On the Relative Reflexivity of Finitely Generated Modules of Operators

Abstract

Let $\mathcal {R}$ be a von Neumann algebra on a Hilbert space $\mathcal {H}$ with commutant $\mathcal {R}’$ 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 $\overline {\mathcal {R}’ \mathcal {S}}$ is $\mathcal {B}(\mathcal {H})$-reflexive, where the bar denotes the closure in the ultraweak operator topology. If $\mathcal {R}$ is a factor, then $\mathcal {R}’ \mathcal {S}$ is closed in the weak operator topology for each finite-dimensional subspace $\mathcal {S}$ of $\mathcal {R}$.