### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 055, 50 pages      arXiv:1304.2284      https://doi.org/10.3842/SIGMA.2014.055
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

### Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry

Berndt Brenken
Department of Mathematics and Statistics, University of Calgary, Calgary, Canada T2N 1N4

Received August 30, 2013, in final form May 22, 2014; Published online May 31, 2014

Abstract
Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and ordered and matricially ordered $*$-semigroups are introduced, along with their universal $C^*$-algebras. The universal $C^*$-algebra generated by a partial isometry is isomorphic to a relative Cuntz-Pimsner $C^*$-algebra of a $C^*$-correspondence over the $C^*$-algebra of a matricially ordered $*$-semigroup. One may view the $C^*$-algebra of a partial isometry as the crossed product algebra associated with a dynamical system defined by a complete order map modelled by a partial isometry acting on a matricially ordered $*$-semigroup.

Key words: $C^*$-algebras; partial isometry; $*$-semigroup; partial order; matricial order; completely positive maps; $C^*$-correspondence; Schwarz inequality; exact $C^*$-algebra.

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