LAUSR

Let ℌ be a complex Hilbert space with dimℌ ≥ 3 and $${\cal B}({\cal H})$$ℬ(ℋ) the algebra of all bounded linear operators on ℌ. Let ≤◇ be the diamond order on $${\cal B}({\cal H})$$ℬ(ℋ), that is, for A, $$B \in {\cal B}({\cal H})$$B∈ℬ(ℋ), we say that A ≤◇B if $$\overline {R(A)} \subseteq \overline {R(B)} ,\;\;\;\;\overline {R(A*)} \subseteq \overline {R(B * )} \;\;\;\;\;{\rm{and}}\;\;\;\;\;A{A^ * }A = A{B^ * }A.$$R(A)¯⊆R(B)¯,R(A*)¯⊆R(B*)¯andAA*A=AB*A..Put $${\rm{\Lambda }}\; = \;{\rm{\{ }}PQ\;:\;P,Q\; \in \;{\cal B}({\cal H})$$Λ={PQ:P,Q∈B(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.