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Disappearance of extreme points. (English) Zbl 0541.46013
An extreme point in the unit ball of a Banach space is called preserved if it is extreme in the unit ball of the second dual. It is asked by R. R. Phelps [Proc. Am. Math. Soc. 12, 291-296 (1961; Zbl 0099.316)] if there were any nonpreserved extreme points at all. It is shown by Katznelson that there is a nonpreserved extreme point in the disk algebra. In this note the author has shown the following theorem:
Let E be a separable Banach space containing an isomorphic copy of \(c_ 0\). Then E is isomorphic to a strictly convex space F such that no extreme point of F is preserved.
Reviewer: L.Koshi

MSC:
46A55 Convex sets in topological linear spaces; Choquet theory
46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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[1] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. · Zbl 0369.46039
[2] R. R. Phelps, Extreme points of polar convex sets, Proc. Amer. Math. Soc. 12 (1961), 291 – 296. · Zbl 0099.31603
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