Jordan And Lie Derivations of ϕ-Johnson Amenable Banach Algebras

Let [Formula: see text] be a [Formula: see text]-Johnson amenable Banach algebra in which [Formula: see text] is a nonzero multiplicative linear functional on [Formula: see text]. Suppose that [Formula: see text] is a Banach [Formula: see text]-bimodule such that [Formula: see text] for all [Formula: see text], [Formula: see text] or [Formula: see text] for all [Formula: see text], [Formula: see text]. We show that every continuous Jordan derivation from [Formula: see text] to [Formula: see text] is a derivation, and every continuous Lie derivation from [Formula: see text] to [Formula: see text] 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 [Formula: see text]-Johnson amenability, especially we give some conditions equivalent to [Formula: see text]-Johnson amenability.