Hereditary and intermediate reflexivity of $W^*$-algebras

An operator algebra $R$ is reflexive if every operator which leaves invariant all $R$-invariant subspaces belongs to $R$. The notion of reflexivity can be extended to linear spaces of operators. An operator algebra is said to be hereditarily reflexive if all its weakly closed subspaces are reflexive. This article presents a criterion for the hereditary reflexivity of a $W^*$-algebra, and also examines the more general problem of conditions for the intermediate reflexivity of a pair of $W^*$-algebras. A number of necessary conditions and sufficient conditions for intermediate reflexivity are also obtained.
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