因子von Neumann代数上的非线性中心化子

Nonlinear centralizers on factor von Neumann algebras

设m,n是任意非零整数,且满足(m+n)(m-n)≠0,M是实或复数域F上的Hilbert空间上的一个因子von Neumann代数.利用代数分解方法证明了M上满足2mφ(AB)+2nφ(BA)=mφ(A)B+mAφ(B)+nφ(B)A+nBφ(A)的非线性映射φ为可加中心化子,并刻画出具体形式φ:A→λA(λ∈F,■A∈M).

Let m,n be non-zero integers with(m+n)(m-n)≠0,M be a factor von Neumann algebra on Hilbert spaces H over the real or complex field F andφbe a nonlinear map from M into itself.By using the algebraic decomposion on M,it is showed that ifφsatisfies 2mφ(AB)+2nφ(BA)=mφ(A)B+mAφ(B)+nφ(B)A+nBφ(A)for all A,B∈M,then the map is an additive centralizer and we characterize the concrete form:there exists aλ∈F such thatφ:A→λA for all A∈M.