Invariant Linear Manifolds for CSL-Algebras and Nest Algebras
  Every invariant linear manifold for a CSL-algebra is a closed subspace if, and only if, each non-zero projection in the projection lattice is generated by finitely many atoms. In the case of a nest, this condition is equivalent to the condition that every non-zero projection in the nest has an immediate predecessor (the nest of orthogonal complements is well ordered).   The invariant linear manifolds of a nest algebra are totally ordered by inclusion if, and only if, every non-zero projection in the nest has an immediate predecessor. 
Alan Hopenwasser