arXiv math/98041361998-04math.FAmath.DGmath.OA

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

Matthias Lesch, Markus J. Pflaum

备注：: 29 pages, LaTeX2e, revised version of 21 Oct 1998, some minor errors corrected, to appear in Trans. Amer. Math. Soc
发表信息：: Trans. Amer. Math. Soc. 352 (2000), No 11, 4911--4936

英文摘要：

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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