Index theory on the Miščenko bundle

Jens Kaad, Valerio Proietti

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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 languageEnglish
JournalKyoto Journal of Mathematics
Volume62
Issue number1
Pages (from-to)103-131
ISSN2156-2261
DOIs
Publication statusPublished - Apr 2022

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