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
DOI:
https://doi.org/10.1090/S0002-9947-00-02480-6
Published electronically:
June 28, 2000
MathSciNet review:
1661258
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space . For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone
, we construct a unique ``symbol valued trace'', which extends the
-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
. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over
. 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
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
DOI:
https://doi.org/10.1090/S0002-9947-00-02480-6
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