Regularity of bounded tri-linear maps and the fourth adjiont of a tri-derivation

Abstract

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z⟶W$ is a bounded tri-linear mapping and $h:W⟶S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint $D^{\ast \ast \ast \ast }$ of a tri-derivation $D$ is again tri-derivation.