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arXiv · 1901.01822

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

Amin A. Khosravi·Hamid Reza Ebrahimi Vishki·Antonio M. Peralta
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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 $(E,π)$, 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.

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Amin A. Khosravi, Hamid Reza Ebrahimi Vishki, Antonio M. Peralta. 2019-04-09. Aron-Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems. https://arxiv.org/abs/1901.01822

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