因子von Neumann代数上的非全局非线性Lie三重可导映射
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摘要
设M是Hilbert空间H上维数大于1的因子von Neumann代数,用代数分解方法证明了:如果非线性映射δ:M→M满足对任意的A,B,C∈M且ABC=0,有δ([[A,B],C])=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)],则存在可加导子d:M→M,使得对任意的A∈M,有δ(A)=d(A)+τ(A)I,其中τ:M→瓘I是一个非线性映射,满足对任意的A,B,C∈M且ABC=0时,有τ([[A,B],C])=0.
Let Mbe a factor von Neumann algebra with dimension greater than 1 on a Hilbert space H.With the help of algebraic decomposition method,we proved that if a nonlinear mapδ:M→Msatisfiedδ([[A,B],C])=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)]for any A,B,C∈M with ABC=0,then there existed an additive derivation d:M→M,such thatδ(A)=d(A)+τ(A)Ifor any A∈M,whereτ:M→瓘Iis a nonlinear map such thatτ([[A,B],C])=0 with ABC=0 for any A,B,C∈M.
Let Mbe a factor von Neumann algebra with dimension greater than 1 on a Hilbert space H.With the help of algebraic decomposition method,we proved that if a nonlinear mapδ:M→Msatisfiedδ([[A,B],C])=[[δ(A),B],C]+[[A,δ(B)],C]+[[A,B],δ(C)]for any A,B,C∈M with ABC=0,then there existed an additive derivation d:M→M,such thatδ(A)=d(A)+τ(A)Ifor any A∈M,whereτ:M→瓘Iis a nonlinear map such thatτ([[A,B],C])=0 with ABC=0 for any A,B,C∈M.
引文
[1]LI Changjing,FANG Xiaochun,LU Fangyan,et al.Lie Triple Derivable Mappings on Rings[J].Comm Algebra,2014,42(6):2510-2527.
[2]LIU Lei.Lie Triple Derivations on Factor von Neumann Algebras[J].Bull Korean Math Soc,2015,52(2):581-591.
[3]YU Weiyan,ZHANG Jianhua.Nonlinear Lie Derivations of Triangular Algebars[J].Linear Algebra Appl,2010,432(11):2953-2960.
[4]JI Peisheng,LIU Rongrong,ZHAO Yingzi.Nonlinear Lie Triple Derivations of Triangular Algebars[J].Linear Multil Algebra,2012,60(10):1155-1164.
[5]BAI Zhaofang,DU Shuanping.The Structure of Nonlinear Lie Derivation on von Neumann Algebras[J].Linear Algebra Appl,2012,436(7):2701-2708.
[6]张芳娟.素环上的非线性Lie导子[J].数学进展,2014,43(1):145-150.(ZHANG Fangjuan.Nonlinear Lie Derivations on Prime Rings[J].Advances in Mathematics(China),2014,43(1):145-150.)
[7]陈琳,张建华.套代数上的零点Lie可导映射[J].数学学报(中文版),2009,52(1):105-110.(CHEN Lin,ZHANG Jianhua.Lie Derivable Mappings at Zero Point on Nest Algebras[J].Acta Mathematics Sinica(Chinese Series),2009,52(1):105-110.)
[8]LU Fangyan,JING Wu.Characterizations of Lie Derivations of B(X)[J].Linear Algebra Appl,2010,432(1):89-99.
[9]QI Xiaofei.Characterizing Lie Derivations on Tringular Algebras by Local Actions[J].Electr J Linear Algebra,2013,26:816-835.
[10]HALMOS P R.A Hilbert Space Problem Book[M].2nd ed.New York:Springer-Verlag,1982.
[2]LIU Lei.Lie Triple Derivations on Factor von Neumann Algebras[J].Bull Korean Math Soc,2015,52(2):581-591.
[3]YU Weiyan,ZHANG Jianhua.Nonlinear Lie Derivations of Triangular Algebars[J].Linear Algebra Appl,2010,432(11):2953-2960.
[4]JI Peisheng,LIU Rongrong,ZHAO Yingzi.Nonlinear Lie Triple Derivations of Triangular Algebars[J].Linear Multil Algebra,2012,60(10):1155-1164.
[5]BAI Zhaofang,DU Shuanping.The Structure of Nonlinear Lie Derivation on von Neumann Algebras[J].Linear Algebra Appl,2012,436(7):2701-2708.
[6]张芳娟.素环上的非线性Lie导子[J].数学进展,2014,43(1):145-150.(ZHANG Fangjuan.Nonlinear Lie Derivations on Prime Rings[J].Advances in Mathematics(China),2014,43(1):145-150.)
[7]陈琳,张建华.套代数上的零点Lie可导映射[J].数学学报(中文版),2009,52(1):105-110.(CHEN Lin,ZHANG Jianhua.Lie Derivable Mappings at Zero Point on Nest Algebras[J].Acta Mathematics Sinica(Chinese Series),2009,52(1):105-110.)
[8]LU Fangyan,JING Wu.Characterizations of Lie Derivations of B(X)[J].Linear Algebra Appl,2010,432(1):89-99.
[9]QI Xiaofei.Characterizing Lie Derivations on Tringular Algebras by Local Actions[J].Electr J Linear Algebra,2013,26:816-835.
[10]HALMOS P R.A Hilbert Space Problem Book[M].2nd ed.New York:Springer-Verlag,1982.