Arens Regularity of Module Actions and the Second Adjoit of a Derivation

TL;DR

This paper provides a criterion for Arens regularity of bilinear mappings, especially in Banach modules, and explores conditions under which the second adjoint of a derivation remains a derivation, offering simplified proofs of existing results.

Contribution

It introduces a straightforward criterion for Arens regularity applicable to Banach module actions and characterizes when the second adjoint of a derivation is also a derivation.

Findings

Established a simple criterion for Arens regularity of bilinear mappings.

Identified conditions for the second adjoint of a derivation to be a derivation.

Provided simplified proofs for several classical results.

Abstract

In this paper, first we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions and then we investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.