Generalized sigma-derivation on Banach algebras

Document Type : Research Paper



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