Additive maps preserving commutativity up to a factor on nest algebras
上传者:曹轶|上传时间:2015-04-24|密次下载
Additive maps preserving commutativity up to a factor on nest algebras
This article was downloaded by: [Beijing Institute of Technology]
On: 28 March 2015, At: 18:26
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
内容需要下载文档才能查看 内容需要下载文档才能查看office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UKLinear and Multilinear Algebra
Publication details, including instructions for authors and
subscription information:Additive maps preserving
commutativity up to a factor on nest
algebras
Meiyan Jiao & Xiaofei Qi
aab Department of Applied Mathematics, Shanxi University of
Finance & Economics, Taiyuan, P.R. China.
b Department of Mathematics, Shanxi University, Taiyuan, P.R.
China.
Published online: 18 Jul 2014.
内容需要下载文档才能查看PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,
and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,
systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &
内容需要下载文档才能查看
Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015
LinearandMultilinearAlgebra,2015
Vol.63,No.6,1242–1256,http://wendang.chazidian.com/10.1080/03081087.2014.927985
Additivemapspreservingcommutativityuptoafactoronnest
algebras
MeiyanJiaoa?andXiaofeiQib
aDepartmentofAppliedMathematics,ShanxiUniversityofFinance&Economics,Taiyuan,
P.R.China;bDepartmentofMathematics,ShanxiUniversity,Taiyuan,P.R.China
Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015 CommunicatedbyB.Kuzma(Received30April2014;accepted5May2014)LetNandMbetwonestswithN∈NandM∈McomplementedinNandMwheneverN?=NandM?=M,respectively.Assumethat??:AlgN→AlgMisaunitaladditivesurjection.Inthispaper,alladditivemaps??preservingcommutativityuptoafactorξ(ξ=0,±1)inbothdirections(i.e.??satis?es??(A)??(B)=ξ??(B)??(A)ifandonlyifAB=ξBAforallA,B∈AlgN)arecharacterized.Keywords:nestalgebras;commutativityuptoafactor;JordanisomorphismsAMSSubjectClassi?cations:Primary:47B49;47A121.IntroductionLetAandBbetwoassociativeringsoralgebras.Recallthatanadditivemap??:A→Bpreserveszero-products(inbothdirections)if,forA,B∈A,??(A)??(B)=0whenever(ifandonlyif)AB=0;preservesJordanzero-products(inbothdirections)if,forA,B∈A,??(A)??(B)+??(B)??(A)=0whenever(ifandonlyif)AB+BA=0;preservesLiezero-products(inbothdirections)if,forA,B∈A,??(A)??(B)???(B)??(A)=0whenever(ifandonlyif)AB?BA=0.Thequestionofcharacterizingadditivemapspreservingzero-products(Jordanzero-productsorLiezero-products)hasbeendiscussed,sayin[1–5](seealsoreferencestherein).LetAandBbetwoalgebrasovera?eldF.Forx,y∈Aandξ∈F,ifxy=ξyx,wesaythatxandyarecommutativeuptoafactorξ.Thecommutativityuptoafactor
forpairsofoperatorshasbeenthesubjectofstudyinthecontextofquantumgroups,[6]andtheirmatrixrealizationsgiveexamplesofoperatorpairscommutinguptoafactor.[7]Recallthatanadditivemap??:A→Bpreservescommutativityuptoafactorξinbothdirectionsif,foreveryA,B∈A,??(A)??(B)=ξ??(B)??(A)ifandonlyifAB=ξBA.ItisclearthattheconceptcorrespondstotheabovepreservingLiezero-productsinbothdirections,preservingzero-productsinbothdirectionsandpreservingJordanzero-productsinbothdirectionsifξ=1,0,?1respectively.Cuietal.in[8]characterizedtheunitaladditivesurjectivemapspreservingcommutativityuptoafactorξ(ξ=0,±1)inbothdirectionsbetweenstandardoperatoralgebrasonrealandcomplexin?nite-dimensional?Correspondingauthor.Email:jmyzgl@http://wendang.chazidian.com
