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

Matthias Lesch·Markus J. Pflaum
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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 $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\R^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

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Matthias Lesch, Markus J. Pflaum. 1999-02-18. Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant. https://arxiv.org/abs/math/9804136

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