Extreme points in spaces of operators and vector-valued measures. (English) Zbl 0579.47048

Werner, Dirk

Rend. Circ. Mat. Palermo, II. Ser., Suppl. 5, 135-143 (1984).

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

Keywords:

nice operator; spaces of vector-valued measures; spaces of; vector-valued functions; spaces of compact operators; extreme points of the unit ball of certain spaces of operators; representation as a space of vector- valued measures; Singer’s theorem; extreme functionals on C(K,E) are point measures

Citations:

Zbl 0198.046