
Journal of Convex Analysis 21 (2014), No. 2, 401413 Copyright Heldermann Verlag 2014 Closedness of the Set of Extreme Points in CalderonLozanovskii Spaces Ewa Kasior Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70451 Szczecin 3, Poland ekasior@univ.szczecin.pl Marek Wisla Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61614 Poznan, Poland marek.wisla@amu.edu.pl [Abstractpdf] \def\Ext{\operatorname{Ext}} \def\R{\Bbb R} It is known [see R. M. Blumenthal, J. Lindenstrauss, R. R. Phelps, {\it Extreme operators into C(K)}, Pacific Journal of Mathematics 15(3) (1965), 747756] 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 CalderonLozanovskii spaces $E_{\varphi}$ generated by the K\"{o}the 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 K\"{o}the space $E$. Keywords: CalderonLozanovskii spaces, extreme points, compact operators, Orlicz spaces, Koethe spaces. MSC: 46B20, 46E30 [ Fulltextpdf (158 KB)] for subscribers only. 