This is an announcement for the paper "Right inverses of surjections from cones onto Banach spaces" by Miek Messerschmidt and Marcel de Jeu.
Abstract: Abstract. We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; an improved version of the usual Open Mapping Theorem is then a special case. As another consequence, a stronger version of the analogue of And^o's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in And^o's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.
Archive classification: math.FA
Mathematics Subject Classification: Primary 47A05, Secondary 46A30, 46B20, 46B40
Submitted from: mmesserschmidt@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1302.2822
or