Advances in Operator Theory

Completely positive contractive maps and partial isometries

Berndt Brenken

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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.

Article information

Adv. Oper. Theory, Volume 3, Number 1 (2018), 271-294.

Received: 1 March 2017
First available in Project Euclid: 5 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras 46L08: $C^*$-modules
Secondary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

completely positive dynamical system partial isometry $C^*$-correspondence Cuntz–Pimsner $C^*$-algebra Morita equivalence


Brenken, Berndt. Completely positive contractive maps and partial isometries. Adv. Oper. Theory 3 (2018), no. 1, 271--294. doi:10.22034/aot.1703-1131.

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