本文選題：中心化子 + 素環(huán)��； 參考：《太原理工大學(xué)》2017年碩士論文

【摘要】：左(右)中心化子、中心化子及Lie導(dǎo)子是算子代數(shù)與算子理論研究中非常重要的內(nèi)容,受到了許多學(xué)者的廣泛關(guān)注.本文主要刻畫三角環(huán),素環(huán)和von Neumann代數(shù)上在某點(diǎn)是中心化子的可加映射,探討可加映射成為中心化子的條件,進(jìn)而得到三角環(huán),素環(huán)和von Neumann代數(shù)上中心化子的新等價刻畫.同時本文刻畫B(X)在值域不稠或非單射算子Lie可導(dǎo)的可加映射.全文結(jié)構(gòu)如下:第一章簡要介紹所研究問題的背景,本文的主要內(nèi)容以及證明過程中所需的結(jié)論和定義.第二章刻畫了三角環(huán)、素環(huán)、von Neumann代數(shù)上的中心化子,主要結(jié)論如下:1.三角環(huán)R上中心化子的刻畫.設(shè)T = Tri(A,M,B)為三角環(huán),T是任意但固定的元.假設(shè)對任意的4 ∈ A,B ∈ B,存在正整數(shù)n1,n2使得n1I1-A,n2I2-B是可逆的,則可加映射Φ:T → T對滿足AB=Z的AB∈ T,有Φ(AB)= Φ(A)B=AΦ B 當(dāng)且僅當(dāng) Φ(AB)= Φ(4)B=AΦ()VA B ∈ T.2.素環(huán)上中心化子的刻畫.設(shè)R是包含非平凡冪等元P且含單位元I的素環(huán),假設(shè)對(?)A11∈ 1,存在整數(shù)n使得nP1-A11在R11中可逆,則可加映射Φ:R→R在Z ∈ R,PZ = Z 點(diǎn)是中心化子,即 Φ(AB)= Φ(A)B = AΦ(B),VA,B ∈ R,Z 當(dāng)且僅當(dāng) Φ(AB)= Φ(A)B=AΦ()(?)A,B ∈ R3.von Neumann代數(shù)上中心化子的刻畫.設(shè)M是沒有I1型中心直和項的von Neumann代數(shù),設(shè)Z ∈ 使得(I-P = 0,其中P ∈ 滿足P = I,P = 0.則可加映射Φ:→ 滿足Φ(AB)= Φ(4)B=AΦ()VA,B∈M,AB = Z當(dāng)且僅當(dāng)Φ(AB)= Φ(A)B = AΦ(B),VA,B ∈ M.第三章刻畫了 B(X)上的Lie導(dǎo)子.主要結(jié)論如下:設(shè)X是維數(shù)至少是2的Banach空間,δ:B(X)→ B(X)是可加映射.本文證明,若存在非平凡冪等算子P ∈ B(X)使得PΩ=Ω,則δ在Ω Lie可導(dǎo),即δ([A,B])=[δ(A],B]+[A,δ(B)],(?)A,B ∈ B(X),ABΩ 當(dāng)且僅當(dāng)存在導(dǎo)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特別地,若X = H是Hilbert空間,Ω ∈ B(H)使得ker(Ω)≠ 0或ran(Ω)≠ H,則δ在Ω Lie可導(dǎo)當(dāng)且僅當(dāng)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，

本文編號：1991069