Nonlinear $$ast$$-Lie-type derivations on von Neumann algebras

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

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

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].