关键词: Skew Lie product, factor von Neumann algebras, preserver problems

Abstract:

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  (A) (B) −  (B) (A)* = AB − BA*  for A, B ∈ A and a real functional h on A with h(0) = 0 such that Φ(A) =  (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.

Key words: Skew Lie product, factor von Neumann algebras, preserver problems