Vector bundles over multipullback quantum complex projective spaces

Abstract

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 $C(\mathbb{P}^{n}(\mathcal{T}))$.