Kadison's transitivity theorem implies that, for irreducible representations
of C*-algebras, every invariant linear manifold is closed. It is known that CSL
algebras have this propery 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 conditon that every invariant linear
manifold is singly generated.
We show that two families of norm closed operator algebras have this
property. First, let L be a CSL and suppose A is a norm closed algebra which is
weakly dense in Alg L and is a bimodule over the (not necessarily closed)
algebra generated by the atoms of L. If L is hyperatomic and the compression of
A to each atom of L is a C*-algebra, then every linear manifold invariant under
A is closed. Secondly, if A is the image of a strongly maximal triangular AF
algebra under a multiplicity free nest representation, where the nest has order
type -N, then every linear manifold invariant under A is closed and is singly
generated.Comment: AMS-LaTeX, 15 pages, minor revision
