2020/02/20 by Francesco D’Andrea, Piotr M. Hajac, D'Andrea, Francesco +7 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2002.09015
openalex publication_date 2020/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The multipullback quantization of complex projective spaces lacks the naive quantum CW-complex structure because the quantization of an embedding of the n-skeleton into the (n+1)-skeleton does not exist. To overcome this difficulty, we introduce the framework of cw-Waldhausen categories, which includes the concept of weak equivalences leading to the notion of a finite weak quantum CW-complex in the realm of unital C*-algebras. Here weak equivalences are unital *-homomorphisms that induce an isomorphism on K-theory. Better still, we construct a noncommutative counterpart of the cup product in K-theory, which is equivalent to its standard version in the classical case. To this end, we define k-topology, a noncommutative version of Grothendieck topology with covering families given by compact principal bundles and bases related by continuous maps, which leads to the much desired idea of multiplicative K-theory for noncommutative C*-algebras. Combining this with cw-Waldhausen structure on the category of compact quantum spaces, we arrive at the multiplicative K-theory type of finite weak quantum CW-complexes. We show that non-isomorphic quantizations of the standard CW-complex structure of a complex projective space enjoy the same multiplicative K-theory type admitting a noncommutative generalization of the Atiyah--Todd calculation of the K-theory ring in terms of truncated polynomials.