Index theory on the Miščenko bundle

We consider the assembly map for principal bundles with fiber a countable discrete group. obtain an index-theoretic interpretation of this homomorphism by providing tensor-product presentation module sections associated to Miščenko line bundle. In addition, we give proof Atiyah’s L2-index theorem in general context flat finitely generated projective Hilbert C∗-modules over compact Hausdorff spaces. thereby also reestablish that surjectivity Baum–Connes implies Kadison–Kaplansky idempotent conjecture torsion-free case. Our approach does not rely on geometric K-homology but rather explicit construction Alexander–Spanier cohomology classes coming from Chern character tracial function algebras.