Suppose $x$ is an extreme point of the unit ball of a Banach space $E$. Embed $E$ in $E^{**}$ in the standard way. Is $x$ an extreme point of the unit ball of $E^{**}$?
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asked Feb 21, 2016 at 14:37
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$\begingroup$ With regard to the title, where does the weak-* topology enter this question? $\endgroup$– Nate EldredgeCommented Feb 21, 2016 at 14:40
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$\begingroup$ The general definition I have in mind is: an extreme point of $K \subset E$ is weak* extreme if it is still an extreme point of the weak* closure of $K$ in $E^{**}$. I guess it's not hard to come up with examples of extreme points that are and that are not weak* extreme. $\endgroup$– Nik WeaverCommented Feb 21, 2016 at 14:48
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$\begingroup$ Come to think of it, that may be a good way to find a counterexample to my question ... $\endgroup$– Nik WeaverCommented Feb 21, 2016 at 14:48
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1 Answer
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The answer is no and $E=\mathscr{K}(\ell_p)$ for $p\in (1,\infty)\setminus \{2\}$ is already a counterexample. See the proof of Proposition 2.3 in
J. Hennefeld, Compact extremal operators, Illinois J. Math. 21 (1997) 61-65.
Here we identifty $\mathscr{K}(\ell_p)^{**}$ with $\mathscr{B}(\ell_p)$.
This question is discussed in:
S. Dutta, T. S. S. R. K. Rao, On Weak-${}^\ast$ Extreme Points in Banach Spaces, Journal of Convex Analysis, 10 (2003), No. 2, 531-539.
where further examples are given.
answered Feb 21, 2016 at 14:53
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1$\begingroup$ Oh great. That answers it nicely! $\endgroup$ Commented Feb 21, 2016 at 15:37