Derivations with a hereditary domain, II

The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies $B[B ∩ S(D)]B ⊂ Rad_B(A)$ provided that BAB ⊂ A and $dim(Rad_J(A) ∩ ⋂_{n=1}^∞ B^n) < ∞$, where $Rad_J(A)$ and $Rad_B(A)$ denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras