Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Noncommutative residues and a characterisation of the noncommutative integral


Authors: Steven Lord and Fedor A. Sukochev
Journal: Proc. Amer. Math. Soc. 139 (2011), 243-257
MSC (2010): Primary 46L51, 47B10, 58B34; Secondary 58J42, 58C35
DOI: https://doi.org/10.1090/S0002-9939-2010-10472-0
Published electronically: July 9, 2010
MathSciNet review: 2729087
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent). Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra. Flat tori, noncommutative tori, and a link with the QUE property of manifolds are given as examples.


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

  • 1. Alan Carey, John Phillips, and Fedor A. Sukochev, Spectral flow and Dixmier traces, Adv. Math. 173 (2003), 68-113. MR 1954456 (2004e:58049)
  • 2. Alan Carey, Adam Rennie, Aleksandr Sedaev, and Fedor A. Sukochev, The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2007), 253-283. MR 2345333
  • 3. Alan Carey and Fedor A. Sukochev, Dixmier traces and some applications in non-commutative geometry, Russ. Math. Surv. 61 (2006), 1039-1099. MR 2330013 (2009i:46132)
  • 4. Yves Colin de Verdiere, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), 497-502. MR 818831 (87d:58145)
  • 5. Alain Connes, The action functional in non-commutative geometry, Comm. Math. Phys. 117 (1988), 673-683. MR 0953826 (91b:58246)
  • 6. -, Noncommutative geometry, Academic Press, San Diego, CA, 1994. MR 1303779 (95j:46063)
  • 7. -, Gravity coupled with matter and the foundations of noncommutative geometry, Comm. Math. Phys. 182 (1996), 155-176. MR 1441908 (98f:58024)
  • 8. Jacques Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris (1966), no. 262, A1107-A1108. MR 0196508 (33:4695)
  • 9. Harold Donnelly, Quantum unique ergodicity, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2945-2951. MR 1974353 (2005a:58048)
  • 10. Jóse M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of noncommutative geometry, Birkhäuser Advanced Texts, Birkhäuser, Boston, 2001. MR 1789831 (2001h:58038)
  • 11. Ricard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras, second ed., vol. I, Graduate Studies in Mathematics, no. 15, AMS, 1997. MR 1468229 (98f:46001a)
  • 12. Steven Lord, Denis Potapov, and Fedor Sukochev, Measures from Dixmier traces and zeta functions, J. Funct Anal. (2010), DOI: 10.1016/j.jfa.2010.06.012.
  • 13. Steven Lord, Aleksandr Sedaev, and Fedor A. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 244 (2005), no. 1, 72-106. MR 2139105 (2006e:46065)
  • 14. Gert K. Pederson, $ C^*$-algebras and their automorphism groups, LMS Monographs, no. 14, Academic Press, London, 1979. MR 548006 (81e:46037)
  • 15. Marc A. Rieffel, $ C^*$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. MR 623572 (83b:46087)
  • 16. Peter J. Sarnak, Arithmetic quantum chaos, Israel Mathematics Conference Proceedings, vol. 8, Bar-Ilan University, 1995, pp. 183-236. MR 1321639 (96d:11059)
  • 17. Barry Simon, Trace ideals and their applications, Mathematical Surveys and Monographs, no. 120, AMS, 2005. MR 2154153 (2006f:47086)
  • 18. A. I. Snirel'man, Ergodic properties of eigenfunctions (Russian), Uspehi Mat. Nauk 29 (1974), 181-182. MR 0402834 (53:6648)
  • 19. Steve Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), no. 4, 919-941. MR 916129 (89d:58129)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L51, 47B10, 58B34, 58J42, 58C35

Retrieve articles in all journals with MSC (2010): 46L51, 47B10, 58B34, 58J42, 58C35


Additional Information

Steven Lord
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
Email: steven.lord@adelaide.edu.au

Fedor A. Sukochev
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
Email: f.sukochev@unsw.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2010-10472-0
Keywords: Dixmier trace, zeta functions, noncommutative integral, noncommutative geometry, normal, noncommutative residue
Received by editor(s): May 27, 2009
Received by editor(s) in revised form: May 28, 2009, and March 1, 2010
Published electronically: July 9, 2010
Additional Notes: This research was supported by the Australian Research Council
Communicated by: Varghese Mathai
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society