Acta mathematica scientia, Series B >
MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS
Received date: 2009-03-04
Revised date: 2010-11-18
Online published: 2012-03-20
Supported by
This work is partially supported by National Natural Science Foundation of China (10871111) and the Specialized Research Fund for Doctoral Program of Higher Education (200800030059) (to Cui); and by Basic Science Research Program through the National Research
Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788) (to Park).
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.
CUI Jian-Lian , Choonkil Park . MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 531 -538 . DOI: 10.1016/S0252-9602(12)60035-6
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