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

Lesch, Matthias; Pflaum, Markus

doi:10.1090/S0002-9947-00-02480-6

Traces on algebras of parameter dependent pseudodifferential operators and the eta–invariant
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by Matthias Lesch and Markus J. Pflaum PDF

Trans. Amer. Math. Soc. 352 (2000), 4911-4936 Request permission

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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