We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C(\mathbb{P}^{n}(\mathcal{T})) $ and $C(\mathbb{S}_{H}^{2n+1})$ of the quantum complex projective spaces $\mathbb{P}^{n}(\mathcal{T})$ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}(\mathcal{T})$, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $C(\mathbb{P}^{n}(\mathcal{T}))$, $C(\mathbb{S}_{H}^{2n+1})$, 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 $C(\mathbb{S}_{H}^{2n+1})$ of rank higher than $\lfloor \frac{n}{2}\rfloor+3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $C(\mathbb{P}^{n}(\mathcal{T}))$, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $C(\mathbb{P}^{n}(\mathcal{T}))$.

中文翻译：

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多重拉背量子复射影空间上的向量束

我们研究C *代数$ C（\ mathbb {P} ^ {n}（\ mathcal {T}））$和$ C（\ mathbb {S} _量子复数射影空间$ \ mathbb {P} ^ {n}（\ mathcal {T}）$和量子球体$ \ mathbb {S} _ {H} ^的{H} ^ {2n + 1}）$美元{2n + 1} $，量子线将$ L_ {k} $捆绑在$ \ mathbb {P} ^ {n}（\ mathcal {T}）$之上，由哈哈克（Hajac）和合作者研究。受Curto，Muhly和Renault的类群方法研究C *-代数结构的影响，我们分析了$ C（\ mathbb {P} ^ {n}（\ mathcal {T}））$，$ C（\ mathbb {S} _ {H} ^ {2n + 1}）$和$ L_ {k} $在类群C *-代数的上下文中，然后应用Rieffel的稳定秩结果来显示所有有限生成的射影模高于$ \ lfloor \ frac {n} {2} \ rfloor + 3 $的$ C（\ mathbb {S} _ {H} ^ {2n + 1}）$是免费模块。此外，