Jordan and Lie derivations of φ-Johnson amenable Banach algebras

Let U be a phi-Johnson amenable Banach algebra in which phi is a nonzero multiplicative linear functional on U. Suppose that X is a Banach U-bimodule such that a.x = phi(a)x for all a is an element of U, x is an element of X or x.a = phi(a)x for all a is an element of U, x is an element of 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. Then we apply our results for character amenable Banach algebras and amenable Banach algebras. We also provide some results about phi-Johnson amenability, especially we give some conditions equivalent to phi-Johnson amenability.