*-代数上的一类线性双射

A class of linear bijections on algebra with involution

目的设A和B是含单位元的*-代数,Φ:A→B是线性双射。揭示了满足Φ(AA*A)=Φ(A)Φ(A*)Φ(A)(A∈A)的映射Φ与Jordan同构的关系;同时也揭示了满足Φ(AA*A)=Φ(A)Φ(A)*Φ(A)(A∈A)的映射Φ与Jordan*-同构的关系。方法从Jordan同构和Jordan*-同构的定义入手,运用Φ的线性性和满性进行了证明。结果如果对任意的A∈A有Φ(AA*A)=Φ(A)Φ(A*)Φ(A),则Φ是一个可逆元乘一个Jordan同构;如果对任意的A∈A有Φ(AA*A)=Φ(A)Φ(A)*Φ(A),则Φ是一个酉元乘一个Jordan*-同构。结论为进一步研究Jordan同构提供了新的思路。

Aim Let A and B be unital algebras with involution, Ф： A→B is a linear bijective mapping. The relation between the map Ф satisfying Ф（AA＊A）=Ф（A）Ф（A＊）Ф（A）（arbitary A∈A） and Jordan isomorphism is shown. Meanwhile, the relation between the map Ф satisfying Ф（AA＊A） = Ф（A）Ф（A）＊Ф（A）（arbitary A∈A） and Jordan ＊ -isomorphism is also shown. Methods The definition of Jordan isomorphism and Jordan ＊-isomorphism, linearity and surjectivity of Ф are used to solve the problem. Results If Ф（AA＊A）=Ф（A）Ф（A＊）Ф（A） for all A∈A, then Ф is an invertible element multiple of a Jordan isomorphism; If Ф（AA＊A） =Ф（A）Ф（A）＊Ф（A） for all A∈A, then Ф is a unitary element multiple of a Jordan ＊-isomorphism. Conclusion A new way is provided to research Jordan isomorphism.