Associated noncommutative vector bundles over the Vaksman–Soibelman quantum complex projective spaces

By a diagonal embedding of $U(1)$ in $SU_q(m)$, we prolongate the diagonal circle action on the Vaksman–Soibelman quantum sphere $S^{2n+1}_q$ to the $SU_q(m)$-action on the prolongated bundle. Then we prove that the noncommutative vector bundles associated via the fundamental representation of $SU_q(m)$, for $m\in\{2,\kern0.5pt.\kern1.5pt.\kern1.5pt.\kern1pt,n\}$, yield generators of the even K-theory group of the C*-algebra of the Vaksman–Soibelman quantum complex projective space $\mathbb{C}{\rm P}^n_q$.