zbMATH — the first resource for mathematics

Traces on pseudodifferential operators and sums of commutators. (English) Zbl 1194.58029
The paper contains proofs of known characterizations, Theorems 2.3 to 2.5, of traces on algebras of classical pseudodifferential operators. These characterizations state that, essentially, the traces on the algebras of integer and of non-integer order pseudodifferential operators are the noncommutative residue and the canonical trace, respectively. The canonical trace is the analytic continuation of the operator trace. The algebra of zeroth order operators possesses, in addition, traces given by linear functionals on the space of principal symbols. The aim of the paper is to give simple proofs of these uniqueness theorems for traces, using only basic pseudodifferential calculus and properties of the Schwartz kernels of pseudodifferential operators. The sums of commutators needed in the arguments are given directly. Finally, the onedimensional case is considered separately and shown to have special properties.

58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
Full Text: DOI
[1] A. A. Albert and B. Muckenhoupt, On matrices of trace zeros, Michigan Math. J. 4 (1957),1–3. · Zbl 0077.24304 · doi:10.1307/mmj/1028990168
[2] R. Beals and P. Greiner, Calculus on Heisenberg Manifolds, Princeton Univ. Press, Princeton, 1988. · Zbl 0654.58033
[3] A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), 174–243. · Zbl 0960.46048 · doi:10.1007/BF01895667
[4] B. V. Fedosov, F. Golse, E. Leichtnam and E. Schrohe, The noncommutative residue for manifolds with boundary, J. Funct. Anal. 142 (1996), 1–31. · Zbl 0877.58005 · doi:10.1006/jfan.1996.0142
[5] J. M. Gracia-Bondía, J. C. Várilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Boston, Boston, MA, 2001.
[6] V. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. Math. 55 (1985), 131–160. · Zbl 0559.58025 · doi:10.1016/0001-8708(85)90018-0
[7] V. Guillemin, Gauged Lagrangian distributions, Adv. Math. 102 (1993), 184–201. · Zbl 0803.58052 · doi:10.1006/aima.1993.1064
[8] V. Guillemin, Residue traces for certain algebras of Fourier integral operators, J. Funct. Anal. 115 (1993), 391–417. · Zbl 0791.35162 · doi:10.1006/jfan.1993.1096
[9] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Springer, Berlin, 1990.
[10] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Springer, Berlin, 1994.
[11] W. Kahan, Only commutators can have trace zero, preprint, June 1999. Available online at http://www.cs.berkeley.edu/_wkahan/MathH110/trace0.pdf .
[12] M. Kontsevich and S. Vishik, Geometry of determinants of elliptic operators, Functional Analysis on the Eve of the 21st Cetury, Vol. 1, Birkhäuser Boston, Boston, 1995, pp. 173–197. · Zbl 0920.58061
[13] M. Lesch, On the noncommutative residue for pseudodifferential operators with logpolyhomogeneous symbols, Ann. Global Anal. Geom. 17 (1999), 151–187. · Zbl 0920.58047 · doi:10.1023/A:1006504318696
[14] J. M. Lescure and S. Paycha, Uniqueness of multiplicative determinants on elliptic pseudodifferential operators, Proc. London Math. Soc. (3) 94 (2007), 772–812. · Zbl 1193.58018 · doi:10.1112/plms/pdm004
[15] L. Maniccia, E. Schrohe and J. Seiler, Uniqueness of the Kontsevich-Vishik Trace, Proc. Amer. Math. Soc. 136 (2008), 747–752. · Zbl 1132.58018 · doi:10.1090/S0002-9939-07-09168-X
[16] V. Mathai, R. Melrose and I. Singer, Fractional index theory, J. Differential Geom. 74 (2006), 265–292.
[17] R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, 1993. · Zbl 0796.58050
[18] S. Paycha, The noncommutative residue and canonical trace in the light of Stokes’ and continuity properties, E-print, arXiv, June 2007.
[19] S. Paycha and S. Rosenberg, Curvature on determinant bundles and first Chern forms, J. Geom. Phys. 45 (2003), 393–429. · Zbl 1028.58030 · doi:10.1016/S0393-0440(01)00079-1
[20] S. Paycha and S. Rosenberg, Traces and characteristic classes on loop groups, IRMA Lect. Math. Theor. Phys. 5, de Gruyter, Berlin, 2004, pp. 185–212. · Zbl 1064.58023
[21] R. Ponge, Noncommutative residue for Heisenberg manifolds and applications in CR and contact geometry, J. Funct. Anal. 252 (2007), 399–463. · Zbl 1127.58024 · doi:10.1016/j.jfa.2007.07.001
[22] E. Schrohe, Noncommutative residues andmanifolds with conical singularities, J. Funct.Anal. 150 (1997), 146–174. · Zbl 0903.58057 · doi:10.1006/jfan.1997.3109
[23] K. Shoda, Über den Kommutator der Matrizen, J. Math. Soc. Japan 3 (1951), 78–81. · Zbl 0045.15402 · doi:10.2969/jmsj/00310078
[24] M. E. Taylor, Partial Differential Equations. II. Qualitative Studies of Linear Equations, Springer-Verlag, New York, 1996. · Zbl 0869.35003
[25] W. J. Ugalde, A Construction of critical GJMS operators using Wodzicki’s residue, Comm. Math. Phys. 261 (2006), 771–788. · Zbl 1104.58006 · doi:10.1007/s00220-005-1384-8
[26] S. Vassout, Feuilletages et résidu non commutatif longitudinal, PhD thesis, University of Paris 7, 2001.
[27] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75 (1984), 143–177. · Zbl 0538.58038 · doi:10.1007/BF01403095
[28] M. Wodzicki, Spectral Asymmetry and Noncommutative Residue (in Russian), Habilitation Thesis, Steklov Institute, (former) Soviet Academy of Sciences, Moscow, 1984.
[29] M. Wodzicki, Noncommutative residue. I. Fundamentals, K-Theory, Arithmetic and Geometry, Lecture Notes in Math. 1289, Springer, Berlin, 1987, pp. 320–399. · Zbl 0649.58033
[30] M. Wodzicki, Report on the cyclic homology of symbols, Preprint, IAS Princeton, Princeton, Jan. 87. Available online at http://math.berkeley.edu/_wodzicki . · Zbl 0635.18010
[31] M. Wodzicki, Personal communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.