Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4664049 | Acta Mathematica Scientia | 2012 | 8 Pages |
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 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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