On the properties of the Aron-Berner regularity of bounded tri-linear maps

TL;DR

This paper investigates the Aron-Berner regularity of bounded tri-linear maps, establishing conditions and relations with Arens regularity of bilinear maps, and provides criteria for regularity.

Contribution

It introduces new criteria and explores the relationship between Arens regularity of bilinear maps and Aron-Berner regularity of tri-linear maps.

Findings

Established a relation between Arens regularity and Aron-Berner regularity.

Provided a simple criterion for Aron-Berner regularity of tri-linear maps.

Analyzed properties of Aron-Berner regular maps in normed spaces.

Abstract

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and $f$ is Aron-Berener regular when all natural extensions are equal. In this manuscript, we have some results on the Aron-Berner regular maps. We investigate the relation between Arens regularity of bounded bilinear maps and Aron-Berner regularity of bounded tri-linear maps. We also give a simple criterion for the Aron-Berner regularity of tri-linear maps.