Inverse semigroups and the Cuntz-Li algebras

Abstract

In this paper, we apply the theory of inverse semigroups to the $C^{\ast }$-algebra $U[\mathbb{Z}]$ considered in \cite{Cuntz}. We show that the $C^{\ast }$-algebra $U[\mathbb{Z}]$ is generated by an inverse semigroup of partial isometries. We explicity identify the groupoid ${\mathcal{G}}_{tight}$ associated to the inverse semigroup and show that ${\mathcal{G}}_{tight}$ is exactly the same groupoid obtained in \cite{Cuntz-Li}.

A section added on Nica covariance and boundary relations. Few typos corrected

References in corpus (2)

• The Regular C*-algebra of an Integral Domain
• Partial crossed product description of the C*-algebras associated with integral domains

Cited by in corpus (2)

• Nuclearity of semigroup C*-algebras and the connection to amenability
• Inverse semigroup C*-algebras associated with left cancellative semigroups