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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.

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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