Finite rank operators in Lie ideals of nest algebras

TL;DR

This paper characterizes finite rank operators within Lie ideals of continuous nest algebras, linking them to order continuous homomorphisms and demonstrating their decomposability into rank one operators.

Contribution

It provides a new characterization of finite rank operators in Lie ideals of nest algebras using order continuous homomorphisms and proves their decomposability.

Findings

Finite rank operators in Lie ideals are characterized by a condition involving order continuous homomorphisms.

Finite rank operators in these ideals can be decomposed into sums of rank one operators within the ideal.

The main theorem offers a complete description of such operators in continuous nest algebras.

Abstract

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.