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Transactions of the American Mathematical Society

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Coarse geometry and Callias quantisation


Authors: Hao Guo, Peter Hochs and Varghese Mathai
Journal: Trans. Amer. Math. Soc. 374 (2021), 2479-2520
MSC (2020): Primary 19K56; Secondary 46L08, 53D50, 46L80
DOI: https://doi.org/10.1090/tran/8202
Published electronically: January 26, 2021
MathSciNet review: 4223023
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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. 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 $\mathrm {Spin}^{c}$ Callias-type operators.


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Additional Information

Hao Guo
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77840
ORCID: 0000-0001-5668-6409
Email: haoguo@math.tamu.edu

Peter Hochs
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia; and Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
MR Author ID: 786204
ORCID: 0000-0001-9232-2936
Email: p.hochs@math.ru.nl

Varghese Mathai
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
MR Author ID: 231404
Email: mathai.varghese@adelaide.edu.au

Received by editor(s): October 8, 2019
Received by editor(s) in revised form: December 20, 2019, and April 14, 2020
Published electronically: January 26, 2021
Additional Notes: The first author was supported in part by funding from the National Science Foundation under grant no. 1564398.
The third author was supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020.
Article copyright: © Copyright 2021 American Mathematical Society