Closedness of the Set of Extreme Points in Calderón-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

into the space of continuous functions

C(Z,\Bbb R) C ( Z , R )

is extreme provided it is nice, i.e.

T^{*}(Z)\subset {\operatorname{Ext}}B(X^{*}) T ∗ ( Z ) ⊂ Ext ⁡ B ( X ∗ )

, where

Z Z

is a compact Hausdorff space and

T^{*}: Z\to X^{*} T ∗ : Z → X ∗

is a continuous function defined by

T^{*}(z)(x)=T(x)(z) T ∗ ( z ) ( x ) = T ( x ) ( z )

. The nice operator condition can be weakened as long as the set of extreme points

\operatorname{Ext}B(X^{*}) Ext ⁡ B ( X ∗ )

is closed, namely it suffices to assume than

T^{*}(Z_0)\subset {\operatorname{Ext}}B(X^{*}) T ∗ ( Z 0 ) ⊂ Ext ⁡ B ( X ∗ )

for some dense subset

Z_0\subset Z Z 0 ⊂ 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} E φ

generated by the Köthe space

E E

and the Orlicz function

\varphi φ

. The main theorem of the paper (Theorem 2.12) gives conditions under which the closedness of the set

\operatorname{Ext}B(E_{\varphi}) Ext ⁡ B ( E φ )

is equivalent to the closedness of the set of extreme points of the unit ball of the corresponding Köthe space

E E

.