Maps Preserving Zero Triple Jordan Products And Idempotency Of Jordan Products On Operator Spaces

The preserver problem is one of important research fields of operator algebras. This paper studies the additive maps preserving zero triple Jordan products on a symmetric operator space and nonzero idempotency of Jordan products on B(X). Firstly, it is shown that an additive map φ on a symmetric operator space preserving zero triple Jordan products preserves rank one operators in both directions and so preserves adjacency in both directions. Consequently, there exist a nonzero scalar c and a linear or conjugate linear invertible operator A：Hâ†'H satisfying AAt=I and φ(T)=cAT At,(?)T∈(?)y(H). Next, we discuss an additive map on B(X) preserving nonzero idempotency of Jordan products. Let X be a complex Banach space and dim(χ)≥3, it is proved the additive map on B(X) preserving nonzero idempotency of Jordan products must be a Jordan isomorphism multiplied by a scalar.