Automorphisms on the Poset of Products of Two Projections

Abstract

Let ℌ be a complex Hilbert space with dimℌ ≥ 3 and ${\cal B}({\cal H})$ the algebra of all bounded linear operators on ℌ. Let ≤^◇ be the diamond order on ${\cal B}({\cal H})$, that is, for A, $B \in {\cal B}({\cal H})$, we say that A ≤^◇B if

$$\overline {R(A)} \subseteq \overline {R(B)} ,\;\;\;\;\overline {R(A*)} \subseteq \overline {R(B * )} \;\;\;\;\;{\rm{and}}\;\;\;\;\;A{A^ * }A = A{B^ * }A.$$

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Put ${\rm{\Lambda }}\; = \;{\rm{\{ }}PQ\;:\;P,Q\; \in \;{\cal B}({\cal H})$ are projections}. In this paper, the relationship between Λ and ≤^◇ is revealed and then the form of automorphisms of the poset (Λ, ≤^◇) is given.