Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$ (Communications in Mathematical Research, 2021)
Let $\mathcal{B}(\mathcal{H})$ be the $C^∗$-algebra of all bounded linear operators on a complexHilbert space $\mathcal{H}$. It is proved that an additive surjective map $φ$ on $\mathcal{B}(\mathcal{H})$ preservingthe star partial order in both directions if and only if one of the following assertionsholds. (1) There exist a nonzero complex number $α$ and two unitary operators $\boldsymbol{U}$ and $\boldsymbol{V}$ on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$. (2)There exist a nonzero $α$ and two anti-unitary operators $\boldsymbol{U}$ and $\boldsymbol{V}$ on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$.
Keywords: linear operator; star partial order; additive map.