Relationships between monotonicity and complex rotundity properties with some consequences - Hudzik, Henryk; Narloch, Agata (Mathematica Scandinavica, 2005)
It is proved that a point f of the complexification E^C 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^C is complex rotund. It is also shown that E is uniformly monotone if and only if E^C is uniformly complex rotund. Next, the fact that |x|S(E^+) is a ULUM-point of E whenever x is a C-LUR-point of S(E^C) is proved, whence the relation that E is a ULUM-space whenever E^C is C-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.