ABSTRACT
In this chapter we return to the problem of characterizing linear isometries from one continuous function space to another. However, unlike in Chapter 2, where we considered continuous scalar-valued functions, we now allow our functions to have values in some Banach space. The goal is to show that an isometry from a space C0(Q, X) of continuous X-valued functions vanishing at infinity, to a space C0(K, Y ), where Q, K are locally compact Hausdorff spaces and X, Y are Banach spaces, must be given by some version of the canonical form given by Theorem 2.2.1 in Chapter 2. In view of the classical results discussed there, we might expect that the spaces Q and K would necessarily be homeomorphic. However, suppose that Q, K are nonhomeomorphic compact Hausdorff spaces and L is a compact Hausdorff space such that Q× L is homeomorphic to K × L. For example, take
Q = {(a, b) : 1/2 ≤ a2 + b2 ≤ 1; or 0 ≤ a ≤ 2, b = 0}, where (a, b) is a point in the Euclidean plane,
K = {(a, b) : 1/2 ≤ a2 + b2 ≤ 1; or 1 ≤ a ≤ 2, b = 0; or a = 0, 1 ≤ b ≤ 2}, and L is the unit interval. Then for any Banach space X, C(L×Q, X) is isometric to C(L×K, X) and so C(Q, C(L, X)) is isometric with C(K, C(L, X)). As a result of all this, we can expect that any extension of the Banach-Stone theorem to continuous functions with values in Banach spaces X, Y will require some special conditions on these spaces.
