Abstract: Topological K-theory of a noncommutative C*-algebra lacks the ring structure and provides a poor invariant which doesn't distinguish very different C*-algebras, even commutative ones tantamount to compact Hausdorff spaces. Even KK-equivalence, the isomorphism in the bivariant Kasparov KK-theory has the same weakness. We introduce a new approach of cw-(cofibration weakening) Waldhausen categories and show that they admit a calculus of left fractions with cofibrations as nominators and K-equivalences as denominators, allowing inductive calculations of K-theory of (quantized) CW-complexes, and of equivariant K-theory of (quantum) principal bundles providing a module structure on the K-theory, generalizing the ring structure of the K-theory of compact Hausdorff spaces. We show that a K-equivalence of the two different quantizations of complex projective spaces, ones obtained by gluing quantum polydiscs and quantum homogeneous ones of Vaksman-Soibelman, and show that the corresponding ring of the module structure on its K-theory coincides with the multiplicative structure obtained by Atiyah and Todd.
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