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

## TL;DR

This paper explores the Aron-Berner extensions of triple maps in Banach spaces, focusing on conditions for the bidual of Jordan Banach triple systems to inherit a compatible triple product structure, with applications to JB*-triples.

## Contribution

It introduces the concept of Aron-Berner regularity for trilinear maps and investigates when the bidual of a Jordan Banach triple system retains its structure under these extensions.

## Key findings

- Conditions for biduals to be Jordan Banach triple systems
- Comparison of Aron-Berner triple products with ultrafilter-based products
- Relation of Aron-Berner extensions to Dineen's theorem

## Abstract

By extending the notion of Arens regularity of bilinear mappings, we say that a bounded trilinear map on Banach spaces id Aron--Berner regular when all its six Aron-Berner extensions to 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 $(E,\pi)$, 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 $\pi.$ 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.