Dimensions and singular traces for spectral triples, with applications to fractals

Journal of Functional Analysis

Volumn 203, Issue 2, 2003, Pages 362-400

  Guido, Daniele ^a   Isola, Tommaso ^a  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

Fractals in the real line; Noncommutative Hausdorff dimensions; Noncommutative Hausdorff Besicovitch; Singular traces; Spectral triples

Indexed keywords

EID: 0141838120     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-1236(03)00230-1     Document Type: Article

References (33)

1. S. Albeverio, D. Guido, A. Ponosov, S. Scarlatti, Singular traces and compact operators, J. Funct. Anal. 137 (1996) 281-302.
2. M.T. Benameur, T. Fack, On von Neumann spectral triples, preprint, math.KT/0012233.
3. C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.
4. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
5. A.L. Carey, J. Phillips, F. Sukochev, Spectral flow and Dixmier traces, preprint, math.OA/0205076.
6. A. Connes, Non Commutative Geometry, Academic Press, New York, 1994.
7. A. Connes, private communication.
8. F. Cipriani, D. Guido, S. Scarlatti, A remark on trace properties of K-cycles, J. Operator Theory 35 (1996) 179-189.
9. J. Dixmier, Existence de traces nonnormales, C.R. Acad. Sci. Paris 262 (1966) 1107-1108.
10. P.G. Dodds, B. de Pagter, E.M. Semenov, F.A. Sukochev, Symmetric functionals and singular traces, Positivity 2 (1998) 47-75.
11. K. Dykema, T. Figiel, G. Weiss, M. Wodzicki, Commutator Structure of Operator Ideals, MSRI preprint no. 2001-013.
12. T. Fack, H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986) 269.
13. H. Furstenberg, Talk at the Conference "Fractals in Graz", June 2001.
14. D. Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc. 352 (2000) 1953-1983.
15. I. Gohberg, M.G. Krein, Introduction to the theory of non-selfadjoint operators, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, RI, 1969.
16. J.M. Gracia-Bondia, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Boston, Inc., Boston, MA, 2001.
17. D. Guido, T. Isola, Singular traces for semifinite von Neumann algebras, J. Funct. Anal. 134 (1995) 451-485.
18. D. Guido, T. Isola, Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds, J. Funct. Anal. 176 (2000) 115-152.
19. D. Guido, T. Isola, Singular traces and Novikov-Shubin invariants, in: A. Gheondea, R.N. Gologan, D. Timotin (Eds.), Operator Theoretical methods, Proceedings of the 17th Conference on Operator Theory Timisoara (Romania) June 23-26, 1998, The Theta Foundation, Bucharest, 2000.
20. D. Guido, T. Isola, Fractals in noncommutative geometry, in: Roberto Longo (Eds.), The Proceedings of the Conference, Mathematical Physics in Mathematics and Physics, Siena 2000, University of Rome II, Italy, Fields Institute Communications, Vol. 30, American Mathematical Society, Providence, RI, 2001.
21. n, in progress.
22. D. Guido, T. Isola, Tangential dimensions for metric spaces and measures, preprint math.FA/0305091.
23. D. Guido, T. Isola, D. Guido, T. Isola, On the domain of singular traces, Internat. J. Math. 13 (2002) 667-674.
24. D. Guido, T. Isola, in progress.
25. C.Q. He, M.L. Lapidus, Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Mem. Amer. Math. Soc. 127 (608) (1997).
26. J.E. Hutchinson, Fractals and self-similarity, Indiana Univ., Math. J. 30 (1981) 711-747.
27. J.P. Kahane, R. Salem, Ensembles Parfaits et séries Trigonométriques, Hermann, Paris, 1994.
28. J. Kigami, M.L. Lapidus, Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 217 (2001) 165-180.
29. M.L. Lapidus, M. van Frankenhuysen, Fractal geometry and number theory, Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, Birkhäuser Boston, Inc., Boston, MA, 2000.
30. M.L. Lapidus, C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993) 41-69.
31. R.D. Mauldin, S.C. Williams, Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986) 325-346.
32. C. Tricot, Curves and Fractal Dimension, Springer, New York, 1995.
33. J.V. Varga, Traces on irregular ideals, Proc. Amer. Math. Soc. 107 (1989) 715.