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arXiv · 1304.2284

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

Berndt Brenken
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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.

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Berndt Brenken. 2014-05-31. Ordered $*$-Semigroups and a $C^*$-Correspondence for a Partial Isometry. https://doi.org/10.3842/sigma.2014.055

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