Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the relative reflexivity of finitely generated modules of operators

Author: Bojan Magajna
Journal: Trans. Amer. Math. Soc. 327 (1991), 221-249
MSC: Primary 47D25; Secondary 46L10, 47A15, 47C15
MathSciNet review: 1038017
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Abstract: Let $ \mathcal{R}$ be a von Neumann algebra on a Hilbert space $ \mathcal{H}$ with commutant $ \mathcal{R}^{\prime}$ and centre $ \mathcal{C}$. For each subspace $ \mathcal{S}$ of $ \mathcal{R}$ let $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})$ be the space of all $ B \in \mathcal{R}$ such that $ XBY= 0$ for all $ X,Y \in \mathcal{R}$ satisfying $ X\,\mathcal{S}\,Y = 0$. If $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})= \mathcal{S}$, the space $ \mathcal{S}$ is called $ \mathcal{R}$-reflexive. (If $ \mathcal{R}= \mathcal{B}(\mathcal{H})$ and $ \mathcal{S}$ is an algebra containing the identity operator, $ \mathcal{R}$-reflexivity reduces to the usual reflexivity in operator theory.) The main result of the paper is the following: if $ \mathcal{S}$ is one-dimensional, or if $ \mathcal{S}$ is arbitrary finite-dimensional but $ \mathcal{R}$ has no central portions of type $ {{\text{I}}_n}$ for $ n > 1$, then the space $ \overline {\mathcal{C}\mathcal{S}} $ is $ \mathcal{R}$-reflexive and the space $ \overline {\mathcal{R}^{\prime}\,\mathcal{S}} $ is $ \mathcal{B}(\mathcal{H})$-reflexive, where the bar denotes the closure in the ultraweak operator topology. If $ \mathcal{R}$ is a factor, then $ \mathcal{R}^{\prime}\,\mathcal{S}$ is closed in the weak operator topology for each finite-dimensional subspace $ \mathcal{S}$ of $ \mathcal{R}$.

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Keywords: Subspaces of operators, reflexivity, elementary operators, von Neumann algebras
Article copyright: © Copyright 1991 American Mathematical Society