A Banach-lattice version of the Josefson-Nissenzweig theorem

Let E, F be two Banach lattices with E order continuous. If F can be mapped positively onto E then the dual F* contains a weak* -null sequence of positive and norm-one elements (Theorem 1). This is a Banach-lattice version of the classical Josefson-Nissenzweig theorem. It is an immediate consequence of the dual characterization of order continuity: E is order continuous iff E is Dedekind complete and every norm-one and pairwise disjoint sequence in E* is weak*-null (Theorem 2).