©2014Taylor&Francis
LinearandMultilinearAlgebra1243
Banachspaces.Molnár[9]gaveacharacterizationofbijectivelinearmapspreservingcommutativityuptoafactorinbothdirectionsonthealgebraofn×ncomplexmatricesandonthespaceofalln×nself-adjointmatrices,respectively.Here,weremarkthat,in
[9],theconceptofcommutativityuptoafactorisslightlydifferentfromwhatwegive.Theauthor[9]saidthatamap??preservescommutativityuptoafactorinbothdirectionsif,foranyAandB,AB=ξBAforsomeξifandonlyif??(A)??(B)=η??(B)??(A)forsomeη,whereξandηmaybedifferent.Obviously,ourde?nitionismorerestrictive.Thepurposeofthepresentpaperistocharacterizetheadditivemapspreservingcom-mutativityuptoafactorξinbothdirectionsonanotherimportantoperatoralgebra:nestalgebras.Notethatnestalgebrasareneitherself-adjointnorsemi-simpleandprime.Itshouldbepointedoutthat,forthecaseξ=0,HouandZhang[5]characterizedadditivesurjections??with??(I)af?ne(thatis,??(I)injectivewithdenserange)betweennestalgebrasonin?nite-dimensionalBanachspaceswhichpreservezero-productsinbothdirections;forthecaseξ=?1,HouandJiao[3]provedthat,undersomemildconditionsonnests,anunitaladditivesurjectivemap??betweennestalgebraspreservesJordanzero-productsinbothdirectionsifandonlyif??iseitheraringisomorphismoraringanti-isomorphism;forthecaseξ=1,Benkovic?andEremita[10]gaveacompletecharacterizationofalladditivebijectivemapspreservingcommutativityinbothdirectionbetweennestalgebras,undertheadditionalassumptionoftheexistenceofacomplementednontrivialelementinthenest.Hence,wedealwiththecaseξ=0,±1inthispaper.
LetXandYbeBanachspacesovertherealorcomplex?eldF.Asusual,B(X)andF(X)denotetheBanachspaceofallboundedlinearoperatorsfromXintoXandthesubspaceofall?niterankoperatorsinB(X),respectively.AnestonXisachainNofclosed(undernormtopology)subspacesofXcontaining{0}??andX,whichisclosedunderthe??formationofarbitraryclosedlinearspan(denotedby)andintersection(denotedby).AlgNdenotestheassociatednestalgebra,whichisthesetofalloperatorsTinB(X)suchthatTN?NforeveryelementN∈N.WhenN={{0},X},wesaythatNisnontrivial.ItisclearthatifNis.ForN∈N,let????trivial,thenAlgN=B(X)⊥N?={M∈N|M?N},N+={M∈N|N?M}andN?=(N?)⊥,whereN⊥={f∈X?|N?ker(f)}andX?isthedualofX.Let{0}?=0andX+=X.Itiswellknownthatarankoneoperatorx?fbelongstoAlgNifandonlyifthereissome⊥.DenotebyAlgNthesetofall?niterankoperatorsN∈Nsuchthatx∈Nandf∈N?FinAlgN.ItisknownthatAlgFNisadensesubsetofAlgNunderthestrongoperatortopology.Formoreinformationonnestalgebras,wereferto[11].
Thepaperisorganizedasfollows.InSection2,welistthemainresultsobtainedinthepresentpaper.InSection3,wegivesomelemmas,whichareneededforprovingourmainresults.Section4isdevotedtoproofsofthemainresults.
2.Mainresults
Inthissection,wewillstatethemainresultsinthispaper.
