Push-outs of derivations

Abstract

Let $\mathfrak{A}$ be a Banach algebra and let X be a Banach $\mathfrak{A}$-bimodule. In studying $\mathcal{H}^1$($\mathfrak{A}$,X) it is often useful to extend a given derivation D: $\mathfrak{A}$ → X to a Banach algebra $\mathfrak{B}$ containing $\mathfrak{A}$ as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators b ↦ D(ba) − b.D(a), a ε $\mathfrak{A}$ in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity and biflatness is inherited to ideals.