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Relationships between monotonicity and complex rotundity properties with some consequences. (English) Zbl 1085.46011
Summary: It is proved that a point $f$ of the complexification $E^{\bbfC}$ of a real Köthe space $E$ is a complex extreme point if and only if $|f|$ is a point of upper monotonicity in $E$. As a corollary, it follows that $E$ is strictly monotone if and only if $E^{\bbfC}$ is complex rotund. It is also shown that $E$ is uniformly monotone if and only if $E^{\bbfC}$ is uniformly complex rotund. Next, the fact that $|x|\in S(E^+)$ is a ULUM-point of $E$ whenever $x$ is a ${\bbfC}$-LUR point of $S(E^{\bbfC})$ is proved, whence the relation that $E$ is a ULUM-space whenever $E^{\bbfC}$ is ${\bbfC}$-LUR is concluded. In the second part of this paper, these general results are applied to characterize complex rotundity of properties Calderón--Lozanovskiĭ spaces, generalized Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces.

46B20Geometry and structure of normed linear spaces
46E99Linear function spaces and their duals