Close-to-regularity and completely 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 we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps. We investigate the relation between Arens regularity of bounded bilinear maps and close-to-regularity bounded tri-linear maps. We give a simple criterion for the completely regularity of tri-linear maps. We provide a necessary and sufficient condition such that the fourth adjoint $D^{\ast \ast \ast \ast }$ of a tri-derivation is again a tri-derivation.