Lie Modules of Banach Space Nest Algebras

Abstract

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A-module. We show that there exist a weakly closed A-bimodule K, a weakly closed subalgebra DK of A, and a largest weakly closed A-bimodule J contained in L,such that J⊆L⊆K+DK, with [K,A]⊆L. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.