The K-theory of the Multipullback Quantum Complex Projective Spaces

06.04.2020 - 08.04.2020 | Online

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

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