Extreme points of weakly closed \(\mathcal{T(N)}\)-modules (Q2758983)

A nest is a chain of closed subspaces of a Hilbert space \(H\) containing \(\{0\}\) and \(H\). The nest algebra \(\mathcal{T}(\mathcal{N})\) of a nest \(\mathcal {N}\) is defined to be the set of all operators \(T\) such that \(TN \subseteq N\) for every element \(N\) in \(\mathcal{N}\) (For more information see \textit{K. R. Davidson} [``Nest algebras. Triangular forms for operator algebras on Hilbert space'' (Pitman Research Notes in Mathematics Series 191, Pitman, New York) (1988; Zbl 0669.47024)]). If \(N\mapsto \widetilde{N}\) is a left order continuous homomorphism from \(\mathcal{N}\) into itself, then \(\mathcal{U}=\{X\in B(H); XN\subseteq \widetilde{N}\) \(\forall N\in \mathcal{N}\}\) is a weakly closed \(\mathcal{T}(\mathcal {N})\)-bimodule. It is known that every weakly closed \(\mathcal{T}(\mathcal {N})\)-bimodule is of this form; cf. \textit{J. A. Erdos} and \textit{S. C. Power} [J. Oper. Theory 7, 219--235 (1982; Zbl 0523.47027)].NEWLINENEWLINEThe authors characterize the rank one operators in the preannihilator \(\mathcal{U}_ {\bot}\) of \(\mathcal{U} \) and then use this characterization to show that a point \(A\) is an extreme point of the closed unit ball \(\mathcal{U}_{1}\) of \(\mathcal{U}\) if and only if for any \(N\in \mathcal{N}\) either \(N\cap \text{ran} (I-AA^{*})^ {\frac{1}{2}}=\{0\}\) or \(N_{\sim}^ {\bot}\cap \text{ran} (I-A^{*}A)^ {\frac{1}{2}}=\{0\}\), where \(N_ {\sim}^{\bot}\) is the annihilator of \(N _{\sim}=\vee \{N^{\prime}; \widetilde{N^{\prime}} < N\}\). In particular, every unitary element in \(\mathcal{U}\) is an extreme point of \(\mathcal{U}_{1}\).NEWLINENEWLINEAn investigation of extreme points of other operator systems is also given.