Maps preserving numerical radius or cross norms of products of self-adjoint operators

Let H be a complex Hilbert space with dim H ≥ 3, B _s (H) the (real) Jordan algebra of all self-adjoint operators on H. Every surjective map Φ: B _s (H) → B _s (H) preserving numerical radius of operator products (respectively, Jordan triple products) is characterized. A characterization of surjective maps on B _s (H) preserving a cross operator norm of operator products (resp. Jordan triple products of operators) is also given.