Generalized sigma-derivation on Banach algebras

Abstract

Let $mathcal{A}$ be a Banach algebra and $mathcal{M}$ be a <br />Banach $mathcal{A}$-bimodule. We say that a linear mapping <br />$delta:mathcal{A} rightarrow mathcal{M}$ is a generalized <br />$sigma$-derivation whenever there exists a $sigma$-derivation <br />$d:mathcal{A} rightarrow mathcal{M}$ such that $delta(ab) = <br />delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{A}$. <br />Giving some facts concerning generalized $sigma$-derivations, we <br />prove that if $mathcal{A}$ is unital and if $delta:mathcal{A} <br />rightarrow mathcal{A}$ is a generalized $sigma$-derivation and <br />there exists an element $a in mathcal{A}$ such that emph{d(a)} is <br />invertible, then $delta$ is continuous if and only if emph{d} is <br />continuous. We also show that if $mathcal{M}$ is unital, has no <br />zero divisor and $delta:mathcal{A} rightarrow mathcal{M}$ is a <br />generalized $sigma$-derivation such that $d(textbf{1}) neq 0$, <br />then $ker(delta)$ is a bi-ideal of $mathcal{A}$ and $ker(delta) = <br />ker(sigma) = ker(d)$, where textbf{1} denotes the unit element of <br />$mathcal{A}$.