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Transactions of the American Mathematical Society

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The canonical trace and the noncommutative residue on the noncommutative torus


Authors: Cyril Lévy, Carolina Neira Jiménez and Sylvie Paycha
Journal: Trans. Amer. Math. Soc. 368 (2016), 1051-1095
MSC (2010): Primary 58B34, 58J42
DOI: https://doi.org/10.1090/tran/6369
Published electronically: July 9, 2015
MathSciNet review: 3430358
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Abstract: Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace-class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the $ \zeta $-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.


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  • [A1] M. S. Agranovič, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 54-56 (Russian). MR 554412 (81e:35096)
  • [A2] M. S. Agranovich, Elliptic pseudodifferential operators on a closed curve, Trudy Moskov. Mat. Obshch. 47 (1984), 22-67, 246 (Russian). MR 774945 (86g:35218)
  • [AGPS] S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal. 137 (1996), no. 2, 281-302. MR 1387512 (97j:46063), https://doi.org/10.1006/jfan.1996.0047
  • [B1] Saad Baaj, Calcul pseudo-différentiel et produits croisés de $ C^*$-algèbres. I, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 11, 581-586 (French, with English summary). MR 967366 (90a:46171)
  • [B2] Saad Baaj, Calcul pseudo-différentiel et produits croisés de $ C^*$-algèbres. II, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 12, 663-666 (French, with English summary). MR 967808 (90a:46172)
  • [C1] Alain Connes, $ C^{\ast } $ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599-A604 (French, with English summary). MR 572645 (81c:46053)
  • [C2] Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779 (95j:46063)
  • [C3] Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257-360. MR 823176 (87i:58162)
  • [CM1] A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174-243. MR 1334867 (96e:58149), https://doi.org/10.1007/BF01895667
  • [CM2] Noncommutative geometry and global analysis, Edited by Alain Connes, Alexander Gorokhovsky, Matthias Lesch, Markus Pflaum and Bahram Rangipour, Contemporary Mathematics, vol. 546, American Mathematical Society, Providence, RI, 2011. MR 2815127 (2012b:58004)
  • [CT] Alain Connes and Paula Tretkoff, The Gauss-Bonnet theorem for the noncommutative two torus, Noncommutative geometry, arithmetic, and related topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, pp. 141-158. MR 2907006
  • [FK1] Farzad Fathizadeh and Masoud Khalkhali, The Gauss-Bonnet theorem for noncommutative two tori with a general conformal structure, J. Noncommut. Geom. 6 (2012), no. 3, 457-480. MR 2956317, https://doi.org/10.4171/JNCG/97
  • [FK2] Farzad Fathizadeh and Masoud Khalkhali, Scalar curvature for the noncommutative two torus, J. Noncommut. Geom. 7 (2013), no. 4, 1145-1183. MR 3148618, https://doi.org/10.4171/JNCG/145
  • [FK3] F. Fathizadeh and M. Khalkhali, Scalar curvature for noncommutative four-tori, arXiv:1301.6135 (2013).
  • [FW] Farzad Fathizadeh and M. W. Wong, Noncommutative residues for pseudo-differential operators on the noncommutative two-torus, J. Pseudo-Differ. Oper. Appl. 2 (2011), no. 3, 289-302. MR 2831659 (2012i:58020), https://doi.org/10.1007/s11868-011-0030-9
  • [FGLS] 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 (97h:58157), https://doi.org/10.1006/jfan.1996.0142
  • [G-BVF] José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1789831 (2001h:58038)
  • [Gu] Victor Guillemin, Gauged Lagrangian distributions, Adv. Math. 102 (1993), no. 2, 184-201. MR 1252031 (95c:58171), https://doi.org/10.1006/aima.1993.1064
  • [K] Christian Kassel, Le résidu non commutatif (d'après M. Wodzicki), Astérisque 177-178 (1989), Exp. No. 708, 199-229 (French). Séminaire Bourbaki, Vol. 1988/89. MR 1040574 (91e:58190)
  • [KV] 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 (96m:58264)
  • [L] 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 (2000b:58050), https://doi.org/10.1023/A:1006504318696
  • [LN-J] Matthias Lesch and Carolina Neira Jiménez, Classification of traces and hypertraces on spaces of classical pseudodifferential operators, J. Noncommut. Geom. 7 (2013), no. 2, 457-498. MR 3054303, https://doi.org/10.4171/JNCG/123
  • [M] Severino T. Melo, Characterizations of pseudodifferential operators on the circle, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1407-1412. MR 1415353 (98f:47058), https://doi.org/10.1090/S0002-9939-97-04016-1
  • [McL] William McLean, Local and global descriptions of periodic pseudodifferential operators, Math. Nachr. 150 (1991), 151-161. MR 1109651 (92f:47055), https://doi.org/10.1002/mana.19911500112
