Automatic Continuity of Higher Derivations

Let A and B be two algebras. A sequence {dn} of linear mappings from A into B is called a higher derivation if dn(a1a2)=∑nk=0^dk(a1)dn−k(a2) for each a1,a2∈A and each nonnegative integer n. In this paper, we show that if {dn} is a higher derivation from A into B such that d0 is onto and ker (d0)⊆ker(dn) (n∈N), then there is a sequence {δn} of derivations on B such that dn=∑i=1n(⎜∑∑ij=1rj=n(∏j=1i1rj+...+ri)δr1...δrid0). As a corollary we prove that a higher derivation {dn} from a Banach algebra into a semisimple Banach algebra is continuous provided that d0 is onto and ker(d0)⊆ker(dn) (n∈N). We also deduce that if A is a semisimple Jordan Banach algebra and {dn} is a higher derivation on A with d0(A)=A and ker(d0)⊆ker(dn) (n∈N) then {dn} is continuous.