Theorems of Krein-Milman type for certain convex sets of operators.

Abstract

Let M M be a real (or complex) Banach space and C ( Y ) C(Y) the space of continuous real (or complex) functions on the compact Hausdorff space Y Y . The unit ball of the space of bounded operators from M M into C ( Y ) C(Y) is shown to be the weak operator (or equivalently, strong operator) closed convex hull of its extreme points, provided Y Y is totally disconnected, or provided M ∗ {M^ \ast } is strictly convex. These assertions are corollaries to more general theorems, most of which have valid converses. In the case M = C ( X ) M = C(X) , similar results are obtained for the positive normalized operators. Analogous results are obtained for the unit ball of the space of compact operators (this time in the operator norm topology) from M M into C ( Y ) C(Y) .