  • [MSS] L. Maniccia, E. Schrohe, and J. Seiler, Uniqueness of the Kontsevich-Vishik trace, Proc. Amer. Math. Soc. 136 (2008), no. 2, 747-752. MR 2358517 (2008k:58062), https://doi.org/10.1090/S0002-9939-07-09168-X
  • [N-J] C. Neira Jiménez, Cohomology of classes of symbols and classification of traces on corresponding classes of operators with non positive order, PhD thesis, Universität Bonn (2010). http://hss.ulb.uni-bonn.de/2010/2214/2214.htm.
  • [NR] Fabio Nicola and Luigi Rodino, Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, vol. 4, Birkhäuser Verlag, Basel, 2010. MR 2668420 (2011k:35001)
  • [P1] S. Paycha, The noncommutative residue in the light of Stokes' and continuity properties, arXiv:0706.2552 [math.OA] (2007).
  • [P2] Sylvie Paycha, Regularised integrals, sums and traces, An analytic point of view. University Lecture Series, vol. 59, American Mathematical Society, Providence, RI, 2012. MR 2987296
  • [P3] Sylvie Paycha, A canonical trace associated with certain spectral triples, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper 077, 17. MR 2769938 (2012c:58047), https://doi.org/10.3842/SIGMA.2010.077
  • [PR] Sylvie Paycha and Steven Rosenberg, Traces and characteristic classes on loop spaces, Infinite dimensional groups and manifolds, IRMA Lect. Math. Theor. Phys., vol. 5, de Gruyter, Berlin, 2004, pp. 185-212. MR 2104357 (2005h:58047)
  • [PS] Sylvie Paycha and Simon Scott, A Laurent expansion for regularized integrals of holomorphic symbols, Geom. Funct. Anal. 17 (2007), no. 2, 491-536. MR 2322493 (2008m:58061), https://doi.org/10.1007/s00039-007-0597-8
  • [RT1] Michael Ruzhansky and Ville Turunen, On the Fourier analysis of operators on the torus, Modern trends in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 172, Birkhäuser, Basel, 2007, pp. 87-105. MR 2308505 (2008b:35165), https://doi.org/10.1007/978-3-7643-8116-5_5
  • [RT2] Michael Ruzhansky and Ville Turunen, Quantization of pseudo-differential operators on the torus, J. Fourier Anal. Appl. 16 (2010), no. 6, 943-982. MR 2737765 (2011m:58043), https://doi.org/10.1007/s00041-009-9117-6
  • [RT3] Michael Ruzhansky and Ville Turunen, Global quantization of pseudo-differential operators on compact Lie groups, $ \rm SU(2)$, 3-sphere, and homogeneous spaces, Int. Math. Res. Not. IMRN 11 (2013), 2439-2496. MR 3065085
  • [RT4] Michael Ruzhansky and Ville Turunen, Pseudo-differential operators and symmetries, Background analysis and advanced topics. Pseudo-Differential Operators. Theory and Applications, vol. 2, Birkhäuser Verlag, Basel, 2010. MR 2567604 (2011b:35003)
  • [Sc] Simon Scott, Traces and determinants of pseudodifferential operators, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2010. MR 2683288 (2011m:58056)
  • [T] V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anwendungen 19 (2000), no. 1, 95-108. MR 1748052 (2001d:35203), https://doi.org/10.4171/ZAA/940
  • [TV] V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwendungen 17 (1998), no. 1, 9-22. MR 1616044 (99b:47077), https://doi.org/10.4171/ZAA/805
  • [W1] M. Wodzicki, Spectral asymmetry and noncommutative residue, PhD thesis, Steklov Mathematics Institute, Moscow, 1984 (in Russian).
  • [W2] Mariusz Wodzicki, Noncommutative residue. I. Fundamentals, $ K$-theory, arithmetic and geometry (Moscow, 1984-1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 320-399. MR 923140 (90a:58175), https://doi.org/10.1007/BFb0078372

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Additional Information

Cyril Lévy
Affiliation: Institut für Mathematik, Am Neuen Palais 10, Universität Potsdam, 14469 Potsdam, Germany
Address at time of publication: Centre Universitaire Jean-François Champollion, Place Verdun 81000 Albi, France – and – Institut de Mathématiques de Toulouse, 118 route de Narbonne 31062 Toulouse Cedex 9, France
Email: levy@math.uni-potsdam.de, cyril.levy@univ-jfc.fr

Carolina Neira Jiménez
Affiliation: Fakultät für Mathematik, Universitätsstrasse 31, University of Regensburg, 92040 Regensburg, Germany
Address at time of publication: Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 30 #45 - 03, Bogotá, Colombia
Email: Carolina.neira-jimenez@mathematik.uni-regensburg.de, cneiraj@unal.edu.co

Sylvie Paycha
Affiliation: Institut für Mathematik, Am Neuen Palais 10, Universität Potsdam, 14469 Potsdam, Germany
Email: paycha@math.uni-potsdam.de

DOI: https://doi.org/10.1090/tran/6369
Received by editor(s): March 12, 2013
Received by editor(s) in revised form: July 20, 2013, July 22, 2013, and December 14, 2013
Published electronically: July 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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