The canonical trace and the noncommutative residue on the noncommutative torus

Lévy, Cyril; Neira Jiménez, Carolina; Paycha, Sylvie

doi:10.1090/tran/6369

The canonical trace and the noncommutative residue on the noncommutative torus
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by Cyril Lévy, Carolina Neira Jiménez and Sylvie Paycha PDF

Trans. Amer. Math. Soc. 368 (2016), 1051-1095 Request permission

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.

References

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
M. S. Agranovich, Elliptic pseudodifferential operators on a closed curve, Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246 (Russian). MR 774945
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, DOI 10.1006/jfan.1996.0047
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
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
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
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360. MR 823176
A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867, DOI 10.1007/BF01895667
Alain Connes, Alexander Gorokhovsky, Matthias Lesch, Markus Pflaum, and Bahram Rangipour (eds.), Noncommutative geometry and global analysis, Contemporary Mathematics, vol. 546, American Mathematical Society, Providence, RI, 2011. MR 2815127, DOI 10.1090/conm/546
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
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, DOI 10.4171/JNCG/97
Farzad Fathizadeh and Masoud Khalkhali, Scalar curvature for the noncommutative two torus, J. Noncommut. Geom. 7 (2013), no. 4, 1145–1183. MR 3148618, DOI 10.4171/JNCG/145
F. Fathizadeh and M. Khalkhali, Scalar curvature for noncommutative four-tori, arXiv:1301.6135 (2013).
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, DOI 10.1007/s11868-011-0030-9
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, DOI 10.1006/jfan.1996.0142
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, DOI 10.1007/978-1-4612-0005-5
Victor Guillemin, Gauged Lagrangian distributions, Adv. Math. 102 (1993), no. 2, 184–201. MR 1252031, DOI 10.1006/aima.1993.1064
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
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
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, DOI 10.1023/A:1006504318696
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, DOI 10.4171/JNCG/123
Severino T. Melo, Characterizations of pseudodifferential operators on the circle, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1407–1412. MR 1415353, DOI 10.1090/S0002-9939-97-04016-1
William McLean, Local and global descriptions of periodic pseudodifferential operators, Math. Nachr. 150 (1991), 151–161. MR 1109651, DOI 10.1002/mana.19911500112
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, DOI 10.1090/S0002-9939-07-09168-X
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.
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, DOI 10.1007/978-3-7643-8512-5
S. Paycha, The noncommutative residue in the light of Stokes’ and continuity properties, arXiv:0706.2552 [math.OA] (2007).
Sylvie Paycha, Regularised integrals, sums and traces, University Lecture Series, vol. 59, American Mathematical Society, Providence, RI, 2012. An analytic point of view. MR 2987296, DOI 10.1090/ulect/059
Sylvie Paycha, A canonical trace associated with certain spectral triples, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper 077, 17. MR 2769938, DOI 10.3842/SIGMA.2010.077
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
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, DOI 10.1007/s00039-007-0597-8
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, DOI 10.1007/978-3-7643-8116-5_{5}
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, DOI 10.1007/s00041-009-9117-6
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, DOI 10.1093/imrn/rns122
Michael Ruzhansky and Ville Turunen, Pseudo-differential operators and symmetries, Pseudo-Differential Operators. Theory and Applications, vol. 2, Birkhäuser Verlag, Basel, 2010. Background analysis and advanced topics. MR 2567604, DOI 10.1007/978-3-7643-8514-9
Simon Scott, Traces and determinants of pseudodifferential operators, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2010. MR 2683288, DOI 10.1093/acprof:oso/9780198568360.001.0001
V. Turunen, Commutator characterization of periodic pseudodifferential operators, Z. Anal. Anwendungen 19 (2000), no. 1, 95–108. MR 1748052, DOI 10.4171/ZAA/940
V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwendungen 17 (1998), no. 1, 9–22. MR 1616044, DOI 10.4171/ZAA/805
M. Wodzicki, Spectral asymmetry and noncommutative residue, PhD thesis, Steklov Mathematics Institute, Moscow, 1984 (in Russian).
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, DOI 10.1007/BFb0078372