Nonlinear $\ast$-Jordan-Type Derivations on von Neumann Algebras

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, $\mathcal{B(H)}$ be the algebra of all bounded linear operators on $\mathcal{H}$ and $\mathcal{A} \subseteq \mathcal{B(H)}$ be a von Neumann algebra without central summands of type $I_1$. For arbitrary elements $A, B\in \mathcal{A}$, one can define their $\ast$-Jordan product in the sense of $A\diamond B = AB+BA^\ast$. Let $p_n(x_1,x_2,\cdots,x_n)$ be the polynomial defined by $n$ indeterminates $x_1, \cdots, x_n$ and their $\ast$-Jordan products. In this article, it is shown that a mapping $\delta: \mathcal{A} \longrightarrow \mathcal{B(H)}$ satisfies the condition $$ \delta(p_n(A_1, A_2,\cdots, A_n))=\sum_{k=1}^n p_n(A_1,\cdots, A_{k-1}, \delta(A_k), A_{k+1},\cdots, A_n) $$ for all $A_1, A_2,\cdots, A_n \in \mathcal{A}$ if and only if $\delta$ is an additive $\ast$-derivation.