Theorem2.1LetNandMbenestsonin?nite-dimensionalBanachspacesXandYovertherealorcomplex?eldF,respectively,withN∈NandM∈McomplementedinXandYrespectivelywheneverN?=NandM?=M.Let??:AlgN→AlgMbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthefollowingsholds.Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015
1244
(1)M.JiaoandX.QiIfξ∈R,thenthereexistadimensionpreservingorderisomorphismθ:N→M
andaninvertibleboundedlinearorconjugate-linearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN.Ifξ∈C\Rand|ξ|=1,thenthereexistadimensionpreservingorderisomorphismθ:N→MandaninvertibleboundedlinearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN.
If|ξ|=1,theneitherthereexistadimensionpreservingorderisomorphismθ:N→MandaninvertibleboundedlinearoperatorT:X→YsuchthatT(N)=θ(N)foreveryN∈Nand??(A)=TAT?1forallA∈AlgN,orthereexistadimensionpreservingorderisomorphismθ:N⊥→Mandaninvertible⊥)=θ(N⊥)forboundedconjugate-linearoperatorT:X?→YsuchthatT(N????1everyN∈Nand??(A)=TATforallA∈AlgN.(2)(3)
Downloaded by [Beijing Institute of Technology] at 18:26 28 March 2015 SinceeverylinearsubspaceofaHilbertspaceiscomplemented,thefollowingcorollaryisimmediatefromTheorem2.1.Corollary2.2LetNandMbenestsonin?nite-dimensionalHilbertspacesHandKoverthe(realorcomplex)?eldF,respectively.Let??:AlgN→AlgMbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthethreeformsinTheorem2.1holds.Forthe?nite-dimensionalcase,itisclearthateverynestalgebraon?nite-dimensionalspacesisisomorphictoanuppertriangularblockmatrixalgebra.LetMn=Mn(F)bethematrixalgebraoverthe?eldF.LetT=T(n1,n2,...,nk)?Mnbeanuppertriangularblockmatrixsubalgebra,i.e.n1+n2+...+nk=n,T={A=(Aij)k×k|Aij∈Mni,njandAij=0ifi>j}.Theorem2.3LetFbetherealorcomplex?eld,andm,nbepositiveintegersgreaterthan1.LetT=T(n1,n2,...,nk)?Mn(F)andS=T(m1,m2,...,mr)?Mm(F)beuppertriangularblockmatrixalgebras,and??:T→Sbeaunitaladditivesurjectivemap.Then??preservescommutativityuptoafactorξinbothdirectionswithξ=0,±1ifandonlyifoneofthefollowingsholds.(1)(n1,n2,...,nk)=(m1,m2,...,mr),thereexistanautomorphismτ:F→Fwithτ(ξ)=ξandaninvertiblematrixT∈Tsuchthat??(A)=TAτT?1forallA∈T;(n1,n2,...,nk)=(mr,mr?1,...,m1),thereexistanautomorphismτ:F→F1andaninvertibleblockmatrixT=(Tij)k×kwithTij∈Mni,njandwithτ(ξ)=?1forallA∈T.Tij=0wheneveri+j>k+1,suchthat??(A)=TAtrτT
HereAτ=(τ(aij))n×nforA=(aij)n×n∈Mn(F)andAtristhetransposeofA.In(1),ifF=R,thenτ=id(i.e.τ(t)=tforallt∈R).(2)
Particularly,if??islinear,wehavethefollowingresults.
