Distinguished bases in the K-theory of multipullback quantum complex projective spaces

We construct distinguished free generators of the $K_0$-group of the C*-algebra $C(\mathbb{CP}^n_\mathcal{T})$ of the multipullback quantum complex projective space. To this end, first we prove a quantum-tubular-neighborhood lemma to overcome the difficulty of the lack of an embedding of $\mathbb{CP}^{n-1}_\mathcal{T}$ in $\mathbb{CP}^n_\mathcal{T}$. This allows us to compute $K_0(C(\mathbb{CP}^n_\mathcal{T} ))$ using the Mayer-Vietoris six-term exact sequence in K-theory. The same lemma also helps us to prove a comparison theorem identifying the $K_0$-group of the C*-algebra $C(\mathbb{CP}^n_q)$ of the Vaksman-Soibelman quantum complex projective space with $K_0(C(\mathbb{CP}^n_\mathcal{T}))$. Since this identification is induced by the restriction-corestriction of a $U(1)$-equivariant \mbox{*-homomorphism} from the C*-algebra $C(S^{2n+1}_q)$ of the $(2n\!+\!1)$-dimensional Vaksman-Soibelman quantum sphere to the C*-algebra $C(S^{2n+1}_H)$ of the $(2n\!+\!1)$-dimensional Heegaard quantum sphere, we conclude that there is a basis of $K_0(C(\mathbb{CP}^n_\mathcal{T}))$ given by associated noncommutative vector bundles coming from the same representations that yield an associated-noncommutative-vector-bundle basis of the $K_0(C(\mathbb{CP}^n_q))$. Finally, using identities in K-theory afforded by Toeplitz projections in $C(\mathbb{CP}^n_\mathcal{T})$, we prove noncommutative Atiyah-Todd identities.