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

作者：Hochs, Peter^*; Wang, Bai-Ling; Wang, Hang

来源：Mathematische Zeitschrift, 2022, 301(2): 1333-1367.

DOI：10.1007/s00209-021-02942-0

• POSITIVE SCALAR CURVATURE
• OPERATOR
• INVARIANTS
• KERNEL

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 is an element of 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) (index(G )(D)) = 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 index(G) (D). It also shows that index(g) (D) is a homotopy-invariant quantity.