Aron–Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems

Abstract

By extending the notion of Arens regularity of bilinear mappings, we say that a bounded trilinear map on the Cartesian product of Banach spaces is Aron–Berner regular when all its six Aron–Berner extensions to the Cartesian product of the bidual spaces coincide. We give some results on the Aron–Berner regularity of certain trilinear maps. We then focus on the bidual, E⁎⁎, of a Jordan Banach triple system (E,π), and investigate those conditions under which E⁎⁎ is itself a Jordan Banach triple system under each of the Aron–Berner extensions of the triple product π. We also compare these six triple products with those arising from certain ultrafilters based on the ultrapower formulation of the principle of local reflexivity. In particular, we examine the Aron–Berner triple products on the bidual of a JB⁎-triple in relation with the so-called Dineen's theorem. Some illuminating examples are included and some questions are also left undecided.