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Innerness of higher derivations. (English) Zbl 1186.47036
Summary: Let $𝒜$ be an algebra. A sequence $\left\{{d}_{n}\right\}$ of linear mappings on $𝒜$ is called a higher derivation if ${d}_{n}\left(ab\right)={\sum }_{k=0}^{n}{d}_{k}\left(a\right){d}_{n-k}\left(b\right)$ 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.
MSC:
 47B47 Commutators, derivations, elementary operators, etc. 16W25 Derivations, actions of Lie algebras (associative rings and algebras)