Some results on $\sigma$-derivations

Abstract

‎Let $\mathcal{A}$ and $\mathcal{B}$ be two Banach algebras and let‎ ‎$\mathcal{M}$ be a Banach $\mathcal{B}$-bimodule‎. ‎Suppose that‎ ‎$\sigma:\mathcal{A} \rightarrow \mathcal{B}$ is a linear mapping and‎ ‎$d:\mathcal{A} \rightarrow \mathcal{M}$ is a $\sigma$-derivation‎. ‎We‎ ‎prove several results about automatic continuity of‎ ‎$\sigma$-derivations on Banach algebras‎. ‎In addition‎, ‎we define a‎ ‎notion for m-weakly continuous linear mapping and show that‎, ‎under‎ ‎certain conditions‎, ‎$d$ and $\sigma$ are m-weakly continuous‎. ‎Moreover‎, ‎we prove that if $\mathcal{A}$ is commutative and $\sigma‎: ‎\mathcal{A} \rightarrow \mathcal{A}$ is a continuous homomorphism‎ ‎such that $\sigma^{2} = \sigma$ then $\sigma d \sigma (\mathcal{A})‎ ‎\subseteq \sigma(Q(\mathcal{A})) \subseteq rad(\mathcal{A})$‎.