DERIVATIONS WITH A HEREDITARY DOMAIN, II

The nilpotency of the separating subspace of an everywhere defined der ivation on a Banach algebra is an intriguing question which remains st ill 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 quantu m systems. We show that the separating subspace S(D) of a Jordan deriv ation defined on a subalgebra B of a complex Banach algebra A satisfie s B[B boolean AND S(D)]B subset of Rad(B) (A) provided that BAB subset of A and dim(Rad(T)(A)boolean AND boolean AND(n=1)(infinity) B-n) < i nfinity, where Rad(T)(A) and Rad(B)(A) denote the Jacobson and the Bae r radicals of A respectively. From this we deduce the closability of p artially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes o f algebras.