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.
Information
Published: 1 January 1988
First available in Project Euclid: 18 November 2014
zbMATH: 0684.47004
Rights: Copyright © 1988, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.
