A Note on Nonadditive Bi-skew Commuting Maps on von Neumann Algebras

Let A be a von Neumann algebra acting on a complex Hilbert space H and ZS(A) be the symmetric center of A . It is shown that, if a map φ : A→ A satisfies [φ (X), P]=[X, φ (P)] for all X∈ A and all projections P∈ A , then there exists a map f: A→ ZS(A) such that φ (X)=Xφ (I)+f(X) for all X∈ A , where φ (I)=φ (I)* and f satisfies $$f(P)=0$$ for all projections P∈ A . Moreover, a characterization of nonadditive bi-skew commuting maps on von Neumann algebras is obtained