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

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$ and $\mathcal{A}\subseteq \mathcal{B}(\mathcal{H})$ 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 $A\diamond B=AB+B{A}^{\ast }$. 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 $\delta :\mathcal{A}⟶\mathcal{B}(\mathcal{H})$ 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.

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