Continuous surjective maps preserving projections of Jordan products on the space of self-adjoint operators

Abstract

Let ${\mathcal {B}}_{s}({\mathcal {H}})$ be the real linear space of all self-adjoint operators on a complex Hilbert space ${\mathcal {H}}$ with $\dim {\mathcal {H}}\ge 3$. It is proved that a continuous surjective map $\varphi$ on ${\mathcal {B}}_{s}({\mathcal {H}})$ preserves nonzero projections of Jordan products of two operators in both directions if and only if there exist a unitary or an anti-unitary operator U on ${\mathcal {H}}$ and a constant $\lambda$ with $\lambda ^2=1$ such that $\varphi (A)=\lambda {U}^*AU$ for all $A\in {\mathcal {B}}_{s}({\mathcal {H}})$.