Coarse geometry and Callias quantisation

Abstract

Consider a proper, isometric action by a unimodular, locally compact group G on a complete Riemannian manifold M. For equivariant elliptic operators that are invertible outside a cocompact subset of M, we show that a localised index in the K-theory of the maximal group C∗-algebra of G is welldefined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group C∗-algebra instead of its reduced counterpart, we can apply the trace given by integration over G to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. This leads to refinements of index-theoretic obstructions to Riemannian metrics of positive scalar curvature on noncompact manifolds, and also on orbifolds and other singular quotients of proper group actions. As a motivating application in another direction, we prove a version of Guillemin and Sternberg’s quantisation commutes with reduction principle for equivariant indices of Spinc Callias-type operators.