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 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. © 2008 Elsevier Inc. All rights reserved.