The Noncommutative Residue for Manifolds with Boundary

Fedosov, Boris; Golse, François; Leichtnam, Éric; Schrohe, Elmar

doi:10.1006/jfan.1996.0142

Cited by 122 publications

(152 citation statements)

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“…It includes the classical differential boundary value problems as well as the parametrices to elliptic elements. Many operatoralgebraic aspects of this algebra (spectral invariance, noncommutative residues and traces, composition sequence, K-theory) have been studied recently [1], [10], [12], [19], [20], [23], [28]. The problem of identifying this algebra as the pseudodifferential algebra (or as an ideal of one) of a Lie groupoid may be the key to an effective application of the methods of noncommutative geometry.…”

Section: Introductionmentioning

confidence: 99%

Boutet de Monvel’s calculus and groupoids I

Aastrup¹,

Melo²,

Monthubert³

et al. 2010

J. Noncommut. Geom.

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Abstract. Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra ‰ 0 .G/ of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C .G/. While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C .G/ possesses an ideal « isomorphic to G . In fact, we prove first that G ' ‰˝K with the C*-algebra ‰ generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both ‰˝K and « are extensions of C.S Y /˝K by K (S Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.Mathematics Subject Classification (2010). 58J32, 19K56, 58H05, 58J40.

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Section: Introductionmentioning

confidence: 99%

Boutet de Monvel’s calculus and groupoids I

Aastrup¹,

Melo²,

Monthubert³

et al. 2010

J. Noncommut. Geom.

Self Cite

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“…Also note that ν| ∆ b ∈ C ∞ (M, Ω b ) as one can check using local coordinates. The local expression (2.2) turns out to be independent of coordinates; a nice proof of this fact can be found in [3]. Thus, the local expressions (2.2) actually define a global density ω(A) ∈ x −p C ∞ (M, Ω b ).…”

Section: Operator Algebras and Trace Functionalsmentioning

confidence: 82%

On the noncommutative residue and the heat trace expansion on conic manifolds

Gil

Loya

2002

manuscripta mathematica

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Abstract. Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 + of the trace Tr P e −tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t) 2 terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals.

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“…It is well-known that the residue trace Res, which was discovered independently by Guillemin [Gui85] and Wodzicki [Wod87b], is up to normalization the unique trace on the algebra CL Z (M) of integer order classical pseudodifferential operators ( [Wod87b], Brylinski and Getzler [BrGe87], Fedosov, Golse, Leichtnam, and Schrohe [FGLS96], Lesch [Les99], for a complete account of traces and determinants of pseudodifferential operators see the recent monograph by Scott [Sco10]). Res is non-trivial only on CL k (M) for integers k ≥ −n, and it is complemented by the canonical trace, TR, of Kontsevich and Vishik [KoVi95].…”

Section: Introduction and Formulation Of The Resultsmentioning

confidence: 99%

“…Therefore, we also obtain as a corollary the precise criterion when a homogeneous function can be written as a sum of partial derivatives of homogeneous functions, cf. [FGLS96], [Les99]. Finally, this criterion is generalized to classical symbol functions, generalizing [Pay07, Prop.…”

Section: Cohomology Of Homogeneous Differential Formsmentioning

confidence: 99%

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Classification of traces and hypertraces on spaces of classical pseudodifferential operators

Lesch

Jiménez

2013

J. Noncommut. Geom.

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Abstract. Let M be a closed manifold and let CL• (M) be the algebra of classical pseudodifferential operators. The aim of this note is to classify trace functionals on the subspacesit is an algebra only if a is a non-positive integer. Therefore, it turns out to be useful to introduce the notions of pretrace and hypertrace. Our main result gives a complete classification of pre-and hypertraces on CL a (M) for any a ∈ R, as well as the traces on CL a (M) for a ∈ Z, a ≤ 0. We also extend these results to classical pseudodifferential operators acting on sections of a vector bundle.As a byproduct we give a new proof of the well-known uniqueness results for the Guillemin-Wodzicki residue trace and for the Kontsevich-Vishik canonical trace. The novelty of our approach lies in the calculation of the cohomology groups of homogeneous and log-polyhomogeneous differential forms on a symplectic cone. This allows to give an extremely simple proof of a generalization of a Theorem of Guillemin about the representation of homogeneous functions as sums of Poisson brackets.

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The Noncommutative Residue for Manifolds with Boundary

Cited by 122 publications

References 10 publications

Boutet de Monvel’s calculus and groupoids I

Boutet de Monvel’s calculus and groupoids I

On the noncommutative residue and the heat trace expansion on conic manifolds

Classification of traces and hypertraces on spaces of classical pseudodifferential operators

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