Jordan Maps on Standard Operator Algebras

DOI：10.3770/j.issn:1000-341X.2010.01.010

 作者 单位 纪培胜 青岛大学数学科学学院, 山东 青岛 266071 周淑娟 青岛大学数学科学学院, 山东 青岛 266071

设$A$是维数 $>1$的Banach空间上的标准算子代数,$B$有理数域$Q$上的代数. 如果满射 $M:A\longrightarrow B$和$M^*:B\longrightarrow A$满足任给$a\in A, x\in B$都有$$\left\{ \begin{array}{c}M(r(aM^*(x) M^*(x)a))=r(M(a)x xM(a)),\\M^*(r(M(a)x xM(a)))=r(aM^*(x) M^*(x)a),\end{array}\right$$其中$r$是给定的非零有

Let $A$ be a standard operator algebra on a Banach space of dimension $>1$ and $B$ be an arbitrary algebra over $Q$ the field of rational numbers. Suppose that $M:A\longrightarrow B$ and $M^*:B\longrightarrow A$ are surjective maps such that $$\left\{ \begin{array}{c}M(r(aM^*(x) M^*(x)a))=r(M(a)x xM(a)),\\M^*(r(M(a)x xM(a)))=r(aM^*(x) M^*(x)a)\end{array}\right.$$ for all $a\in A, x\in B$, where $r$ is a fixed nonzero rational number. Then both $M$ and $M^*$ are additive.