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

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 ⊂ Rp, we construct a unique “symbol valued trace”, which extends the L2–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 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. 1991 Mathematics Subject Classification. 58G15