Jordan Maps on Standard Operator Algebras

Abstract:

      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.