Vector bundles over multipullback quantum complex projective spaces

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras C(P-n(T)) and C(S-H(2n+1)) of the quantum complex projective spaces P-n(T) and the quantum spheres S-H(2n+1), and the quantum line bundles L-k over P-n(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(P-n(T)), C(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(S-H(2n+1)) of rank higher than [n/2] + 3 are free modules. Furthermore, besides identifying a large portion of the positive cone of the K-0-group of C(P-n(T), we also explicitly identify L-k with concrete representative elementary projections over C(P-n(T)).