# An extreme point of the ball of the space of compact operators

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 $p\in (1,\infty)$?

• 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. – Bill Johnson Nov 20 '13 at 19:04
• There is also a more general fact that the unit ball of any non-unital C*-algebra does not have any extreme points. – Mateusz Wasilewski Nov 20 '13 at 19:11

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