The existence of maximal \(w^*\)-closed submodules in nest algebra modules (Q884877)

A nest \({\mathcal N}\) is a chain of closed subspaces of a complex Hilbert space \({\mathbb H}\), which is ordered by inclusion, contains \((0)\) and \({\mathbb H}\), and is closed under intersection and closed linear span of its elements. The corresponding nest algebra is the set of operators \({\mathcal T}({\mathcal N}):=\{T:TN\subseteq N\;\forall N\in {\mathcal N}\}\). If the map \(N\mapsto \widetilde{N}\) is a left-continuous order homomorphism of a nest \({\mathcal N}\) into itself, then the set \({\mathcal U}:=\{X: XN\subseteq \widetilde{N}\;\forall N\in {\mathcal N}\}\) is a weakly closed \({\mathcal T}({\mathcal N})\)-module. Indeed, \textit{J.\,A.\thinspace Erdos} and \textit{S.\,C.\thinspace Power} [J.~Oper.\ Theory 7, 219--235 (1982; Zbl 0523.47027)] showed that every weakly closed \({\mathcal T}({\mathcal N})\)-module has this form. For a finite nest \({\mathcal F}\) consisting of \((0)=N_0<N_1<\cdots <N_n={\mathbb H}\) and an operator \(T\), define \(\widetilde{\mathcal D}_{\mathcal F}(T) := \sum_{i=1}^{n}{(P(\widetilde{N_i})-P(\widetilde{N_{i-1}}))T (P({N_i})-P({N_{i-1}}))}\). For a weakly closed \({\mathcal T}({\mathcal N})\)-module \({\mathcal U}\), define the corresponding finite-partition submodule \({\mathcal R}({\mathcal N},\widetilde{}\;)\) to be the set of operators \(T\) in \({\mathcal U}\) such that, for every \(\epsilon>0\), there exists a finite partition \({\mathcal F}\) of the nest \({\mathcal N}\) for which \(\|\widetilde{\mathcal D}_{\mathcal F}(T)\|<\epsilon\). In the event that the order homomorphism on \({\mathcal N}\) is the identity, then \({\mathcal R}({\mathcal N},\widetilde{}\;)\) is the Jacobson radical of the nest algebra \({\mathcal T}({\mathcal N})\). In the final section of this well crafted paper, the author provides a characterization of the pre-annihilator in the trace class of a finite-partition submodule. This leads to a characterization of the weak* closure of \({\mathcal R}({\mathcal N},\widetilde{}\;)\) as the set of operators \(\{X:XN\subseteq \widetilde{N_-}\;\forall N\in {\mathcal N}\}\), where \(N_-:=\vee\{N^\prime\in {\mathcal N}:N^\prime <N\}\) (Theorem~3.10.). Hence, the module \({\mathcal U}\) has a maximal weak*-closed submodule exactly when \(N\neq N_-\) for some \(N\) in the nest \({\mathcal N}\). Implications for the Jacobson radical are also mentioned.