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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.
pdf (691 kb)
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