Push-outs of derivations

TL;DR

This paper presents an algebraic framework for extending derivations in Banach algebras, clarifying the analytic and algebraic aspects, and applies it to inheritance properties of biprojectivity and biflatness.

Contribution

It introduces an algebraic scheme for derivation extension that simplifies existing methods and clarifies the analytic versus algebraic components involved.

Findings

Provides a unified algebraic approach to derivation extension

Improves and simplifies proofs of known results

Establishes necessary and sufficient conditions for inheritance of biprojectivity and biflatness

Abstract

Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a given derivation D: A-> X to a Banach algebra B containing 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 in 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 are inherited to ideals.