Maps Preserving Commutativity up to a Factor on Standard Operator Algebras

DOI：10.3770/j.issn:2095-2651.2013.06.007

 作者 单位 焦美艳 山西财经大学应用数学学院, 山西 太原 030006

设$X$和$Y$是实数域或复数域上维数大于2的Banach空间, ${\mathcal A}$和${\mathcal B}$分别是$X$和$Y$上的标准算子代数,$\Phi:{\mathcal A}\rightarrow {\mathcal B}$是保单位的满射. 本文中，我们给出了满足对任意的$A, B, R\in {\mathcal A}$以及某个数$\xi$, $(A-B)R=\xi R(A-B)\Leftrightarrow (\Phi(A)-\Phi(B))\Phi(R)=\xi\Phi(R)(\Phi(A)-\Phi(B))$ 的一般映射$\Phi$的具体形式.

Let $X$, $Y$ be real or complex Banach spaces with dimension greater than 2 and ${\mathcal A}$, ${\mathcal B}$ be standard operator algebras on $X$ and $Y$, respectively. Let $\Phi:\mathcal A \rightarrow \mathcal B$ be a unital surjective map. In this paper, we characterize the map $\Phi$ on $\mathcal A$ which satisfies $(A-B)R=\xi R(A-B)\Leftrightarrow (\Phi(A)-\Phi(B))\Phi(R)=\xi\Phi(R)(\Phi(A)-\Phi(B))$ for $A,B,R\in \mathcal A$ and for some scalar $\xi$.