算子代数上某些映射的刻画
发布时间：2018-09-04 06:31
【摘要】：本文刻画了算子代数上的一些线性映射.我们所研究的映射包括：左导子,Jordan左导子,(m,n)-Jordan导子,广义导子以及广义Jordan导子；我们所研究的代数包括：C*-代数,von Neumann代数,套代数,完全分配的子空间格代数,J-子空间格代数,P-子空间格代数,交换子空间格代数以及广义矩阵代数.全文共分为七个章节.在第一章中,我们介绍了本文的研究背景,回顾了国内外学者之前的研究进展以及所取得的一些重要成果,同时介绍了本文所涉及的一些基本概念.在第二章中,我们证明了如果代数A和左A-模M满足下列三个条件之一,那么每一个从A到MM的Jordan左导子恒等于零：(1A是一个C*-代数且M是一个Banach左A-模；(2)A=A1gl满足∩[L_:L∈Jl)=(0)且M=B(X),其中L是Banach空间X上的一个子空间格；(3)A=B∩Algl且M=B(H),其中B是Hilbert空间H上的一个von Neumann代数,L∈B是H上的一个交换子空间格.在第三章中,我们证明了如果A是复数域C上的单位代数,M是一个单位A-双边模,并且M含有一个由A中幂等元代数生成的左(右)分离集,那么当m,n0且m≠n时,每一个从A到M的(m,n)-Jordan导子恒等于零.同时我们也证明了如果m,n0且是一个|mn(m-n)(m+n)I-毛挠的广义矩阵代数,并且M是一个忠实的单位(A,B)-双边模,那么每一个从U到自身的(m,n)-Jordan导子恒等于零.在第四章中,我们证明了由幂等元代数生成的单位代数是零Jordan乘积确定的,从而对M.Bresar和M.Kosan等分别在2009年和2014年提出的两个问题给予了肯定的回答.同时我们也研究了含有非平凡幂等元的单位代数何时是零Jordan乘积确定的,并给出了三角代数是零Jordan乘积确定的充要条件.作为应用,我们刻画了零Jordan乘积确定代数上Jordan左导子,(m,n)-Jordan导子,Jordan导子,Lie导子,Jordan同态以及Lie同态的局部性质.在第五章中,我们通过零乘积和零Jordan乘积刻画了广义导子和广义Jordan导子的性质.设A是复数域C上的一个单位代数,M是一个单位A-双边模,δ是从A到M的线性映射.首先,我们证明了如果A包含一个由幂等元代数生成的理想J满足{M∈M：对任意J,K∈J,JMK=0}={0},且δ满足对任意A,B,C∈A,AB=BC=0蕴含Aδ(B)C=0,那么δ是一个广义导子.特别的,如果δ是一个A到M局部导子,那么δ是一个导子.接下来,我们证明了如果A包含一个由幂等元代数生成的理想J满足{M∈M：对任意J∈J,JM=MJ=0}={0},且δ在零点可导,即满足对任意A,B∈A,AB=0蕴含Aδ(B)+δ(A)B=0,那么δ是一个广义导子.显然,如果M含有一个由A中幂等元代数生成的分离集J,那么J同时满足上述两个条件.最后,我们证明了如果A包含一个由幂等元代数生成的理想J满足{M∈M：对任意J∈J,JMJ=0}={0},且δ满足对任意A,B∈A,AB=BA=0蕴含A o 6(B)+δ(A)(?)B=0(特别的,δ是零点可导或者零点Jordan可导映射),那么δ是一个广义Jordan导子.在第六章中,我们证明了如果A是复数域C上的一个单位代数,M是一个单位左A模,并且M含有一个由A中幂等元代数生成的右分离集,δ是一个从A到M的线性映射满足对任意A,B∈A, AB=BA=0蕴含Aδ(B)+Bδ(A)=0,那么对任意A∈A,δ(A)=Aδ(I)我们也证明了如果A是一个因子von Neumann代数,那么每个在右分离元或者非零自伴元处左可导的映射恒等于零.在第七章中,我们对全文进行了总结和概括,并提出了一些我们想要解决但还尚未解决的问题.
