Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras

Abstract

Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.

Keywords

• mixed bi-skew jordan triple derivation
• $ \ast $-derivation
• $ \ast $- algebra
• involution