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Index theory on the Mi\v{s}\v{c}enko bundle. (arXiv:1807.05757v1 [math.KT])
来源于:arXiv
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 Mi\v{s}\v{c}enko line bundle. In addition, we give a
proof of Atiyah's $L^2$-index theorem in the general context of principal
bundles 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. 查看全文>>