Extreme operator-valued continuous maps

Abstract

Let ℒ(H) denote the space of operators on a Hilbert space H. We show that the extreme points of the unit ball of the space of continuous functions C(K, ℒ(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dim H<∞ and card K<∞, (b) exposed if and only if H is separable and K carries a strictly positive measure.