Document Type : Research Paper

Authors

• Lila Naranjani ¹
• Mahmoud Hassani ¹
• Madjid Mirzavaziri ²

¹ Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

² Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran

Abstract

Let $\mathfrak{A}$ be a Banach algebra. We say that a sequence $\{D_n{\}}_{n=0}^{\mathrm{\infty }}$ 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 $\{d_{a,n}{\}}_{n=0}^{\mathrm{\infty }}$ such that $D_n(a)=d_{a,n}(a)$ for each non-negative integer $n$. We show that if $\mathfrak{A}$ is a $C^{\ast }$-algebra then each local higher derivation on $\mathfrak{A}$ is a higher derivation. We also prove that each local higher derivation on a $C^{\ast }$-algebra is automatically continuous.

Keywords

• Higher derivation
• local higher derivation
• Derivation
• local derivation