Hilbert C*-module

Abstract: Hilbert C*-modules are objects like Hilbert spaces, except that the inner product, instead of being complex valued, takes its values in a C*-algebra. The theory of these modules, together with their bounded and unbounded operators, is not only rich and attractive in its own right but forms an infrastructure for some of the most important research topics in operator algebras. This book is based on a series of lectures given by Professor Lance at a summer school at the University of Trondheim. It provides, for the first time, a clear and unified exposition of the main techniques and results in this area, including a substantial amount of new and unpublished material. It will be welcomed as an excellent resource for all graduate students and researchers working in operator algebras.

Abstract: This paper is an investigation of right modules over a B*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*-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*-algebras. The extension of an inner product module over B by a B*-algebra A containing B as a *-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.