TY - JOUR
T1 - Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$
AU - Xi , Cui
AU - JI , Guoxing
JO - Communications in Mathematical Research 
VL - 1
SP - 89
EP - 96
PY - 2021
DA - 2021/05
SN - 31
DO - http://doi.org/10.13447/j.1674-5647.2015.01.10
UR - https://global-sci.org/intro/article_detail/cmr/18951.html
KW - linear operator, star partial order, additive map.
AB - <p style="text-align: justify;">Let $\mathcal{B}(\mathcal{H})$ be the $C^∗$-algebra of all bounded linear operators on a complex
Hilbert space $\mathcal{H}$. It is proved that an additive surjective map $φ$ on $\mathcal{B}(\mathcal{H})$ 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 <span style="text-align: justify;"></span><span style="text-align: justify;">$\boldsymbol{U}$&nbsp;and&nbsp;$\boldsymbol{V}$</span><span style="text-align: justify;"></span> on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$&nbsp;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&nbsp;$\boldsymbol{U}$&nbsp;and&nbsp;$\boldsymbol{V}$&nbsp;on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$.</p>