Strong Skew Commutativity Preserving Maps on Rings with Involution  

作　　者：Chang Jing LI Quan Yuan CHEN 

机构地区：[1]School of Mathematical Sciences, Shandong Normal University [2]College of Information, Jingdezhen Ceramic Institute

出　　处：《Acta Mathematica Sinica,English Series》2016年第6期745-752,共8页数学学报（英文版）

基　　金：Supported by Natural Science Foundation of Shandong Province,China(Grant No.ZR2015Item PA010);National Natural Science Foundation of China(Grant Nos.11526123 and 11401273)

摘　　要：Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф： R →R is strong skew commutativity preserving （that is, satisfies Ф（A）Ф（B） - Ф（B）Ф（A）＊ ： AB- BA＊ for all A, B ∈R） if and only if there exist a map f ： R → ZSz（R） and an element Z ∈ ZS（R） with Z^2 =I such that Ф（A） =ZA ＋ f（A） for all A ∈ R, where ZS（R） is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime ＊-rings and von Neumann algebras with no central summands of type I1 are characterized.Let R be a unital ＊-ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP= 0 implies A = 0 and AR（I - P） = 0 implies A = 0. In this paper, it is shown that a surjective map Ф： R →R is strong skew commutativity preserving （that is, satisfies Ф（A）Ф（B） - Ф（B）Ф（A）＊ ： AB- BA＊ for all A, B ∈R） if and only if there exist a map f ： R → ZSz（R） and an element Z ∈ ZS（R） with Z^2 =I such that Ф（A） =ZA ＋ f（A） for all A ∈ R, where ZS（R） is the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime ＊-rings and von Neumann algebras with no central summands of type I1 are characterized.

关 键 词：Strong skew commutativity preserving von Neumann algebras prime rings 

分 类 号：O177[理学—数学] O153.3[理学—基础数学]