(untitled) (Communications in Mathematical Research, 2021)
Let B(H) be the C^∗-algebra of all bounded linear operators on a complex Hilbert space H. It is proved that an additive surjective map φ on B(H) preserving the star partial order in both directions if and only if one of the following assertions holds. (1) There exist a nonzero complex number α and two unitary operators U and V on H such that φ(X) = αUXV or φ(X) = αUX^∗V for all X ∈ B(H). (2) There exist a nonzero α and two anti-unitary operators U and V on H such that φ(X) = αUXV or φ(X) = αUX^∗V for all X ∈ B(H).