Closedness of the Set of Extreme Points in Calderon-Lozanovskii Spaces

It is known [see R. M. Blumenthal, J. Lindenstrauss, R. R. Phelps, Extreme operators into C(K), Pacific Journal of Mathematics 15(3) (1965), 747-756] that a compact linear operator from a Banach space $X$ into the space of continuous functions $C(Z,\R)$ is extreme provided it is nice, i.e. $T^{*}(Z)\subset \Ext B(X^{*})$, where $Z$ is a compact Hausdorff space and $T^{*}: Z\to X^{*}$ is a continuous function defined by $T^{*}(z)(x)=T(x)(z)$. The nice operator condition can be weakened as long as the set of extreme points $\Ext B(X^{*})$ is closed, namely it suffices to assume than $T^{*}(Z_0)\subset \Ext B(X^{*})$ for some dense subset $Z_0\subset Z$ in that case. The aim of this paper is to characterize the closedness of the set of extreme points of the unit ball of Calderon-Lozanovskii spaces $E_{\varphi}$ generated by the Köthe space $E$ and the Orlicz function $\varphi$. The main theorem of the paper (Theorem 2.12) gives conditions under which the closedness of the set $\Ext B(E_{\varphi})$ is equivalent to the closedness of the set of extreme points of the unit ball of the corresponding Köthe space $E$.