B(H)上的可乘ξ-Lie同构
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摘要
设H,K为实数域或复数域F上的无限维Hilbert空间,B(H),B(K)分别表示H和K上的全体有界线性算子构成的代数。若双射φ:B(H)→B(K)满足对任意的A,B_φ([A,B]_ξ)=[φ(A),φ(B)]_ξ成立,则称φ为B(H)上的一个可乘ξ-Lie同构。显然,当ξ=0,-1,1时的可乘ξ-Lie同构分别对应可乘同构,Jordan可乘同构以及Lie可乘同构。本文利用Peirce分解的方法证明了B(H)上的每个可乘ξ-Lie(ξ∈F且ξ≠0,±1)同构是可加的,从而存在非零数c∈F以及可逆算子T∈B(H,K),使得对任意的A∈B(H),有φ(A)=cTAT~(-1)。
Let H and K be the infinite-dimensional Hilbert space over real number field or complex number field F,and B( H) and B( K) be the algebra of all bounded linear operators on H and K. If a bijective mapping φ: B( H) →B( K) meets arbitrary A,B___ φ( [A,B]_ξ) =[φ( A),φ( B) ]_ξ,φ is called multiplicative ξ-Lie isomorphism over B( H). Obviously,when ξ = 0,-1,1,the multiplicative ξ-Lie isomorphism respectively matches to multiplicative isomorphism,Jordan multiplicative isomorphism and Lie multiplicative isomorphism. With the help of Peirce decomposition,we have proved that every multiplicative ξ-Lie( ξ ∈ F and ξ ≠ 0,± 1) isomorphism on B( H) is additive,and there exist a nonzero c ∈ F and an invertible operator T ∈ B( H,K),making that there is φ( A) = cTAT~(-1) for arbitrary A ∈ B( H).
Let H and K be the infinite-dimensional Hilbert space over real number field or complex number field F,and B( H) and B( K) be the algebra of all bounded linear operators on H and K. If a bijective mapping φ: B( H) →B( K) meets arbitrary A,B___ φ( [A,B]_ξ) =[φ( A),φ( B) ]_ξ,φ is called multiplicative ξ-Lie isomorphism over B( H). Obviously,when ξ = 0,-1,1,the multiplicative ξ-Lie isomorphism respectively matches to multiplicative isomorphism,Jordan multiplicative isomorphism and Lie multiplicative isomorphism. With the help of Peirce decomposition,we have proved that every multiplicative ξ-Lie( ξ ∈ F and ξ ≠ 0,± 1) isomorphism on B( H) is additive,and there exist a nonzero c ∈ F and an invertible operator T ∈ B( H,K),making that there is φ( A) = cTAT~(-1) for arbitrary A ∈ B( H).
引文
[1]Kassel C.Quantum Groups[M].New York:Springer-Verlag,1995.
[2]Cui J L,Hou J C,PARK Choonkil.Additive maps preserving commutativity up to a factor[J].Chin.Ann.Math.,2008,29A(5):583-590.
[3]Qi X F,Hou J C.Additive Lie(ξ-Lie)derivations and generalized Lie(ξ-Lie)derivations on nest algebras[J].Lin.Alg.Appl.,2009(431):843-845.
[4]Zhang J H,Zhang F J.Nonlinear maps preserving Lie products on factor von Neumann algebras[J].Lin.Alg.Appl.,2008(429):18-30.
[5]安润玲.算子代数上的约当同构和初等映射[D].太原:山西大学,2007.
[6]Ji P S.Additivity of Jordan maps on Jordan algebras[J].Lin.Alg.Appl.,2009(431):197-188.
[7]Lu F Y.Multiplicative mappings of operator algebras[J].Lin.Alg.Appl.,2002(347):283-291.
[8]Bunch H L.Local multiplications on algebras spanned by idempotents[J].Lin.Mul.Alg.,1994(37):259-263.
[2]Cui J L,Hou J C,PARK Choonkil.Additive maps preserving commutativity up to a factor[J].Chin.Ann.Math.,2008,29A(5):583-590.
[3]Qi X F,Hou J C.Additive Lie(ξ-Lie)derivations and generalized Lie(ξ-Lie)derivations on nest algebras[J].Lin.Alg.Appl.,2009(431):843-845.
[4]Zhang J H,Zhang F J.Nonlinear maps preserving Lie products on factor von Neumann algebras[J].Lin.Alg.Appl.,2008(429):18-30.
[5]安润玲.算子代数上的约当同构和初等映射[D].太原:山西大学,2007.
[6]Ji P S.Additivity of Jordan maps on Jordan algebras[J].Lin.Alg.Appl.,2009(431):197-188.
[7]Lu F Y.Multiplicative mappings of operator algebras[J].Lin.Alg.Appl.,2002(347):283-291.
[8]Bunch H L.Local multiplications on algebras spanned by idempotents[J].Lin.Mul.Alg.,1994(37):259-263.