在vN代数的可逆元处可导映射的特征及Spin因子上的Jordan可乘同构

【作者】 孙琳；

【导师】 纪培胜；

【作者基本信息】 青岛大学 ， 基础数学， 2012， 硕士

【摘要】 M是一个无限维复Hilbert空间H上的vN代数,ψ为M上一个线性映射,z∈M,称ψ在Z处可导,如果ψ满足ψ(ST)=ψ(S)T+Sψ(T)对任意S,T∈M并且ST=Z成立.现令Z∈M是一个可逆元,本文证明了若M上范数连续的映射ψ在Z处可导,则ψ在M的单位元I处可导,从而可得ψ是M的一个内导子.设R是实数域,H是维数大于1的实的Hilbert空间,A=H(?)R是相应于H的Spin因子.如果A上的双射φ满足任给x,y∈A都有φ(x·y)=φ(x)·φ(y),并且任给α,β∈R有φ(a+β)=φ(a)+φ(β),则H上存在酉算子U使得任给α∈H,α∈R都有φ(α+α)=Uα+α.更多还原

【Abstract】 Let M be a vN algebra on an infinite-dimensional complex Hilbert space H.Z∈M, ψ is a derivable mapping at Z only if ψ is a linear mapping and ψ(ST)=ψ(S)T+Sψ(T)for any S,t∈M with ST=Z.Let Z∈M be an invertible element,in this paper the result that ψ is a derivable mapping at I which is the unit of M,so ψ is an inner derivation,is proved provided that ψ is a norm-continuous derivable mapping at Z on M.Let R be the field of real numbers and H be a real Hilbert space of dimension at least2.Let A=H(?)R be the Spin factor conesponding to H.In this note,it is proved that if a bijective map φ from A onto itself satisfiesφ(x(?)y)=φ(x)(?)φ(y)for all x,y∈A, and φ(α+β)=φ(α)+φ(β)for all α,β∈R,then there is a unitary operate U on H such thatφ(a+α)=Ua+α for every a∈H,α∈R.更多还原