  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.
