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

Abstract

Let $f:X\times Y\times Z⟶W$ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t\ast \ast \ast \ast s}=f^{s\ast \ast \ast \ast t}$ 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.

8 pages