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

Abstract

AbstractConsider 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 $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) in the K-theory of the reduced group $$C^*$$ C ∗ -algebra $$C^*_rG$$ 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 $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) was defined for an element $$g \in G$$ 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, $$\begin{aligned} \tau _g({{\,\mathrm{index}\,}}_G(D)) = {{\,\mathrm{index}\,}}_g(D), \end{aligned}$$ τ g ( index G ( D ) ) = index g ( D ) , for a trace $$\tau _g$$ 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 $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) . It also shows that $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) is a homotopy-invariant quantity.