A decomposition theorem for Lie ideals in nest algebras

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})}) .