Theorem2.4LetFbetherealorcomplex?eld,andm,nbepositiveintegersgreaterthan1.LetT=T(n1,n2,...,nk)?Mn(F)andS=T(m1,m2,...,mr)?Mm(F)beuppertriangularblockmatrixalgebras.Assumethat??:T→Sisalinearsurjective
下载文档
热门试卷
- 2016年四川省内江市中考化学试卷
- 广西钦州市高新区2017届高三11月月考政治试卷
- 浙江省湖州市2016-2017学年高一上学期期中考试政治试卷
- 浙江省湖州市2016-2017学年高二上学期期中考试政治试卷
- 辽宁省铁岭市协作体2017届高三上学期第三次联考政治试卷
- 广西钦州市钦州港区2016-2017学年高二11月月考政治试卷
- 广西钦州市钦州港区2017届高三11月月考政治试卷
- 广西钦州市钦州港区2016-2017学年高一11月月考政治试卷
- 广西钦州市高新区2016-2017学年高二11月月考政治试卷
- 广西钦州市高新区2016-2017学年高一11月月考政治试卷
- 山东省滨州市三校2017届第一学期阶段测试初三英语试题
- 四川省成都七中2017届高三一诊模拟考试文科综合试卷
- 2017届普通高等学校招生全国统一考试模拟试题(附答案)
- 重庆市永川中学高2017级上期12月月考语文试题
- 江西宜春三中2017届高三第一学期第二次月考文科综合试题
- 内蒙古赤峰二中2017届高三上学期第三次月考英语试题
- 2017年六年级(上)数学期末考试卷
- 2017人教版小学英语三年级上期末笔试题
- 江苏省常州西藏民族中学2016-2017学年九年级思想品德第一学期第二次阶段测试试卷
- 重庆市九龙坡区七校2016-2017学年上期八年级素质测查(二)语文学科试题卷
- 江苏省无锡市钱桥中学2016年12月八年级语文阶段性测试卷
- 江苏省无锡市钱桥中学2016-2017学年七年级英语12月阶段检测试卷
- 山东省邹城市第八中学2016-2017学年八年级12月物理第4章试题(无答案)
- 【人教版】河北省2015-2016学年度九年级上期末语文试题卷(附答案)
- 四川省简阳市阳安中学2016年12月高二月考英语试卷
- 四川省成都龙泉中学高三上学期2016年12月月考试题文科综合能力测试
- 安徽省滁州中学2016—2017学年度第一学期12月月考高三英语试卷
- 山东省武城县第二中学2016.12高一年级上学期第二次月考历史试题(必修一第四、五单元)
- 福建省四地六校联考2016-2017学年上学期第三次月考高三化学试卷
- 甘肃省武威第二十三中学2016—2017学年度八年级第一学期12月月考生物试卷
网友关注
- 园林景观工程设计取费标准
- 家装过程中甲醛的主要来源
- 居住区景观设计说明
- 单 位 工 程 质 量 评 估 报 告18#
- 标签的背后_绿色的承诺_艾利丹尼森GlobalMDO的绿色足迹_朱伟
- 主导产业选择与结构优化模型分析_以河北省为例
- 垂直绿化在城市园林中的应用
- LED日光灯管
- 动态图像技术领先 等离子还需解决功耗问题
- 土力学(重点看计算) (1)
- 脱硝装置对袋式除尘器运行的影响及其防治措施
- 国内生活方式购物中心交往空间研究——以北京蓝色港湾为例
- 小论水污染的治理
- 西澳大学城市设计硕士专业
- 水电站 (2)
- 外部规模经济、拥挤效应与城市发展+一个新经济地理学城市模型
- 人们怎样才能喝上健康水
- 公共设施等换乘枢纽
- 国内外工程造价计价模式比较研究_王星
- 三网整合中的FTTH网络设计与实践
- 水火箭接口的两种制作方法
- 绿色建筑
- 光纤到户(FTTH)的ODN工程设计及测试
- 家用纯水机有罐和无罐的区别表现
- 新立河公园
- 《设计家》杂志官网征稿 森林公园设计初探
- 矿山沉陷的模型识别和模型优化
- 场地分析图常用技巧大列举
- 怎么装修除甲醛 改善室内空气质量
- 连续梁预应力管道压浆施工技术交底
网友关注视频
