Mathematics

 Among the popular and well analyzed quantum spaces $X_{q}$ with their $K$-groups $K^{i}\left(  X_{q}\right)  \equiv K_{i}\left(  C\left(X_{q}\right)  \right)  $ already computed, are the multipullback quantum odd-dimensional spheres $\mathbb{S}_{H}^{2n+1}$ and the associated quantum complex projective spaces $\mathbb{P}^{n}\left(  \mathcal{T}\right)  $, introduced and studied by Hajac, Kaygun, Nest, Pask, Sims, and Zielinski.                                                                                                                                                                                  In noncommutative geometry, finitely generated projective modules (f.g.p.m.) over $C\left(  X_{q}\right)$, efficiently encoded by projections over $C\left(  X_{q}\right)  $, are viewed as (quantum) vector bundles over $X_{q}%$, and are classified up to stable isomorphism by the positive cone of $K_{0}\left(  C\left(  X_{q}\right)  \right)  $. With the $K_{0}$-groups of $C\left(  \mathbb{S}_{H}^{2n+1}\right)  $ and $C\left(  \mathbb{P}^{n}\left(\mathcal{T}\right)  \right)  $ known, it is natural to seek the classification of vector bundles over $\mathbb{S}_{H}^{2n+1}$ and $\mathbb{P}^{n}\left(\mathcal{T}\right)  $ up to isomorphism.                                                                                                                                                                 Following an approach popularized by Curto, Muhly, and Renault, we first realize $C\left(  \mathbb{P}^{n}\left(  \mathcal{T}\right)  \right)  $ and $C\left(  \mathbb{S}_{H}^{2n+1}\right)  $ as groupoid $C^*$-algebras to better understand their structures in the framework of groupoid $C^*$-algebras, which also provides a convenient context for discussing projections over them. Then we apply Rieffel's theory of stable ranks to derive some answers regarding the classification of f.g.p.m. over the $n$-dimensional Toeplitz algebra $\mathcal{T}^{\otimes n}$, $C\left(  \mathbb{S}_{H}^{2n+1}\right)  $, and $C\left(  \mathbb{P}^{n}\left(  \mathcal{T}\right)  \right)  $. In particular, as concrete direct sums of elementary projections over $C\left(\mathbb{P}^{n}\left(  \mathcal{T}\right)  \right)  $, we can identify those distinguished quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left(\mathcal{T}\right)  $ for $k\in\mathbb{Z}$ that were constructed from quantum principal $U\left(  1\right)  $-bundles by Hajac and his collaborators,rendering their module structures transparent.