Dimensions and singular traces for spectral triples, with applications to fractals
| dc.creator | Guido, Daniele | |
| dc.creator | Isola, Tommaso | |
| dc.date | 2002-02-12 | |
| dc.date | 2002-11-12 | |
| dc.date.accessioned | 2026-07-07T04:46:24Z | |
| dc.date.available | 2026-07-07T04:46:24Z | |
| dc.description | Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|^(-1), and that the set of t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega. | |
| dc.description | latex, 36 pages, no figures, to appear on Journ. Funct. Analysis | |
| dc.identifier | https://arxiv.org/abs/math/0202108 | |
| dc.identifier | http://arxiv.org/abs/math/0202108 | |
| dc.identifier | Journ. Funct. Analysis, 203 (2003), 362-400 | |
| dc.identifier | doi:10.1016/S0022-1236(03)00230-1 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63319 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46Lxx; 28A80 | |
| dc.title | Dimensions and singular traces for spectral triples, with applications to fractals | |
| dc.type | text |