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 , 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-Poincaré characteristic is proved. L 2 -Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.

A Spectral Triple for a Solenoid Based on the Sierpinski Gasket

“…Unfortunately such trace is not canonical, since it depends on a generalized limit procedure. However, in the case of infinite self-similar CW-complexes, it was observed in [13] that such trace becomes canonical when restricted to the C * -algebra of geometric operators.…”

“…the renormalized trace. We refer to [13,26] for an analogous construction of the C * -algebra and of a canonical trace based on the self-similarity structure.…”