Generalized derivations on certain Banach algebras

TL;DR

This paper investigates the behavior of derivations and generalized derivations on specific Banach algebras, especially those related to group algebras, revealing conditions under which these mappings target the radical of the algebra.

Contribution

It characterizes when derivations and generalized derivations map into the radical for certain Banach algebras, including applications to Fourier algebras of discrete amenable groups.

Findings

Derivations map into the radical under specified conditions.

Necessary and sufficient conditions for the square of a generalized derivation to remain generalized.

Applications to the structure of Fourier algebras of discrete amenable groups.

Abstract

Let ${\mathcal A}$ be a Banach algebra with the properties that $\mathrm{rad}({\mathcal A})={\rm rann}({\mathcal A})$ and the algebra ${\mathcal A}/\mathrm{rad}({\mathcal A})$ is commutative. We show that a derivation of ${\mathcal A}$ maps ${\mathcal A}$ into ${\rm rad}({\mathcal A})$. Using this, we determine among other things when a generalized derivation of ${\mathcal A}$ maps ${\mathcal A}$ into ${\rm rad}({\mathcal A})$. We also study $k$-centralizing generalized derivations of ${\mathcal A}$. Then, for a generalized derivation $(\delta, d)$ of ${\mathcal A}$ we obtain a necessary and sufficient condition for $(\delta^2, d^2)$ to be still a generalized derivation of ${\mathcal A}$. The main applications are concerned with the algebras over locally compact groups. In particular, we deduce these results for bidual of Fourier algebras of discrete amenable groups as an application of our approach.