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Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Berndt Brenken},
     title = {Ordered $*${-Semigroups} and a~$C^*${-Correspondence} for {a~Partial} {Isometry}},
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     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a54/}
}
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Berndt Brenken. Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a54/

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