On nilpotency of the separating ideal of a derivation

Abstract

Abstract. We prove that the separating ideal S(D) of any derivation D on a commutative unital algebra B is nilpotent if and only if S(D) n (f) R") is a nil ideal, where R is the Jacobson radical of B . Also we show that any derivation D on a commutative unital semiprime Banach algebra B is continuous if and only if f)(S(D))" = {0} . Further we show that the set of all nilpotent elements of S(D) is equal to (~){S(D)nP), where the intersection runs over all nonclosed prime ideals of B not containing S{D). As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.

On continuity of derivations and epimorphisms on some vector-valued group algebras

“…For relevant information on separating ideals, their relation to the prime ideals of the Banach algebra, and for related results on automatic continuity theory we refer to [1,2,3,4,7,9,12].…”