The reflexive closure of the adjointable operators

Abstract

Given a Hilbert $C^{*}$-module $E$ over a $C^{*}$-algebra $\mathcal{A}$, we give an explicit description for the invariant subspace lattice $\operatorname{lat} \mathcal{L} (E)$ of all adjointable operators on $E$. We then show that the collection $\operatorname{End}_{\mathcal{A} }(E)$ of all bounded $\mathcal{A}$-module operators acting on $E$ forms the reflexive closure for $\mathcal{L} (E)$, i.e., $\operatorname{End}_{\mathcal{A} }(E)=\operatorname{alg} \operatorname{lat} \mathcal{L} (E)$. Finally, we make an observation regarding the representation theory of the left centralizer algebra of a $C^{*}$-algebra and use it to give an intuitive proof of a related result of H. Lin.