Completely positive contractive maps and partial isometries

Abstract

Associated with a completely positive contractive map $\varphi$ of a $C^*$-algebra $A$ is a universal $C^*$-algebra generated by the $C^*$-algebra $A$ along with a contraction implementing $\varphi$. We prove a dilation theorem: the map $\varphi$ may be extended to a completely positive contractive map of an augmentation of $A$. The associated $C^*$-algebra of the augmented system contains the original universal $C^*$-algebra as a corner, and the extended completely positive contractive map is implemented by a partial isometry.