[Abstract]:In this paper, we characterize some linear mappings on operator algebras. The mappings we study include: left derivation, Jordan left derivation, (m, n) - Jordan derivation, generalized derivation and generalized Jordan derivation. The algebras we study include: C* - algebra, von Neumann algebra, nested algebra, fully allocated subspace lattice algebra, J - subspace lattice algebra, P - subspace lattice algebra. Subspace lattice algebra, commutative subspace lattice algebra and generalized matrix algebra. The whole paper is divided into seven chapters. In the first chapter, we introduce the research background of this paper, review the research progress and some important achievements of scholars at home and abroad, and introduce some basic concepts involved in this paper. We prove that if algebra A and left A-module M satisfy one of the following three conditions, then every Jordan left derivative from A to MM is invariably zero: (1A is a C*-algebra and M is a Banach left A-module; (2) A = A1gl satisfies [L: L < Jl]= (0) and M = B (X), where L is a subspace lattice on Banach space X; (3) A = B Algl and M = B (H), whose In Chapter 3, we prove that if A is a unit algebra over a complex field C, M is a unit A-bilateral module, and M contains a left (right) separation set generated by idempotent algebras in A, then when m, N0 and m_n, each of them is from A to M. At the same time, we prove that if m, N0 is a generalized matrix algebra with | m n (m-n) (m + n) I-deflection and M is a faithful unit (A, B) - bilateral module, then every (m, n) - Jordan derivative from U to itself is equal to zero. I n Chapter 4, we prove that the unit generation generated by idempotent Algebras is zero. The number is determined by zero Jordan product, so we give positive answers to two questions raised by M. Bresar and M. Kosan in 2009 and 2014 respectively. We also study when the unit algebra containing nontrivial idempotents is determined by zero Jordan product, and give the necessary and sufficient conditions for the trigonometric algebra to be determined by zero Jordan product. For application, we characterize the local properties of Jordan left derivatives, (m, n) - Jordan derivatives, Jordan derivatives, Lie derivatives, Jordan homomorphisms and Lie Homomorphisms on a zero Jordan product deterministic algebra. In algebra, M is a unit A-bilateral module and delta is a linear mapping from A to M. Firstly, we prove that if A contains an ideal J generated by idempotent algebra satisfying {M < M: for any J, K < J, JMK = 0}={0}, and delta satisfies any A, B, C < A, AB = BC = 0 containing A Delta (B) C = 0, then delta is a generalized derivation. Next, we prove that if A contains an ideal J generated by idempotent algebra satisfying {M < M: for any J < J, JM = MJ = 0} = {0}, and delta is derivable at zero, that is to say, satisfies for any A, B < A, AB = 0 containing A Delta (B) + delta (A) B = 0, then delta is a generalized derivation.Obviously, if M contains a power from A. Finally, we prove that if A contains an ideal J generated by idempotent algebras satisfying {M < M: for any J < J, JMJ = 0}={0}, and delta satisfies for any A, B < A, AB = BA = 0 containing A o 6 (B) + delta (A) (?) B = 0 (in particular, delta is zero derivable or zero Jordan derivable). In Chapter 6, we prove that if A is a unit algebra over a complex field C, M is a unit left A module, and M contains a right separation set generated by idempotent algebras in A, and delta is a linear mapping from A to M that satisfies the implication of A Delta (B) + B Delta (A) = 0 for any A, B <, A = 0. We also prove that if A is a factor von Neumann algebra, then every left-derivable mapping at the right separator or non-zero self-adjoint element is invariably zero.
【学位授予单位】：华东理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O153.3
本文编号：2221292
[Abstract]:In this paper, we characterize some linear mappings on operator algebras. The mappings we study include: left derivation, Jordan left derivation, (m, n) - Jordan derivation, generalized derivation and generalized Jordan derivation. The algebras we study include: C* - algebra, von Neumann algebra, nested algebra, fully allocated subspace lattice algebra, J - subspace lattice algebra, P - subspace lattice algebra. Subspace lattice algebra, commutative subspace lattice algebra and generalized matrix algebra. The whole paper is divided into seven chapters. In the first chapter, we introduce the research background of this paper, review the research progress and some important achievements of scholars at home and abroad, and introduce some basic concepts involved in this paper. We prove that if algebra A and left A-module M satisfy one of the following three conditions, then every Jordan left derivative from A to MM is invariably zero: (1A is a C*-algebra and M is a Banach left A-module; (2) A = A1gl satisfies [L: L < Jl]= (0) and M = B (X), where L is a subspace lattice on Banach space X; (3) A = B Algl and M = B (H), whose In Chapter 3, we prove that if A is a unit algebra over a complex field C, M is a unit A-bilateral module, and M contains a left (right) separation set generated by idempotent algebras in A, then when m, N0 and m_n, each of them is from A to M. At the same time, we prove that if m, N0 is a generalized matrix algebra with | m n (m-n) (m + n) I-deflection and M is a faithful unit (A, B) - bilateral module, then every (m, n) - Jordan derivative from U to itself is equal to zero. I n Chapter 4, we prove that the unit generation generated by idempotent Algebras is zero. The number is determined by zero Jordan product, so we give positive answers to two questions raised by M. Bresar and M. Kosan in 2009 and 2014 respectively. We also study when the unit algebra containing nontrivial idempotents is determined by zero Jordan product, and give the necessary and sufficient conditions for the trigonometric algebra to be determined by zero Jordan product. For application, we characterize the local properties of Jordan left derivatives, (m, n) - Jordan derivatives, Jordan derivatives, Lie derivatives, Jordan homomorphisms and Lie Homomorphisms on a zero Jordan product deterministic algebra. In algebra, M is a unit A-bilateral module and delta is a linear mapping from A to M. Firstly, we prove that if A contains an ideal J generated by idempotent algebra satisfying {M < M: for any J, K < J, JMK = 0}={0}, and delta satisfies any A, B, C < A, AB = BC = 0 containing A Delta (B) C = 0, then delta is a generalized derivation. Next, we prove that if A contains an ideal J generated by idempotent algebra satisfying {M < M: for any J < J, JM = MJ = 0} = {0}, and delta is derivable at zero, that is to say, satisfies for any A, B < A, AB = 0 containing A Delta (B) + delta (A) B = 0, then delta is a generalized derivation.Obviously, if M contains a power from A. Finally, we prove that if A contains an ideal J generated by idempotent algebras satisfying {M < M: for any J < J, JMJ = 0}={0}, and delta satisfies for any A, B < A, AB = BA = 0 containing A o 6 (B) + delta (A) (?) B = 0 (in particular, delta is zero derivable or zero Jordan derivable). In Chapter 6, we prove that if A is a unit algebra over a complex field C, M is a unit left A module, and M contains a right separation set generated by idempotent algebras in A, and delta is a linear mapping from A to M that satisfies the implication of A Delta (B) + B Delta (A) = 0 for any A, B <, A = 0. We also prove that if A is a factor von Neumann algebra, then every left-derivable mapping at the right separator or non-zero self-adjoint element is invariably zero.
【学位授予单位】：华东理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O153.3
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3 张建华，，社鸿科;Von Neumann代数中套子代数上的导子(Ⅱ)[J];数学学报;1997年03期
本文编号：2221292
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