HILBERT ${ C}^*$-MODULES : A USEFUL TOOL

Abstract

In this article, we show how the concept of Hilbert $C^*$-module can be used to investigate completely positive linear maps. We show when two unital pure completely positive linear maps of a $C^*$-algebra into $M_n$ are unitarily equivalent. We also develop and characterize a concept of weak containment between two completely positive linear maps of a $C^*$-algebra into a von Neumann algebra. In preparation, we exhibit some basic known properties of Hilbert $C^*$-modules. In addition, we explore the norm of the standard Hilbert column $C^*$-modules and show it is the Haagerup tensor norm of two operator spaces.