Inner product modules over 𝐵*-algebras

Abstract

This paper is an investigation of right modules over a B ∗ {B^\ast } -algebra B which posses a B-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a W ∗ {W^\ast } -algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and W ∗ {W^\ast } -algebras. The extension of an inner product module over B by a B ∗ {B^\ast } -algebra A containing B as a ∗ ^\ast -subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.