Vector bundles over multipullback quantum complex projective spaces

• Published in: Journal of noncommutative geometry Vol. 15; no. 1; pp. 305 - 345
• Main Author: Sheu, Albert Jeu-Liang
• Format: Journal Article
• Language: English
• Published: European Mathematical Society Publishing House 21.04.2021
• Subjects: Algebra; Mathematical research; Vector spaces; United States
• ISSN: 1661-6952; 1661-6960; 1661-6960
• DOI: 10.4171/jncg/401

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})) .