\(\sigma\)-derivations in Banach algebras (Q2471505)

Let \(\mathcal D\) be a dense subalgebra of a Banach algebra \(\mathcal A\), and let \(\sigma, d: {\mathcal D} \to {\mathcal A}\) be linear. The authors call \(d\) a \(\sigma\)-derivation if \(d(ab) = \sigma(a) d(b) + d(a) \sigma(b)\) holds for all \(a,b \in \mathcal D\). Of course, every derivation is a \(\sigma\)-derivation (with \(\sigma\) as the identity map), but so is every homomorphism from \(\mathcal D\) into \(\mathcal A\): take \(\sigma = \frac{d}{2}\). The authors then go on and define \(\sigma\)-endomorphisms, one parameter groups of such, etc. They show that some results for derivations on Banach algebras, such as the Leibniz rule and the Kleinecke--Shirokov theorem, remain true for \(\sigma\)-derivations under suitable hypotheses.