Close-to-regularity and completely regularity of bounded tri-linear maps

TL;DR

This paper studies the properties of bounded tri-linear maps in normed spaces, introducing concepts of close-to-regularity and complete regularity, and explores their relations to Arens regularity and tri-derivations.

Contribution

It introduces the notions of close-to-regular and completely regular tri-linear maps, providing criteria and conditions for these properties and their relation to Arens regularity.

Findings

Established a criterion for complete regularity of tri-linear maps.

Connected Arens regularity of bilinear maps with tri-linear map regularity.

Provided conditions under which the fourth adjoint of a tri-derivation remains a tri-derivation.

Abstract

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 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 $D^{****}$ of a tri-derivation is again a tri-derivation.