An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M / G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index index G ( D ) in the K -theory of the reduced group C ∗ -algebra C r ∗ G of G . This is a common generalisation of the Baum–Connes analytic assembly map and the (equivariant) Atiyah–Patodi–Singer index. In part I of this series, a numerical index index g ( D ) was defined for an element g ∈ G , in terms of a parametrix of D and a trace associated to g . An Atiyah–Patodi–Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, τ g ( index G ( D ) ) = index g ( D ) , for a trace τ g defined by the orbital integral over the conjugacy class of g . This implies that the index theorem from part I yields information about the K -theoretic index index G ( D ) . It also shows that index g ( D ) is a homotopy-invariant quantity.