素环上强保持2-Jordan乘积的映射

对于给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}_k={{x,y}_(k-1),y}_1,其中{x,y}_0=x,{x,y}_1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f(x),f(y)}2={x,y}_2对所有x,y∈R成立当且仅当存在λ∈l(R的可扩展中心)且λ~3=1,使得下列之一成立:(1)若R的特征不为2,则f(x)=λx对所有x∈R成立;(2)若R的特征为2,则f(x)=λx+μ(x)对所有x∈R成立,其中μ:R→l是一个映射.作为应用,得到了因子von Neumann代数上保持上述性质映射的结构.

Strong 2-Jordan Product Preserving Maps on Prime Rings

For any k ≥ 1, k-Jordan product of two elements x,y in a ring R is defined by {x, y}_k={{x, y}_k-1, y}₁, where {x, y}0=x and {x, y}1=xy + yx. Assume that R is a unital prime ring with a nontrivial idempotent. It is shown that a surjective map f:R → R satisfies {f(x), f(y)}₂={x, y}₂ for all x, y ∈ R if and only if there exists some λ ∈ C (the extended centroid of R) with λ³=1 such that one of the following statements holds:(1) if the characteristic of R is not 2, then f(x)=λx holds for all x ∈ R; (2) if the characteristic of R is 2, then f(x)=λx + μ(x) holds for all x ∈ R, where μ:R → C is a map. As an appication, such maps on factor von Neumann algebras are characterized.