Local higher derivations on C*-algebras are higher derivations

Let \mathfrak{A} be a Banach algebra. We say that a sequence \{D_n\}_{n=۰}^\infty of continuous operators form \mathfrak{A} into \mathfrak{A} is a \textit{local higher derivation} if to each a\in\mathfrak{A} there corresponds a continuous higher derivation \{d_{a,n}\}_{n=۰}^\infty such that D_n(a)=d_{a,n}(a) for each non-negative integer n. We show that if \mathfrak{A} is a C^*-algebra then each local higher derivation on \mathfrak{A} is a higher derivation. We also prove that each local higher derivation on a C^*-algebra is automatically continuous.

Keywords:

Higher derivation , local higher derivation , Derivation , local derivation