Isomorphic Banach-Stone theorems and isomorphisms which are close to isometries (Q1822299)

A Banach space X is said to have the isomorphic Banach-Stone property if for locally compact Hausdorff spaces K and L one always can conclude that K and L are homeomorphic provided that the Banach spaces \(C_ 0(K,X)\) and \(C_ 0(L,X)\) \((=\) the continuous X-valued functions on K resp. L which vanish at infinity) are isomorphic with sufficiently small Banach- Mazur distance. Our main results are a characterization of the finite-dimensional spaces with this property, and we also get an abundance of new finite- and infinite-dimensional examples. These results appear as corollaries to general theorems about isomorphisms between certain spaces of continuous vector-valued functions. They enable us also to conclude that, for certain spaces X, and all compact K all isomorphisms T on \(C_ 0(K,X)\) with \((1/(1+\tau))\| f\| \leq \| Tf\| \leq (1+\tau)\| f\|\) for small \(\tau\) can be approximated by isometries.