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 $E^{**}$?

  • $\begingroup$ With regard to the title, where does the weak-* topology enter this question? $\endgroup$ – Nate Eldredge Feb 21 '16 at 14:40
  • $\begingroup$ 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 Weaver Feb 21 '16 at 14:48
  • $\begingroup$ Come to think of it, that may be a good way to find a counterexample to my question ... $\endgroup$ – Nik Weaver Feb 21 '16 at 14:48

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.

  • 1
    $\begingroup$ Oh great. That answers it nicely! $\endgroup$ – Nik Weaver Feb 21 '16 at 15:37

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