Kadison's transitivity theorem implies that, for
irreducible representations of \cstar algebras, every
invariant linear manifold is closed. It is known that CSL
algebras have this property if, and only if, the lattice is
hyperatomic (every projection is generated by a finite number
of atoms). We show several other conditions are equivalent,
including the condition that every invariant linear manifold
is singly generated. \par We show that two families of norm
closed operator algebras have this property. First, let $\LL$
be a CSL and suppose $\AA$ is a norm closed algebra which is
weakly dense in $\operatorname{Alg} \LL$ and is a bimodule
over the (not necessarily closed) algebra generated by the
atoms of $\LL$. If $\LL$ is hyperatomic and the compression of
$\AA$ to each atom of $\LL$ is a \cstar algebra, then every
linear manifold invariant under $\AA$ is closed. Secondly, if
$\AA$ is the image of a strongly maximal triangular AF algebra
under a multiplicity free nest representation, where the nest
has order type $-\mathbb{N}$, then every linear manifold
invariant under $\AA$ is closed and is singly generated.