A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers
arXiv:math/0607603 · doi:10.1016/j.jfa.2008.10.013
Abstract
A class of CW-complexes, called self-similar complexes, is introduced, together with C*-algebras A_j of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian Delta_j belongs to A_j, L^2-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincare' characteristic is proved. L^2-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.
arXiv comment30 pages, 7 figures
References in corpus (9)
- Dimensions and singular traces for spectral triples, with applications to fractals
- A trace on fractal graphs and the Ihara zeta function
- Bose Einstein condensation on inhomogeneous amenable graphs
- Fractals in Noncommutative Geometry
- Dimensions and spectral triples for fractals in R^N
- Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds
- Tangential dimensions I. Metric spaces
- Tangential dimensions II. Measures
- Aperiodic order, integrated density of states and the continuous algebras of John von Neumann