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Abstract reflexive sublattices and completely distributive collapsibility. (English) Zbl 0920.47005
Summary: 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 ${\cal L}$ is completely distributive, then ${\cal L}$ is reflexive. In this paper, we study the more general situation of complete lattices for which the least complete congruence $\Delta$ on ${\cal L}$ such that ${\cal L}/\Delta$ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

MSC:
47A15Invariant subspaces of linear operators
06D10Complete distributivity of lattices
06B15Representation theory of lattices
47C05Operators in topological algebras
47L30Abstract operator algebras on Hilbert spaces
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