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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.

##### MSC:
 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