Maps preserving zero Jordan products on Hermitian operators

Abstract

Let $H$ be a separable complex Hilbert space and $B_s(H)$ the Jordan algebra of all Hermitian operators on $H$. Let $\theta:B_s(H)\to B_s(H)$ be a surjective ${\mathbb R}$-linear map which is continuous in the strong operator topology such that $\theta(x)\theta(y)+\theta(y)\theta(x)=0$ for all $x,y\in B_s(H)$ with $xy+yx=0$. We show that $\theta(x)=\lambda uxu^*$ for all $x\in B_s(H)$, where $\lambda$ is a nonzero real number and $u$ is a unitary or anti-unitary operator on $H$.