Automatic closure of invariant linear manifolds for operator algebras

Abstract

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.