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

Abstract

Let f : X × Y × Z −→ 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.