算子代数上的中心化子和Lie可导映射

发布时间：2018-06-07 11:49

  本文选题：中心化子 + 素环　； 参考：《太原理工大学》2017年硕士论文

【摘要】：左(右)中心化子、中心化子及Lie导子是算子代数与算子理论研究中非常重要的内容,受到了许多学者的广泛关注.本文主要刻画三角环,素环和von Neumann代数上在某点是中心化子的可加映射,探讨可加映射成为中心化子的条件,进而得到三角环,素环和von Neumann代数上中心化子的新等价刻画.同时本文刻画B(X)在值域不稠或非单射算子Lie可导的可加映射.全文结构如下:第一章简要介绍所研究问题的背景,本文的主要内容以及证明过程中所需的结论和定义.第二章刻画了三角环、素环、von Neumann代数上的中心化子,主要结论如下:1.三角环R上中心化子的刻画.设T = Tri(A,M,B)为三角环,T是任意但固定的元.假设对任意的4 ∈ A,B ∈ B,存在正整数n1,n2使得n1I1-A,n2I2-B是可逆的,则可加映射Φ:T → T对满足AB=Z的AB∈ T,有Φ(AB)= Φ(A)B=AΦ B 当且仅当 Φ(AB)= Φ(4)B=AΦ()VA B ∈ T.2.素环上中心化子的刻画.设R是包含非平凡幂等元P且含单位元I的素环,假设对(?)A11∈ 1,存在整数n使得nP1-A11在R11中可逆,则可加映射Φ:R→R在Z ∈ R,PZ = Z 点是中心化子,即 Φ(AB)= Φ(A)B = AΦ(B),VA,B ∈ R,Z 当且仅当 Φ(AB)= Φ(A)B=AΦ()(?)A,B ∈ R3.von Neumann代数上中心化子的刻画.设M是没有I1型中心直和项的von Neumann代数,设Z ∈ 使得(I-P = 0,其中P ∈ 满足P = I,P = 0.则可加映射Φ:→ 满足Φ(AB)= Φ(4)B=AΦ()VA,B∈M,AB = Z当且仅当Φ(AB)= Φ(A)B = AΦ(B),VA,B ∈ M.第三章刻画了 B(X)上的Lie导子.主要结论如下:设X是维数至少是2的Banach空间,δ:B(X)→ B(X)是可加映射.本文证明,若存在非平凡幂等算子P ∈ B(X)使得PΩ=Ω,则δ在Ω Lie可导,即δ([A,B])=[δ(A],B]+[A,δ(B)],(?)A,B ∈ B(X),ABΩ 当且仅当存在导子 T:B(X)→ B(X)和可加映射f:B(X)→F,使得 δ(A)= T(A)+f(A)I,(?)A∈B(X),其中 f([A,B])= 0,VA,B∈B(X),AB = Ω特别地,若X = H是Hilbert空间,Ω ∈ B(H)使得ker(Ω)≠ 0或ran(Ω)≠ H,则δ在Ω Lie可导当且仅当δ有上述分解式.
[Abstract]:Left (right) centroids, centroids and Lie derivations are very important contents in the study of operator algebra and operator theory, which have been paid more and more attention by many scholars. In this paper, we mainly characterize the additive mappings on triangular rings, prime rings and von Neumann algebras which are centralizers at a certain point. We discuss the conditions under which additive mappings become centralizers, and then obtain new equivalent characterizations of centralizers on triangular rings, prime rings and von Neumann algebras. At the same time, in this paper, we characterize the additive mappings of the Lie derivative of BX) in the range of indense or non-monojective operators. The structure of the paper is as follows: chapter 1 briefly introduces the background of the research, the main contents of this paper and the necessary conclusions and definitions in the process of proof. In chapter 2, we characterize the centroids of triangular rings, prime rings and von Neumann algebras. The main results are as follows: 1. Characterization of centroids over triangular rings R. Let T = Trigna Agni M B) be a triangulated annulus T is an arbitrary but fixed element. Assuming that for any 4 鈭，

本文编号：1991069