Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68229120180901Local higher derivations on C*-algebras are higher derivations111115309810.22075/ijnaa.2018.3098ENLilaNaranjaniDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, IranMahmoudHassaniDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, IranMadjidMirzavaziriDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad
91775, IranJournal Article20160513Let $\mathfrak{A}$ 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.https://ijnaa.semnan.ac.ir/article_3098_2dd5a1ec2b9eb291b3144ecc1e96595e.pdf