## Proceedings of the Centre for Mathematics and its Applications

### Non-reflexive realizations of non-distributive subspace lattices

W. E. Longstaff

#### Abstract

Let $F$ be a subspace lattice on a (complex) Hilbert space H . A subspace lattice $F$ on a Hilbert space K is a realization of $F$ on K if $G$ is lattice-isomorphic to $F$. In 1975 it was proved that if $F$ is completely distributive every realization $G$ of it is reflexive (that is, $F$ is the set of invariant subspaces of a family of operators). A partial converse has recently been found: If every realization of $F$ is reflexive and $F$ has a finite--dimensional realization, then $F$ is completely distributive. This is proved by showing that every non-distributive subspace lattice on a finite-dimensional space has a non-reflexive realization on the same space.

#### Article information

Dates
First available in Project Euclid: 18 November 2014