Research group

**The scope of the meeting:**

We construct distinguished free generators of the *K*_{0}-group of the C*-algebra *C*(CP^{n}_{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 CP^{n-}^{1}_{T} in CP^{n}_{T}. This allows us to compute *K*_{0}(*C*(CP^{n}_{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*(CP^{n}_{q}) of the Vaksman-Soibelman quantum complex projective space with *K*_{0}(*C*(CP^{n}_{T})). Since this identification is induced by the restriction-corestriction of a *U*(1)-equivariant *-homomorphism from the C*-algebra *C*(*S*^{2}^{n}^{+}^{1}* _{q}*) of the (2

*n*+1)-dimensional Vaksman-Soibelman quantum sphere to the C*-algebra

*C*(

*S*

^{2}

^{n}^{+}

^{1}

*) of the (2*

_{H}*n*+1)-dimensional Heegaard quantum sphere, we conclude that there is a basis of

*K*

_{0}(

*C*(CP

^{n}_{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*(CP

^{n}_{q})). Finally, using identities in K-theory afforded by Toeplitz projections in

*C*(CP

^{n}_{T}), we prove noncommutative Atiyah-Todd identities.