- 第4章 幂函数、指数函数和对数函数(下)_六 指数方程和对数方程_4.7 简单的指数方程_第一课时(沪教版高一下册)_T1566237
- 化学九年级下册全册同步 人教版 第25集 生活中常见的盐(二)
- 8.对剪花样_第一课时(二等奖)(冀美版二年级上册)_T515402
- 【部编】人教版语文七年级下册《泊秦淮》优质课教学视频+PPT课件+教案,天津市
- 三年级英语单词记忆下册(沪教版)第一二单元复习
- 冀教版小学数学二年级下册第二单元《有余数除法的整理与复习》
- 外研版英语七年级下册module3 unit2第二课时
- 外研版八年级英语下学期 Module3
- 【部编】人教版语文七年级下册《泊秦淮》优质课教学视频+PPT课件+教案,湖北省
- 人教版二年级下册数学
- 冀教版小学英语五年级下册lesson2教学视频(2)
- 【部编】人教版语文七年级下册《老山界》优质课教学视频+PPT课件+教案,安徽省
- 沪教版八年级下次数学练习册21.4(2)无理方程P19
- 沪教版八年级下册数学练习册20.4(2)一次函数的应用2P8
- 苏教版二年级下册数学《认识东、南、西、北》
- 河南省名校课堂七年级下册英语第一课(2020年2月10日)
- 【部编】人教版语文七年级下册《老山界》优质课教学视频+PPT课件+教案,安徽省
- 19 爱护鸟类_第一课时(二等奖)(桂美版二年级下册)_T3763925
- 七年级英语下册 上海牛津版 Unit9
- 第五单元 民族艺术的瑰宝_16. 形形色色的民族乐器_第一课时(岭南版六年级上册)_T1406126
- 外研版英语三起5年级下册(14版)Module3 Unit2
- 19 爱护鸟类_第一课时(二等奖)(桂美版二年级下册)_T502436
- 沪教版八年级下册数学练习册21.3(3)分式方程P17
- 冀教版英语五年级下册第二课课程解读
- 冀教版小学英语四年级下册Lesson2授课视频
- 沪教版牛津小学英语(深圳用) 五年级下册 Unit 12
- 《小学数学二年级下册》第二单元测试题讲解
- 3月2日小学二年级数学下册(数一数)
- 第19课 我喜欢的鸟_第一课时(二等奖)(人美杨永善版二年级下册)_T644386
- 每天日常投篮练习第一天森哥打卡上脚 Nike PG 2 如何调整运球跳投手感?
精品推荐
- 2016-2017学年高一语文人教版必修一+模块学业水平检测试题(含答案)
- 广西钦州市高新区2017届高三11月月考政治试卷
- 浙江省湖州市2016-2017学年高一上学期期中考试政治试卷
- 浙江省湖州市2016-2017学年高二上学期期中考试政治试卷
- 辽宁省铁岭市协作体2017届高三上学期第三次联考政治试卷
- 广西钦州市钦州港区2016-2017学年高二11月月考政治试卷
- 广西钦州市钦州港区2017届高三11月月考政治试卷
- 广西钦州市钦州港区2016-2017学年高一11月月考政治试卷
- 广西钦州市高新区2016-2017学年高二11月月考政治试卷
- 广西钦州市高新区2016-2017学年高一11月月考政治试卷
分类导航
- 互联网
- 电脑基础知识
- 计算机软件及应用
- 计算机硬件及网络
- 计算机应用/办公自动化
- .NET
- 数据结构与算法
- Java
- SEO
- C/C++资料
- linux/Unix相关
- 手机开发
- UML理论/建模
- 并行计算/云计算
- 嵌入式开发
- windows相关
- 软件工程
- 管理信息系统
- 开发文档
- 图形图像
- 网络与通信
- 网络信息安全
- 电子支付
- Labview
- matlab
- 网络资源
- Python
- Delphi/Perl
- 评测
- Flash/Flex
- CSS/Script
- 计算机原理
- PHP资料
- 数据挖掘与模式识别
- Web服务
- 数据库
- Visual Basic
- 电子商务
- 服务器
- 搜索引擎优化
- 存储
- 架构
- 行业软件
- 人工智能
- 计算机辅助设计
- 多媒体
- 软件测试
- 计算机硬件与维护
- 网站策划/UE
- 网页设计/UI
- 网吧管理