Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Coarse geometry and Callias quantisation
HTML articles powered by AMS MathViewer

by Hao Guo, Peter Hochs and Varghese Mathai PDF
Trans. Amer. Math. Soc. 374 (2021), 2479-2520 Request permission


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.

Similar Articles
Additional Information
  • Hao Guo
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77840
  • ORCID: 0000-0001-5668-6409
  • Email:
  • 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:
  • Varghese Mathai
  • Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
  • MR Author ID: 231404
  • Email:
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 2479-2520
  • MSC (2020): Primary 19K56; Secondary 46L08, 53D50, 46L80
  • DOI:
  • MathSciNet review: 4223023