MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

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Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear bijective map Ψ:A - A satisfying Ψ(A)Ψ(B) - Ψ(B)Ψ(A)* =AB - BA* for A,B ∈ A and a real functional h on A with h(0) =0 such that Φ(A) =Ψ(A) + h(A)I for every A ∈ A.In particular,if A is a type Ⅰ factor,then,Φ(A) =cA + h(A)I for every A ∈ A,where c =±-1.
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