Nonlinear Maps Preserving the Mixed Triple Products between Factors

DOI：10.3770/j.issn:2095-2651.2022.03.008

 作者 单位 张芳娟 西安邮电大学理学院, 陕西 西安 710121 朱新宏 西安现代控制技术研究所, 陕西 西安 710065

设$\mathcal{A}$, $\mathcal{B}$是两个因子且$\dim\mathcal{A}>4$.本文证明了双射$\phi:\mathcal{A}\rightarrow\mathcal{B}$ 满足对所有的$A,B,C\in\mathcal A$有$\phi([A,B]\bullet C)=[\phi(A),\phi(B)]\bullet\phi(C)$当且仅当$\phi$是线性*-同构, 共轭线性*- 同构,负的线性*-同构, 负的共轭线性*-同构.

Let $\mathcal{A}$ and $\mathcal{B}$ be two factors with $\dim\mathcal{A}>4$. In this paper, it is proved that a bijective map $\phi:\mathcal{A}\rightarrow\mathcal{B}$ satisfies $\phi([A,B]\bullet C)=[\phi(A),\phi(B)]\bullet\phi(C)$ for all $A,B,C\in\mathcal A$ if and only if $\phi$ is a linear $*$-isomorphism, or a conjugate linear $*$-isomorphism, or the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism.