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 Gamma subset of 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 R2k-1. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over R2k-1. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.