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

It is known (see [1]) 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 of Ext B(X*), where Z is a compact Hausdorff space and T* : Z -> 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 of Ext B(X*) for some dense subset Z(0) subset of 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-phi generated by the Kothe space E and the Orlicz function phi. The main theorem of the paper (Theorem 2.14) gives conditions under which the closedness of the set Ext B(E-phi) is equivalent to the closedness of the set of extreme points of the unit ball of the corresponding Kothe space E.