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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)

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Traces on algebras of parameter dependent pseudodifferential operators and the eta–invariant
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by Matthias Lesch and Markus J. Pflaum
Trans. Amer. Math. Soc. 352 (2000), 4911-4936
DOI: https://doi.org/10.1090/S0002-9947-00-02480-6
Published electronically: June 28, 2000

Abstract:

We identify Melrose’s suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\mathbb {R}$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $\Gamma \subset \mathbb {R}^p$, we construct a unique “symbol valued trace”, which extends the $L^2$–trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher–dimensional eta–invariants on algebras with parameter space $\mathbb {R}^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\mathbb {R}^{2k-1}$. The eta–invariant of this family coincides with the spectral eta–invariant of the operator.
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Bibliographic Information
  • Matthias Lesch
  • Affiliation: Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany
  • Address at time of publication: Department of Mathematics, The University of Arizona, Tucson, Arizona 85721-0089
  • Email: lesch@math.arizona.edu
  • Markus J. Pflaum
  • Affiliation: Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany
  • Email: pflaum@mathematik.hu-berlin.de
  • Received by editor(s): September 15, 1998
  • Received by editor(s) in revised form: November 1, 1998
  • Published electronically: June 28, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 4911-4936
  • MSC (2000): Primary 58G15
  • DOI: https://doi.org/10.1090/S0002-9947-00-02480-6
  • MathSciNet review: 1661258