Generalized Derivations on Modules

Abstract

Let $A$ be a Banach algebra and $M$ be a Banach right $A$-module. A linear map $\delta :M\to M$ is called a generalized derivation if there exists a derivation $d:A\to A$ such that $$\delta (xa)=\delta (x)a+xd(a)(a\in A,x\in M).$$ In this paper, we associate a triangular Banach algebra $\mathcal{T}$ to Banach $A$-module $M$ and investigate the relation between generalized derivations on $M$ and derivations on $\mathcal{T}$. In particular, we prove that the so-called generalized first cohomology group of $M$ is isomorphic to the first cohomology group of $\mathcal{T}$.

11 pages, minor changes, to appear in Bull. Iran. Math. Soc

References in corpus (2)

• Deformations of Lie Algebras using $\sigma$-derivations
• Hyers-Ulam-Rassias Stability of Generalized Derivations