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

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 allows 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.

29 pages, LaTeX2e, revised version of 21 Oct 1998, some minor errors corrected, to appear in Trans. Amer. Math. Soc