EXTREME-POINTS IN SPACES OF CONTINUOUS-FUNCTIONS

We study the lambda-property for the space C(T, X) of continuous and b ounded functions from a topological space T into a strictly convex Ban ach space X. We prove that the lambda-property for C(T, X) is equivale nt to an extension property for continuous functions of the pair (T, X ). We show also that, when X has even dimension, the lambda-property i s equivalent to the fact that the unit ball of C(T, X) is the convex h ull of its extreme points and that this last property is true if X is infinite dimensional. As a result we get that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space can be expressed as the average of four retractions of the unit ball o nto the unit sphere.