Abstract
We consider the assembly map for principal bundles with fiber a countable discrete group. We obtain an index-theoretic interpretation of this homomorphism by providing a tensor-product presentation for the module of sections associated to the Miščenko line bundle. In addition, we give a proof of Atiyah's L2-index theorem in the general context of flat bundles of finitely generated projective Hilbert C∗-modules over compact Hausdorff spaces. We thereby also reestablish that the surjectivity of the Baum-Connes assembly map implies the Kadison-Kaplansky idempotent conjecture in the torsion-free case. Our approach does not rely on geometric K-homology but rather on an explicit construction of Alexander-Spanier cohomology classes coming from a Chern character for tracial function algebras.
Original language | English |
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Journal | Kyoto Journal of Mathematics |
Volume | 62 |
Issue number | 1 |
Pages (from-to) | 103-131 |
ISSN | 2156-2261 |
DOIs | |
Publication status | Published - Apr 2022 |