Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Uniqueness of the Kontsevich-Vishik trace


Authors: L. Maniccia, E. Schrohe and J. Seiler
Journal: Proc. Amer. Math. Soc. 136 (2008), 747-752
MSC (2000): Primary 58J40, 58J42, 35S05
Published electronically: November 1, 2007
MathSciNet review: 2358517
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Abstract: Let $ M$ be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on $ M$, whose (complex) order is not an integer greater than or equal to $ - \dim M$, is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the $ L^2$-operator trace on trace class operators.

Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique.


References [Enhancements On Off] (What's this?)

  • 1. A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867, 10.1007/BF01895667
  • 2. Boris V. Fedosov, François Golse, Eric Leichtnam, and Elmar Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), no. 1, 1–31. MR 1419415, 10.1006/jfan.1996.0142
  • 3. Gerd Grubb and Elmar Schrohe, Traces and quasi-traces on the Boutet de Monvel algebra, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 5, 1641–1696, xvii, xxii (English, with English and French summaries). MR 2127861
  • 4. Gerd Grubb, A resolvent approach to traces and zeta Laurent expansions, Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 67–93. MR 2114484, 10.1090/conm/366/06725
  • 5. G. Grubb. The local and global parts of the basic zeta coefficient for pseudodifferential boundary operators. Preprint arXiv math.AP/0611854.
  • 6. Victor Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. in Math. 55 (1985), no. 2, 131–160. MR 772612, 10.1016/0001-8708(85)90018-0
  • 7. M. Kontsevich and S. Vishik. Determinants of elliptic pseudo-differential operators. Preprint, Max-Planck-Institut für Math., Bonn, 1994.
  • 8. Maxim Kontsevich and Simeon Vishik, Geometry of determinants of elliptic operators, Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993) Progr. Math., vol. 131, Birkhäuser Boston, Boston, MA, 1995, pp. 173–197. MR 1373003
  • 9. Matthias Lesch, On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom. 17 (1999), no. 2, 151–187. MR 1675408, 10.1023/A:1006504318696
  • 10. L. Maniccia, E. Schrohe and J. Seiler. Determinants of SG-pseudodifferential operators. In preparation.
  • 11. K. Okikiolu, Critical metrics for the determinant of the Laplacian in odd dimensions, Ann. of Math. (2) 153 (2001), no. 2, 471–531. MR 1829756, 10.2307/2661347
  • 12. S. Paycha and S. Scott. An explicit Laurent expansion for regularized integrals of holomorphic symbols. To appear in Geom. and Funct. Anal., arXiv math.AP/0506211.
  • 13. M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75 (1984), no. 1, 143–177. MR 728144, 10.1007/BF01403095

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Additional Information

L. Maniccia
Affiliation: Università di Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
Email: maniccia@dm.unibo.it

E. Schrohe
Affiliation: Leibniz Universität Hannover, Institut für Analysis, Welfengarten 1, 30167 Hannover, Germany
Email: schrohe@math.uni-hannover.de

J. Seiler
Affiliation: Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany
Email: seiler@ifam.uni-hannover.de

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09168-X
Keywords: Kontsevich-Vishik canonical trace, pseudodifferential operators
Received by editor(s): February 9, 2007
Published electronically: November 1, 2007
Dedicated: Dedicated to Boris V. Fedosov on the occasion of his 70th birthday
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.