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 $\mathbb{R}$. 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.