Weakly closed Lie modules of nest algebras

TL;DR

This paper characterizes weakly closed Lie modules over nest algebras by explicitly constructing maximal bimodules and describing their relation to the modules and diagonal subalgebras.

Contribution

It provides explicit constructions of maximal bimodules associated with weakly closed Lie modules in nest algebras, clarifying their structure and inclusions.

Findings

Constructed the largest weakly closed bimodule J(L)

Identified a bimodule K(L) with specific properties

Established the inclusion relations involving J(L), K(L), and the diagonal D_{K(L)}

Abstract

Let $\mathcal{T}(\mathcal{N})$ be a nest algebra of operators on Hilbert space and let $\mathcal{L}$ be a weakly closed Lie $\mathcal{T}(\mathcal{N})$-module. We construct explicitly the largest possible weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{J}(\mathcal{L})$ and a weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{K}(\mathcal{L})$ such that \[ \mathcal{J}(\mathcal{L})\subseteq \mathcal{L} \subseteq \mathcal{K}(\mathcal{L}) +\mathcal{D}_{\mathcal{K}(\mathcal{L})}, \] $[\mathcal{K}(\mathcal{L}), \mathcal{T}(\mathcal{N})]\subseteq \mathcal{L}$ and $\mathcal{D}_{\mathcal{K}(\mathcal{L})}$ is a von Neumann subalgebra of the diagonal $\mathcal{T}(\mathcal{N})\cap \mathcal{T}(\mathcal{N})^*$.