Nonlinear $\ast$-Lie-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 nonzero central abelian projections. Let ${p_n(x_1,x_2 ,\ldots ,x_n)}$ be the commutator polynomial defined by n indeterminates ${x_1, \ldots , x_n}$ and their skew Lie products. It is shown that a mapping ${\delta \colon \mathcal{A} \longrightarrow \mathcal{B(H)}}$ satisfies

$$ \delta(p_n(A_1, A_2 ,\ldots , A_n))=\sum_{k=1}^n p_n(A_1 ,\ldots , A_{k-1}, \delta(A_k), A_{k+1} ,\ldots , A_n) $$

for all ${A_1, A_2 ,\ldots , A_n \in \mathcal{A}}$ if and only if ${\delta}$ is an additive *-derivation. This gives a positive answer to Conjecture 4.2 of [14].