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

A class of CW-complexes, called self-similar complexes, is introduced, together with C∗-algebras Aj of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian Δj belongs to Aj, 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.