Non-reflexive pentagon subspace lattices

Volume 125 / 1997

M. S. Lambrou, W. E. Longstaff Studia Mathematica 125 (1997), 187-199 DOI: 10.4064/sm-125-2-187-199


On a complex separable (necessarily infinite-dimensional) Hilbert space H any three subspaces K, L and M satisfying K∩M = (0), K∨L = H and L⊂M give rise to what has been called by Halmos [4,5] a pentagon subspace lattice $P={(0),K,L,M,H}$. Then n = dim M ⊖ L is called the gap-dimension of P. Examples are given to show that, if n < ∞, the order-interval $[L,M]_{Lat Alg P} = {N ∈ Lat Alg P: L ⊆ N ⊆ M}$ in Lat Alg P can be either (i) a nest with n+1 elements, or (ii) an atomic Boolean algebra with n atoms, or (iii) the set of all subspaces of H between L and M. For n > 1, since Lat Alg P = P∩[L,M]_{Lat Alg P}$, all such examples of pentagons are non-reflexive, the examples in case (iii) extremely so.


  • M. S. Lambrou
  • W. E. Longstaff

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