|
SIGMA 6 (2010), 077, 17 pages arXiv:1003.1210
https://doi.org/10.3842/SIGMA.2010.077
Contribution to the Special Issue “Noncommutative Spaces and Fields”
A Canonical Trace Associated with Certain Spectral Triples
Sylvie Paycha
Laboratoire de Mathématiques, 63177 Aubière Cedex, France
Received March 11, 2010, in final form September 13, 2010; Published online September 29, 2010
Abstract
In the abstract pseudodifferential setup of
Connes and Moscovici, we prove a general
formula for the discrepancies of zeta-regularised traces associated with
certain spectral triples, and we introduce a
canonical trace on operators, whose order lies outside (minus) the dimension
spectrum of the spectral triple.
Key words:
spectral triples; zeta regularisation; noncommutative residue; discrepancies.
pdf (291 kb)
ps (194 kb)
tex (20 kb)
References
- Berline N., Getzler E., Vergne M.,
Heat kernels and Dirac operators,
Springer-Verlag, Berlin, 2004.
- Connes A.,
Noncommutative geometry,
Academic Press, San Diego, CA, 1994.
- Cardona A., Ducourtioux C., Magnot J.P., Paycha S.,
Weighted traces on algebras of pseudo-differential operators and geometry on loop groups,
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 503-540,
math.OA/0001117.
- Connes A., Moscovici H.,
The local index formula in noncommutative geometry,
Geom. Funct. Anal. 5 (1995), 174-243.
- Dunford N., Schwartz J.T.,
Linear operators. Part III. Spectral operators,
John Wiley & Sons, Inc., New York, 1988.
- Higson N.,
The residue index theorem of Connes and Moscovici,
in Surveys in Noncommutative Geometry, Clay Math. Proc., Vol. 6, Amer. Math. Soc., Providence, RI, 2006, 71-126.
- Hörmander L.,
The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis,
Grundlehren der Mathematischen Wissenschaften, Vol. 256, Springer-Verlag, Berlin, 1983.
- Kassel Ch.,
Le résidu non commutatif (d'après M. Wodzicki),
Astérisque no. 177-178 (1989), exp. no. 708, 199-229.
- Kontsevich M., Vishik S.,
Geometry of determinants of elliptic operators,
in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), Progr. Math., Vol. 131, Birkhäuser Boston, Boston, MA, 1995, 173-197,
hep-th/9406140.
Kontsevich M., Vishik S.,
Determinants of elliptic pseudo-differential operators, hep-th/9404046.
- Lesch M.,
On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols,
Ann. Global Anal. Geom. 17 (1998), 151-187,
dg-ga/9708010.
- Melrose R., Nistor V.,
Homology of pseudo-differential operators. I. Manifolds with boundary,
funct-an/9606005.
- Maniccia L., Schrohe E., Seiler J.,
Uniqueness of the Kontsevich-Vishik trace,
Proc. Amer. Math. Soc. 136 (2008), 747-752,
math.FA/0702250.
- Okikiolu K.,
The Campbell-Hausdorff theorem for elliptic operators and a related trace formula,
Duke Math. J. 79 (1995), 687-722.
- Paycha S.,
Regularised integrals, sums and traces; an analytic point of view, Monograph in preparation.
- Paycha S.,
Noncommutative Taylor expansions and second quantised regularised traces,
in Combinatorics and Physics, Clay Math. Proc., to appear.
- Paycha S., Scott S.,
A Laurent expansion for regularised integrals of holomorphic symbols,
Geom. Funct. Anal. 17 (2007), 491-536,
math.AP/0506211.
- Seeley R.T.,
Complex powers of an elliptic operator,
in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, 288-307.
- Shubin M.A.,
Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001.
- Taylor M.E., Pseudodifferential operators,
Princeton Mathematical Series, Vol. 34, Princeton University Press, Princeton, N.J., 1981.
- Trèves F.,
Introduction to pseudodifferential and Fourier integral operators, Vol. 1. Pseudodifferential operators,
The University Series in Mathematics, Plenum Press, New York - London, 1980.
- Varilly J.C.,
An introduction to noncommutative geometry,
EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2006.
- Wodzicki M.,
Spectral asymmetry and noncommutative residue,
PhD Thesis, Steklov Mathematics Institute, Moscow, 1984 (in Russian).
- Wodzicki M.,
Noncommutative residue. I. Fundamentals, in K-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, 320-399.
|
|