Local higher derivations on C*-algebras are higher derivations

Naranjani, Lila; Hassani, Mahmoud; Mirzavaziri, Madjid

doi:10.22075/ijnaa.2018.3098

Local higher derivations on C*-algebras are higher derivations

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

10.22075/ijnaa.2018.3098

Abstract

Let $\mathfrak{A}$ be a Banach algebra. We say that a sequence $\{D_n\}_{n=0}^\infty$ 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 $\{d_{a,n}\}_{n=0}^\infty$ 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