# Completely positive contractive maps and partial isometries

Document Type: Article dedicated to Uffe Haagerup

Author

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.

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### History

• Receive Date: 01 March 2017
• Revise Date: 20 July 2017
• Accept Date: 16 August 2017