The Banach-Stone property and the weak Banach-Stone property in three-dimensional spaces

Abstract:

Let X and Y be compact Hausdorff spaces, E a Banach space, and $C(X,E)$ the space of continuous functions on X to E. E has the weak Banach-Stone property if, whenever $C(X,E)$ and $C(Y,E)$ are isometric, then X and Y are homeomorphic. E has the Banach-Stone property if the descriptive as well as the topological conclusions of the Banach-Stone theorem for scalar functions remain valid in the case of isometries of $C(X,E)$ onto $C(Y,E)$. These two properties were first studied by M. Jerison, and it we later shown that every space E found by Jerison to have the weak Banach-Stone property actually has the Banach-Stone property, thus raising the question of whether the two properties are distinct. Here we characterize all three-dimensional spaces with the weak Banach-Stone property, and, in so doing, show the properties to be distinct.