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

TL;DR

This paper establishes a connection between the $K$-theoretic index of equivariant Dirac operators on manifolds with boundary and a numerical index defined via traces, generalizing the Atiyah-Patodi-Singer index theorem in the context of proper group actions.

Contribution

It demonstrates that under certain conditions, the $K$-theoretic index can be recovered from a numerical index via a trace, linking abstract $K$-theory to concrete index calculations.

Findings

The $K$-theoretic index equals the trace-based numerical index under specific conditions.

The numerical index $ ext{index}_g(D)$ is homotopy-invariant.

The results unify the Baum-Connes assembly map with Atiyah-Patodi-Singer index theory.

Abstract

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