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Finite rank operators in Lie ideals of nest algebras

The main theorem provides a characterisation of the finite rank operators lying in a norm closed Lie ideal of a continuous nest algebra. These operators are charaterised as those finite rank operators in the nest algebra satisfying a condition determined by a left order continuous homomorphism on the nest. A crucial fact used in the proof of this theorem is the decomposability of the finite rank operators. One shows that a finite rank operator in a norm closed Lie ideal of a continuous nest algebra can be written as a finite sum of rank one operators lying in the ideal.