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

Abstract

Let U be a $\varphi$-Johnson amenable Banach algebra in which $\varphi$ is a non-zero multiplicative linear functional on U. Suppose that X is a Banach U-bimodule such that $a.x=\varphi (a)x$ for all a in U and x in X or $x.a=\varphi (a)x$ 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.