Coarse geometry and Callias quantisation

TL;DR

This paper develops a new framework for defining and analyzing equivariant indices of elliptic operators on noncompact manifolds using maximal group C*-algebras, enabling refined obstructions to positive scalar curvature and a version of quantization commutes with reduction.

Contribution

It introduces maximal equivariant Roe algebras for proper group actions and applies them to define localized indices, refining previous scalar curvature obstructions and establishing a quantization principle.

Findings

Defined localized indices in the K-theory of maximal group C*-algebras.

Refined obstructions to positive scalar curvature on noncompact Spin manifolds.

Proved a version of the quantization commutes with reduction principle for Callias-type operators.

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 well-defined. 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. As a very special case, this allows one to refine numerical obstructions to positive scalar curvature on a noncompact $\operatorname{Spin}$ manifold $X$ defined via Callias index theory, to obstructions in the $K$-theory of the maximal $C^*$-algebra of the fundamental group $\pi_1(X)$. 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 $\operatorname{Spin}^c$ Callias-type operators.