Non-linear $\lambda $-Jordan triple $\ast $-derivation on prime $\ast $-algebras

Abstract

Let $\mathcal {A}$ be a prime $\ast $-algebra and $\Phi $ a $\lambda $-Jordan triple derivation on $A$, that is, for every $A,B,C \in \mathcal {A}$, $$\Phi (A\diamond _{\lambda } B \diamond _{\lambda }C)=\Phi (A)\diamond _{\lambda }B\diamond _{\lambda } C+A\diamond _{\lambda }\Phi (B)\diamond _{\lambda }C+A\diamond _{\lambda } B\diamond _{\lambda }\Phi (C),$$ where $A\diamond _{\lambda } B = AB + \lambda BA^{\ast }$ such that a complex scalar $|\lambda |\neq 0,1$, and $\Phi $ is additive. Moreover, if $\Phi (I)$ is self-adjoint, then $\Phi $ is a $\ast $-derivation.