## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1988)

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416340109

**Zentralblatt MATH identifier**

0684.47004

#### Citation

Longstaff, W. E. Non-reflexive realizations of non-distributive subspace lattices. Workshop/Miniconference on Functional Analysis and Optimization, , Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1988. https://projecteuclid.org/euclid.pcma/1416340109