Jordan and Lie derivations of $\phi $-Johnson amenable Banach algebras

TL;DR

This paper investigates the structure of derivations in $mbda$-Johnson amenable Banach algebras, showing that Jordan and Lie derivations have specific decompositions under certain conditions.

Contribution

It establishes that in $mbda$-Johnson amenable Banach algebras, all continuous Jordan derivations are derivations, and Lie derivations decompose into derivations plus center-valued traces.

Findings

Every continuous Jordan derivation is a derivation.

Every continuous Lie derivation decomposes into a derivation and a center-valued trace.

Results apply to Banach algebras with a non-zero multiplicative linear functional.

Abstract

Let U be a $\phi $-Johnson amenable Banach algebra in which $\phi$ is a non-zero multiplicative linear functional on U. Suppose that X is a Banach U-bimodule such that $a.x=\phi(a)x$ for all a in U and x in X or $x.a=\phi(a)x$ for all a in U and x in X. We show that every continuous Jordan derivation from U to X is a derivation, and every continuous Lie derivation from U to X decomposed into the sum of a continuous derivation and a continuous center-valued trace.