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Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

Authors: Matthias Lesch and Markus J. Pflaum
Journal: Trans. Amer. Math. Soc. 352 (2000), 4911-4936
MSC (2000): Primary 58G15
Published electronically: June 28, 2000
MathSciNet review: 1661258
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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.

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Additional 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

Markus J. Pflaum
Affiliation: Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany

Received by editor(s): September 15, 1998
Received by editor(s) in revised form: November 1, 1998
Published electronically: June 28, 2000
Article copyright: © Copyright 2000 American Mathematical Society