On Lattices Whose Every Realization on Hilbert Space is Reflexive
W. E. Longstaff · Journal of the London Mathematical Society · 1988
Let L be an abstract complete lattice. Call a subspace lattice ℒ on a complex Hilbert space H a realization of L on H if ℒ is lattice-isomorphic to L. The author has previously observed that if L is completely distributive, then every realization of it is reflexive. Here the converse is proved under the additional assumption that L has a realization on a finite-dimensional space. This is done by showing that every non-distributive subspace lattice on a finite-dimensional space has a non-reflexive realization on the same space.