Bulletin of the Australian Mathematical Society

Research Article

Abstract reflexive sublattices and completely distributive collapsibility

W. E. Longstaffa1, J. B. Nationa1 and Oreste Panaiaa2

a1 Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia e-mail: longstaf@maths.uwa.edu.au, oreste@maths.uwa.edu.au

a2 Department of Mathematics, University of Hawaii, Honolulu, HI 96822-2273, United States of America e-mail: jb@math.hawaii.edu

Abstract

There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice xs2112 is completely distributive, then xs2112 is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on xs2112 such that xs2112/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

(Received February 11 1998)