Orthogonal complementing in Hilbert C∗-modules

Abstract

We characterize orthogonally complemented submodules in Hilbert $C^{\ast }$-modules by their orthogonal closures. Applying Magajna’s characterization of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras of compact operators by the complementing property of submodules, we give an elementary proof of Schweizer’s characterization of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras of compact operators. Also, we prove analogous characterization theorems for $C^{\ast }$-algebras of compact operators related to topological properties of submodules of strict completions of Hilbert modules over a nonunital $C^{\ast }$-algebra.