算子代數(shù)上某些映射的刻畫
發(fā)布時間：2018-09-04 06:31
【摘要】：本文刻畫了算子代數(shù)上的一些線性映射.我們所研究的映射包括：左導子,Jordan左導子,(m,n)-Jordan導子,廣義導子以及廣義Jordan導子；我們所研究的代數(shù)包括：C*-代數(shù),von Neumann代數(shù),套代數(shù),完全分配的子空間格代數(shù),J-子空間格代數(shù),P-子空間格代數(shù),交換子空間格代數(shù)以及廣義矩陣代數(shù).全文共分為七個章節(jié).在第一章中,我們介紹了本文的研究背景,回顧了國內(nèi)外學者之前的研究進展以及所取得的一些重要成果,同時介紹了本文所涉及的一些基本概念.在第二章中,我們證明了如果代數(shù)A和左A-模M滿足下列三個條件之一,那么每一個從A到MM的Jordan左導子恒等于零：(1A是一個C*-代數(shù)且M是一個Banach左A-模；(2)A=A1gl滿足∩[L_:L∈Jl)=(0)且M=B(X),其中L是Banach空間X上的一個子空間格；(3)A=B∩Algl且M=B(H),其中B是Hilbert空間H上的一個von Neumann代數(shù),L∈B是H上的一個交換子空間格.在第三章中,我們證明了如果A是復數(shù)域C上的單位代數(shù),M是一個單位A-雙邊模,并且M含有一個由A中冪等元代數(shù)生成的左(右)分離集,那么當m,n0且m≠n時,每一個從A到M的(m,n)-Jordan導子恒等于零.同時我們也證明了如果m,n0且是一個|mn(m-n)(m+n)I-毛撓的廣義矩陣代數(shù),并且M是一個忠實的單位(A,B)-雙邊模,那么每一個從U到自身的(m,n)-Jordan導子恒等于零.在第四章中,我們證明了由冪等元代數(shù)生成的單位代數(shù)是零Jordan乘積確定的,從而對M.Bresar和M.Kosan等分別在2009年和2014年提出的兩個問題給予了肯定的回答.同時我們也研究了含有非平凡冪等元的單位代數(shù)何時是零Jordan乘積確定的,并給出了三角代數(shù)是零Jordan乘積確定的充要條件.作為應用,我們刻畫了零Jordan乘積確定代數(shù)上Jordan左導子,(m,n)-Jordan導子,Jordan導子,Lie導子,Jordan同態(tài)以及Lie同態(tài)的局部性質(zhì).在第五章中,我們通過零乘積和零Jordan乘積刻畫了廣義導子和廣義Jordan導子的性質(zhì).設A是復數(shù)域C上的一個單位代數(shù),M是一個單位A-雙邊模,δ是從A到M的線性映射.首先,我們證明了如果A包含一個由冪等元代數(shù)生成的理想J滿足{M∈M：對任意J,K∈J,JMK=0}={0},且δ滿足對任意A,B,C∈A,AB=BC=0蘊含Aδ(B)C=0,那么δ是一個廣義導子.特別的,如果δ是一個A到M局部導子,那么δ是一個導子.接下來,我們證明了如果A包含一個由冪等元代數(shù)生成的理想J滿足{M∈M：對任意J∈J,JM=MJ=0}={0},且δ在零點可導,即滿足對任意A,B∈A,AB=0蘊含Aδ(B)+δ(A)B=0,那么δ是一個廣義導子.顯然,如果M含有一個由A中冪等元代數(shù)生成的分離集J,那么J同時滿足上述兩個條件.最后,我們證明了如果A包含一個由冪等元代數(shù)生成的理想J滿足{M∈M：對任意J∈J,JMJ=0}={0},且δ滿足對任意A,B∈A,AB=BA=0蘊含A o 6(B)+δ(A)(?)B=0(特別的,δ是零點可導或者零點Jordan可導映射),那么δ是一個廣義Jordan導子.在第六章中,我們證明了如果A是復數(shù)域C上的一個單位代數(shù),M是一個單位左A模,并且M含有一個由A中冪等元代數(shù)生成的右分離集,δ是一個從A到M的線性映射滿足對任意A,B∈A, AB=BA=0蘊含Aδ(B)+Bδ(A)=0,那么對任意A∈A,δ(A)=Aδ(I)我們也證明了如果A是一個因子von Neumann代數(shù),那么每個在右分離元或者非零自伴元處左可導的映射恒等于零.在第七章中,我們對全文進行了總結(jié)和概括,并提出了一些我們想要解決但還尚未解決的問題.
[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.
【學位授予單位】：華東理工大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O153.3
本文編號：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.
【學位授予單位】：華東理工大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O153.3
【參考文獻】
相關期刊論文 前3條
1 宋曉輝;張建華;;一類零積與Jordan零積確定的代數(shù)[J];數(shù)學進展;2014年02期
2 張建華;套代數(shù)上的Jordan導子[J];數(shù)學學報;1998年01期
3 張建華，，社鴻科;Von Neumann代數(shù)中套子代數(shù)上的導子(Ⅱ)[J];數(shù)學學報;1997年03期
本文編號：2221292
本文鏈接：http://sikaile.net/shoufeilunwen/jckxbs/2221292.html
最近更新
教材專著
