Extreme nuclear-valued continuous maps

Abstract

For a Banach space X and a compact Hausdorff space I, we let C (I, X) denote the Banach space of continuous functions on I with values in X equipped with the supremum norm $ | f | = \sup \{| f(t) |: t\in I\} $ . The unit ball of X is denoted by B 1 (X). If f is an extreme element in B 1 (C (I, X) ), must f (t) be extreme in B 1 (X) for all $ t \in I $ ? That this is not the case in general, has been settled by Blumenthal and Lindenstrauss. However, the question has been answered positively for many Banach spaces X. Let N (H) be the space of nuclear operators on a Hilbert space H and I be a compact Hausdorff space. It was shown that the answer to the above question is positive when H is finite dimensional. The authors show, for any Hilbert space H, that the extreme points of the unit ball of C (I, N (H) ) are precisely the functions with extremal values. They also establish a general theorem on extreme points in C (I, X),X a Banach space, that gives many known results in this direction.