Jewell Theorem for Higher Derivations on C*-Algebras

Abstract

Let A be an algebra. A sequence {d n } of linear mappings on A is called a higher derivation if d n (ab) = P n j=0 d j (a)d n−j (b) for each a, b ∈ A and each nonnegative integer n. Jewell [Pacific J. Math. 68 (1977), 91-98], showed that a higher derivation from a Banach algebra onto a semisimple Banach algebra is continuous provided that ker(d 0) ⊆ ker(d m), for all m ≥ 1. In this paper, under a different approach using C *-algebraic tools, we prove that each higher derivation {d n } on a C *-algebra A is automatically continuous, provided that it is normal, i. e. d 0 is the identity mapping on A.