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

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.