Product of derivations on C$^*$-algebras

Document Type: Research Paper

Authors

1 Department of Mathematics, Payame Noor University

2 Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Struc-tures (CEAAS), Ferdowsi University of Mashhad

3 Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Struc-tures (CEAAS), Ferdowsi University of Mashhad,

Abstract

Let $\mathfrak{A}$ be an algebra. A linear mapping $\delta:\mathfrak{A}\to\mathfrak{A}$ is called a \textit{derivation} if $\delta(ab)=\delta(a)b+a\delta(b)$ for each $a,b\in\mathfrak{A}$. Given two derivations $\delta$ and $\delta'$ on a $C^*$-algebra $\mathfrak A$, we prove that there exists a derivation $\Delta$ on $\mathfrak A$ such that $\delta\delta'=\Delta^2$ if and only if either $\delta'=0$ or $\delta=s\delta'$ for some $s\in\mathbb{C}$.

Keywords