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Published online by Cambridge University Press: 20 November 2018
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
$C\left( S_{q}^{2\ell +1} \right)$ is generated by an inverse semigroup
$T$ of partial isometries. We show that the groupoid
${{\mathcal{G}}_{tight}}$ associated with the inverse semigroup
$T$ by Exel is exactly the same as the groupoid considered by Sheu.