On the relative reflexivity of finitely generated modules of operators

Abstract

Let R \mathcal {R} be a von Neumann algebra on a Hilbert space H \mathcal {H} with commutant R ′ \mathcal {R}’ and centre C \mathcal {C} . For each subspace S \mathcal {S} of R \mathcal {R} let ref R ( S ) \operatorname {ref}_\mathcal {R}\,(\mathcal {S}) be the space of all B ∈ R B \in \mathcal {R} such that X B Y = 0 XBY= 0 for all X , Y ∈ R X,Y \in \mathcal {R} satisfying X S Y = 0 X\,\mathcal {S}\,Y = 0 . If ref R ( S ) = S \operatorname {ref}_\mathcal {R}\,(\mathcal {S})= \mathcal {S} , the space S \mathcal {S} is called R \mathcal {R} -reflexive. (If R = B ( H ) \mathcal {R}= \mathcal {B}(\mathcal {H}) and S \mathcal {S} is an algebra containing the identity operator, R \mathcal {R} -reflexivity reduces to the usual reflexivity in operator theory.) The main result of the paper is the following: if S \mathcal {S} is one-dimensional, or if S \mathcal {S} is arbitrary finite-dimensional but R \mathcal {R} has no central portions of type I n {{\text {I}}_n} for n > 1 n > 1 , then the space C S ¯ \overline {\mathcal {C}\mathcal {S}} is R \mathcal {R} -reflexive and the space R ′ S ¯ \overline {\mathcal {R}’\,\mathcal {S}} is B ( H ) \mathcal {B}(\mathcal {H}) -reflexive, where the bar denotes the closure in the ultraweak operator topology. If R \mathcal {R} is a factor, then R ′ S \mathcal {R}’\,\mathcal {S} is closed in the weak operator topology for each finite-dimensional subspace S \mathcal {S} of R \mathcal {R} .