A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L2-Betti numbers

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_j belongs to A_j , L2-Betti numbers and Novikov–Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler–Poincaré characteristic is proved. L2-Betti and Novikov–Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.