A characterization of locally compact spaces with homeomorphic one-point compactifications

Let X and Y be locally compact noncompact spaces. For a large class of closed linear subspaces, A and B, of C0(X,R) and C0(Y,R), respectively, we show that there exist basically two types of diameter-preserving linear bijections Ψ :A→B: those (type 1) which can be written as Ψ(f)(y)=τ(f∘ϕ)(y), where τ∈{−1,1} and ϕ :Y→X is a surjective homeomorphism, and those (type 2) which can be represented as Ψ(f)(y)=τ(f∘ϕ)(y)−τf(x0),y≠y0,−τf(x0),y=y0 for some x0∈X and y0∈Y, where ϕ :Y⧹{y0}→X⧹{x0} is a homeomorphism which can be extended to the one-point compactifications of Y and X. All these facts are proved as a consequence of a full description of the extreme points of the closed unit ball of the dual of C0(X) and such subspaces endowed all with the diameter norm. Finally, we characterize the locally compact spaces with homeomorphic one-point compactifications as those which admit diameter-preserving linear bijections like the ones described above.