Inverse Semigroups and Sheu's Groupoid for Odd Dimensional Quantum Spheres

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.