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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: P. D. Morris and R. R. Phelps
Journal: Trans. Amer. Math. Soc. 150 (1970), 183-200
MSC: Primary 46.10
DOI: https://doi.org/10.1090/S0002-9947-1970-0262804-3
MathSciNet review: 0262804
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Abstract: Let $ M$ be a real (or complex) Banach space and $ C(Y)$ the space of continuous real (or complex) functions on the compact Hausdorff space $ Y$. The unit ball of the space of bounded operators from $ M$ into $ C(Y)$ is shown to be the weak operator (or equivalently, strong operator) closed convex hull of its extreme points, provided $ Y$ is totally disconnected, or provided $ {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)$, 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$ into $ C(Y)$.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0262804-3
Keywords: Bounded linear operators, convex sets of operators, extreme operators, Banach space-valued functions, linear operators into $ C(X)$, compact linear operators
Article copyright: © Copyright 1970 American Mathematical Society