Vector Bundles over Multipullback Quantum Complex Projective Spaces

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right)$ and $C\left( \mathbb{S}_{H}^{2n+1}\right)$ of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T} \right)$ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left( \mathcal{T}\right)$, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right)$, $C\left( \mathbb{S}_{H}^{2n+1}\right)$, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $C\left( \mathbb{S}_{H} ^{2n+1}\right)$ of rank higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right)$, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $C\left( \mathbb{P} ^{n}\left( \mathcal{T}\right) \right)$.