Nonlinear mixed bi-skew Jordan-type derivations on prime ∗-algebras

Abstract

Let $\mathcal A$ be a unite prime $\ast$-algebra containing a non-trivial projection. Assume that $\phi:\mathcal A\rightarrow\mathcal A$ satisfies $\phi(A_{1}\diamond_{1} A_{2}\diamond_{2}\cdots\diamond_{n} A_{n+1})=\sum^{n+1}_{h=1}A_{1}\diamond_{1} \cdots\diamond_{h-2} A_{h-1}\diamond_{h-1}\phi(A_{h})\diamond_{h} A_{h+1}\diamond_{h+1}\cdots\diamond_{n} A_{n+1}(n\geq2)$
for any $A_{1}, A_{2},\cdots ,A_{n+1}\in\mathcal A$ and $\diamond_{r}$ is $\bullet$ or $\circ$ with $1\leq r\leq n$, where $A\bullet B=AB^{\ast}+BA^{\ast}$ and $A\circ B=AB+BA$. In this article, we prove that if $n$ is even and $\diamond_{2u-1}=\bullet$, $\diamond_{2u}=\circ$ with $1\leq u\leq\frac{n}{2}$, then there exists an element $\lambda\in\mathcal{Z}_{S}(\mathcal{A})$ such that $\phi(A)=\delta(A)+\mathrm i \lambda A$, where $\delta$ is an additive $\ast$-derivation. Otherwise, $\phi$ is an additive $\ast$-derivation. In particular, the nonlinear mixed bi-skew Jordan-type derivation on factor von Neumann algebras and standard operator algebras are
characterized.