STRUCTURE TOPOLOGY AND EXTREME OPERATORS

We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$-space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$. We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$. The proof of our results relies on the structure topology.