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It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a non-commutative version of $c_0$) but without success.

Does the unit ball of $K(\ell_p)$ have any extreme points for some $p\in (1,\infty)$?

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    $\begingroup$ For a Hilbert space there are none for basically the same reason as for $c_0$. Use e.g. the fact that a compact operator maps some orthonormal basis to an orthogonal sequence. $\endgroup$
    – Bill Johnson
    Nov 20, 2013 at 19:04
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    $\begingroup$ There is also a more general fact that the unit ball of any non-unital C*-algebra does not have any extreme points. $\endgroup$
    – Mateusz Wasilewski
    Nov 20, 2013 at 19:11

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For $p\neq 2$ there are plenty of extreme points in $K(\ell_p)$.

J. Hennefeld, Compact extremal operators, Il. J. Math. 21 (1977) 61-65.

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