Innerness of higher derivations

Abstract

Let $A$ be an algebra. A sequence $\{d_n\}$ of linear mappings on $A$ is called a higher derivation if $d_n(ab)=\sum_{k=0}^n d_k(a)d_{n-k}(b)$ for each $a,b\in A$ and each nonnegative integer $n$. In this paper a notion of an inner higher derivation is given. We characterize all uniformly bounded inner higher derivations on Banach algebras and show that each uniformly bounded higher derivation on a Banach algebra $A$ is inner provided that each derivation on $A$ is inner.