Derivations on ρ-banach Algebras

Abstract

Let A be a ρ-Banach algebra with radical R and let D be a derivation on A. We generalize Johnson's theorem, Sinclair's theorem and Thomas's theorem (Singer-Wermer's conjecture) to ρ-Banach algebras. If K = {x ∈ R: for every n ∈N, D ⁿ x ∈ R}, then (for commutative A) DA ⊆ R if and only if K is closed. We also prove that if A is a ρ -Banach algebra, then D preserves primitive ideals if and only if K is closed.