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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.

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On finitely summable Fredholm modules from Smale spaces
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by Dimitris Michail Gerontogiannis PDF
Trans. Amer. Math. Soc. 375 (2022), 8885-8944 Request permission

Abstract:

We prove that all $K$-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead $K$-duality between the Ruelle algebras. Finally, we obtain general results on stability under holomorphic functional calculus and construct Lipschitz algebras on étale groupoids.
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Additional Information
  • Dimitris Michail Gerontogiannis
  • Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
  • MR Author ID: 1511578
  • ORCID: 0000-0002-0764-9174
  • Email: d.gerontogiannis@hotmail.com
  • Received by editor(s): December 13, 2021
  • Received by editor(s) in revised form: June 7, 2022
  • Published electronically: October 3, 2022
  • Additional Notes: The research was supported by EPSRC (grants NS09668/1, M5086056/1) as well as the London Mathematical Society and Heilbronn Institute for Mathematical Research (Early Career Fellowship).
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8885-8944
  • MSC (2020): Primary 37D20, 19K33, 58B34; Secondary 54E15
  • DOI: https://doi.org/10.1090/tran/8768