Non-linear mixed Jordan bi-skew Lie-type derivations on $*$-algebras

Abstract

Let $\mathcal{M}$ be a unital $*$-algebra. For any $M_1, M_2\in\mathcal{M}$ the Jordan and bi-skew Lie product of $M_1$ and $M_2$ are defined as $M_1 \circ M_2 = M_1 M_2 + M_2 M_1$ and $[M_1, M_2]_\diamond = M_1 M_2^* - M_2 M_1^*$, respectively. A product defined as $p_n\Big(M_1, M_2, \ldots, M_n\Big) = [M_1\circ M_2\circ\ldots\circ M_{n-1}, M_n]_\diamond$ for all $M_1, M_2,\ldots, M_n\in\mathcal{M}$, is called a mixed Jordan bi-skew Lie $n$-product of $M_1, M_2,\ldots, M_n$. In this article, we prove that a map $\Psi:\mathcal{M}\rightarrow \mathcal{M}$, satisfies $\Psi\Big(p_n\Big(M_1, M_2,\ldots,M_n\Big)\Big)=\sum_{k=1}^{n} p_n\Big(M_1, M_2,\ldots, M_{k-1}, \Psi(M_k), M_{k+1},\ldots, M_n\Big)$ for all $M_{1}, M_{2},\\ \ldots , M_{n}\in \mathcal{M}$, if and only if $\Psi$ is an additive $*$-derivation. We apply the above result to prime $*$-algebras, factor von Neumann algebras, von Neumann algebras with no central summands of type $I_1$ and standard operator algebras.