Let \({\mathcal {B}}_{Id}({\mathcal {H}})\) be the set of all idempotents on a Hilbert space \({\mathcal {H}}.\) We give characterizations of the star partial order on \({\mathcal {B}}_{Id}({\mathcal {H}})\) with respect to a particular space decomposition, which is related to Halmos’ two projections theory. Using this, we investigate the lattice properties of \({\mathcal {B}}_{Id}({\mathcal {H}})\) endowed with the star partial order.

中文翻译：

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$$ {\ mathcal {B}} _ {Id}（{\ mathcal {H}}）$$ B Id（H）上的星形偏序

让\（{\ mathcal {B}} _ {ID}（{\ mathcal {H}}）\）是集合所有幂等Hilbert空间\（{\ mathcal {H}} \），我们给予表征关于特定空间分解的\（{\ mathcal {B}} _ {Id}（{\ mathcal {H}}）\）上的恒星偏序，这与Halmos的两个投影理论有关。使用此方法，我们研究了具有星形偏序的\（{\ mathcal {B}} _ {Id}（{\ mathcal {H}}）\）的晶格属性。