A Unified Approach to Various Notions of Derivation

Abstract

The main goal of this article is introducing a fairly general framework to treat several types of derivations simultaneously. Let ${{\mathcal{A}}}$ and ${\mathcal{B}}$ be Banach algebras, $\alpha$ and $\beta$ be homomorphisms from ${{\mathcal{A}}}$ onto ${\mathcal{B}}$, and ${\mathcal{X}}$ be a Banach ${\mathcal{B}}$-bimodule. A map $D\in {{\mathcal {B}}}({{\mathcal {A}}},{{\mathcal {X}}})$ is called an $(\alpha , \beta )$-derivation if $D(ab) =\alpha (a)D(b)+D(a)\beta (b)$. All homomorphisms, ordinary derivations, skew derivations, and point derivations are certain types of $(\alpha , \beta )$-derivations. We define $(\alpha , \beta )$-analog of notions of amenability and weak amenability. Amenability modulo a closed ideal in the sense of Rahimi (Acta Math Hungarica 144:407–415, 2014) and generalized weak amenability in the sense of Bodaghi et al. (Banach J Math Anal 3:131–142, 2009) are special cases of our concepts. We provide several characterizations of $(\alpha ,\beta )$-weak amenability under some mild conditions. Moreover $({\alpha }^{**},{\beta }^{**})$-weak amenability of ${{\mathcal{A}}}^{**}$ implies $(\alpha ,\beta )$-weak amenability of ${{\mathcal{A}}}$.