = min { 2 1∕ , 2 1−1∕ } . On the other hand, it is well known that this result is not true for = , even though and are compact Hausdorff spaces. K E Y W O R D S 0 ( , ) spaces, spaces, Schäffer constant, strictly convex spaces, vector-valued Banach-Stone theorems M S C ( 2 0 1 0 ) Primary: 46B03, 46E15; Secondary: 46B25, 46E40 996

The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

“…The above theorem establishes the Banach-Stone theorem for 𝐶 0 (𝐾, 𝑋) spaces, with dimension of 𝑋 greater than or equal to 2, which is obtained through of linear isomorphisms 𝐿 with the highest distortion ‖𝐿‖ ‖𝐿 −1 ‖ known so far, see [5,9,10]. Moreover, our method of proving Theorem 1.1 does not work if we change the digit 8 to 9 in the statement of Theorem 1.1, see Remark 9.1.…”