Projective Modules over Quantum Projective Line

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right)$ constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right)$ realized as a concrete groupoid C*-algebra, and find its $K$-groups. Furthermore after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right)$, we identify those quantum principal $U\left( 1\right)$-bundles introduced by Hajac and collaborators among the projections classified.