Generalized \(\sigma\)-derivation on Banach algebras (Q1953951)

Let \(\mathcal A\) be a unital Banach algebra and \(\delta: \mathcal A\to \mathcal A\) be a generalized \(\sigma\)-derivation associated with a \(\sigma\)-derivation \(d:\mathcal A\to \mathcal A\). The authors show that, if there exists \(a\in \mathcal A\) such that \(d(a)\) is invertible, then \(\delta\) is continuous if and only if \(d\) is continuous. Furthermore, if \(\mathcal M\) is a unital Banach \(\mathcal A\)-bimodule and \(\delta: \mathcal A\to \mathcal M\) a generalized \(\sigma\)-derivation associated with a \(\sigma\)-derivation \(d:\mathcal A\to \mathcal M\) such that \(d(1)\neq 0\), then \(\mathrm{ker} (\delta)\) is a bi-ideal of \(\mathcal A\) and \(\mathrm{ker} (\delta)=\mathrm{ker} (\sigma)=\mathrm{ker} (d)\).