Left derivable or Jordan left derivable mappings on Banach algebras

TL;DR

This paper investigates the conditions under which linear mappings on Banach algebras are Jordan left derivations or left derivable, establishing equivalences and exploring their properties in the context of Banach modules.

Contribution

It establishes the equivalence between Jordan left derivations and left derivability at a point in Banach algebras, and explores generalized derivations in this setting.

Findings

Equivalence of Jordan left derivation and left derivability at a point.

Characterization of generalized (Jordan) left derivations.

Analysis of derivations in Banach algebras with property (B).

Abstract

Let d be a linear mapping from a unital Banach algebra A into a unital left A-module M, and w in Z(A) be a left separating point of M. We show that the following three conditions are equivalent: (i) d is a Jordan left derivation; (ii) d is left derivable at w; (iii) d is Jordan left derivable at w. Let A be a Banach algebra with the property (B), and M be a Banach left A-module. We consider the relations between generalized (Jordan) left derivations and (Jordan) left derivable mappings at zero from A into M.