Vector bundles over multipullback quantum complex projective spaces

We work on the classification of isomorphism classes finitely generated projective modules over C*-algebras $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and \mathbb{S}_{H}^{2n+1}\right) quantum complex spaces $\mathbb{P}^{n}\left( \mathcal{T} spheres $\mathbb{S}_{H}^{2n+1}$, line bundles $L_{k}$ $, studied by Hajac collaborators. Motivated groupoid approach Curto, Muhly, Renault to study C*-algebraic structure, we analyze in context C*-algebras, then apply Rieffel's stable rank results show that all \mathbb{S}_{H} ^{2n+1}\right) higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$ are free modules. Furthermore, besides identifying a large portion positive cone $K_{0}$-group also explicitly identify with concrete representative elementary projections \mathbb{P} ^{n}\left( $.