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Extreme points in spaces of operators and vector-valued measures. (English) Zbl 0579.47048
This note is concerned with characterizing the extreme points of the unit ball of certain spaces of operators. Extending a result due to P. D. Morris and R. R. Phelps a little bit [Trans. Am. Math. Soc. 150, 183-200 (1970; Zbl 0198.046)] we prove: If T is an extreme point of the unit ball of the space of compact operators from C(K,E) to C(L), where K and L are compact Hausdorff spaces and E is a Banach space, then \(T^*\) maps point measures onto point measures. (Here we used the fact that \(C(K,E)^*\) has a representation as a space of vector-valued measures.) As a corollary one obtains that such a T is necessarily a finite rank operator if \(\dim (E)<\infty\). Moreover, a very simple proof of Singer’s theorem that extreme functionals on C(K,E) are point measures is included.

47L07 Convex sets and cones of operators
47L05 Linear spaces of operators
46E40 Spaces of vector- and operator-valued functions
46E27 Spaces of measures
46A55 Convex sets in topological linear spaces; Choquet theory
Zbl 0198.046