Volume 31, Issue 1
Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$

Commun. Math. Res., 31 (2015), pp. 89-96.

Published online: 2021-05

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• Abstract

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 $\boldsymbol{U}$ 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.

• AMS Subject Headings

47B49, 47B47

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COPYRIGHT: © Global Science Press

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@Article{CMR-31-89, author = {Cui and Xi and and 18605 and and Cui Xi and Guoxing and JI and and 18606 and and Guoxing JI}, title = {Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {31}, number = {1}, pages = {89--96}, abstract = {

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 $\boldsymbol{U}$ 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 $X ∈ \mathcal{B}(\mathcal{H})$.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2015.01.10}, url = {http://global-sci.org/intro/article_detail/cmr/18951.html} }
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 -

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 $\boldsymbol{U}$ 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 $X ∈ \mathcal{B}(\mathcal{H})$.

Cui Xi & Guoxing JI. (2021). Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$. Communications in Mathematical Research . 31 (1). 89-96. doi:10.13447/j.1674-5647.2015.01.10
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