On the Invariance of Primitive Ideals via phi-derivations on Banach Algebras

The noncommutative Singer-Wermer conjecture states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of (sic)-derivations to be nilpotent. Moreover, we show that the spectral boundedness of (sic) derivations implies that they leave each primitive ideal of Banach algebras invariant.