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

TL;DR

This paper explores the algebraic structure of $*$-semigroups related to partial isometries, introducing ordered structures and showing their universal $C^*$-algebras relate to Cuntz-Pimsner algebras, with implications for dynamical systems.

Contribution

It introduces ordered and matricially ordered $*$-semigroups and establishes their universal $C^*$-algebras as isomorphic to relative Cuntz-Pimsner algebras, linking algebraic and dynamical system perspectives.

Findings

Universal $C^*$-algebra of a partial isometry is isomorphic to a Cuntz-Pimsner algebra.

Ordered $*$-semigroups are fundamental in understanding the structure of these $C^*$-algebras.

Partial isometries can be viewed as crossed products from dynamical systems.

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.