On Φ-derivation in Banach algebras

Using Bresar and Semrl approach, we give a proof of the extended Jacobson density theorem for phi-derivations. Further, some applications on Banach algebras will be given. Precisely, for d being a continuous phi-derivation on a given Banach algebra B, we show that: d(B) subset of rad(B) double left right arrow [b,[a,d(a)]] is an element of rad (B) for all a, b is an element of B and d leaves invariant all maximal ideals of codimension one double left right arrow for every a is an element of B there exists a positive integer n such that (d(a))(n) is quasi-nilpotent double left right arrow [d, phi](B) subset of rad (B) and d(2) (a) is an element of rad (B) for all a is an element of B. Finally, we characterize all pairs d , delta of continuous phi-derivations such that d delta(a) is quasi-nilpotent for all a is an element of B and [d ,delta](B), [d,phi](B), [delta,phi](B) are subsets of rad (B).