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
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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.References
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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