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• W1550161840 abstract "We identify Melrose’s suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma subset-of double-struck upper R Superscript p> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Gamma subset mathbb {R}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct a unique “symbol valued trace”, which extends the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L squared> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript 2 k minus 1> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^{2k-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript 2 k minus 1> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^{2k-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The eta–invariant of this family coincides with the spectral eta–invariant of the operator." @default.
• W1550161840 created "2016-06-24" @default.
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• W1550161840 date "2000-06-28" @default.
• W1550161840 modified "2023-09-25" @default.
• W1550161840 title "Traces on algebras of parameter dependent pseudodifferential operators and the eta–invariant" @default.
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• W1550161840 doi "https://doi.org/10.1090/s0002-9947-00-02480-6" @default.
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