SOME NOTES ON (σ, τ)-AMENABILITY OF BANACH ALGEBRAS

Abstract. Let A be a Banach algebra and σ, τ be continuous homomor-phisms on A. Suppose that X be a Banach A-bimodule. A linear mapping d: A − → X is a (σ, τ)-derivation if d(ab) = d(a)σ(b) + τ(a)d(b) (a, b ∈ A), and is a (σ, τ)-inner derivation if there exists x ∈ X such that d(a) = xσ(a) − τ(a)x (a ∈ A). The Banach algebra A is called (σ, τ)-amenable if every (σ, τ)-derivation is (σ, τ)-inner. In this paper, we investigate the relation between amenabil-ity and (σ, τ)-amenability of Banach algebras and also hereditary prop-erties of (σ, τ)-amenability. We give the notion σ-virtual diagonal and σ-approximate diagonal and apply them in study of σ-amenability. 1. Introduction an