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