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Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry
Berndt Brenken Department of Mathematics and Statistics, University of Calgary,
Calgary, Canada T2N 1N4
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.
Keywords:
$C^*$-algebras; partial isometry; $*$-semigroup; partial order; matricial order; completely positive maps;
$C^*$-correspondence; Schwarz inequality; exact $C^*$-algebra.
Received: August 30, 2013; in final form May 22, 2014; Published online May 31, 2014
Citation:
Berndt Brenken, “Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry”, SIGMA, 10 (2014), 055, 50 pp.
Linking options:
https://www.mathnet.ru/eng/sigma920 https://www.mathnet.ru/eng/sigma/v10/p55
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Abstract page: | 108 | Full-text PDF : | 34 | References: | 34 |
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