A decomposition theorem for Lie ideals in nest algebras

Abstract

Let ${\mathcal {N}}$ be a nest and let ${\mathcal {L}}$ be a weakly closed Lie ideal of the nest algebra ${\mathcal {T} (\mathcal {N})}$ . We explicitly construct the greatest weakly closed associative ideal ${\mathcal {J} (\mathcal {L})}$ contained in ${\mathcal {L}}$ and show that ${\mathcal {J} (\mathcal {L}) \subseteq \mathcal {L} \subseteq \mathcal {J} (\mathcal {L})\oplus {\breve{\mathcal{D}}} (\mathcal {L})}$ , where ${{\breve{\mathcal{D}}}} (\mathcal {L})$ is an appropriate subalgebra of the diagonal ${\mathcal {D} (\mathcal {N})}$ of the nest algebra ${\mathcal {T} (\mathcal {N})}$ . We show that norm-preserving linear extensions of elements of the dual of ${\mathcal {L}}$ , satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal ${\mathcal {J} (\mathcal {L})}$ .