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 indexG(D) in the K-theory of the reduced group C^∗-algebra Cr∗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 indexg(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, τg(indexG(D))=indexg(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 indexG(D). It also shows that indexg(D) is a homotopy-invariant quantity.