RANK-ONE OPERATORS IN REFLEXIVE A-SUBMODULES OF OPERATOR ALGEBRAS

Abstract

In this paper, we first show that any reflexive $\mathcal{A}$-submodule $\mathcal{U}$ of a unital operator algebra $\mathcal{A}$ in $\mathcal{B(H)}$ is precisely of the following form: $$\mathcal{U} = \{ T \in \mathcal{B(H)}: TE \subseteq \phi(E) \quad \forall E \in \mathrm{Lat}\mathcal{A} \},$$ where $\phi$ is an order homomorphism of $\mathrm{Lat}\mathcal{A}$ into itself. Furthermore we investigate the density of the rank-one submodule of a reflexive $\mathcal{A}$-submodule in the $w^{*}$-topology and in certain pointwise approximation, and obtain several equivalent conditions by means of the order homomorphism $\phi$.