Extreme operator-valued continuous maps (Q1174918)

Suppose \(K\) denotes a compact Hausdorff space, \(E\) a Banach space and \(f\) an extreme point of the unit ball of \(C(K,E)\). A question considered in many papers is whether \(f(k)\) must be an extreme point of the unit ball of \(E\) for all \(k\in K\). In the present note the author settles this question in the affirmative for \(E=L(\ell^ 2)\) and in the negative for \(E=L(\ell^ p)\), \(1<p<\infty\), \(p\neq 2\). Also, the exposed points of the unit ball of \(C(K,L(\ell^ 2))\) are discussed.