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

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