## Advances in Operator Theory

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

### 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.

#### Article information

**Source**

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

**Dates**

Received: 1 March 2017

First available in Project Euclid: 5 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512497962

**Digital Object Identifier**

doi:10.22034/aot.1703-1131

**Mathematical Reviews number (MathSciNet)**

MR3730349

**Zentralblatt MATH identifier**

06804326

**Subjects**

Primary: 46L05: General theory of $C^*$-algebras 46L08: $C^*$-modules

Secondary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

**Keywords**

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

#### Citation

Brenken, Berndt. Completely positive contractive maps and partial isometries. Adv. Oper. Theory 3 (2018), no. 1, 271--294. doi:10.22034/aot.1703-1131. https://projecteuclid.org/euclid.aot/1512497962