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A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers

Fabio Cipriani·Daniele Guido·Tommaso Isola
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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 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.

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Fabio Cipriani, Daniele Guido, Tommaso Isola. 2006-07-24. A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers. https://doi.org/10.1016/j.jfa.2008.10.013

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