Non-reflexive pentagon subspace lattices

M. S.  Lambrou; W. E.  Longstaff

ArticleOriginal scientific text

Title

Non-reflexive pentagon subspace lattices

Authors ¹, ²

Affiliations

Department of Mathematics, University of Crete, Iraklion, Crete Greece
Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia

Abstract

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]}_{L a t A l g P} = {N \in L a t A l g P : L \subseteq N \subseteq 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.

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Title

Non-reflexive pentagon subspace lattices

Affiliations

Abstract

Bibliography