  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.
