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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Limit operator theory for groupoids
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by Kyle Austin and Jiawen Zhang PDF
Trans. Amer. Math. Soc. 373 (2020), 2861-2911 Request permission

Abstract:

We extend the symbol calculus and study the limit operator theory for $\sigma$-compact, étale, and amenable groupoids, in the Hilbert space case. This approach not only unifies various existing results which include the cases of exact groups and discrete metric spaces with Property A, but also establish new limit operator theories for group/groupoid actions and uniform Roe algebras of groupoids. In the process, we extend a monumental result by Exel, Nistor, and Prudhon, showing that the invertibility of an element in the groupoid $C^*$-algebra of a $\sigma$-compact amenable groupoid with a Haar system is equivalent to the invertibility of its images under regular representations.
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Additional Information
  • Kyle Austin
  • Affiliation: Weizmann Institute of Science and Technology, Department of Mathematics, 234 Herzl Street, POB 26, Rehovot 7610001, Israel
  • MR Author ID: 1074025
  • Email: ksaustin88@gmail.com
  • Jiawen Zhang
  • Affiliation: University of Southampton, School of Mathematics, Building 54, Highfield, Southampton SO17 1BJ, United Kingdom
  • MR Author ID: 1108765
  • Email: jiawen.zhang@soton.ac.uk
  • Received by editor(s): April 25, 2019
  • Received by editor(s) in revised form: September 11, 2019
  • Published electronically: January 28, 2020
  • Additional Notes: The first author was supported by the Israeli Science Foundation grants ISF-Moked grant 2095/15 and grant No. 522/14.
    The second author was supported by the Sino-British Trust Fellowship by Royal Society, International Exchanges 2017 Cost Share (China) grant EC$\backslash$NSFC$\backslash$170341, NSFC11871342, and NSFC11811530291.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 2861-2911
  • MSC (2010): Primary 47A53; Secondary 22A22, 46L55
  • DOI: https://doi.org/10.1090/tran/8005
  • MathSciNet review: 4069235