Research group    

     The scope of the meeting:

We construct distinguished free generators of the K₀-group of the C*-algebra C(CPⁿ_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 CPⁿ_T. This allows us to compute K₀(C(CPⁿ_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 K₀-group of the C*-algebra C(CPⁿ_q) of the Vaksman-Soibelman quantum complex projective space with K₀(C(CPⁿ_T)). Since this identification is induced by the restriction-corestriction of a U(1)-equivariant *-homomorphism from the C*-algebra C(S²ⁿ⁺¹_q) of the (2n+1)-dimensional Vaksman-Soibelman quantum sphere to the C*-algebra C(S²ⁿ⁺¹_H) of the (2n+1)-dimensional Heegaard quantum sphere, we conclude that there is a basis of K₀(C(CPⁿ_T)) given by associated noncommutative vector bundles coming from the same representations that yield an associated-noncommutative-vector-bundle basis of the K₀(C(CPⁿ_q)). Finally, using identities in K-theory afforded by Toeplitz projections in C(CPⁿ_T), we prove noncommutative Atiyah-Todd identities.