Extreme points in spaces of continuous functions

Bogachev, V. I.; Mena-Jurado, J. F.; Navarro Pascual, J. C.

doi:10.1090/S0002-9939-1995-1204371-6

Extreme points in spaces of continuous functions
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by V. I. Bogachev, J. F. Mena-Jurado and J. C. Navarro Pascual PDF

Proc. Amer. Math. Soc. 123 (1995), 1061-1067 Request permission

Abstract:

We study the $\lambda$-property for the space $\mathfrak {C}(T,X)$ of continuous and bounded functions from a topological space T into a strictly convex Banach space X. We prove that the $\lambda$-property for $\mathfrak {C}(T,X)$ is equivalent to an extension property for continuous functions of the pair (T, X). We show also that, when X has even dimension, the $\lambda$-property is equivalent to the fact that the unit ball of $\mathfrak {C}(T,X)$ is the convex hull 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 onto the unit sphere.

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