On 2-local derivations of von Neumann algebras

Let $ \mathcal {A} $ A be a semi-finite factor von Neumann algebra. We prove that if a map $ \delta : \mathcal {A}\rightarrow \mathcal {A} $ delta:A -> A satisfies that for any $ A, B\in \mathcal {A} $ A,B is an element of A there is a linear derivation $ \delta _{A, B}: \mathcal {A}\rightarrow \mathcal {A} $ delta A,B:A -> A such that $ \delta (A)B + A\delta (B) = \delta _{A, B}(AB) $ delta(A)B+A delta(B)=delta A,B(AB), then delta is a linear derivation. It is a positive answer to the problem on semi-finite factor von Neumann algebra posed by Moln & aacute;r in the paper [A new look at local maps on algebraic structures of matrices and operators, New York J Math 28 (2022)].